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FACULITY OF TECHNOLOGY
Modern fatigue analysis methodology for laser welded
joints
Kokko Rami
PROGRAMME OF MECHANICAL ENGINEERING
Master’s Thesis
February 2018
FACULITY OF TECHNOLOGY
Modern fatigue analysis methodology for laser welded
joints
Kokko Rami
Supervisors: Jari Laukkanen and Joona Vaara
PROGRAMME OF MECHANICAL ENGINEERING
Master’s Thesis
February 2018
Tiivistelmä
Moderni väsymisanalyysimetodiikka laserhitsatuissa rakenteissa
Kokko Rami
Oulun yliopisto, Konetekniikan koulutusohjelma
Diplomityö 2018, 92 s.
Työn ohjaajat: Jari Laukkanen ja Joona Vaara
Hitsausliitos aiheuttaa aina väsymisominaisuuksien huonontumista dynaamisen
kuormituksen alaisissa rakenteissa. Hitsiliitos aiheuttaa epäjatkuvuutta ja vikoja
rakenteisiin, joista väsyminen ydintyy kuormituksessa. Työn teoriaosassa laser-
hitsauksen prosessi käydään kattavasti läpi väsymiseen johtavien seikkojen va-
lossa.
Työssä esitetään hitsauksen numeerisen mallinnuksen mahdollisuuksia. Nu-
meerisella mallinnuksella saadaan esimerkiksi jäännöjännitykset, jäännösmuodon-
muutos tai materiaalin faasimuutokset, joita voidaan käyttää hyväksi rakenne-
ja toleranssisuunnittelussa.
Diplomityössä tutkittiin laserhitsatun liitoksen väsymismitoituksen sopivuutta
ohjeiden ja standardien esittämiin menetelmiin. Kirjallisuudesta kerättyjä laser-
hitsien väsymistestituloksia sovitettiin standardien antamiin mitoitusohjeisiin.
Sovituksen tuloksen pohjalta on pohdittu standardimitoituksen haasteita laser-
hitsatun liitoksen väsymisiän arvioinissa.
Työssä tutkittiin kuinka paljon jäännösjännitykset selittävät laserhitsatun liitok-
sen väsymiskäyttäytymistä. Numeerisessa analyysissä testisauvalle laskettiin
jäännösjännitystila. Sauvan väsymiskäyttäytimistä tutkittiin perusaineen suh-
teen. Tulokset osoittivat, että jäännösjännityksillä on huomattava merkitys hit-
siliitoksen väsymisessä. Testitulosten tilastolliselle hajoamalle johdettiin vian
todennäköisyysjakauma kirjallisuuden tulosten perusteella.
Asiasanat: laserhitsaus, väsyminen, väsymismitoitus, jäännösjännitykset,
materiaalimalli
Abstract
Modern fatigue analysis methodology for laser welded joints
Kokko Rami
University of Oulu, Degree Programme of Mechanical Engineering
Master’s thesis 2018, 92 p.
Supervisors: Jari Laukkanen and Joona Vaara
Welding has always a deteriorating effect on fatigue strength in structures under
dynamic loading. Weld joints induce discontinuity and defects where potential
fatigue cracks initiate. In the theory part of this thesis, the laser welding process
is discussed in sense of fatigue.
The simulation possibilities for welding are introduced. With simulation, such
effects as residual stresses, initial distortion and phase changes can be obtained.
These residual states can be used in structure and tolerance design.
In this thesis, the suitability of fatigue assessment offered in rules and regula-
tions is discussed. A large number of test results was collected from literature
in order to determine the common fatigue assessment suitability for the fatigue
design of laser-welded joints. Common assessment suitability for the fatigue
design of laser-welded joint is discussed on the basics of the results.
In this thesis work, the effect of residual stresses on fatigue of welded joints was
studied. In the study, the effect of residual stresses on the fatigue behaviour of
the base material was studied. The results suggested that residual stresses play
a significant role in fatigue. The statistical analysis of the probable defect size
was done with test results from literature.
Keywords: laser welding, fatigue, residual stresses, material model
Preface
This thesis was made for Wärtsilä Finland Oyj Structural and dynamic team
under employment at Global Boiler Works Oy. Wärtsilä manufactures and offers
complete lifecycle solutions for marine and energy markets.
First, I would like to thank my supervisor Joona Vaara from Wärtsilä Finland Oyj
for overall guidance with the technical part of the thesis, and research sugges-
tions that led to essential role in the results. I would also like to thank Wärtsilä
Structural Analysis and Dynamics team manager Tero Frondelius for the oppor-
tunity to work with Wärtsilä and for the scope of work in the thesis
I would like to thank my girlfriend Laura for supporting my studies and thesis
writing. I would also like to thank GBW and its employees for support and help
with practical arrangements.
And finally I would like to thank my son Akseli, who helped by filling any
possible free time left.
Oulu, 15.01.2018
Kokko Rami
Contents
1 Introduction 1
2 Laser welding process 2
2.1 Laser and applicability for welding . . . . . . . . . . . . . . . . . . . 2
2.1.1 CO2 laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.2 Nd:YAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.3 Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.4 Fiber laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Benefits of laser welding . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Challenges in laser welding . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Visuals of a laser welded joint . . . . . . . . . . . . . . . . . . . . . . 4
2.5 Physics of laser welding . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5.1 Forming of keyhole . . . . . . . . . . . . . . . . . . . . . . . 5
2.5.2 Energy absorption . . . . . . . . . . . . . . . . . . . . . . . . 6
2.6 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.6.1 Stability of keyhole welding . . . . . . . . . . . . . . . . . . . 7
2.6.2 Molten pool flows . . . . . . . . . . . . . . . . . . . . . . . . 7
2.6.3 Parameters in keyhole laser welding . . . . . . . . . . . . . . 7
2.7 Heat conduction welding . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Weldable metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8.1 Dissimilar material welding . . . . . . . . . . . . . . . . . . . 10
3 Mechanics of welded joint 12
3.0.1 Fatigue propagation of welded joint . . . . . . . . . . . . . . 12
3.1 Laser welding fatigue in general . . . . . . . . . . . . . . . . . . . . 13
3.2 Fusion zone and heat-affected zone . . . . . . . . . . . . . . . . . . 14
3.3 Residual stresses and strains . . . . . . . . . . . . . . . . . . . . . . 16
3.3.1 Forming of residual stresses . . . . . . . . . . . . . . . . . . 16
3.3.2 Effect of residual stresses . . . . . . . . . . . . . . . . . . . . 17
3.3.3 Controlling residual stresses . . . . . . . . . . . . . . . . . . 18
3.4 Distortion of joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4.1 Effect of plate thickness . . . . . . . . . . . . . . . . . . . . . 19
3.4.2 Impurity and other imperfections . . . . . . . . . . . . . . . 19
3.5 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5.1 Porosity formation . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5.2 Effect of gap between weldable parts . . . . . . . . . . . . . 21
3.5.3 Effect of parameters . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.4 Porosity preventing . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.5 Standards and regulations . . . . . . . . . . . . . . . . . . . . 23
3.6 Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.7 Imperfections on keyhole form . . . . . . . . . . . . . . . . . . . . . 24
4 Material behaviour in cyclic loading 25
4.1 Material fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.1 Damage models . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5 Common methods for weld fatigue calculations 30
5.1 General presuppositions . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Common models introduced . . . . . . . . . . . . . . . . . . . . . . 33
5.2.1 Nominal stress concept . . . . . . . . . . . . . . . . . . . . . 35
5.2.2 Structural hot spot approach . . . . . . . . . . . . . . . . . . 36
5.2.3 Notch stress concept . . . . . . . . . . . . . . . . . . . . . . . 37
5.2.4 Notch strain concept . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.5 Novel notch stress approach (3R) . . . . . . . . . . . . . . . . 43
5.2.6 Continuum damage mechanism approach . . . . . . . . . . 43
5.2.7 Stress intensity factors (SIF) . . . . . . . . . . . . . . . . . . . 44
5.3 Comparison of fatigue models . . . . . . . . . . . . . . . . . . . . . 45
5.3.1 Conclusion of the comparison . . . . . . . . . . . . . . . . . 50
6 Modelling and simulating laser welding 51
6.1 Simulation according HFF approach . . . . . . . . . . . . . . . . . . 53
6.1.1 Fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.1.2 Modelling of the laser beam absorption . . . . . . . . . . . . 56
6.2 Simulation according TMM approach . . . . . . . . . . . . . . . . . 57
6.2.1 Heat source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2.2 Material behaviour . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2.3 Phase transformation . . . . . . . . . . . . . . . . . . . . . . 60
6.3 Welding residual stress simulation for fatigue behaviour analysis . 61
6.3.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3.2 Temperature dependent material mode . . . . . . . . . . . . 62
6.3.3 Virfac simulation . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3.4 Residual stresses . . . . . . . . . . . . . . . . . . . . . . . . . 65
7 Commercial fatigue program 67
7.1 FEMFAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8 Fatigue calculation with residual stresses 69
8.1 Cyclic loading analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.1.1 Fatigue analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.3 Further processing of results . . . . . . . . . . . . . . . . . . . . . . 77
9 Discussion 82
References 84
Nomenclature
: double contraction
Ck kinematic hardening modulus
Cp heat capacity
D damage parameter
Fb Buoyance force
Fg gravity force
Hv Vickers hardness
K f fatigue notch factor
L f latent heat of fusion
N number of load cycles
N f total number of cycles
Pv surface tension
Papl ablation pressure
Pint initial power
Q∞ maximun hardening
R stress ratio
Rlocal local stress ratio
Tre f reference temperature
∆ range
Φ heat source
α backstress
δij cronecker delta
∂γ∂T surface tension gradient
γk rate of change
S deviatoric stress tensor
C fourth order stiffens tensor
I identity tensor
S compliance tensor
∇ gradient operator
ν Poisson’s ratio
ν dynamic viscosity
νs tangential velocity
σ effective notch stress
εpl equivalent plastic strain
ρ density
ρ real notch radius
ρ∗ microstructural length
ρ f fictitious notch radius
σ′f fatigue strength coefficient
σi stress in i load cycles
σy yield strength
τ delay time
σ Cauchy stress tensor
ε′f fatigue ductility coefficient
εa strain amplitude
εel elastic strain
εpl plastic strain
εth thermal strain
a0 averaging distance
b fatigue strength exponent
c fatigue ductility exponent
fL fraction of liquid
g gravity
k thermal expansion coefficient
km stress magnification factor
rre f reference radius
s support factor
sR ratio between residual stresses and ultimate strength
E Young’s modulus
G shear modulus
G thermal gradient
H hardening law coefficient
HAZ heat affected zone
K bulk modulus
N hardening law exponent
R growth rate
SAW submerged arc welding
T temperature
b hardening slope
k thermal conductivity
m slope of S-N curve
r radius
t time
v velocity
yeg phase proportion
1
1 Introduction
The cyclic-loading-induced fatigue is one of the main restricting design features
for dynamic welded structures. Welded joints have a deteriorating effect on the
fatigue life of structures. Joints induce discontinuities in geometry, microstruc-
ture and stress state, which leads to a reduced fatigue strength. Weld fatigue
assessments are traditionally based on an idealized geometry because it is prac-
tically impossible to predict the actual weld geometry. The geometry idealization
is verified to work well with thick weldable parts and traditional welding meth-
ods, but it suits poorly thin weldable parts and laser welding. For laser welded
joints, there are no straightforward fatigue strength assessment methods.
Laser welding demand is rising in the industry, and that is why there is a need
for a precise fatigue assessment method for laser-welded joints. Nominal stress
methods lead to conservative results, and more precise methods, such as the
notch stress method, require additional modelling of the joint. Moreover, mod-
elling of the idealized geometry of laser welded joints is more challenging than
traditional because the laser welded joints do not have clearly noticeably weld
bead form.
The forming and characteristic features of deep penetrating laser welding con-
cerning fatigue are dealt for understanding the fatigue of the joint. The effect
of forming microstructure, residual stresses and strains, distortion of joint and
impurities are discussed from the perspective of fatigue. The common fatigue
assessment modes are presented, and the suitability for laser welding is eval-
uated. The suggested fatigue model is based on simulation of welding, and
thus the modeling and simulation of laser welding is presented. A commercial
program for fatigue assessment of laser joints is shown.
In fatigue tests, a large dispersion of fatigue strength was noted for laser welded
joints. Multiple authors explain this dispersion with a geometry variation where
other defects that affects the variation are excluded. In the study, in addition
to the effect of residual stresses, the size and probability for defects that have a
deteriorating effect on fatigue are statistically obtained.
2
2 Laser welding process
2.1 Laser and applicability for welding
Laser is applicable to welding if the sufficient power density and wavelength are
reached (Kujanpää et al., 2005; Ion, 2005). The power density needs to be high
enough to melt the weldable material. The laser welding can be performed in
two different modes: deep penetrating and heat conduction welding. The deep
penetrating mode, or keyhole welding, is the more common mode. Absorption
goodness of the laser is dependent on the metal and wavelength (Katayama,
2013). The most common laser welding lasers are the carbon dioxide laser,
Nd:YAG, diode, and fiber laser (Kujanpää et al., 2005).
2.1.1 CO2 laser
The carbon dioxide laser, CO2 laser, is one of the most common high power lasers
for welding. The laser beam is developed on gas containing carbon dioxide, ni-
trogen and helium (Kujanpää et al., 2005). The laser is produced by stimulation
of laser active medium gas mixture by an electrical discharge in the gas. The
nitrogen is excited by electron impact and collision energy between N2 and CO2
transfer energy in carbon dioxide. Carbon dioxide loses energy by photo emis-
sion, and thus the population inverse that is necessary for laser is achieved. The
CO2 laser is widely used in industrial laser welding due to its simplicity and
reliability (Katayama, 2013).
2.1.2 Nd:YAG
Nd:YAG is a solid-state laser (Ion, 2005). In Nd:YAG, the active medium is a solid
host material of ytterium-aluminium-garnet, YAG, doped with neodymium ions,
Nd3+ (Kujanpää et al., 2005; Ion, 2005). The forming laser has a short wavelength
and thus can be delivered through optical fibers unlike in CO2 where mirrors
need to be used (Katayama, 2013). The Nd:YAG laser beam is formed within
laser rods which can be connected in parallel or in a chain to provide sufficient
power.
3
2.1.3 Diode
A diode laser is a semiconductor-based laser (Ion, 2005). In the last decade,
diode lasers have been an object of development (Katayama, 2013). The active
medium of a diode laser is a semiconductor p-n junction (Kujanpää et al., 2005).
The power of the laser can be increased with joining multiple diode lasers. The
beam can be transferred within optical fibers.
2.1.4 Fiber laser
In a fibre laser, the laser beam is emitted directly inside an optical fibre (Kujan-
pää et al., 2005). The active medium is the optical fibre doped with ions. The
advantage in a fibre laser is that the laser beam is emitted directly in an optical
fibre.
2.2 Benefits of laser welding
Small and focused energy input which is characteristic to laser welding, leads
to a high power density. The most influencial parameters in welding are speed
and power which can be written in form of heat input as energy per length
unit, J/mm. High power density and high penetration lead to low heat input
and narrow heat-affected zone. The heat affected zone, HAZ, is smaller in laser
welding than in traditional welding and has a better microstructure (Ion, 2005;
Moraitis and Labeas, 2008; Cho et al., 2004; Liu et al., 2017). Low heat input
leads to rapid solidification which imposes smaller grain size (Ion, 2005; Remes,
2008). Moreover, fast solidification reduces embrittlement segments like sulfur.
Laser welded joints have more beneficial large homogeneous islands of high
yield strength martensite. Low heat input leads also to low residual stresses and
distortions (see chapter 3.3).
One main benefit of laser welding is the welding speed which is significantly
higher compared to traditional welding processes (Cho et al., 2004; Yang and
Lee, 1999; Gao et al., 2016). In additional, the laser welding process is easily
automatized (Cho et al., 2004; Caiazzo et al., 2017). With laser-welded joints,
such as lap joints, are easy to control, and therefore joints such as rivet joints
and spot weld joints can be made with continuous welding. The joints can be
welded through from one side making a complete seam with one pass. This
4
provides many opportunities for the design of lightweight structures such as
sandwich panels. (Remes, 2008; Zain-ul Abdein et al., 2009). Sandwich panels
generally have a very good weight to strength relation.
2.3 Challenges in laser welding
Laser welding is almost always done by an automated process, which is why
the process needs to be preconfigured and programmed. In order to welding
sufficiently, the process programmer needs to have good expertise. With good
parameters, the quality of welding is significantly better (Jiang et al., 2016; Ai
et al., 2016).
