temperature profiles and hardness estimation of laser welded

Temperature profiles and hardness estimation of laser welded heat affected zone in low carbon steel Axel Lundberg One year Master's Degree Computational Materials Science June 2014

MT624A Temperature profiles and hardness of laser weld HAZ Axel Lundberg

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Avdelningen för Materialvetenskap och Tillämpad Matematik

Malmö Högskola

205 06 MALMÖ

Division of Material Science and Applied Mathematics

Faculty of Technology and science

Malmö University

S-205 06 MALMÖ

Sweden

Temperature profiles and hardness

estimation of laser welded heat affected

zone in low carbon steel

Axel Lundberg

Examiner: Christina Bjerkén

Supervisor: John C. Ion

One year Master's Degree

Computational Materials Science June 2014


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Abstract Thermal modelling of hardness in the heat-affected zone (HAZ) in a laser welded steel plate

is a cumbersome process both in calculation and simulation. The analysis is however

important as the microstructural phase transformations induced by welding may cause

unwanted hardness levels in the HAZ compared with that of the parent material. In this

thesis analytical equations have been implemented and checked for validity against

simulations made by other authors and against experimental values.

With such a large field as thermal modelling, the thesis had to be narrowed down to

make the analysis more subject focused. Limitations made were for mathematical modelling

only looking at a two-dimensional heat flow in welded plates; in this thesis only the

analytical solution to the heat flow is considered. The work was also directed towards steel;

such a material as used largely all over the globe. As laser welding is a fast and cost-

effective process, an analysis of hardness is of great importance.

Work was divided into three overlapping parts; the first was to derive and understand the

work done in the field of thermal modelling of welds, thus understanding the mathematics

behind the basic problem. This modelling provides a number of curves and parameters from

a thermal cycle, thus enabling one to do the hardness analysis correctly.

Secondly, this mathematical modelling was applied to a number of cases, simulating

different circumstances. This was done using self-programmed Graphical User Interfaces

(GUI) for convenience. This enables engineers to easily plug in the materials and processing

properties and thus simulate the required parameters and curves for further analysis.

Lastly, a GUI for simulating the hardness of any point in the HAZ was programmed and

used, thus implementing and validating the equations. A theoretical introduction of the

phases induced in the HAZ is also included, in order of understanding the problems of

unwanted hardness in the HAZ of laser-welded steel.

Main conclusions of this thesis:

Mathematical modelling of heat transfer in welds by Rosenthal (1946) is still

applicable for modern laser welding apparatus.

The empirical model presented by Ion et al. (1984) is not applicable with

experimental results of hardness in the HAZ of the steels investigated here.

Equations by Ion (2005) are accurate for simulating the hardness.

The analytical solutions investigated are superior to numerical solutions with regard

to quick, simple simulations of thermal cycles and hardness. Numerical solutions

allows for more advanced modelling, which can be lengthy.

Preheating the steel prior to welding is favourable in reducing hardness levels,

especially with steel of higher carbon equivalent.

Keywords: Laser welding, HAZ, heat-affected zone, hardness, heat equation, thermal

modelling, thermal cycle, Rosenthal


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Sammanfattning Termisk modellring av hårdhet genom beräkning och simulering av den värmepåverkade

zonen i en lasersvetsad stålplatta är en omfattande process. Dock är analysen viktig då

mikrostrukturella fastransformationer förorsakade av svetsningen kan ge oönskade

hårdhetsnivåer av den värmepåverkade zonen jämfört med hårdeheten i basmaterialet. I

denna avhandling har analytiska ekvationer implementerats och testats för validitet mot

simuleringar gjorda av andra författare och mot experimentella värden.

Eftersom termisk modellering av svetsar är ett omfattande område var avhandlingen

tvungen att smalnas av för att göra analysen mer fokuserad. Begränsningar gjordes för den

matematiska modelleringen genom att endast titta på två-dimensionellt värmeflöde i

svetsade plattor där endast den analytiska lösningen är av intresse. Arbetet har också

inriktats mot stål då detta material är vida använt över hela världen. Då lasersvetsning är en

snabb och kostnadseffektiv process så är hårdhetsanalysen av största vikt.

Avhandlingen är uppdelad i tre övergripande delar; den första är att ta fram och förstå

arbetet som gjorts inom termisk modellering av svetsar, alltså förstå matematiken bakom

problemet. Modelleringen är till för att producera diagram parametrar från en termisk cykel,

för att kunna fortgå med korrekt hårdhets analys.

För det andra så sätts den matematiska modelleringen på prov i ett antal situationer som

var och en simulerar olika förutsättningar. Detta gjordes i ett grafiskt användargränssnitt av

ren bekvämlighet. Detta gör att ingenjörer lätt kan implementera olika egenskaper för

materialet och få fram diagram och kurvor.

Sist, ett liknande grafisk användargränssnitt för att simulera hårdheten i valfri punkt i

den värmepåverkade zonen programmerades och därigenom implementerades ekvationerna

som denna avhandling handlar om i grund och botten. En teoretisk bakgrund till

fasomvandlingen är också inkluderad som förklaring till grundproblemet med oönskad

hårdhet i den värmepåverkade zonen i lasersvetsat stål.

Huvudslutsatser i avhandlingen:

Matematisk modellering av värmeöverföring i svetsar genomförd av Rosenthal är

fortfarande applicerbar på modern lasersvetsningsapparatur.

Den empiriska modellen från Ion et al. (1984) är ej applicerbar med godkänt resultat

för hårdhetsuppskattning.

Ekvationerna från Ion (2005) är statistiskt godkända för att simulera hårdhet.

Den analytiska lösningen är överlägsen den numeriska när det gäller snabb och enkel

implementering för att simulera termiska cykler och hårdhet, medan den numeriska

lösningen kan ta i beaktning mera avancerade egenskaper.

Förvärming av stålet innan svetsning kan vara mycket fördelaktigt för hårdheten i

den värme-påverkade zonen, speciellt vid högre kolekvivalent.

Nyckelord: laser-svetsning, värme påverkad zon, hårdhet, värmeledningsekvationen, termisk

modellering, termisk cykel, Rosenthal


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Preface This master´s thesis was written as the last step toward a one-year master´s degree in

Computational Materials Science at Malmö University. Work was initiated in February 2014

under the supervision of John C. Ion of Malmö University and was finished in June the same

year. The total extent of this dissertation is 15 credits.

The motivation for this thesis and the introduction to the field was made by the supervisor.

Thanks to my supervisor for his interest in advising me and thus making this possible. He

created the foundation on which to build further. Many hours have been spent behind the

MacBook, on which this has been written, with programming, reading and writing. I hope

that it one day will be worth the effort. A really special thanks to my family and especially

my better half, Guðný, who put up with me during this time of life…

Kristianstad

June 2014

Axel Lundberg


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Nomenclature Symbol Definition Unit

Absorptivity ---

Transformation temperature K

Carbon equivalent wt%

Hb Vickers hardness number of bainite HV

Hfp Vickers hardness number of ferrite-pearlite mixture HV

Hm Vickers hardness number of martensite HV

Hmax Vickers hardness number of HAZ HV

K0 Bessel function ---

Nhet Heterogeneous nucleation rate ---

T Temperature K

T0 Initial temperature K

Tm Melting temperature K

Tp Peak temperature K

TMs Temperature at which martensite starts to form K

TM50 Temperature at which martensite formation is 50% complete K

TMf Temperature at which martensite formation is complete K

V’ Cooling rate at 923 K K h-1

Vb Volume fraction of bainite ---

Vfp Volume fraction of ferrite-pearlite mixture ---

Vm Volume fraction of martensite ---

a Thermal diffusivity m2 s

-1

c Specific heat capacity J kg-1

K-1

d Thickness m

e Base of natural logarithms, 2.718 ---

f Matrix volume fraction available ---

k Boltzmann’s constant, 1.381 J K-1

q Beam power J s-1

(W)

r Lateral distance from centre of a through-thickness heat source M

t Time s

x,y,z,ξ Spatial coordinates ---

w Width M

Laplace operator ---

Differential operator ---

λ Thermal conductivity J s-1

m-1

K-1

ρ Density kg m-3

ΔGm Activation energy for atomic migration per atom J mol-1

ΔG* Activation energy barrier for nucleation of the critical nucleus radius J mol-1

Δt8-5 Time to cool from 800 to 500°C s

Cooling time for 50 % martensite formation s

Cooling time for 50 % bainite formation s

Cooling time for 0 % ferrite-pearlite mixture s

Cooling time for 0 % bainite formation s


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Contens

1 Introduction ...................................................................................................................... 1

1.1 Background ............................................................................................................................. 1

1.2 Purpose .................................................................................................................................... 2

1.3 Objectives ................................................................................................................................ 2