Laser welding restrains the design of the structure. The most advantageous con-
nection type for laser-welded joints is the overlap joint. The challenge with joints
like T-joints is to focus the welding line on the right place. Jig or equivalent is
needed in laser welding because the process is automated. Gaps between weld-
able parts affect highly the overall quality of welding, which creates challenges
for the jig design. The challenges are even greater with large structures. When
designing laser-welded products, the designer needs to take manufacturing into
account.
The machine base is more demanding for laser welding compared to traditional
welding processes (Kujanpää et al., 2005). The laser needs to be controlled by
a CNC machine or equivalent. In laser welding, safety factors are also different
from conventional welding. A laser beam is harmful especially for eyes and
special protection glasses are needed. For protection, access to the welding area
needs to be restricted for outside people.
From the fatigue point of view, the laser welding challenge is the very large
dispersion in fatigue strength of joints.
2.4 Visuals of a laser welded joint
A laser-welded joint is more visually sound and unnoticeable compared to a
traditional weld joint. The weld bead is minimized because additional material
is not generally added in laser welding. Laser welding in consistently performed
as deep penetrating welding. In figure 1, an example of weld bead’s cross-section
shape is shown for submerged arc welding and laser welding.
5
Figure 1: Difference in heat-affected zones between SAW and laser welding.
Modified from Remes (2008)
2.5 Physics of laser welding
For deep penetrating laser welding, the term key hole welding is used (Ion,
2005). The term keyhole comes from the shape of the weld where the laser beam
penetrates deeply in to the material and a molten pool is formed on top of the
weldable part. The principle of the key hole is shown in figure 2.
Figure 2: Principle of deep penetrating welding. Figure taken from Ion (2005).
2.5.1 Forming of keyhole
In laser welding, the heat is inducted to a part with a laser beam leading to
a small and penetrating, focused spot of energy (Ion, 2005). The laser beam’s
energy is absorbed into a surface that is in contact even when the keyhole cavity
starts to form. Laser beam heat initially melts the material to be welded. The
material starts to vaporize when a sufficient amount of heat is conducted to the
metallic material to break atomic bonds and further generates plasma (ionized
6
vapour) of vaporized metal atoms and the surrounding gases when electrons are
removed (Ola and Doern, 2015).
The energy density needs to be around 1 MW/cm2 in welding (Moraitis and
Labeas, 2008; Ion, 2005). The penetrating cavity is formed by the depression of
the liquid surface as the laser “drills” trough the part (Lin et al., 2017; Moraitis
and Labeas, 2008; Ion, 2005). The joint is formed when the laser beam and thus
the keyhole move through a welding line followed by a molten the pool where
the material is not yet solidified. The keyhole is held open by recoil pressure of
evaporation of particles that pushes the surrounding molten material (Ola and
Doern, 2015). The characteristic mode of deep penetrating welding is beneficial
compared to traditional welding where only a shallow molten pool is formed.
Laser welding involves multiple physical phenomena, starting from absorption
of surface, metal melting, keyhole formation, keyhole plasma formation, laser-
plasma interaction, laser-cavity hole wall interaction, vapour and liquid pres-
sure, metal solidification, etc. (Katayama, 2013). Laser welding can be done
without protection gas, but it makes the process more unstable. Keyhole welding
can be done with filler material, and the process is called hybrid laser welding.
2.5.2 Energy absorption
The energy of the laser is absorbed within an evaporated region through two
mechanism: inverse Bremsstrahlung and Frensnell absorption (Ion, 2005). At a
low welding speed, the inverse Bremsstrahlung is dominant where energy absorp-
tion takes place in the plasma formed over the keyhole. In Fresnel absorption,
multiple reflections at the walls of the keyhole transfer energy from the beam.
Absorption is dependent on the polarization of the beam and its efficiency is
dependent on how well the beam, is in touch with the surface (Lin et al., 2017).
2.6 Efficiency
The total efficiency is dependent on the energy conduction of the material. Pro-
tective gas forms plasma in the laser-gas interaction, which makes the keyhole
more stable and radiates heat energy to the weldable part (Moraitis and Labeas,
2008; Ion, 2005). The plume (vaporized material) and plasma have a defocus-
ing effect on the laser beam, and thus can affect the efficiency of the process,
7
even changing the welding mode from deep penetrating to head conduction (Lin
et al., 2017). The material reflectivity of metals have an effect on keyhole origi-
nation (Sun and Ion, 1995). For butt-joint laser welding, the energy losses due
to weldable part imperfections and reflections in approximately 30 % (Tsirkas
et al., 2003).
2.6.1 Stability of keyhole welding
The stability of the keyhole is dependent on two major forces affecting the key-
hole region: ablation pressure, Pabl, and surface tension, Pv (Ola and Doern,
2015; Ion, 2005). The ablation force tends to open the keyhole and surface ten-
sion pressure to close it. Stability is also dependent on welding speed (Alcock
and Baufeld, 2017). With low welding speeds, the heat input is larger, leading
to a relatively large width of the molten pool. The width of the molten pool
decreases when the welding speed is increased. With a low welding speed, the
larger heat input may cause unstable welding and spatter. The stability of the
solidification front, the back of the molten pool where liquid metal starts to
solidify, determines the kind of microstructure that forms (Liu et al., 2017).
2.6.2 Molten pool flows
Welding forms a molten pool which, when solidified, forms a welding joint.
The high temperature differences and temperature gradients induce flows in
the molten pool. Flows have an effect on the bead form, porosity formation,
inclusion, etc., and thus also on quality of welding. More on molten pool flows
and modelling of molten pool flows in chapters 3.5 and 6.1.1.
2.6.3 Parameters in keyhole laser welding
Parameters have a significant effect on the quality of the weld in laser welding
(Jiang et al., 2016). Parameters affect the quality and profile of the weld bead
as well as the whole welding process. Parameters are often determined by the
welder’s professional skills, charts or by the method of trial and error (Ai et al.,
2016; Jiang et al., 2016). The method of trial and error is a waste of resources,
and it can often lead only to a sub-optimal solution. The laser welding process
is automatic, therefore the use of parameters is efficient through structure if the
8
material and thickness are constant. Parameters can be optimized with numeri-
cal simulation. The parameter optimization is based on visual observation of the
weld, and it can lead to a visually sound weld. This can still include porosity,
collapse, undercut, root humping, etc. (Jiang et al., 2016). Inner defects affect
highly the quality of the weld.
Laser beam quality is evaluated with parameters such as Rayleigh length, inten-
sity, divergence, coherence, etc. (Ion, 2005) (Kujanpää et al., 2005). The foremen-
tioned parameters are not modified by the welder and therefore not processed.
The three main parameters are the laser power, LF, welding speed, WS, and
focal position, FP. Laser power increases the weld bead depth and width as the
welding speed decreases as mentioned before (Jiang et al., 2016). This is due to a
simple analogy: if the brought heat amount is great and heat has time to absorb
to the material, the molten region increases. The focal point is a point where
the laser beam’s focal point is focused. The focal point is usually inside of the
welded part. The focal point affects the depth of the bead more strongly than
its width. The effect of parameters on weld bead width, BW, and depth, DP, for
stainless steel 316L are shown in figure 3.
Figure 3: The effects of focal point (LF), laser power (LP) and welding speed
(WS) on bead profile when welding stainless steel 316 L. Figure taken from:
Jiang et al. (2016).
Heat input affects the bead geometry (Liu et al., 2017; Tsirkas et al., 2003; Caiazzo
et al., 2017). With less heat input, the weld bead takes a nail-shaped form where
a relatively small amount of metal is melted, which results in a narrow transverse
form. With increasing energy, the width of the molten pool increases, resulting
in a V-shaped form. The Marangoni effect starts to appear when the molten
pool widens (see chapter 2.6.2). With an even larger heat input, a peanut-shaped
9
bead forms, where the welded pool extends to the root side of the weld. The
root side welded pool widens the weld bead as the metallic vapours emit from
the bottom. The mentioned bead shapes are shown in figure 4.
Figure 4: Different weld bead geometries: Peanut shape, nail-shape and V-
shaped. Figure modified from Liu et al. (2017)
Weld bead soundness improves the quality of the weld. Poor welding param-
eters might lead to an incomplete weld joint. The heat input effect on fatigue
strength is shown in figure 5. The figure shows that, with a sufficient heat input
a better fatigue strength was obtained in the low cycle region.
Figure 5: The effect of heat input on fatigue strength in S355 MC and Raex 400
steels. The test data is collected from KeKeRa, 2017
10
2.7 Heat conduction welding
If the energy density is below 1 MW/cm2, the keyhole does not form but the
surface of the material still melts and welding can be performed (Kujanpää et al.,
2005). Heat conduction welding is similar to submerged arc welding where the
heat is transfered in to a material by conduction (Ion, 2005). The molten pool
and welding bead is shallower and wider than in keyhole welding. The lower
energy density can be managed by reducing power or increasing the laser spot
area.
2.8 Weldable metals
Laser welding suits all metals that can be welded by melting part (Sun and Ion,
1995; Hiltunen, 2012). Common steel grades, like low-carbon, high-strength low-
alloy and austenitic stainless steels, are readily laser-weldable. Non-ferrous met-
als, like aluminum and copper, can also be welded. The reflectivity of aluminum
and copper complicates keyhole origination. Aluminum welding is challenging
because porosity governs the weld bead easily.
The physical properties of weldable materials affect weldability. Reflectivity of
the material is more critical in laser welding than with convectional arc welding
processes (Sun and Ion, 1995).
2.8.1 Dissimilar material welding
Laser welding suits dissimilar material welding due to its characteristic low and
focused heat input (Yuce et al., 2016). A filler material can be added to the
minimize material mismatch effect (Sun and Ion, 1995).
The use of dissimilar material in structures is in the scope of interest as it offers
new possibilities. The availability of dissimilar material welding is continuously
growing (Ai et al., 2016). The main advantages in the use of dissimilar materi-
als in welded structure are weight reduction, cost saving and greater flexibility
(Yuce et al., 2016; Ai et al., 2016). The fatigue strength of structures can be im-
proved with material choices.
The use of dissimilar material welding brings new difficulties and is more chal-
lenging than similar material welding (Ai et al., 2016; Yuce et al., 2016; Sun and
Ion, 1995). Unsuccessful welding leads to defects like lack of fusion, underfill,
11
porosity, spatter, and others. The strength of a weld bead determines the rigidity
and durability of the weld which needs complete fusion between two materials
at the joint. The welding suitability of dissimilar materials is shown in table 1.
Table 1: Weldability between dissimilar materials. (E=excellent, G = good, F =
fair, P = poor, * = no data available) (Sun and Ion, 1995)
W Ta Mo Cr Co Ti Be Fe Pt Ni Pd Cu Au Ag Mg Al Zn Cd Pd
Ta EMo E ECr E P ECo F P F GTi F E E G FBe P P P P F PFe F F G E E F FPt G F G G E F P GNi F G F G E F F G EPd F G G G E F F G E ECu P P P P F F F F E E EAu * * P F P F F F E E E EAg P P P P P F P P F P E P EMg P * P P P P P P P P P F F FAl P P P P F F P F P F P F F F FZn P * P P F P P F P F F G F G P GCd * * * P P P * P F F F P F G E P PPd P * P P P P * P P P P P P P P P P PSn P P P P P P P P F P F P F F P P P P F
12
3 Mechanics of welded joint
Weld joints have always a deteriorating effect on the fatigue strength of an struc-
ture, and they are main reason for fatigue failure (Shen et al., 2017). Weld-
joint-induced discontinuity is the reason for fatigue strength deteriorating (Hob-
bacher, 2009; Remes, 2008; Cho et al., 2004; Radaj and Vormwald, 2013; Nykänen
and Björk, 2016).
3.0.1 Fatigue propagation of welded joint
A weld joint induces discontinuity in geometry and in microstructure. This dis-
continuity can be divined in to defects in the surface, microstructure and inside
of a welded joint. A fatigue crack initiates from the defects and imperfections
(Cho et al., 2004; Remes, 2008; Radaj and Vormwald, 2013; Liu et al., 2017). The
defects and imperfections lead to fatigue of the material because of the induced
stress concentrations that allow crack initiation early in cyclic loading. Geo-
metrical defects are defects such as porosity, cracks, surface roughness and the
weld-geometry-induced notch effect. Grain boundaries, grain size, impurities
and other material properties are microstructure defects. Cracks leading to a
fatigue failure of welded joint usually originates from the area between the base
material and heat-affected zone. In this region, the effect of defects is at its
highest.
In addition to weld bead geometry, fatigue strength is affected by the misalign-
ment of the joint, plate thickness, material, geometry of joint, etc. (Remes, 2008).
The reason for the large variation of fatigue stress in thin plates is the empha-
sized effect of weld geometry and axial misalignment of the joint (Liinalampi
et al., 2016).
Welded joint fatigue strength behaves as materials in general when subjected
to variable loading: periodic overloads increase fatigue life and underloads de-
crease it. Underloads increase the crack growth rate because of tensile residual
stresses. Periodic overloads have a beneficial effect on fatigue strength because
of compressive residual stresses that decrease the crack growth rate.
The applied load affects fatigue durability as the fatigue phenomena originate
from cyclic loading. The load type varies the crack driving force, and therefore
the fatigue strength is different between normal, shear and bending loads (Gallo
13
et al., 2017). For a convectional welding joint, the bending fatigue strength is
better but, in case of a laser welded T-joint, the strength is worse by up to around
two million cycles (Lazzarin and Livieri, 2001; Gallo et al., 2017).
3.1 Laser welding fatigue in general
It has been claimed in the early state of research in laser welding that the fatigue
strength is 50 % better than in conventional arc welding (Remes, 2008). Exper-
iments have exposed differences in fatigue strength between laser welding and
arc welding. The differences are due to different geometry and microstructure.
Laser welding fatigue strength is found to have a large scatter in fatigue tests.
This has been claimed to result from surface roughness induced sharp notches
(Remes, 2008; Alam et al., 2009; Liinalampi et al., 2016; Schork et al., 2017). The
effect is evaluated with a sharp fictitious notch with a radius of ρ f = 0.05 mm
within notch stress assessment (see chapter 5.2.3 ). The usage of a small radius
is justified with the schematics of surface roughness where surface ripples have
a notch-like effect (figure 6) (Liinalampi et al., 2016; Schork et al., 2017). Laser
welding is traditionally used with thin sheets which also have a fatigue strength
scattering effect (Lillemäe et al., 2013).
Figure 6: The schematics of surface roughness in weld root region. Modified
from Schork et al. (2017)
Laser weld bead geometry fits poorly in geometry idealization as such. Remes
(2008) suggested a geometry that is presented figure 7 for butt joints. In the sug-
gestion, the effective radius and sharp V-notch of the welded joint are combined.
14
Figure 7: The idealization of weld bead geometry in the case of defining the
effect of weld dimension to notch stresses. Modified from Remes (2008)
3.2 Fusion zone and heat-affected zone
Microstructure changes have an effect the on fatigue behavior (Leppänen et al.,
2017; Remes, 2008; Kumpula et al., 2017). The terminal load needed to make
welding take place leads inexorably to microstructure changes in the heat-affected
zone. The material hardness rises in the heat affected-zone 8 (Remes, 2008;
Sowards et al., 2017; Cerný and Sís, 2016). The strength of material can be ex-
pressed by a function of material hardness (Vaara et al., 2017; Liinalampi et al.,
2016; Murakami, 2002). The yield strength can be approximated with equation
1 (Liinalampi et al., 2016).
σy = −90 + 2.9Hv (1)
Welds have two separated zones: a fusion zone and heat-affected zone (Ion,
2005). The fusion zone is the region where material is melted and solidified
rapidly. In the heat-affected zone, where are multiple sub-zones that can be rec-
ognized as the result of different peak temperatures and cooling rates. The heat
affected-zone microstructure is dependent on the base material properties. The
heat input form is narrower in laser welding compared to traditional welding,
which leads to narrower HAZ and smaller grain size (Remes, 2008). Different
heating leads to a differing hardness distribution (figure 8). With a narrower
weld and HAZ in a laser weld, also the hardness distribution is narrow and has
a strong variation in distribution.
15
Figure 8: The hardness distribution between convectional arc welded joint and
laser welded joint Remes (2008)
The average grain size in laser welding is smaller than with submerged arc weld-
ing (figure 9) (Remes, 2008). The thermal gradient, G, and growth rate, R, have a
significant effect on the mode of solidification and growth rate (Liu et al., 2017).
The G/R ratio the controls solidification mode and has an impact on the forming
microstructure types, such as columnar dendrites, and fine equiaxed grains. The
forming microstructure is dependent on the welding process quality.
Figure 9: The difference in microstructure in heat affected zone between sub-
merged arc welding and laser welding. Modified from Remes (2008)
In case of high strength steels, the welding can lower the quality of the mi-
crostucture in high strength dual phase, DP, steels, and thus reducing fatigue
strength (Sowards et al., 2017). But for high strength low-alloy (HSLA) steels,
the fatigue strength is significantly increased compared to the base material. The
better fatigue strength with HSLA is due to the formation of martensite caused
by rapid cooling. The advantageous microstructure forming in HSLA steel can
lead to base metal failures in a fatigue test. The high strength for DP steels is due
16
to the mixture of martensite and ferrite in the base metal, and the composition
is softened by welding in HAZ.
3.3 Residual stresses and strains
Welding induces very strong thermal variations that cause thermal expansion
which yields residual thermal stresses and strains in the structure (Tsirkas et al.,
2003; Macwood and Crafer, 2005; Zain-ul Abdein et al., 2009). Thermal stresses
and strains yields residual stresses and strains in the restrained region. In laser
welding, residual distortions and stresses are smaller comparing to traditional
welding processes because of the lower heat input (Moraitis and Labeas, 2008).
The residual stress effect is noticed in literature but ignored in most fatigue
assessments. The effect of residual stresses are bypassed as they are included in
S-N curves. Commonly, residual stresses are assumed to be in the material yield
limit. Cyclic loading decreases the residual stress level due to the combined
effect of the plastic deformation and fatigue damage (Shen et al., 2017).
3.3.1 Forming of residual stresses
In the welding, a process high amount of heat is brought in to the welding
zone, resulting in elevated temperature gradients during both heating and cool-
ing (Tsirkas et al., 2003; Cho et al., 2004; Zain-ul Abdein et al., 2009). The
non-homogeneous heating leads to non-homogeneous thermal expansion fields.
The cooling process leads to non-homogeneous strain fields that yields residual
stresses that remain in structure without any external loads. Residual stresses
are a result of structural self-balancing of non-homogeneous thermal expansion
fields. Residual stress magnitude and distribution is a sum of material composi-
tion, thickness of welded parts, welding parameters, and applied restraints (Cho
et al., 2004). The applied restraint may further increase residual stresses (Zain-ul
Abdein et al., 2009).
The stresses are zero in a molten pool, but the high temperature of the molten
pool results in high compressive stresses around the fusion zone. The stresses in
the molten pool region are shown in figure 10. The heat is absorbed by the sur-
rounding material which radiates heat to the surrounding space. A weld cools
down quickly as the surrounding material acts like a heat sink. The cooling-
17
induced shrinkage results in tensile stresses.
Figure 10: The stresses in the molten pool region that yields residual stresses
in restrained regions. The high temperature of the molten pool results in high
compressive stresses.
3.3.2 Effect of residual stresses
Residual stresses increase the maximum stress and mean stress levels, thus re-
ducing the fatigue life (Shen et al., 2017). The presence of residual stresses has an
effect on material behavior in cyclic loading. The residual stresses affect stress
distribution. The residual stress decreases the fatigue limit if the stresses are
close to the tensile yield limit (Radaj et al., 2006). Usually, residual stresses are
assumed to be the tensile stresses in material yield limit (Radaj et al., 2006). The
residual stresses close to tensile yield limit decrease the fatigue strength because
the ultimate tensile strength is decreased (Nykänen et al., 2017). The assumption
of zero residual stresses would lead to fatigue problems (Carmignani et al., 1999;
Nykänen and Björk, 2016). In view of crack growth, tensile residual stresses in-
crease crack driving force while compressive residual stress decreases it (Ninh
and Wahab, 1995).
Residual stresses can be calculated by modeling welding (see chapter 6) (Ninh
and Wahab, 1995; Zain-ul Abdein et al., 2009). The residual stress distribution
over a line in transverse direction over a weld line is plotted in figure 11. The
longitudinal stresses, σxx, have the strongest effect on fatigue strength (Zain-ul
Abdein et al., 2009). Longitudinal stresses are entirely tensile, and transverse
stresses, σyy, are mostly compressive. The trough thickness stresses, σzz, have a
very small magnitude. The shear components, σxy, σxz, and σyz have a negligible
magnitude and thus are not plotted.
18
Figure 11: vonMises stress field and main stress components in transverse direc-
tion. Residual stress components taken from Carmignani et al. (1999)
3.3.3 Controlling residual stresses
Residual stresses can be reduced with external treatments like preheating and
hammer-, needle- and shot-peening (Macwood and Crafer, 2005; Nykänen et al.,
2017). The preheating reduces thermal strains and thus also distortions and
residual stresses. With impact methods, plastic deformation is produced in or-
der to relieve formed residual strains. The aim is to improve fatigue strength
of welds by mechanically modifying the residual stress state. The IIW standard
allows fatigue class stress limit increase if residual stresses are reduced (Hob-
bacher, 2007).
3.4 Distortion of joint
The welding-induced thermal strains result in the distortions in as-welded state
(Remes, 2008; Lillemäe et al., 2013). Axial and angular misalignment decreases
the fatigue strength of the weld joint. Misalignment of welded joints causes
stress rising by an additional bending component and the decrease of nomi-
nal surface in normal direction. The angular misalignment induces additional
stresses as a secondary bending stress. The effects of misalignment are most
evident in butt-joint welds and cruciform welds where the increase in stress can
be 30 % to 45 % (Hobbacher, 2009).
The IIW regulations offer a stress magnification factor km to deal with misalign-
ment. The magnification factor takes axial and angular misalignment and plate
thickness into account. Some axial allowance for misalignment is already in-
duced in the FAT classes.
19
3.4.1 Effect of plate thickness
Thin-plate welding leads to larger different initial distortions in comparison of
thicker plates (Lillemäe et al., 2013). Due to the lower bending stiffness of thin
plates, the initial distortions close to the weld are curved. The curved shape
in the weld region makes angular misalignment determination difficult. In tra-
ditional rule-based fatigue assessments, the welded geometry is idealized and
thus misalignments are obsolete. The idealization is suitable for thicker plates
but suits poorly thin plates. The response of thin plates is strongly and nonlin-
early depended on the distortions and magnitudes. The amount of distortions
can be reduced by adding axial tensile loading or by sucking plates into stiff
surface (Lillemäe et al., 2013; Zain-ul Abdein et al., 2009). In the IIW recommen-
dations the plate thickness can be managed by using a shallower slope in the
S-N curve. The S-N curve slope of m=5 for normal and m=7 for shear stress is
suggested in literature for thin and flexible structures (t<5 mm) (see chapter 5)
(Nykänen and Björk, 2016; Malikoutsakis and Savaidis., 2014; Hobbacher, 2009).
3.4.2 Impurity and other imperfections
As stated above, the micro crack initiation leading to macro crack and fatigue
failure begins from imperfections in the weld region. Although the surface de-
fects are more critical, the imperfections have an effect. Porosity is more severe
in metals like aluminium that are poorly suitable for welding (see chapter 2.8).
Material properties and welding conditions may lead to hot or cold cracking in
the weld region (Katayama, 2013). Hot cracking is called solidification crack-
ing (Remes, 2008; Katayama, 2013). Solidification cracking can occur along
grain boundaries. Cracking is a result of low solidification temperature films
along grain boundaries. Cold cracking occurs below 300◦C (Katayama, 2013).
Cold cracking or delayed cracking is affected by the hydrogen content, residual
stresses and different hardness regions in the weld.
3.5 Porosity
In the deep penetrating mode of laser welding porosity defects are frequent
(Katayama, 2013). Porosity has an effect on fatigue strength as it reduces the
effective bearing volume and causes stress concentrations with irregular poros-
20
ity shapes (Lin et al., 2017; Katayama, 2013). Keyhole-induced macro porosity
reduces the quality of weld significantly, as it can result in aforementioned loss
of mechanical strength, creep and corrosion failures (Ola and Doern, 2015). The
presence of pores leads to notch-like defects inside the material that provide po-
tential spots for crack initiation (Shen et al., 2017). Fatigue cracks initiate from
pores with the maximum size regardless the distribution. The S-N curves in-
clude the influence of porosity in general as they are the combined result of
multiple experiments.
3.5.1 Porosity formation
Porosity is the result of keyhole fluctuation and molten pool flows (Meng et al.,
2013; Lin et al., 2017; Ola and Doern, 2015; Katayama, 2013). The fluctuation
of a keyhole leads to bubble formation at the bottom of the keyhole (Lin et al.,
2017; Meng et al., 2013; Ola and Doern, 2015). Porosity is formed when induced
bubbles are being captured by a solidification front. Porosity formation can be
prevented by interrupting one of the aforementioned tree steps
The size and shape or more closely the depth of a keyhole is related with the
stability of the keyhole. When a keyhole fluctuates violently, evaporation at
the keyhole walls does not occur uniformly but rather concentrates on bumps
formed in the keyhole wall (Zhang et al., 2004). Very focused evaporation is
formed in the keyhole wall leading to strong evaporation jets and bubble form-
ing at the bottom of the keyhole. The violent melt flow of the molten pool behind
the keyhole is pursuing to close the keyhole and intense evaporation is enabling
partial collapsing.
Melt flow is downward in front of the keyhole and rearward at the bottom of
the keyhole. In a stable keyhole, a clockwise vortex is formed by the impact of
liquid metal coming from the bottom of the keyhole. A downward flow behind
the keyhole and on the top of the molten pool is driven by the resultant force
of recoil pressure, gravity and hydrodynamic pressure (Lin et al., 2017). The
collapse of the keyhole and bubble formation is shown in figure 12. The molten
pool flows are floating bubbles from the bottom to the surface of the molten
pool. The strong vortex in the molten pool flows have an effect on the bubbles
trajectory and make escaping more difficult (Meng et al., 2014).
21
Figure 12: Bubble forming in bottom of the keyhole. Modified from Lin et al.
(2017)
Surface treatment such as zinc-coating of steel, leads to porosity problems (Zhang
et al., 2004). A visually sound weld can be obtained with surface treated steels,
but zinc gas gets trapped in the fusion zone, forming porosity. Stainless steel
and aluminium welding easily leads to porosity, spatter and other defects. Alu-
minium is porosity sensitive because of the low boiling point of the Al alloy (Lin
et al., 2017).
3.5.2 Effect of gap between weldable parts
A gap between welded parts has great influence on porosity formation as the
beam energy is not uniformly absorbed because of the gap (Meng et al., 2013).
The effect of a gap is more severe with thin plates where even small gaps are a
large portion of plate thickness. The results according to Meng et al. (2013) are
shown in figure 13 from a study where they studied the effect of gap in T-joints.
A zero gap has no effect on porosity formation, while a larger gap has a strong
effect. With even a larger gap, porosity is absent in joint. Porosity is formed by
the fluctuation and instability of a molten pool which are disturbed by a gap.
Porosity decreases when a gap is increased because the gap allows the keyhole
plumes to escape.
A molten pool is more stable but shallower with a large gap because a large
amount of heat escapee with plumes from the gap. Also bubble forming is
difficult, therefore no porosity is formed. With a large gap, the molten pool and
weld bead “drops” in to the gap.
22
Figure 13: The effect of gap in T-joins. Figure shows gap effect on forming
porosity and weld bead shape. Figure taken from Meng et al. (2013).
3.5.3 Effect of parameters
Porosity is found to be related to the keyhole depth-to-width ratio (Katayama,
2013). The heat input is the most influential parameter in the formation of poros-
ity (Caiazzo et al., 2017). The increase of heat input (ratio between welding
power and welding speed) increases the porosity. The focal position has a de-
creasing effect in aluminium welding (Ola and Doern, 2015).
3.5.4 Porosity preventing
Heat input is in the key role in porosity prevention. Low heat input decreases
porosity formation (Meng et al., 2014; Lin et al., 2017). Low heat input with a low
speed lead to a shallow molten pool where bubbles easily escape, and therefore
porosity is reduced. But the shallow molten pool welding mode is against laser
welding advantages. On the other hand, a high welding speed also reduces
porosity (Lin et al., 2017). A long and large molten pool is beneficial for bubbles
to escape because flows are more laminar. Turbulent flows promote porosity
formation. A long and large molten pool has a large momentum to resist any
interrupting forces, protecting from turbulent flows. With lower welding speeds,
the beam energy evaporates to both the front and back wall of the keyhole,
leading to strong turbulent flows.
Shielding gases have an effect on the stability of the keyhole and thus on poros-
ity formation. The shielding gases may alter material properties, and different
shielding gases work better for some metals (Han et al., 2005). For example, N2
promotes porosity formation in an aluminium alloy because of formation of AlN
23
in the weld metals, but N2 reduces porosity significantly for 304 Stainless steel
(Sun et al., 2017). N2 bubbles dissolve in steel and thus increase the nitrogen
content negligible.
The laser beam angle has an effect on keyhole dynamics and therefore porosity
formation (Sun et al., 2017). The lower inclination angle promotes a deep and
narrow shape of the molten pool, which makes it easy for bubbles to escape. Lin
et al. (2017) simulated the angle of laser beam and stated that, with high angles,
the porosity forming is decreasing.
Other factors to be considered in porosity formation preventing are double-
spot welding, oscillation of beam, pulse modulation, laser-hybrid processes and
welding under vacuum (Lin et al., 2017). Porosity can be reduced to minimum
with correct welding parameters.
3.5.5 Standards and regulations
In regulations, acceptable levels for porosity and other imperfections are in-
cluded in tables of classified structural details and S-N curve norms. In the IIW
standard, porosity is combined with other imperfections and dealt as a single
large imperfection (Hobbacher, 2009). For crack-like imperfections, IIW uses
idealized elliptical cracks for which stress intensity factors are calculated. Poros-
ity is given as the maximum length of inclusion for fatigue classes. For example,
the maximum length of inclusion for fatigue class 100 is 1.5 mm and the porosity
area is limited to 3 %, and with a lower fatigue class FAT 63 it is 35 mm with
5 % of area. The permitted porosity is included in fatigue tests and a designer
needs to trust the welder’s professional skills and that the porosity is within the
standard tolerances.
3.6 Inclusions
Inclusions, such as oxides or nitrides, are formed from an oxidized surface or
from shielding gas reactions in welding (Katayama, 2013). Oxides are some-
times harmful to ductility in a weld of martensitic and austenitic stainless steel.
Oxide formation improves the ductility in a fusion zone by easing ferrite phase
formation. Nitrites have a reducing effect on cold crack formation.
24
3.7 Imperfections on keyhole form
Keyhole defects such as incomplete penetration, incomplete fusion and lack of
fusion, are controlled with welding conditions and parameters (Katayama, 2013;
Liu et al., 2017). Incomplete penetration is a consequence of incorrect heat input.
A sufficient weld is more difficult to obtain with materials with high reflectivity
or high thermal conductivity. The heat input affects the shape of the forming
weld and HAZ (Liu et al., 2017).
Notable welding defects can be geometrical defects such as humping, undercut-
ting or underfilling. Humping forms due to a periodic backward flow of the
melt and can be visually observed as periodic humps on the surface of a weld
bead (Katayama, 2013). The backward flows leading to humping are a result of
uneven plume ejection on keyhole walls by vaporation. Undercutting is caused
by a slow welding speed with a high heat input and the properties of the metal
to be welded (Katayama, 2013). In an undercut, the weld bead is not formed
soundly on the surface of the welded parts but rather forms grooves alongside
weld bead. The grooves are formed at the border of the fusion zone. In under-
filling, the weld has a concave surface form (Katayama, 2013). Underfilling can
be a result from spatter, gap or shortage of filler wire. Underfilling occurs easily
if a gap between welded sheets is present (Meng et al., 2013).
25
4 Material behaviour in cyclic loading
In static linear finite element analysis, material behavior is assumed to follow
Hooke’s law (equation 2). The linear material behavior is acceptable when dis-
placements are assumed to be infinitesimal. When the applied load is sufficient,
displacements increase and the material starts to behave non-linearly. The non-
linear behavior of the material affects strongly the stress and strain behavior.
Nonlinear behavior must be taken into account in the static analysis if stresses
exceed the yield limit. In cyclic fatigue analysis, the nonlinear behavior needs to
be taken into account.