1.4 Limitations............................................................................................................................... 2

1.5 Method..................................................................................................................................... 3

2 Laser welding and related welding processes ................................................................ 5

2.1 Regions of the weld-zone ........................................................................................................ 5

2.2 Why study the HAZ-microstructure? ...................................................................................... 6

3 Mathematical modelling .................................................................................................. 7

3.1 The equations of heat flow in the HAZ ................................................................................... 7

3.2 Temperature-time profile in the HAZ ................................................................................... 10

3.2.1 Time constants derived from temperature-time profile.................................................. 12

3.3 Peak temperature-distance relationship ................................................................................. 13

3.4 Input energy – HAZ width relationship ................................................................................. 14

3.5 Verification Rosenthal thermal modelling ............................................................................ 16

3.5.1 Rosenthal modelling for different materials ...................................................................... 18

4 Evolution of microstructure in the HAZ ..................................................................... 19

4.1 Eutectoid transformation – pearlite, bainite or martensite formation .................................... 19

4.1.1 Pearlite formation ........................................................................................................... 20

4.1.2 Bainite formation ........................................................................................................... 21

4.1.3 Martensite formation ...................................................................................................... 22

4.2 Transformation rates to TTT-diagrams – theoretical approach ............................................. 23

5 Hardness in HAZ ........................................................................................................... 29

5.1 Analytical equations of phase volume fraction in low carbon steels ..................................... 29

5.2 Hardness calculations by rule of mixtures ............................................................................. 31


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6 Results and discussion – thermal modelling ................................................................ 33

6.1 Thermal modelling simulations ............................................................................................. 33

6.1.1 MATLAB® implemented GUI for thermal simulation ................................................. 33

6.1.2 Graphical simulations of thermal modelling .................................................................. 34

6.2 Discussion of thermal modelling ........................................................................................... 45

7 Results and discussion – empirical hardness estimation ............................................ 49

7.1 Empirical hardness estimation using calculated volume fractions ........................................ 49

7.1.1 MATLAB® implemented GUI for hardness simulation ............................................... 49

7.1.2 Graphical results of hardness simulations ...................................................................... 50

7.2 Discussion of empirical hardness simulation ........................................................................ 63

8 Results and discussion – graphical hardness estimation ............................................ 65

9 Conclusion ...................................................................................................................... 67

9.1 Conclusions ........................................................................................................................... 67

9.2 Future work – Possible improvements .................................................................................. 69

10 References ..................................................................................................................... 71

Appendix A: MATLAB® GUI-code for temperature profiles ......................................... 73

Appendix B: MATLAB® GUI-code for hardness estimation .......................................... 79

Appendix C: Table for hardness simulation comparison ................................................. 83


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1 Introduction

1.1 Background

Two or more pieces of metal can be joined together, using some type of welding apparatus.

This welding apparatus might contain a high powered laser beam, thus being a laser welding

apparatus. When this weld is made, the weld fuses the two pieces together by heating the

base-material to melt. When this happens, both a weld bead (the area of fused and molten

base material) and the heat affected zone (HAZ) are formed. The HAZ is formed when the

heat radiates from the bead via conduction in the base-material, thus the heat will transform

the material in this zone and in the welding situation induce phase transformations. When

phase transformations occur, the mechanical properties of the HAZ will differ slightly from

those of the base material, if the welded piece is of steel or other weldable metals.

This problem has long been studied, in which the foundation is one of many differential

equations derived for certain problems. This heat transfer problem, as welding is, looks into

the solutions for the heat equation as the basis for further mathematical modelling. When this

differential equation is solved for the selected problem, specifically for the analytical

solutions, as this work describes, one can produce temperature profiles for specified energy

input and desired mechanical properties of base material.

By then looking at what is produced by this analytical solution and its temperature

profiles, the phases of the HAZ may be studied. By producing a time-temperature-

transformation (TTT) diagram, the phase volume fractions in the HAZ are derived and

compared with selected calculations. These volume fractions then help determine the

hardness.

Phase transformations are important to study because when they occur, the base material

will possess different properties to the HAZ. In industry there is a certain measure of

hardness that is considered to be maximum allowed, a value that if exceeded might lead to

cracks or failure of any construction. It is therefore of great importance to study the weld

HAZ in order to fabricate constructions that will withstand the forces put on them at all

times. Cyclic loads are one of the greatest threats to a brittle weld, as these tests its strength

over a timespan and a large number of load cycles until fatigue of the weld cause it to break.