σ = C : εel (2)
In the theory of plasticity, strain is decomposed in an elastic part, εel, and a
plastic part, εpl. The increments are rate-dependent. The rate and direction
of a plastic strain are determined with the flow rule. With the assumption of
associated flow, the direction of the plastic flow is perpendicular to the yield
surface. The plastic flow is the dependent on magnitude of the yield surface.
The strain decomposition in rate form is presented in equation 3 (Abaqus 2016
Theory guide).
εa = εel + εpl (3)
The transformation from the elastic to plastic part is defined with a yield func-
tion. A yield function defines the yield surface: the boundary between elas-
tic and plastic regions. A yield function is dependent on the stress and yield
stress of the material. The yield surface definition by von Mises is in equation 4
(Abaqus 2016 Theory guide).
f (σ, σy) =
√32
S : S− σy (4)
S = σ− 13
tr(σ)I (5)
where S is the deviatoric part of stress tensor, : is the double contraction, σy is
the yield stress, tr is a trace operator and I is an identity tensor.
26
In the flow rule, the direction of the plastic flow is determined with the Yield
surface. The direction of flow, ∂ f∂σ , is the normal direction in relation to the yield
surface. The equivalent plastic strain rate, εpl, is defined as follow:
εpl = λ∂ f∂σ
(6)
where λ is the plastic multiplier. The plastic multiplier corresponds with the
magnitude of the plastic strain rate.
The material yield strength exceeding develops plasticization. Plasticization af-
fects material properties, such as the yield strength, by decreasing or increasing
it. Hardening increases and softening decreases yield strength. The hardening
is accounted by two models: isotropic and kinematic hardening.
Isotropic hardening describes the change of the yield surface size and kinematic
hardening the change of the yield surface location. Schematics of von Mises
isotropic and kinematic hardenings are shown in figure 14. In the figure, the
red line represents the original yield surface and the dashed line the surface
transformation due to hardening. Hardening is a complex phenomenon, and its
modelling requires a combination of isotropic and kinematic hardening.
Figure 14: von Mises yield surface with isotropic and kinematic hardening.
The isotropic hardening increases the yield surface (fig. 14(a)). In isotropic hard-
ening, the material yield strength increases in both tension and compression. In
isotropic hardening, the material elasto-plastic behavior can be modeled with
the exponential laws after Voce or equivalent (Abaqus 2016 Theory guide). The
increase of the yield surface is dependent on the plastic part of strain.
σ0 = σ|0 + Q∞(1− e−bεpl)
(7)
27
where the σ|0 is the yield surface at zero point plastic strains, Q∞ is the maxi-
mum hardening, b is the hardening slope and εpl are the equivalent plastic strain.
The kinematic hardening account for the movement of the yield surface (fig.
14(b)). In the kinematic hardening affect of Bauschinger, the effect and ratcheting
are taken into account. The kinematic hardening is modeled by taking into
account the plastic behaviour in respect of "kinematic shift", α (Abaqus 2016
Theory guide). The pressure-independent yield surface is defined by function
(Abaqus 2016 Analysis User’s guide)
f (σ− α) = σ0 (8)
where σ is the stress tensor, α is the backstresses and σ0 is the yield surface. The
function with respect to von Mises yield surface is defined as follows (Abaqus
2016 Theory guide)
f (σ− α) =
√32(S− αdev) : (S− αdev) (9)
where S is the deviatoric part of stress tensor and αdev is the deviatoric part of
backstress tensor.
The overall backstress includes multiple backstresses. The evolution for each
backstress is defined after equation 10 where temperature and field variables
are omitted.
αk = Ck ˙εpl 1σ0(σ− α)− γkαk ˙εpl (10)
where Ck and γy are material parameters and εpl is the equivalent plastic strain
rate. Ck is the initial kinematic hardening modulus. γy defines the decreasing
rate of kinematic hardening modulus when plastic strain increases. The "recal"
term, γkαk εpl introduces nonlinearity in the evolution law. The overall back-
stresses are defined as follows:
α =N
∑k=1
αk (11)
where N is the number of backstresses.
28
4.1 Material fatigue
Fatigue is material failing due to cyclic loading. Multiple cycles of loading in-
duce flaws, such as cracks, in a material, and they can propagate to a total failure
of a part. Fatigue deteriorates the strength of a part or structure, and therefore
fatigue behaviour needs to be considered for parts or structures under cyclic
loading.
Cyclic loading propagates nucleation of micro-voids and micro-cracks which
develop into macrocracks (Shen et al., 2017). A micro-crack initiate to macro-
cracks that propagate up to the failure of the structure when crack grows up to
a critical size (Nykänen et al., 2017). The macro crack initiation period can take
a significant amount of fatigue life, but the initiation cracks are hard to observe
(Remes, 2008).
The fatigue strength is usually presented as a number of cycles acceptable for a
stress amplitude or S-N curve. The fatigue resistance of materials is determined
with fatigue tests (Korhonen et al., 2017a; Väntänen et al., 2017; Korhonen et al.,
2017b)
4.1.1 Damage models
Fatigue strength determines the lifecycle in cycles. Fatigue propagation can be
calculated with the relation of the number of cycles to the initiation of a macro
crack (de Jesus et al., 2012). This relation is called the damage parameter.
Damage models propose the correction of the damage parameter with a number
of cycles to initiate macroscopic crack (de Jesus et al., 2012). The most well-
known relations are proposals by Basquin (equation 12), Coffin and Manson
(equation 14) and Morrow (equation 14) (de Jesus et al., 2012). The proposal
by Basquin predicts fatigue with elastic behavior and by Coffin-Manson with
plastic behaviour. The Morrow’s relation is a combination of elastic and plastic
relations. The principle of relations is shown in figure 15.
∆σ
2= σ′f
(2N f
)b (12)
∆εpl
2= ε′f
(2N f
)c (13)
29
∆σ
2=
σ′fE(2N f
)b+ ε′f
(2N f
)c (14)
where ∆σ is the stress range, σ′f the fatigue strength coefficient, N f is the total
number of cycles, b is the fatigue strength exponent, ∆εpl is the plastic strain
range, ε′f is the fatigue ductility coefficient, and c is the fatigue ductility expo-
nent.
Figure 15: The principle of damage relations in elastic and plastic domains (eFa-
tigue, 2017).
SWT and Morrow’s relations take into account the elastic and plastic behaviors.
The Smith-Watson-Topper relation is the most used damage parameter. The
mean stress effect can also be taken into account with Smith-Watson-Topper
(de Jesus et al., 2012; Nykänen and Björk, 2016; Malikoutsakis and Savaidis.,
2014).
σn,max∆ε
2E = σ′f
2(2N f)2b
+ σ′f ε′f E(2N f
)b+c (15)
where σn,max is the maximum stress, and other nomenclatures as before. The
σn,max is the maximum stress in a maximum strain plane.
30
5 Common methods for weld fatigue calculations
The fatigue models are roughly divided in global concepts and local concepts
(Malikoutsakis and Savaidis., 2014; Bruder et al., 2012; Sowards et al., 2017). In
the global concepts, the material nominal stresses are compared to tables for fa-
tigue strength resistance values for different structures. Global concepts require
a nominal stress to be defined. In local concepts, fatigue durability is assessed
within stress concentrations, and therefore local concepts require a more precise
modeling of the weld.
In standards and regulations, such as IIW, EUROCODE 9 and British Standard
7608, the fatigue of welded joints is designed on the basic of standardized S-N
curves corresponding to FAT classes (Hobbacher, 2007; Shen et al., 2017). FAT
classes correspond to permissible stress ranges for different structures. The S-N
curves and FAT classes are defined on the basic of fatigue endurance tests. The
fatigue strength studies of welded joints rely heavily on experimental investiga-
tion. The basic S-N curves corresponding to FAT classes are shown in figure 16.
FAT curves are generally defined with a permitted range at a number of cycles
to potential fatigue cracking. Different FAT classes are offered for nominal, hot
spot and notch stress approaches. A convectional assumption has been that the
fatigue strength has alimit that prescribes the level below which fatigue will not
occur (Hobbacher, 2009). Experiments have shown that fatigue occurs also in
the high cycle region, but the slope of curve in high cycle region is not as steep
as in the region below the knee-point (107 cycles).
31
Figure 16: Fatigue resistance S-N curves for steel according to IIW Recommen-
dations for Fatigue Design of Welded Joints and Components. Figure taken from
Hobbacher (2009)
The fixed values of the S-N curve slope of m=3, m=5 and m=22 are used by norms
and regulations (Hobbacher, 2009; Nykänen and Björk, 2016). The slopes are for
normal and shear stress where the shear stress slope is shallower m=5. The slope
of m=22 is used in the high cycle region after the knee point (107 cycles) in IIW
recommendations, excect for shear stresses with m=5 where the knee point is
assumed to correspond with 108 cycles. The fixed slopes can be justified and ex-
plained with a fracture mechanism, and the slope value is consistent with Paris´
crack growth law exponent. The slope, m, shown in Wöhler curve equation 16
can be fitted to test data with a fixed value or free slopes (Nykänen and Björk,
2016). In standard fitting the curve is fitted with a fixed slope. The design curves
have 95 % survival probability level.
∆σ1
∆σ2=
(N1
N2
)− 1m
(16)
5.1 General presuppositions
In rules and regulations, all similar welded fatigue class joints are assumed to
behave in the same way even though materials and methods vary. Rules and reg-
ulations do not take defects and imperfections individually into consideration.
Also, the same fatigue strength is assumed all steels irrespective of their tensile
32
strength. The stress ratio is also thought to be negligible. These assumptions are
justified because the curves are based on numerous fatigue test results. Design
by standards often leads to conservative results (Bruder et al., 2012; Nykänen
and Björk, 2016).
The common concept is that weld fatigue is due to the geometrical notch effect.
The weld bead forms geometrical discontinuity that has a notch-like effect. The
notch-like effect of the geometry creates considerable stresses in a small area in
the notch region as shown in figure 17. The weld damage is an extremely local
phenomenon where mechanical and geometrical properties have an important
role. Local concepts with the notch stress approach lead to better results than
the nominal stress approach (Nykänen and Björk, 2016; Pedersen et al., 2010).
Figure 17: The stress concentration due to weld bead notch effect.
In notch-based approaches, the influence of the notch brings uncertainties as the
geometry is always an idealization (Liinalampi et al., 2016). A notch represents
a defect on the joint that causes stress concentration. Notch analysis leads to
infinite stresses with a linearly elastic material model. The effect of different
base materials is assumed to be insignificant as the higher notch sensitivity of
high strength steels is assumed to decrease fatigue strength (Nykänen and Björk,
2016).
The suggested range of different notch-based fatigue assessments is shown in
figure 18. The notch stress assessment is made trough a cycle range because the
evaluation is based on S-N curves fitted to test data. The assessment is binary:
the fatigue either occurs or does not occur. Others mentioned are based on prop-
agation of the crack and therefore divided in crack initiation and propagation
periods.
33
Figure 18: The range of notch stress, notch strain and crack propagation ap-
proaches in cyclic loading. The range of assessment is shown for crack size and
for number of load cycles. Figure taken from Radaj et al. (2006).
Fatigue classes recommended by guidelines and standards agree reasonably well
with thicker plates and traditional arc-welds (Pedersen et al., 2010; Bruder et al.,
2012). Thinner plates have a higher sensitivity to the weld bead geometry and
initial distortions than thicker plates (Lillemäe et al., 2013). The raised sensitiv-
ity generates challenges in fatigue design as dispersion is much higher in S-N
curves. The behaviour of thin plates is harder to predict than the behaviour of
thicker plates. The nominal stress method is unsuitable for thin plates with ini-
tial distortions. For thin plates, a shallower slope in the S-N curve is suggested
in literature. In S-N curve slope of m=5 for normal and m=7 for shear stress is
suggested for thin and flexible structures (t<5 mm) (Nykänen and Björk, 2016;
Malikoutsakis and Savaidis., 2014; Hobbacher, 2009).
5.2 Common models introduced
In this section common and some recent fatigue models are introduces. Most
assessments are based on maximum stresses including notch strain approach
where the evaluation for fatigue strength is the maximum stress based on ma-
terial strain data (Radaj, 1990; Remes, 2008; Hobbacher, 2009). The fatigue as-
sessment evaluations are based on the nominal stress in a structure, notch stress,
stress intensity factors, material behaviour or linear fracture mechanics and a
combination of the aforementioned (Radaj, 1990; Ninh and Wahab, 1995; Laz-
zarin and Livieri, 2001; Cho et al., 2004; Radaj et al., 2006; Remes, 2008).
34
Linear fracture mechanics can be applied to weld fatigue with an approximation
of crack in the weld region. Ninh and Wahab (1995) studied residual the stress
effect analytically on laser welds fatigue with the linear elastic fracture mecha-
nism (LEFM). Yang and Lee (1999) studied the weld joint area effect on fatigue
strength and residual stress distributions. Cho et al. (2004) applied residual
stresses calculated with a thermo–mechanical model to the fatigue assessment
with linear fracture mechanics. Dong et al. suggested modernized structural
stress in which the stress field was taken into consideration across the plate
thickness (Radaj et al., 2009). A special approach for thin welded sheets was
suggested by Dermer and Svensson (2001) in the early 2000’s. In this approach,
extra elements were used to act as a weld bead. Lately slopes of S-N curves are
suggested to alter in rules and regulations (Atzori et al., 2009; Baumgartner et al.,
2015; Bruder et al., 2012; Nykänen and Björk, 2016; Liinalampi et al., 2016). At-
zori et al. (2009) referred to fatigue data published within notch stress approach
not to fit the suggested slope of m=3. Bruder et al. (2012) re-analysed a large
number of SAW welded test data with both nominal and local approaches. They
discovered that the slopes of S-N curves vary from 3 to 8 in a low cycle region,
depending on the sheet thickness. Nykänen and Björk (2015) suggested a new
method for curve fitting with a free slope value. Gallo et al. (2017) conducted
that the T-joint has a steeper slope with bending loading. Alam et al. (2009) stud-
ied comprehensively the fatigue cracking of laser hybrid joints. In their study
they concluded that cracks initiate from defects in the weld and on the surface.
They stated that the plate surface in the weld region is rough and allows poten-
tial crack initiation places. Malikoutsakis and Savaidis. (2014) showed the usage
of damage parameters in weld material states. Damage parameters are also used
in a novel notch tress approach (3R) by Nykänen and Björk (2016). The 3R ap-
proach takes into account material plasticity and residual stresses. Shen et al.
(2017) used a continuum fatigue damage model (CDM) for fatigue assessment
that takes into account material defects, like porosity and residual stresses. The
inclusion effect on a weld was also studied by Yates et al. (2002) with the means
of stress concentration.
The notch stress analysis with rre f = 1mm was applied in 1969 within photoelas-
tic materials and in 1975 first with element analysis (Radaj, 1990, p.page). The
notch stress concept was first applied to welded joints by Mattos and Lawrence
35
in 1977 (Radaj et al., 2009). For general use surface notch stress rising effect was
transformed as a notch factors for use in operative fatigue assessments. With
the analytical formulaes, the notch stresses can be defined, but it is significantly
easier to do so with FEM. The rules and regulations incapacity to describe thin
welded parts was recognized in early 2000´s, and the usage of rre f = 0.05mm
was successfully applied to thinner plates and laser welded joints by the middle
of 2000´s (Radaj et al., 2009). Later small reference radia have been validated by
multiple authors (Baumgartner et al., 2015; Bruder et al., 2012; Liu et al., 2017;
Liinalampi et al., 2016; Marulo et al., 2017). A FAT class corresponding to notch
stress analysis was added to IIW fatigue assessment regulations in late 2000´s
(Hobbacher, 2007). The FAT 225 if K f ≤ 1.6 is suggested for notches with radia
rre f = 1mm or, if K f ≥ 1.6, then the FAT class 1.6 · FAT160 is suggested. Peder-
sen et al. (2010) did re-analysis of a large number of fatigue strength results by
using notch stress approach. In their study they concluded that the notch stress
factor is changed to K f = 2.0 for butt joints to have more conservative safety
factors. Pedersen et al. (2010) also stated that parent the material limit K f · FAT
160 is unnecessary low if high strength steels are applied. Marulo et al. (2017)
did re-analysis for a large number of thin laser-welded joints to compare stress
averaging methods. They stated that the Taylor critical distance approach gives
a better scatter band in the results than the Neuber´s stress averaging.
5.2.1 Nominal stress concept
The nominal stress approach is the most common global approach (Malikout-
sakis and Savaidis., 2014). The nominal stress approach is based on the permissi-
ble nominal stresses for an endurance limit given in diagrams (Radaj, 1990). The
concept design is based on the assumption of cyclic design load with constant
amplitude. The nominal stress approach leads to over-dimensioning if the am-
plitude includes large numbers of smaller loads or the component is designed to
be fatigue-resistant in infinite life. This assessment leads to conservative results
(Nykänen and Björk, 2016). The experiments to validate stress ranges have been
conducted with thicker-walled components. This assessment suits thin plates
poorly (Lillemäe et al., 2013). An example of FAT classes and the structural de-
tails corresponding to the FAT classes are shown in figure 19. With different FAT
36
classes for joint types, the stress concentration and joint type durability is dealt
with.
Figure 19: Design S-N curves for nominal stress concept. The (a) shows curves
for different FAT classes and (b) structural details and FAT classes corresponding
to those according to IIW fatigue design recommendations. Figure taken from
Radaj et al. (2006).
The nominal stresses are calculated for a sectional area of the weld region (Hob-
bacher, 2009). The local tress rise due to the welded joint is not taken into
account, but the effect of structural details like large cutouts, shape of structure,
pressure and other macrogeometric factors that have an effect on stress distribu-
tion need to be considered. The axial or angular misalignments can be taken into
account with a stress multiplier factor. The FAT classes include some amount of
acceptable misalignment and other imperfections. Stress determination is based
on linear material and analysis with overall elastic behaviour (Nykänen and
Björk, 2016; Hobbacher, 2009).
5.2.2 Structural hot spot approach
The structural hot spot or geometric stress is an improved version of the nom-
inal stress concept (Malikoutsakis and Savaidis., 2014; Hobbacher, 2009). The
structural hot stress was developed to cover the geometry effects and disconti-
nuities that the nominal stress approach cannot define (Radaj, 1990; Hobbacher,
2009). In the hot spot stress approach, a weld joint is considered to have a
stress concentration on the weld toe. The welds causes non-linear stress peaks
37
in the structure, and the notch effect may lead to singularity in computation.
The hotspot stress is extrapolated from stresses in the weld region. With stress
extrapolation, the non-linear stress peak can be excluded. Thus the approach
includes a weld stress rising effect in the structure but removes the weld profile
itself. The principle is explained in figure 20.
Figure 20: The schematics of structural hot spot stress
Originally structural hot spot stresses were determined by measuring a ready
structure (Radaj, 1990). The structural hot spot evaluation is done with the finite
element method where shell and plate elements can be used. (Hobbacher, 2009).
The analytical evaluation of structural stresses is difficult because parametric for-
mulas are rarely available. Extrapolation can be easily done from element nodes
with a fine mesh and from element integration points with a relatively coarse
mesh. Imperfections can be taken into account with stress multitier factors. The
stresses may vary in the structure, and therefore the hot spot stresses may vary
from the chosen extrapolation path (Bruder et al., 2012). The assessment identi-
fies only failures starting from the weld toe.
5.2.3 Notch stress concept
The notch stress concept is based on modelling a fictitious notch in the weld toe
or root (Radaj, 1990; Hobbacher, 2009). The modelling of the notch requires more
modelling and knowledge on idealization, especially with thin-plates (Pedersen
et al., 2010; Bruder et al., 2012; Lillemäe et al., 2017). The notch stress approach
has been dominating in welded joints local fatigue assessments with increased
computation power, mainly because the results are more precise (Malikoutsakis
and Savaidis., 2014; Pedersen et al., 2010). The notch stress approach corre-
sponds to a single S-N curve with a fictitious rounding, rre f , of 1 mm according
38
to IIW, which make the utilization simple. The FAT 225 is suggested by the
IIW fatigue design recommendations (Hobbacher, 2009). The selected FAT class
is dependent on the chosen stress criteria where von Mises is suggested (Ma-
likoutsakis and Savaidis., 2014). The notch stress approach has been validated
in IIW only for thicknesses over 5 mm (Hobbacher, 2009). The surroundings
of welds are not considered in the present guidance, and therefore a size effect
has to be given with separate concentration factors. Baumgartner et al. (2015),
Liinalampi et al. (2016) and Marulo et al. (2017) have validated that a smaller
reference radius in idealization corresponds well with a laser welded joint. The
proposal, made by IIW guidelines, for thinner plates (t < 5 mm) is to use a
reference radius of 0.05 mm (Hobbacher, 2007). The difficulties with thinner
plates was recognized over two decades ago as a consequence of a relatively
large radius (Radaj, 1990).
In the notch stress approach, the principle is that the actual weld contour is
replaced with an effective reference radius which is suggested to be 1 mm in the
IIW recommendations. The suggested radius of 1 mm is estimated according
to Neuber’s work and have proved to be realistic for structural steels with a
thickness over 5 mm (Radaj, 1990). The IIW rules and recommendations define
an effective notch as in figure 21.
Figure 21: Fictitious rounding of weld toes and roots according to IIW guide-
lines. Figure taken from Malikoutsakis and Savaidis. (2014).
The reference radius is based on Neuber’s notch idealization where the material
microstructural length and the actual notch curvature are taken into account.
Neuber´s hypothesis of microstructural support was that a macro crack initiates
in a notch crack as soon as the stresses exceed the endurance limit of the material
(Radaj and Vormwald, 2013). The reference radius, ρ f , or fictitious rounding is
depended on the microstructural length, ρ∗, multiplier factor, s, and actual notch
39
curvature ρ. The microstuctural length is based on material properties. Radaj
offered a graph for evaluating the microstructural length (figure 22) for different
materials (Radaj, 1990; Nykänen et al., 2017). The multiplier factor is depended
on the stress or strain hypothesis. The definition of the case-spesific support
effect is shown in table 2. The equation for fictitious rounding ρ f is following:
ρ f = ρ∗s + ρ (17)
Figure 22: Substitute microstructural length for metallic materials dependent on
static strength according to Taylor. Figure taken from Radaj (1990).
Table 2: Support factor, s of the microstructural support effect after Neuber
(Radaj, 1990)
Tensile and bending load
Flat bar Round bar Shear and torsional load
Normal stress hypothesis 2 2 1
Shear stress hypothesis 2 2−ν1−ν 1
Octahedral shear hypothesis & Distortion energy hypothesis 52
5−2ν+2ν2
2−2ν+2ν2 1
Strain hypothesis 2 + ν 2−ν1−ν 1
Strain energy hypothesis 2 + ν 2+ν1−ν 1
The basis of fictitious rounding is that material properties, load type and plate
thickness are taken into account. In the suggested ρ f = 1 mm, the actual notch
radius is assumed to be zero, microstructural length is assumed to be 0.4 mm,
and support factor 2.5. The reference radius of 1mm has been used in fatigue as-
sessment for over two decades, and it is included in commercial software (Baum-
gartner et al., 2015). The microstuctural length corresponds approximately with
40
the cast iron with a yield limit of 240 MPa. The fictitious rounding of 1 mm suits
poorly thin plates where the reference radius is large portion of plate thickness.
The assumption is that a weld has a sharp angle-like notch in the root. The fa-
tigue notch can also be modelled as a V-notch (Remes, 2008). With thinner plates
and especially with laser welded joints, it is justified to use the V-shaped notch.
The stress value of the notch will be finite even if the notch is not V-shaped, and
therefore it is necessary to use a reduced “effective” stress value, σ (Baumgartner
et al., 2015; Hobbacher, 2009). The stress value σ is commonly calculated with
Neuber´s stress averaging or with Taylor´s critical distance (Marulo et al., 2017;
Baumgartner et al., 2015).
Neuber´s stress averaging approach
The notch stresses are singular by nature and therefore approximations are
needed (Hobbacher, 2009; Remes, 2008). The singular nature of linear elastic
stresses at the notch mean that using stresses as such suits poorly structural fa-
tigue assessments. Neuber suggested that stresses should be averaged over a
thin layer from the surface instead of using the maximum stress at the notch
(Liinalampi et al., 2016). The basic idea is shown in figure 23.
Figure 23: Principle of stress averaging with Neuber´s line medhod for effective
stress calculation. Figure taken from Remes (2008).
The support effect is defined in a plane perpendicular to the notch root over
distance a0 (Radaj and Vormwald, 2013). The distance a0 is called material char-
acteristic length, and it should correlate with the material grain size according
to experimental results (Remes, 2008). The effective notch stress σ is defined via
41
Neuber´s line method (Remes, 2008; Radaj and Vormwald, 2013; Baumgartner
et al., 2015):
σ =1a0
∫ a0
0σdy (18)
where a0 is the distance defining area of averaging and σ is the stress distribution
along a line. Length a0 should be about three times the averaged grain size based
on measurable smallest crack length (Remes, 2008).
rre f = ρ + s · ρ∗ (19)
Taylor’s critical distance approach
Taylor’s critical distance approach according to Peterson and Taylor uses a criti-
cal distance for effective stress defining (Baumgartner et al., 2015; Baudoin et al.,
2016; Atzori et al., 2009). Taylor’s critical distance approach is based on a concept
created by Peterson and Taylor. Effective stresses are calculated from a certain
distance away from a singular notch tip. As seen in figure 23, the stress distribu-
tion is a function of the distance from a notch, and therefore the right distance
needs to be choosen.
σ = σ(a) (20)
5.2.4 Notch strain concept
The notch strain approach is a material-mechanism-based local approach (Ma-
likoutsakis and Savaidis., 2014). The fatigue behaviour is based on the material
cyclic loading behaviour in the notch region. The idea of the notch strain ap-
proach is that the mechanical behaviour is comparable in experimental speci-
mens (Radaj et al., 2006). This approach is similar to the notch stress approach
but takes into account material elasticity and crack propagation. The notch strain
approach was originally developed to cover a plainly notched specimen lifecycle
up to initiation of fatigue crack and later applied to welded joints (Malikoutsakis
and Savaidis., 2014; Remes, 2008).
In the notch strain approach, material properties dominate the fatigue life (Ma-
likoutsakis and Savaidis., 2014). The fatigue endurance is investigated with ma-
terial elasto-plastic stress-strain response and failure criteria. The fatigue life of
42
a welded joint is divided in crack initiation and crack propagation in the notch
strain approach (Nykänen and Björk, 2016). As stated in the common model
introduction (see figure 18), the notch strain approach predicts fatigue life only
up to the initiation of a crack as it is, but the fatigue damage can be described
with the linear elastic fracture mechanics approach (Malikoutsakis and Savaidis.,
2014; Radaj et al., 2006). Macro crack propagation can be modelled with the Paris
law or equivalent. The macro crack initiation phase, which is dominant in terms
of fatigue and have significant effect on the total fatigue life, can be described
with fatigue damage models (de Jesus et al., 2012). The result of damage models
is the number of cycles until macro crack initiation (Remes, 2008). Stresses as-
sociated with elasto-plastic strains and parameters relating to fatigue need to be
determined from a comparison specimen that should have the same microstruc-
ture, volume and conditions (Radaj et al., 2006).
The calculation process is more complex than in the notch stress approach. The
material model elasto-plastic properties need to be defined (Radaj et al., 2006;
Malikoutsakis and Savaidis., 2014). The definition of the damage model that
corresponds to crack initiation in cyclic loading, such as Smith-Watson-Topper,
SWT or Morrow, needs to be defined. In the notch strain concept, the mean
stress can be taken into account with a mean stress sensitivity factor, M, or with
a damage model that corresponds to the mean stress. Application of the mean
stress sensitivity factor is done with Haigh diagrams, such as shown in figure
24.
Figure 24: Example of Haigh diagram. Taken from Malikoutsakis and Savaidis.
(2014)
43
5.2.5 Novel notch stress approach (3R)
The novel notch stress (3R) approach is a recently developed method by Nykä-
nen and Björk (2015, 2016) that takes material behaviour into account in cyclic
loading. The approach takes into account the combined effect of residual stresses
(σres), a applied stress ratio R, and the material ultimate strength Rm. The resid-
ual stresses have a significant effect on low-cycle-region fatigue strength which
covers a wide range of the total fatigue life (Remes, 2008). The fatigue strength
can be calculated more precise by taking residual stresses into account (Nykänen
and Björk, 2015). The method is based on a local stress ratio, Rlocal, that depends
on material properties, the residual stress level, and the applied stress level and
range (Nykänen and Björk, 2016). The stress ratio is based on the stress range in
the reference notch (Nykänen and Björk, 2015). The Rlocal is calculated with the
notch strain approach. The effect of Rlocal is obtained with the Smith-Watson-
Topper approach. With the Smith-Watson-Topper damage rule (PSWT), the mean
stress effect is calculated to be is used for assessing the effect of the local stress
ratio to fatigue life. The mean stress correlation factor according to the novel
notch stress method depends on the stress range, stress ratio and ratio between
residual stresses and ultimate strength (Nykänen and Björk, 2015):
f (∆σ, R, sR) =√
1− Rlocal (21)
where ∆σ is the applied stress range, R the applied stress ratio, and s is the ratio
between residual stresses and ultimate strength σres/Rm.
5.2.6 Continuum damage mechanism approach
The continuum damage mechanism approach, CDM, is an approach based on
the mechanical behaviour of material and macroscopic progressive damage (Do
et al., 2015; Shen et al., 2017; Jussila et al., 2017). This continuum-based approach
has been recently applied to welded joints (Do et al., 2015). With CDM, the effect
of residual stresses can be undoubtedly introduced. In the framework of the
CDM approach, the material defects, effect such as porosity, can be taken into
account (Shen et al., 2017).
The approach deals with the mechanical behaviour of a material in a macro-
scopic scale (Do et al., 2015; Shen et al., 2017). The approach is based on the prin-
ciple that plastic deformation occurs in low cycle regime due to cyclic loading
44
and therefore an elasto-plastic constitutive material model needs to be utilized.
The fatigue behaviour is controlled by a damage model. In the CDM approach,
micro-cracks and micro-voids in material are defined as damage variable D. The
damage variable is defined as a measurement of defects like porosity. The dam-
age variable is integrated in a material plasticity model in the CDM approach.
For example, the damage variable is utilized in the elastic law as follows (Shen
et al., 2017):
σ = (1− D)Cel : εel (22)
where D is the damage parameter, Cel is the fourth-order tensor of elastic mod-
uli, and εel is the elastic strain. The term (1-D) is utilized also in material be-
haviour, hardening laws and fatigue damage evolution laws. In case of porosity,
the damage variable D is taken into account with the ratio of defects SD to the
total area S in an isotropic material. The schematic is shown in figure 25.
Figure 25: Schematics of representative volume element (RVE) where the defect
area is taken into account in functional area. Figure taken from Shen et al. (2017)
5.2.7 Stress intensity factors (SIF)
The stress intensity factor (SIF) method is based on the linear fracture mech-
anism (Lazzarin and Livieri, 2001). For welds the notch stress intensity factor
NSIF method is used. In the NSIF method, the weld toe is assumed to be a sharp
V-notch with an opening angle of 135◦. The notch tip is singular as in the notch
stress method. The NSIF method takes into account plane modes I and II. The
stain energy density SED can be utilized in NSIF´s with coarse meshes (Berto
et al., 2017).
IIW recommendations include correlation functions and parameters that depend
on the size and shape of different materials and stress types (Hobbacher, 2009).
45
5.3 Comparison of fatigue models
As stated before, laser welded joints have different properties than those welded
with traditional methods. The laser-welded joint is being reported to have a
wide scatter in the nominal stress range (Liinalampi et al., 2016; Lillemäe et al.,
2017). In laser welding, the additive material is not used, except in hybrid laser
welding. The lack of additional material leads to the fact that the correct param-
eters are in an important role. Usually laser welding is utilized for thin plates.
The effect of plate thickness on fatigue strength and scatter is shown in chapter
3.2. With thin plates, distortions and surface defects have a strong effect.
The fatigue test results collected from literature are shown in table 3. The fatigue
test data is divined in nominal, hot spot, and notch stress. Specimen stress types
are shown in the table. In tube specimens, the stress is parallel shear stress across
weld. The IIW fatigue design regulations assume that all materials have the
same fatigue strength in welded joints regardless the yield strength or ultimate
strength, and thus materials are not displayed separately in the graphs.
46
Table 3: Test results collected from literature
Author Stress Spec. Material Sy Rm R Process Ref.