This project is aimed at determining whether or not one can use models derived in 1984

and 1996 (Ion 2005, p.532-535) and if they are applicable on modern low carbon steels. If

not, further studies are required.


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1.2 Purpose

Investigating the phases within the weld HAZ is very important in assessing the mechanical

properties of a selected weld. By knowing these properties, constructions can be made more

durable. Verifying if one may or may not use the equations described by Ion et al. (1984) or

the equations in Ion (2005, p.534) is the main purpose. The aim is to be able to establish if

new equations are needed in order to initiate further work in this field. Other objectives are

to find if the thermal modelling by Rosenthal (1946) may be used as validation for hardness

simulations and laser welding applications.

1.3 Objectives

The main objective of this thesis is to evaluate the hardness of a weld HAZ, using

experimental data and analytical modelling. The first step is to analyse the mathematics

behind the heat-problem using thermal modelling, further on into the solutions to produce

temperature profiles. Then different diagrams will be produced for certain steel compositions

that will be used when investigating hardness. Values will then be compared to experimental

values obtained for modern steels, thus confirming or falsifying the old equations and

methods.

1.4 Limitations

The mathematical modelling will only be valid for 2-dimensional heat flow for the analytical

solution of laser welded low carbon steel plates. This means that the laser beam will

penetrate the entire plate of base-material. This is done in order to narrow the amount of

analytical equations stated in the modelling part, but also because 3-dimensional heat flow,

as found in partial penetration welds, is a much more computer demanding operation.

Limitation to the carbon equivalent of the parent material will be made so that all the low-

carbon steels analysed are eutectoid or hypo-eutectoid, thus the carbon equivalent will be at

maximum CEq = 0.78 wt %. The analysis is only about the HAZ, so what happens in the weld

bead or otherwise in the base-material is neglected.


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1.5 Method

A literature study aims to establish the state of the art of the modelling methods. These

topics will be mathematical modelling of heat transfer problems, the solutions to this

differential equation, TTT-diagrams, phase transformations in steels and model based

hardenability. Other subjects that will be addressed are how this mathematical modelling

generates the temperature profiles needed for further analysis, but also some experimental

literature values. Thus no physical experiment will be performed, but the values obtained

from an experiment performed in literature will be used as validation further on in the thesis.

The first analysis, described in Ch. 3 below, is how the mathematics are formed from the

initial differential equation to the final, specified, analytical solutions to the limitations of the

problem. This deduces the thermal/mathematical modelling part of this thesis.

The second analysis, described in Ch. 4, will consist of how the phases are formed in the

HAZ and how to construct the necessary diagrams in order for interpretation of the results

from temperature profiles. These profiles are brought forth in the first analysis and then used

for understanding in the phase transformation process of the base material in the HAZ.

The last analysis, described in Ch. 5 is how the hardness is calculated. This chapter will

also contain the part where the theories behind interpreting the hardness diagrams and in

what region of the diagram one want to recognize.

This will be rounded off by a results part where results are divided into the same

disposition as the theoretical analysis. This ends with a discussion and interpretation of the

results, were the main objective be considered and further work proposed.

To handle the data in a correct and smooth way and produce plots, MATLAB® will be

used. Inside MATLAB®

, small graphical user interfaces (GUI:s) will be constructed. This

will assist the author and others to follow, to interpret data and plot the necessary curves and

calculate data easily. The algorithm of these GUI:s will be presented inside the thesis,

although the computer code for them may be found in App. A and App. B.


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2 Laser welding and related welding processes

The basic physics behind the laser and how it can be used in the materials processing area

will not be dealt with in this thesis. Although if interested is awaken, many books like Ion

(2005) have been written on the subject of laser apparatus and involved processes.

During the process, when the high-powered laser beam is impinged upon the surface of

the adjacent base material, the beam has to be of sufficient power and focused, all in order to

initiate vaporization. When this is achieved the material will start to melt and then fuse (Ion

2005, p.396-397). When the beam of the welding apparatus is travelling transversely to the

work piece, a narrow channel will be formed, called the keyhole. This keyhole is what

eventually forms the weld bead. This keyhole effect is what makes the laser welding process

so efficient according to Fabbro et al. (2000), thus modelling of this process is of highest

interest. The efficiency can be traced to that there is a narrow bead in the centre, adjacent to

a relatively small HAZ and also that there is a high aspect ratio of the welded zone

(depth/width) (Ion 2005, p. 435).