[MPa] [MPa]
Yang and Lee (1999) Nominal Shear steel 250 320 0.1 CO2 Yan
Cho et al. (2004) Nominal Shear ateel 210 320 0 CO2 Cho
Sharifimehr et al. (2016) Nominal Shear Steel 790 829 0.1 - Sh0
Sharifimehr et al. (2016) Nominal Tensile Steel 790 829 0.1 - Sh1
Liinalampi et al. (2016) Nominal Tensile Steel 355 - 0 L-H(1 Ll0
Lillemäe et al. (2017) Nominal - Steel 320 458 0 L-H(1 Ll1
Lillemäe et al. (2017) Nominal - Steel 320 458 0.1 L-H(1 Lm0
KeKeRa, 2017 Nominal Shear RAEX400 1000 1250 - Nd:YAG Ke0
KeKeRa, 2017 Nominal Shear S355 MC 355 430 - Nd:YAG Ke1
Liinalampi et al. (2016) Hot spot stress Tensile Steel 355 - 0 L-H(1 Ll2
Baumgartner et al. (2015)(4 max n-s(2 ρ f = 0.05 - several(2 - - 0.5 MAG Ba0
Liinalampi et al. (2016)(5 Mean n-s(2 ρ∗ = 0.4 Tensile Steel 355 - 0 L-H(1 Ll3
Liinalampi et al. (2016)(6 Mean n-s(2 ρ∗ = 0.4 Tensile Steel 355 - 0 L-H(1 Ll4
Liinalampi et al. (2016)(5 Mean n-s(2 ρ∗ = 0.05 Tensile Steel 355 - 0 L-H(1 Ll6
Baumgartner et al. (2015)(4 Mean n-s(2 ρ∗ = 0.5 - several(2 - - 0.5 MAG Ba1
Liinalampi et al. (2016) 95% n-s(2 ρ∗ = 0.4 Tensile Steel 355 - 0 L-H(1 Ll7
Liinalampi et al. (2016) 95% n-s(2 ρ∗ = 0.05 Tensile Steel 355 - 0 L-H(1 Ll8
Liinalampi et al. (2016)(3 95% n-s(1 ρ∗ = 0.05 Tensile Steel 355 - 0 L-H(1 Ll9
Marulo et al. (2017)(7 n-s ρ f = 0.05 Shear St14 210 313 0 - Ma0
Marulo et al. (2017)(7 n-s ρ f = 0.05 Peel St14 210 313 0 - Ma1
Marulo et al. (2017)(7 n-s ρ f = 0.05 Shear 22MnB5 - 1500 0.1 - Ma2
Marulo et al. (2017))(7 n-s ρ f = 0.05 Peel 22MnB5 - 1500 0.1 - Ma3
Marulo et al. (2017)(7 n-s ρ f = 0.05 Shear DC04 210 313 0 - Ma4
Marulo et al. (2017)(7 n-s ρ f = 0.05 Peel DC04 210 313 0 - Ma5
Marulo et al. (2017)(7 n-s ρ f = 0.05 Tube St35 235 313 -1 - Ma6
Marulo et al. (2017)(7 n-s ρ f = 0.05 Shear 2340 300Y 315 415 0 CO2 Ma7
Marulo et al. (2017)(7 n-s ρ f = 0.05 Tube S235 G2T 235 405 -1 - Ma8
Marulo et al. (2017)(7 n-s ρ f = 0.05 Shear Dx52D+Z - 343 0.1 - Ma9
Marulo et al. (2017)(7 n-s ρ f = 0.05 Shear XIP 1000 - 1500 -1 - M10
Lillemäe et al. (2017)(8 Hot spot stress - Steel 320 458 0 L-H(1 Lm2
Lillemäe et al. (2017)(9 Hot spot stress - Steel 320 458 0.1 L-H(1 Lm31) L-H: Laser-hybrid welding2) n-s: notch stress3) several: several materials: ZStE340, S355 MC4) multiple test results collected from literature5) test series HY3a with root and toe failures6) test series HY3c7) test series collected from literature and notch stress values are calculated by Marulo et al (2017)8) small-scale specimens 9) full-scale specimens
The nominal stress results for laser welded joints are shown in figure 26. Laser
welded joints are in poor agreement with the nominal stress method. The nom-
inal stress method may lead to very conservative fatigue results.
Yang and Lee (1999) (Yan) tested laser spot welded joints for lap joint fatigue.
47
In their study they concluded that the fatigue strength increases when the spot
weld area increases. Cho et al. (2004) (Cho) studied the residual stress effect on a
laser welded lap joint. The results are with transverse and longitudinal welds. In
the KeKeRa, 2017 (Ke0,Ke1) project, the effect of heat input and effect of number
of welds was studied. Liinalampi et al. (2016) (Li0) studied a wide scatter in
S-N results for laser welded joint. They concluded that the that the scatter is
due to a strongly altering surface roughness and scatter can be reduced by using
the notch stress method. Lillemäe et al. (2017) (Lm0,Lm1) tested the difference
between small-scale and full-scale specimens. The results between small-scale
and full-scale were similar in their studies. Sharifimehr et al. (2016) (Sh0,Sh1)
researched the effect of periodic overloads and underloads. In the figure 26, the
FAT class corresponding to the IIW fatigue design regulations is shown. The IIW
fatigue resistance values don’t offer any structural details that would correspond
with a through-welded lap joint. Thus structural details with transverse loaded
overlap with filled welds is used (No. 614). The FAT 36 value correspond with
the stress in the weld throat (Hobbacher, 2009). It need to be mentioned that IIW
corresponds only to thick plates with a thickness of over 5 mm.
Figure 26: Laser-welded joint nominal stresses from literature with FAT 63 and
FAT 36 curves.
The nominal stresses are given for sheets with a thickness from 0.8 to 3 mm.
The wide scatter can be explained with thin sheets and different welding and
48
testing methods. Test points that are below the FAT limit are shear specimens.
The results by Lillemäe et al. (2017) were in agreement with FAT 80.
The structural hot spot results are shown in figure 27. The structural hot spot
stress suits better laser welded joints than the nominal stress method but is not
widely used for laser-welded joints. The geometry of the joint needs to be known
for fatigue critical point evaluation.
Lillemäe et al. (2017) calculated results also for hot spot stress in their research.
They used the IIW suggestions as extrapolation point distances. Results both by
Lillemäe et al. (2017) (Lm2,Lm3) and (Liinalampi et al., 2016) (LI2) are below the
IIW design curve FAT 100, corresponding to structural hot spot stress.
Figure 27: Structural hot spot S-N results from literature with FAT 100 corre-
sponding to structural hot spot stress method
The most used method for evaluating fatigue strength is the notch stress method.
The results for notch stresses from literature for laser welded joints are shown
in figure 28. The notch stress assessment suits poorly laser welded joints with
a reference radius of r=1 mm. The suggested radius r=0.05 mm for thin plates
and laser welded joints suits moderately.
Liinalampi et al. (2016) studied the scatter of laser-welded joints with differ-
ent stress averaging distances in stress concentration regions. In the study, Li-
inalampi et al. (2016) did not use the notch stress method but the actual weld
geometry that had a notch-like shape. The scope was to study the stress aver-
49
aging distance effect on scatter of S-N results. It is shown here as it falls within
the notch stress result region. Marulo et al. (2017) re-analysed a large number
of laser welded joint specimens from literature. They re-analysed data with a
fictitious notch radius of 0.05 mm. The notch stress approach is independent
from the joint type.
Figure 28: Notch stress method S-N results from literature with FAT 225 corre-
sponding to notch stress method
The difference between laser welded joints and MAG welded joints is plotted
in figure 29. Baumgartner et al. (2015) studied MAG-welded thin sheets with
a thickness of less than 5 mm with a suggested fictitious radius ρ f = 0.05 mm.
The notch stresses for laser-welded joints are higher, suggesting a better fatigue
strength.
50
Figure 29: Notch stresses with ρ f = 0.05 mm for laser welded joints (Marulo
et al., 2017) and MAG welded thins sheets (Baumgartner et al., 2015)
5.3.1 Conclusion of the comparison
The scatter of laser welded joint fatigue is very wide. The stress range dif-
ference is almost 1000 MPa in nominal stress and even with the notch stress
approach. Reasons for this scatter are discussed in literature and it is concluded
to be related to surface deformation, surface ripples and the effect of thin plate
thickness.
The notch stress approach is recommended because weld fatigue originates from
stress concentration. Notch stress concentration is assumed to include the effect
of all defects and imperfections and thus the cover the whole range of phenom-
ena leading to failures. It is proposed by multiple authors that, with the notch
stress approach, the scatter can be reduced due to the aforementioned reason.
The suggested linear elastic-material model may also lead to inaccuracy in some
cases.
In rules and regulations, the fatigue strength of a welded joint is assumed to be
independent from the base material effect, stress ratio and plate thickness (with
some corrections). When comparing the scatter of nominal stresses (fig. 26) and
the scatter of notch stresses (fig. 28), it can be noted that the concentration is
not significant. From this, it can be concluded that the scatter of fatigue is not
entirely the effect of a surface notch.
51
6 Modelling and simulating laser welding
With comprehensive simulation of welded joints residual stresses, deformations
and surface deformations can be approximated (Traidia, 2011; Lillemäe et al.,
2013; Lin et al., 2017; Tsirkas et al., 2003). Surface deformations and residual
stresses play an important role in the fatigue of a welded joint and thus need
to be taken into account. FEA is used in solving physical problems, such as
welding (Frondelius and Aho, 2017; Könnö et al., 2017; Rapo et al., 2017; Remes,
2008; Traidia, 2011).
In welding, four states of material are present: solid, liquid, gas and plasma
(Traidia, 2011). For comprehensive simulation, multiple material variables and
phenomena need to be taken into account. Material behaviour is dependent on
material temperature and state. The material behaviour and parameters describ-
ing behaviour vary greatly over a temperature range, thus simulation needs to
be temperature-dependent. For example, the steel modulus of elasticity drops
constantly towards the melting point as the steel becomes more elastic. The mod-
ulus of elasticity, thermal expansion coefficient, and poisson’s ratio temperature
depedency are shown in figure 31 for AH36 the steel. In liquid state, material
properties vary greatly due to alloying elements. For example, the sulfur con-
tent for AISI 304 can change the flow direction of the molten pool surface, which
affects the weld bead profile.
Figure 30: Temperature dependency of properties for AH36 shipbuilding steel.
Figure taken from Tsirkas et al. (2003).
Laser welding can be modelled with two different main approaches: the HFF
approach and TMM approach (Traidia, 2011). The heat and fluid flow (HFF) ap-
52
proach is more comprehensive as it takes more phenomena into consideration.
The HFF approach takes heat transfer, fluid flow, surface deformations and elec-
tromagnetic fields into consideration. The fluid flow alone takes wide a range
of dynamic factors into account, like surface tension, multiphase flows and va-
porization. The HFF approach can be expanded to include solid mechanics and
metallurgy.
The thermo-mechanical and metallurgical approach, TMM, is coupling heat
transfer, metallurgical transformations and solid mechanics (Traidia, 2011). The
TMM approach is more widely used in practise as it easier to obtain (Traidia,
2011; Lin et al., 2017; Sun et al., 2017)
Figure 31: Coupling of physical phenomena. Taken from (Traidia, 2011)
Welding defects, such as porosity and undercuts, can be covered with the com-
prehensive heat and fluid flow approach (Lin et al., 2017). Porosity is a signifi-
cant problem in laser welding of aluminium and high strength steels (Lin et al.,
2017; Sun et al., 2017). Porosity can be reduced with optimized welding param-
eters that can be obtained with numerical simulation. With the HFF approach,
a wide range of physical phenomena, in addition to the weld pool dynamics
can be obtained, for example residual stresses. As the HFF approach is widely
multiphysical and takes into account multiple phenomena, the calculation with
it is slower than with the TMM approach. TMM approach in its simplest is a
coupling with solid mechanism and heat transfer (Traidia, 2011).
53
6.1 Simulation according HFF approach
The heat and fluid flow approach is a multiphysical approach that is used for
welding parameter optimization, in defect, like porosity formation, studies and
in scientific studies. The fluid flow in the molten pool affects in the weld pool
shape and thus the quality of the weld (Traidia, 2011; Lin et al., 2017; Le Guen
et al., 2008). For SAW, the heat input is modelled with a cathode-anode region
alongside with the plasma section originating from a protective gas in the HFF
approach (Traidia, 2011). In laser welding, heat input can be modelled with an
approaching laser beam or beams that are in reaction with the material surface
or keyhole surface (Lin et al., 2017). This method gives the HFF approach an ad-
vantage over the TMM method where heat input needs to be manually adjusted
when simulating the weld bead.
In figure 32, the results of a HFF -approach simulation are shown. In the study,
the intent was to investigate the origin of residual stresses and molten pool flows.
The simulation was made with the Comsol Multiphysics program.
Figure 32: Simulation results with Comsol Multiphysics program with HFF ap-
proach. The temperature is plotted with legend and the molten pool region is
outlined. Fluid flow presented with arrow field.
6.1.1 Fluid flow
The heat transfer and fluid flow approach takes into account heat transfer in
the solid and fluid flow and heat transfer in the fluid. The weld pool molten
metal flow is usually considered to be laminar and incompressible due to the
small size of the molten pool (Traidia, 2011). In order to evaluate the tempera-
ture and hydrodynamics phenomena as a function of time, classical incompress-
ible Navier-Stokes equations are deployed (Traidia, 2011; Le Guen et al., 2008;
Bruyere et al., 2014). The classical conservation equations for velocity, pressure
54
and temperatures are conducted in simulation. In a solid domain, flow terms
are obsolete.
∇ · −→v =∂u∂x
+∂v∂y
+∂w∂z
= 0 (23)
where ∇ is a gradient operator and −→v is the velocity field in a weld pool. The
equation is a conservation of mass for Newtonian viscous fluids.
ρ(∂−→v
∂t+−→v · ∇−→v
)= −∇p +∇ · µ
(∇−→v + (∇−→v )T)+−→Fv (24)
where ρ is the density, p is the pressure, µ is the dynamic viscosity, and−→Fv is the
sum of effective forces like gravity forces. This equation is a classic conservation
of momentum after Navier-Stokes equations. Forces affecting the momentum
are the gravity force−→Fg and Buoyancy force
−→Fb . In a liquid flow, the Buoyance
forces are governing (Traidia, 2011)
ρCp∂T∂t
+ ρCp−→v · ∇T −−→∇ · (k−→∇T) = Φ + ρL f fL (25)
Where ρ is the density, Cp is the heat capacity, T is the temperature, Φ is the
volumetric heat source in the weld pool, L f is the latent heat of fusion, and fL
is the fraction liquid. This equation is an equation of energy conservation. The
liquid fraction fL depends on liquidus temperature, Tl, of the material.
Unevenly distributed heat in a molten pool creates fluid flows between tempera-
ture zones (Traidia, 2011; Lin et al., 2017). The density of a molten metal depends
on the temperature, and thus high terminal gradients induce natural a convec-
tion flow (Buoyancy effect) (Traidia, 2011). Buoyancy forces tend to create an
outward flow from a heat source, increasing the width of a weld pool (fig. 33).
Velocity fields induced by surface forces, for example the Maragnoni effect, are
stronger than velocity fields induced by natural convection. Other dominating
forces in welding are induced by surface and electromagnetic forces in SAW. In
laser welding, electromagnetic forces are obsolete.
55
Figure 33: Different liquid flows in molten pool originating from different forces.
The arc drag and Loretz force are strongly present in arc welding, but their
occurrence in laser welding is a matter of discussion. Figure taken from Traidia
(2011)
Gravity forces depend on material density Fg = ρg. The gravity force is signifi-
cantly larger than the Buoyancy force, but it does not contribute to the creation
of flows as it is a constant. Buoyancy forces are a result of material density
dependence on temperature, which creates natural convection forces (Traidia,
2011). The Buoyancy effect can be modelled with additional force Fb (Traidia,
2011):
Fb = −ρ0k(T − Tre f )g (26)
where ρ is the density, k is the thermal expansion and Tre f is the reference tem-
perature.
The Marangoni effect is one of the main components in molten pool flow forces.
The Marangoni effect is formed when a high surface tension pulls the surround-
ing liquid metal more strongly than the low surface tension region. High ther-
mal gradients occur in the top surface of the molten pool, depending on the
highly concentrated heat source in laser welding. The surface tension of the liq-
uid becomes non-uniform and the liquid starts to flow from low to high surface
tension regions. A surface tension gradient (direction field) can be caused by
a concentration gradient or temperature gradient. The Marangoni effect on the
free surface can be described by the following equation (Traidia, 2011):
µ∇nνs =∂γ
∂T∇sT (27)
56
where µ is the dynamic viscosity, ∇n is the normal gradient operator, νs is the
tangential velocity, ∂γ∂T is the surface tension gradient, ∇s is the tangential gradi-
ent operator, and T is the surface temperature.
The urface tension gradient ∂γ∂T has a great effect on the weld pool shape as it
determines the flow direction of the Marangoni flows (Traidia, 2011). When
the coefficient is negative, the flow is outward as the surface tension is highest
at the edge of the weld pool. The flow is inward if the coefficient is positive.
For pure metals the coefficient is negative, but the presence of sulfur or oxygen
can alternate it to positive. It needs to be mentioned that the coefficient is also
dependent on temperature.
6.1.2 Modelling of the laser beam absorption
The laser beam energy is absorbed with Fresnell absorption in the keyhole. The
Fresnell absorption can be represent by a function of the incident angle, laser-
dependent coefficients and material-dependent coefficients (Lin et al., 2017).
α(φ(n)) = 1− 12
(1 + (1− εcosφ(n))
2
1 + (1 + εcosφ(n)+
ε2 − 2εcpsφ + 2cos2φ(n)
ε2 + 2cosφ + 2cos2φ(n)
)(28)
where φ(n) is the angle between the beam incident ray and the surface normal,
ε is the coefficient determined by laser type and material properties. The beam
is distributed into discrete form. The absorption rate α is calculated for each
reflection point, and the assorted energy is reduced from the total beam energy.
The schematics of multiple reflection is shown in figure 34.
Figure 34: Scematics of Fresnel absorption (multiple absorptions) in keyhole
welding. Figure taken from Lin et al. (2017)
57
6.2 Simulation according TMM approach
The thermo-mechanical and metallurgical approach is usually used in solv-
ing weld bead (molten area) dimensions, metallurgic distributions and residual
stresses and distortions (Lillemäe et al., 2013; Jiang et al., 2016; Tsirkas et al.,
2003; Zain-ul Abdein et al., 2009; Carmignani et al., 1999).
With the TMM approach, usually residual stresses and phase fractions are cal-
culated. An example of a TMM simulation is shown in figure 35 where phase
fractions are calculated in the edge of HAZ. The temperature rises suddenly,
leading to ferrite transformation to austenite. The surrounding material quickly
starts to cool down the welding zone, which leads back to forming of ferrite.
The martensite starts to form when temperature decreases.
Figure 35: Simulation result with TMM approach. The temperature field from
20 ◦C to 1400 ◦C is plotted at 1.42 s. In figure fractions and temperatures are
plotted over time.
6.2.1 Heat source
In the TMM approach, the most commonly used heat source in laser welding
simulation is Gaussian distribution or a combination of multiple Gaussian dis-
tributions (Jiang et al., 2016; Tsirkas et al., 2003; Le Guen et al., 2008). Jiang et al.
(2016) used a combination of a double ellipsoid, rotating Gaussian and cone
to control the weld width, depth and parameters of the focal position of laser
when investigating the welding parameter effect on the weld bead. Le Guen
et al. (2008) used a more simple heat source which was a combination of a cylin-
drical uniform temperature, according to Lankalapalli et al. (1996), combined
58
with a Gaussian heat flux on the surface of the part when investigating weld-
ing process and surface deformations. It can be concluded here that the bead
shape in the TMM approach is a result of a given heat source distribution, and
its validation requires testing.
Figure 36: The schematics of multiple heat sources. Figure taken from Jiang et al.
(2016).
The heat source is modelled as a moving heat source (Lillemäe et al., 2017; Jiang
et al., 2016; Tsirkas et al., 2003; Zain-ul Abdein et al., 2009; Carmignani et al.,
1999). Gaussian heat flux can be given in the following form (equation 29):
ql =Pinp
2πr20
exp(−r2
r20
)(29)
where Pinp is the power conducted to part, r0 is the initial radius, and r is the
current distance from the weld center.
The simulation is temperature-dependent by nature and therefore all phenom-
ena are results of temperature changes. Temperature evolution can be evaluated
in the entire domain with a modified classical conservation equation of energy
(VIRFAC 1.4.1 Material constitutive laws, 2016).
ρCp∂T∂t−−→∇ · (k−→∇T) = Φ (30)
where ρ is the density, Cp is the heat capacity, ∂T∂t is the temporal gradient of
temperature, k is the thermal conductivity,−→∇T is the spatial gradient of temper-
ature, and Φ is the source term. The equation is modified by removing terms
concerning flow. The course term Φ contains boundary conditions like heat flux,
59
convection, radiation and contact conductivity (VIRFAC 1.4.1 Material constitu-
tive laws, 2016).
6.2.2 Material behaviour
The heat source is the only boundary condition importing energy. In the TMM
approach, temperature is conducted directly to the part. The welding energy
transfers into the welded part as heat, which leads to thermal strains. In thesis
studies, we noticed that it is crucial to model metal liquefaction so that thermal
strains and stresses will reset. The residual stresses are a result from thermal
strains. Convection and displacement boundary convection need to be defined
in the welding model geometry. Exceeding the melting temperature in the seam
results in remarkable strains in the welding area. The total strain is the sum of
all strains (VIRFAC 1.4.1 Material constitutive laws, 2016):
εa = εel + εpl + εth (31)
where εa is the total strain, εel is the elastic strain, εpl is the plastic strain, and
εth is the thermal strain. When temperature are excluded from the analysis, the
term of thermal strain can be omitted.
Elastic strain is the stress dependency on Young´s modulus ε = σ/E. Elastic
strain is evaluated with equation (VIRFAC 1.4.1 Material constitutive laws, 2016):
εeij =
1E[σij(1 + ν)− νδijσkk] (32)
where ε is the elastic strain tensor, E is the Young´s modulus, σ is the stress
tensor, ν is Poisson´s ratio, and δij is Kronecker delta.
Plastic strains are dependent on the material model. The elastoplastic model cos-
titutive law captures the effect of strain hardening behaviour. The elastoplastic
model is the most commonly used material model to describe static behaviour,
as it represent most materials (VIRFAC 1.4.1 Material constitutive laws, 2016).
σ = σy + H(εpl)N (33)
where σ is the total stress, σy is the initial yield stress, H is the hardening law
coefficient, and N is the hardening law exponent. Plastic strain is in 1D εpl =
εa− (σel/E) without thermal strains. It needs to be mentioned that there are also
60
other elastoplastic models, but the power law is presented here as it was used in
the simulations. Thermal strains can be calculated using equation:
εth = αth(∆T)I (34)
where αth is the temperature dependent coefficient of thermal expansion and I
is the identity tensor.
6.2.3 Phase transformation
In the welding, phase transformation is always present in steel welding due
to the high thermal input. Phase transformation involves notable changes in
the kinetics of a metal which are not always associated solely with the external
thermal impact but also with the rate of thermal changes (Domanski et al., 2016;
Magnabosco et al., 2006). The phase fractions in solid state after cooling can
be obtained with mathematical models, such as Jonson-Mehl-Avrami, Leblond-
Devaux and Koistinen-Marburger (Domanski et al., 2016).
The form of phase fraction models is presented in equation 35 (VIRFAC 1.4.1
Materials, 2016). The phase transformation is dependent on the temperature
change and rate of temperature change. The fractions between phases, such as
ferrite, austenite and martensite, are calculated with a value between 0 and 1
so that the sum of phase fractions is 1. The Leblond-Devaux models control
transformations involving austenite, ferrite, pearlite and bainite. If more that
two phases are calculated, transformations of phases are included with CCT
diagrams.
dYdt
=Yeq(T)− p
τ(T)F(T) (35)
where Y is the phase proportion, T is the temperature, Yeq(T) is the phase pro-
portion function depending on time, τ is the delay time of the transformation
due to the heating or cooling, and F is a function of the heating and cooling rate.
Phase proportions Yeq and τ are temperature dependent material parameters. F
controls the rate.
61
6.3 Welding residual stress simulation for fatigue behaviour anal-
ysis
The fatigue model of this thesis is based on the material behaviour in cyclic
loading after welding-induced residual stresses. Residual stresses were simu-
lated with GEONX’s Virfac program. The material and geometry were chosen
according to the test rod geometry used by Lillemäe et al. (2017) and Liinalampi
et al. (2016) in order to repeat the fatigue behaviour as accurately as possible.
For the material, S355 steel was chosen as it is used in multiple tests in literature.
The phase fractions were not calculated in the analysis because Remes (2008)
stated in his doctoral thesis that hardness distribution does not have any signif-
icant effect on fatigue in analysis. Remes (2008) used the notch strain approach
with a 2D model of idealized notch geometry. The assumption of evaluating
stresses in the notch region and the assumption of 2D simplification may affect
the effect of hardens distribution on fatigue behaviour.
6.3.1 Mesh
The mesh was created with the Gmsh program. The Gmsh is a free finite element
generator (Gmsh Reference Manual). The mesh was exported in the .msh form
for Virfac. The mesh was created with Gmsh because a proper mesh generation
with the Virfac mesh generator is challenging. The mesh is much more dense in
the weld region than in the restrain region due to the physics of calculating: the
transformations in the weld region are intense and non-linear, whereas in the
restrain region they are not. The used mesh is shown in figure 37. The element
length on the welding line was 0.3 mm.
Figure 37: Mesh of test rod used in analysis
62
First order elements were used in the mesh because Virfac does not support
quadratic elements in welding simulation. In addition, Virfac did not support
hexahedron elements on the welding line. The welding line is set on the existing
line in geometry in Virfac, and therefore the mesh geometry is divided.
6.3.2 Temperature dependent material mode
As mentioned before, the welding is highly temperature dependent, and there-
fore the material model needed to be temperature dependent. The temperature
dependency of the chosen material S355 is based on a model by Schenk et al.
(2009). Schenk et al. (2009) used the power law (equation 7 in chapter 4) to
represent the material’s plastic behaviour.
In Virfac, material plasticity is modelled with the power law (equation 33). The
parameters associated with hardening were fitted from the exponent law to
power law form. The fitting was done for each temperature region by mini-
mizing the square of difference between curves. Material parameter fitting was
done with Microsoft Excel.
The temperature dependency of the used material model is shown if figures 38,
39 and 40. In figure 38, thermodynamic material parameters are plotted. As
shown in the figure, density and thermal conductivity decrease when tempera-
ture and heat capacity increase. The thermodynamic parameters of the material
do not vary as much as the parameters related to elasticity and elastoplacticity at
temperatures below melting point (see fig 40). This shows the high temperature
dependency of the elasticity of the metal. Parameters associated with hardening
in figure 40 are fitted to the power law.
63
Figure 38: Thermodynamic material parameters for structural steel S355. Pa-
rameters taken from Schenk et al. (2009)
Figure 39: Expansion coefficient and elastoplastic material parameters related
to hardening for structural steel S355. Paremeters related to hardening fitted to
power law form and expansion coefficient taken from Schenk et al. (2009)
Figure 40: Temperature dependent parameters related to material elasticity and
elastoplacticity. Parameters taken from Schenk et al. (2009)
64
6.3.3 Virfac simulation
The welding simulation was done with GEONX’s Virfac. The program was used
because it offers a ready physics package for welding simulation. Virfac is a
manufacturing design and modelling program including modules for additive
manufacturing, welding, heat treatment, machining and crack propagation. The
welding module was used in this thesis work. Virfac uses two different solvers:
Morfeo or JWELD (VIRFAC 1.4.1 Runs manager, 2017). The Morfeo solver was
used in this thesis work because the welding designer module is optimized to it.
In Virfac, welding module geometry, material, boundary conditions, physics
etc. are determined in the user interface. Virfac is a novel program where the
basic usability is easy with a clear interface. While using the program, some
difficulties were found in the welding model: quadratic mesh does not work,
but it can be assessed in the program’s own meshing tool; the double ellipsoid
heat source model does not work, and heat source cannot be set to hexahedron
elements.
The simulation model determination in Virfac begins by defining the model. In
this study, the geometry was imported as a Gmsh mesh. The material model
was specified in the program by defining parameters for each variable. The
temperature-dependent parameters, such as density, young’s modulus, thermal
expansion coefficient, initial yield stress, hardening coefficient, hardening expo-
nent, thermal conductivity and specific heat capacity, were set to the parame-
ters shown in figures 38, 39 and 40. The effect of metal liquefaction was ex-
pressed with an annealing option. The annealing option in Virfac defines the
temperature beyond which plastic strains are annulled (VIRFAC 1.4.1 Material
constitutive laws, 2016). The option allows stresses and strains to drop to low
values when the annealing temperature, here used as temperature of fusion, is
exceeded. The annealing temperature was set to 1400 ◦C.
The mechanical and thermal boundary conditions were set to describe welding.
The displacement restraints used in the welding simulation are shown in figure
41. The thermal boundary condition was set to the uniform thermal convection
of 20 W/(m2K) on free surfaces.
65
Figure 41: The restraints used in Virfac. The thermal convection was set on every
free surface.
The welding line was set as shown in figure 41. The welding speed was set to
0.022 m/s which matches the speed used with laser welding. The heat source
was a combination of double ellipsoid and double cone shape. The widths of
the double ellipsoid and double cone shape were 1.8 mm and 1.2 mm. The
power of the weld used was 2000 W with an efficiency rate of 65 %. With these
parameters, the weld bead shape and heat distribution of the part coincide with
the research by Cho et al. (2004); Jiang et al. (2016); Tsirkas et al. (2003); Zain-ul
Abdein et al. (2009). A cooling stage lasting 50 s was added after the welding
stage.
For further processing of welding simulation results, Morfeo input file was mod-
ified so that a separate output file for element stresses was printed. A modifi-
cation was done by adding a separate section on the input file that exports the
element stresses related to the elements in the .msh file. With these parameters,
the calculation lasted for 18.8 h in multiple core run.
6.3.4 Residual stresses
The residual stresses included in welding simulation analysis are shown in fig-
ure 42. The results are calculated with the aforementioned material parameters,
process parameters and geometry. The localized distribution of residual stress
and a high level of stress were anticipated. The stresses are partly exceeding
the original pulling test yield strength. The residual stresses are within yield
strength as stated in literature, but the yield surface has changed due to the
hardening. Same kind of residual stress distribution is shown by Carmignani
et al. (1999) and Zain-ul Abdein et al. (2009)
66
Figure 42: Calculated main components and vonMises stress field of residual
stresses.
The linear element mesh of Virfac inevitably led to discontinuity in strain and
stress. Linear elements work well with heat transfer problems but describe
poorly strains and stresses. The result of the analysis is very discontinuous and
high stress peaks occur in individual elements. Moreover, very high and very
low element values occurs in neighbour elements. The effect of discontinuity
was emphasized when residual stresses were calculated in the initial phase for
the Abaqus CAE program. The discontinuity of stresses is shown in figure 43.
The averaging of 100 % was used to rectify bad mesh.
Figure 43: The effect of linear mesh: the stress field is considerably discontinu-
ous and the stress peaks are in individual elements
67
7 Commercial fatigue program
7.1 FEMFAT
FEMFAT (Finite Element Method Fatigue) is a commercial program developed
for fatigue life prediction and, among other applications, it includes a weld ap-
plication (femfat.magna.com). FEMFAT includes weld assessment which covers
different standards such as EUROCODE, British Standard 7608, IIW, DIN13001,
DVS1612, FKM and DVS1608. These regulations cover a wide range from civil
engineering to railway regulations. The FEMFAT weld approach module covers
a data base of material and structural fatigue and automatic recognition of weld
regions (FEMFAT 5.1 -Weld, 2014)
The geometry can be imported in FEMFAT. Parts or structures need to be im-
ported with defined stresses. The program supports both shell and solid struc-
ture models. In solid FEM, the notch stress approach in needed in modelling.
A suggestion for a program for a solid modelled T-joint is shown in figure 44.
Welded structures are usually thin-walled, which leads to the use of shell ele-
ments in modelling structures for Finite element analysis. If a structure is mod-
elled with shell elements the details of weld can be left un-modelled as the type
of joint can be defined in the FEMFAT program. When modelling a joint element
for a lap joint with shell elements, a connection element is needed between the
parts for definition of weld seam (FEMFAT 5.1 -Weld, 2014).
Figure 44: example of solid modelled T-seam. Taken from FEMFAT 5.1 -Weld,
(2014).
The automated FEM weld assessment of FEMFAT recognises several joint types,
weld types and weld executions (FEMFAT 5.1 -Basic, 2014). The user can select
different weld executions when defining the course of weld. Joint types include,
68
for example, butt joints, T-joints, lap joints, etc. Weld types include different
types, for example a square butt weld, filled weld, single-V butt weld, laser
weld, etc. For a laser welded joint only the lap joint weld type is selectable.
FEMFAT uses the mean value/variance approach based on notch stress consid-
erations (FEMFAT 5.1 -Weld, 2014). The basis for defining fatigue is represented
by notch stress endurance limits which are provided by tests. The structural
stresses evaluated with FEM analysis are scaled to notch stresses that are used
to evaluate fatigue. Stresses at the weld are derived from elements, and notch
factors are utilized to corresponding weld types. Stresses in the weld are linearly
interpolated from distance the from weld.
69
8 Fatigue calculation with residual stresses
The geometry used was the idealized smooth test rod. The intend was to study
the effect of residual stresses on fatigue behaviour. The dispersion of the fatigue
strength of laser welded joints is claimed to result from surface defects. It can be
easily concluded that other defects and imperfections also have an effect on the
fatigue strength. In literature the discontinuity if geometry is well established to
have an effect on fatigue, but the effect of microstructure and stress equilibrium
are not. The assumption in this study was that residual stresses have an effect on
fatigue strength of welded joints on the basis of material behaviour. In section
8.3, the statistical behaviour and dispersion of S-N results are studied.
8.1 Cyclic loading analysis
The residual stresses were based on the fatigue analysis of the new model. Resid-
ual stresses were calculated with the GEONX’s Virfac program. The effect of
cyclic, loading i.e. fatigue analysis, was investigated with the Abaqus 6.14 CAE
program. The workflow of the study is presented in figure 45.
GMSHmesh.msh
VIRFACwelding simulation
VIRFAC resultsinode, ielem, σ, εpl
.msh
Geometry & boundary conditions
.inp
Residual stresses
ielem, σ
Kinematic hardening
ielem,εpli ,α1 , α2
Initial conditions
Abaqus
input file
Material properites
E, σy, C1, γ1, C2, γ2
Abaqus 6.14analysis
Figure 45: The work flow of calculation. Green blocks are done with commercial
programs, blue block is output file and yellow blocks are data parsing and re-
writing
70
8.1.1 Fatigue analysis
The mesh of Virfac simulation was used also in the Abaqus analysis. The mesh
was converted in to the Abaqus input file form (.inp) with Gmsh. The boundary
conditions in Abaqus analysis are shown in figure 46. The restrain and force
boundary conditions were written in a geometry input file. For the surface
where the force affects, the nodes were coupled with a distributed coupling in
order to distribute the effect evenly.
Figure 46: The boundary conditions on Abaqus analysis. The blue area corre-
sponds with loading.
Material model and initialization
The residual stresses, exported from Virfac, were written as the initial condi-
tions. The material model used was the elasto-visco plastic material presented
in chapter 4. The result of virfac simulation was in the form of stress tensor for
element. The initialization and material model required that the stress tensor
was divined in to elastic and plastic components.
The backstresses were calculated from the plastic part of the stress tensor. The
Kinematic hardenings were solved with the Secant Method. The iteration for
backstresses was made if they were out of the yield surface. The kinematic
hardening backstresses αi j were solved with the assumption shown in equation
36.
∆αi =
αi+1k − αi
k
αi+1j − αi
j
(36)
The iteration returned the stress, σ, equivalent plastic strain rate, εpl, and back-
stress rate α. From where the equivalent plastic strains were calculated.
71
The analysis in Abaqus CAE was done with the static, general method. Three
steps were used in analysis: initial, fatigue and release. In the initial step, resid-
ual stresses and hardenings were initialized in the test rod. In this step, Abaqus
calculated the equilibrium on the part. The initial distortions are included in the
stress balance state and thus not separately taken into account. In the fatigue
step, a cyclic force was added.
The initial values were given in the form prescribed in the Abaqus manual. The
stress component was written for each element. Backstresses were given in the
following form (Abaqus 2016 Keywords reference guide).
First line:
1. Element number
2. Initial equivalent plastic strain, εpl|03. First value of the initial backstresses, α10
11
4. Second value of the initial backstresses, α1022
5. Etc., up to six backstress components.
Subsequent lines:
1. First value of the initial backstresses, α2011
2. Second value of the initial backstresses, α2022
3. Etc., up to six backstress components.
Material model fitting
The parameters C1, C2, γ1 and γ2 related to kinematic hardening were fitted from
the Virfac material model to Abaqus. The material model was fitted from the
power law form with parameters H and N to the elasto-plastic form with two
backstresses. The material parameters C1 and C2 fitting was done by minimizing
the square of difference between the Virfac stress-strain curve and the fitted
curve. The parameters γ1 and γ2 were set to constant 500.0 and 20.0. The
elastic strains in the parameter fitting were calculated on the basis of generalized
Hooke’s law. The stiffness tensor, C, can be expressed with bulk modulus, K,
and shear modulus, G. The relation can be also expressed as a function of stress
εel = S : σ (37)
72
where S is the compliance tensor.
The parameter fitting in the σ11 direction is shown in figure 47. The Virfac model
is shown with a black line and the fitted model with kinematic hardening with
a blue line. The backstresses are plotted with a dashed line.
Figure 47: The fitted uni-axial stress-strain behaviour for σ11 in in relation to
backstresses
Base material properties and loading
The geometry, parameters for material, initial conditions and boundary condi-
tions were written in the input file (.inp). In analysis, base material fatigue prop-
erties were used. The parameters are shown in table 4. The Young’s modulus,
E, and Poisson’s ratio, µ, are general values for steel. The hardening parameter
C1 and C2 are a result of parameter fitting. Six different forces were used, cor-
responding to stresses 300 MPa, 200 MPa, 150 MPa, 100 MPa, 60 MPa, 30 MPa
and 25 MPa in the cross section of the surface.
Table 4: The parameters used in the cyclic loading analysis. Fatigue parameters
taken from de Jesus et al. (2012)
Material σ0 [MPa] C1 C2 γ1 γ2 σ′f [MPa] ε′f b c
S355 329.0 52165.2 28609.5 500.0 20.0 952.2 0.7371 -0.089 -0.664
The cyclic force was defined as a periodic amplitude that follows the sin curve.
The cyclic loading analysis was done with R=0 and with R=-1. The principle of
73
used loading with different R ratios is shown in figure 48. In the release step,
the external loads were removed.
Figure 48: The different loads used, corresponding to the stress ratios R = 0 and
R = -1.
Lifetime analysis
The lifetime analysis was done with a SWT damage model. Damage models
correspond with crack initiation and therefore stresses, σn,max, are calculated
in the maximum strain plane direction. In the residual stress analysis, it was
found that residual stresses cause a very strong variation in the strain and stress
gradient fields. The strain and stress field directions and magnitudes are shown
in figure 49 for the test rod with initialized residual stresses. For comparison,
in figure 50, the same gradients are plotted for the test rod without initialized
stresses. It can clearly be seen that residual stresses cause great changes in
the direction and magnitudes of stresses and strains. The high redistribution
is due to the non-uniform stress and strain fields due to thermal strains. The
high variation of the direction of strains and stresses lead to the damage model
working poorly with stresses calculated in the maximum strain fields.
Figure 49: Gradient field with welding induced residual stresses
74
Figure 50: Gradient field without welding induced residual stresses
The stresses were calculated by the stress definition of Sines. The Sines method is
a multiaxial fatigue strength criteria (Fojtik et al., 2010). In the method, the stress
criterium is the square root of the second invariant of Cauchy stress tensor. The
stress tensor is determined from stress amplitudes. The term of mean hydrostatic
stress is added as shown in equation 38 (Fojtik et al., 2010).
as(√
J2)
a + bsσH,m ≤ f−1 (38)
where coefficients as and bs are defined as:
as =f−1t−1
,
bs = 6 · f−1f0−√
3 f−1t−1
(39)
where J2 is the second invariant of stress tensor, f−1 is the fatigue limit in fully
reversed axial loading, σH,m is the mean value of hydrostatic stress, t−1 is the
fatigue limit of fully reversed torsion, and f0 is the fatigue limit in cyclic bending.
In the calculations, the stresses were defined as√
J2 + σH,m.
8.2 Results
The results were well in line with the results from the literature. The comparison
was made with fatigue tests with a similar steel and similar stress ratio. For more
information, see table 3. Curve fitting was done with the least square method.
The SN curves for the test rod with residual stresses as the initial condition
is plotted in figure 51. The test data collected from literature is plotted for
comparison.
75
Figure 51: The S-N results from SWT analysis. Laser welded S355 steel fatigue
test results with R = 0 . . . 0.1 from literature for comparison
The SWT results are in the top section of dispersion. The test data from the
KeKeRa project (Ke1) was shear data and thus multiplied with√
3. The test data
slope did also fit well with the literature test data as shown in figure 52.
Figure 52: The comparison of SN curves between test rod analysis with SWT
and data from literature.
From the results, it can be concluded that residual stresses have significant effect
on fatigue behaviour. The slope of the curves coincide well with the curves
fitted from literature test data. The analysis was done with base material fatigue
76
parameters (see table 4). The test rod with no initialized stresses had a better
fatigue strength and the S-N curve was less steep.
Cyclic loading altered the material stress-strain behaviour. The hysteresis loop
of stress and strain moved as a result of hardening. The stresses were levelled
at cyclic loading as shown in figure 53. In the figure, the hysteresis of stress
component σ11 and the post cyclic loading of von Mises stresses for the test rod
are shown. The test rod was loaded with a nominal stress of 200 MPa.
Figure 53: The hysteresis loop of σ11 and vonMises stresses after hysteresis sta-
bilization on point in weld region.
The poor mesh caused stress concentrations to some particular elements. In
these elements, the number of cycles for failure was close to zero. This effect was
corrected by excluding 5 % of the N f results in the calculation of the number of
cycles for S-N curve. The histogram for N f is shown in figure 54.
Figure 54: N f histogram for SWT results with R = -1.
In figure 55 it is shown the area where the probability of fatigue damage is
77
likely. The results shown are for test rod with 200 MPa initial load. The high
probability of a fatigue failure to occur is in the region where the residual stress
effect is a its highest.
Figure 55: Area that is likely to experience fatigue failure after 1E6 cycles.
8.3 Further processing of results
All defects and imperfections have am effect on the fatigue behavior of a weld
joint. In respect of residual stresses, the analysis was in good agreement with
test results from literature but with a better fatigue strength. In the suggested
notch stress methods, the effect of the defects are united as one defect on the
surface with ρ f = 0.05 mm.
The defects have a deteriorating effect on fatigue strength, and the dispersion
can be explained by them. The statistical analysis of defect size was done with
Murakami’s theory of critical area. Murakami suggested and proved by tests that
the defect size has an effect on fatigue ductility 56 (Murakami, 2002). According
to the theory, bigger defects have a bigger impact. In Murakami’s theory, a
critical areaa√
area, instead of crack length is used.√
area describes the effect for
small surface defects, small surface cracks, and nonmetalllic surface inclusions.
78
Figure 56: The deteriorating effect of defects on fatigue strength (Murakami and
Miller, 2005)
The statistical analysis was done on the basis of Murakami’s theory that a defect
affects curve transformation. Stress differences, ∆σi between the analysis SWT
curve, σi(N), and literature test data for S355, σ∗i , were calculated for each point.
The same relative transition was assumed in the knee point as in the measured
point. The knee point was assumed to be 107 as in rules and regulations. The
value of the stress of the SWT curve on the knee point, σre f , is supposed to
correspond to√
areare f as the actual defect distribution is not known. The fatigue
test point stress value at the knee point, σw, correspond to defect size that is
known to have a deteriorating effect on fatigue strength. The schematics of
analysis is shown in figure 57.
Figure 57: Schematics of calculation
In figure 58, the relative distribution is shown. The fitting was done in a log-
normal distribution. Relative stresses fitted poorly in the normal distribution
because some values were significantly outside the normal distribution region.
79
Figure 58: Log norm distribution for stress differentials
In figure 60, survival probabilities of 5 % , 50 % and 95 % are shown. The slope
is based on the analysis with residual stresses and distribution on the statistical
analysis of test results. The survival probability of 5 % coincides approximately
with the FAT 80 class and the average percentile coincides with FAT 120. Even
though the results are with only S355 steel and R=0, the scatter is high. In
the slope of the survival percentile of 95 %, the stress range is twice as high
compared to the stress range the survival percentiles 5 %. Regardless the distri-
bution scatter, the fatigue stress of the survival probability of 5 % is higher than
the rules and regulations suggest for welded joints.
Figure 59: S-N curve fitted with the calculated fatigue strength difference range.
The relative log-normal distribution can be shown as a cumulative density func-
tion. In the rules and regulations, a fictitious notch with a radius of ρ f = 0.05
80
mm is suggested. It can be assumed that a defect with the size of ρ f = 0.05 mm is
present in a laser welded joint. Assuming that a defect with the size of ρ f = 0.05
mm is present with a 50 % probability, the cumulative density can be presented
with a function of the defect size√
area. In figure 60, the expected defect size
according to the statistical study is shown.
Figure 60: Assumed probable defect size.
Liinalampi et al. (2016) studied the defects induced by surface geometry. They
measured the actual welding geometry and studied the stress concentration on
a model based on the actual geometry with FEA. They averaged the stress con-
centrations with Neuber’s approach (see 5.2.3) with a averaging distance 50 µm.
They analysed joints in 2D simplification and with a linearly elastic material
model.
Liinalampi et al. (2016) presented the effect of stress concentration with averag-
ing percentiles of 50 % and 95 % as well as nominal stresses. From the results,
a relative stress rise, ki = σ%/σnom, was calculated. Relative stresses were fit-
ted in a cumulative probability form in order to unify the results with the ones
calculated from the distribution of test data.
σa f = σnomki (40)
81
Figure 61: Cumulative density for stress rise factors. Blue wide line illustrates
the stress rise factor calculated from literature test results. The rest are fitted
from Liinalampi et al. (2016).
The stress magnifications presented with results by Liinalampi et al. (2016) are
greater than those calculated from test results. This may suggest that a linearly
elastic material model suits poorly stress magnification factor calculation.
82
9 Discussion
In this thesis work, the study intention was to research the suitability of common
weld fatigue evaluation methods on laser welded joints. Fatigue test results of
laser welded joints were collected from literature for comprehensive analysis of
the suitability of common methods offered by rules and regulations, such as IIW,
EUROCODE 9 and BS 7608. The basis of common methods, such as the nominal
stress method, hot spot method and notch stress method, was introduced, con-
sidering their suitability for laser welding. Some recent weld fatigue assessment
from literature was also introduced. The fatigue design of laser-welded joint
with common methods offered by rules and regulations lead to conservative
design. The laser-welded joints have a better fatigue strength compared to tradi-
tional welded joints. IN notch stress method, laser-welding has been taken into
account by smaller reference radius. The smaller reference radius is expected to
explain the fatigue strength scatter by the stress concentration. It is assumed that
welded joint fatigue occurs when stress value in a the stress concentration ex-
ceeds a certain value. Thus using a smaller reference radius, the fatigue strength
scatter is expected to be smaller. After analysing a large number of results, the
consequence was not visible for laser-welded joints.
The process of laser welding was profoundly addressed in order to understand
the mechanics of the joint. Laser welding leads to a better microstructure and
a more sound weld compared to traditional welding. Welding always intro-
duces discontinuity in geometry and microstructure, defects and imperfections
that have a deteriorating effect on fatigue strength. However, these defects are
not studied extensively with fatigue testing and no measured defect sizes and
distributions are given in order to make accurate conclusions.
Residual stresses were found to have a great effect on the fatigue behaviour of the
weld. Analysis of base material fatigue properties and the elastic-plastic material
model with kinematic hardening revealed that stress discontinuity in the weld
region due to residual stresses affect the behaviour of the material greatly. From
the results of the analysis, it can be concluded that residual stresses define the
slope of the S-N curve. The dispersion of test results is due to different defects
and imperfections. The effect of the aforementioned factors can be estimated
with statistical analysis.
83
The traditional welding fatigue assessment is based on testing and the S-N
curves deduced from the results. Welding fatigue assessment differs from nor-
mal fatigue development. In nominal fatigue studies, the direction of develop-
ment is toward material and microstructure behaviour. This study presented
the possibility to approach welding fatigue within normal fatigue assessments.
The analysis was well in line with normal fatigue tests and the assumptions of
the insignificant effect of mean stress was confirmed. The effect of hardness
distribution, material and joint type are interesting for future studies.
84
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