This laser process is also of great improvement over traditional arc welding due to that it

has a lower energy input per unit length that produces this relatively narrow HAZ. This

means that thermal distortion of the work piece is not that significant. This is of course more

applicable the thinner the plate, which has a higher degree of warping during greater amount

of energy input (Sokolov et al. 2011).

2.1 Regions of the weld-zone

What is important is to distinguish between the two zones of the weld, seen clearly in

Fig. 2.1, where the present work is concentrated upon the HAZ. This is due to that the bead

material will somewhat resemble the base-material. The grains grow in a columnar

morphological way, something that is also observed of rapidly cooling base-materials.

Therefore the physical properties and microstructure of the weld bead can be stated as less of

interest in the hardness investigation due to the resembling to the base material (Yilbas et al.

2010 and Ion 1984, p.48-49)


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Figure 2.1: Welding zones: WM-weld metal and BM-Base material, for 3-dimensional heat flow

(Adapted from Poorhaydari et al. 2005).

2.2 Why study the HAZ-microstructure?

In the HAZ, the rapid thermal cycles will not result in the same microstructural grain

growth as in the bead, but instead phase transformations of the low carbon steel that occur in

the HAZ of the weld are induced by the high cooling rates of the weld passing. These phase

transformations in the HAZ of the welded steel may induce a hardness that is higher than

preferred, thus making the weld more brittle and less ductile compared to the parent

material. Very fast cooling rates may also induce martensite formation in the HAZ, which

also induces unwanted embrittlement (Yilbas et al. 2010) and (Ion et al. 1984). One good

way of reducing the unwanted hardness levels in the HAZ of the weld is to raise the preheat

temperature. The problem will then be a much more complicated work process, whilst only

gaining approximately 20 % difference in the HAZ microstructure compared to an unheated

work piece (Sokolov et al. 2011). This thesis starts with the mathematical modelling and

continues looking at the formation of various microstructural changes in hypo-eutectoid and

eutectic low carbon steels to begin with. In the hypo-eutectoid region most of the induced

transformations are results of changing from austenite ( ) into mainly martensite, pearlite-

ferrite mixture and bainite. These phases and constituents are in turn built up different but

they all contain the chemical compound cementite (Fe3C) in various amounts, see Ch. 4.


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3 Mathematical modelling

As stated before, the welding process induces melting and vaporization by a high-powered

laser beam in a base-material. How is heat flow caused by a moving line heat source

modelled? This heat flow development can be described mathematically using appropriate

heat transfer equations. Note that experimental procedures are necessary in order to verify

the equations (Poorhaydari et al. 2005). One example is that bringing forth a model

predicting the fusion depth of the weld within 0.01 mm would be unreasonable as the

limitation in fabrication welding would be approximately 0.15 mm even during the best of

conditions (Ion 1984, p.32).

3.1 The equations of heat flow in the HAZ

Heat flow in welding, whether it is arc or laser welding, is a very complex mathematically

descriptive situation. This process can be divided into three different situations: Transient-,

quasi- and steady state (Bass 1983, p.183).

Quasi-steady state heat flow is presenting a situation in which the observed temperature

field from a chosen moving heat source is constant. In order to find the analytical solutions

to this complex welding heat flow problem, solution to the partial differential equation of

energy conservation seen as Eq. (3.1) is needed (Darmadi et al. 2011).

(3.1)

Where is the thermal diffusivity in either spatial coordinate. If the material that is of

interest is an isotropic homogenous material, the thermal diffusivity a will be constant in all

space-coordinates (x,y,z). If then the Gaussian-distributed temperature field varies in both

space and time, the differential equation becomes Eq. (3.2) (Nunes 1983):

(3.2)

If the heat is supplied to the weld with a constant speed v, moving along the x-axis like in

Fig. 3.1, Eq. (3.2) may be rewritten with the point heat source as the origin of the problem.

By defining a variable , where is the specific length from the origin to the

specified point along the x-axis and then differentiating Eq. (3.2) with respect to the new

variable, Eq. (3.3) is developed (Goldak et al.1986).

(3.3)


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Figure 3.1: Co-ordinate system and geometry of plate welding (Ion et al. 1984).

Now in order to show that Eq. (3.3) can be reduced to Eq. (3.4), one needs to show that the

solid is of infinite length, compared with the extent of the point heat source and the heat

sources extent (Ion et al. 1984). Then the temperature distribution around this particular

source will be constant. This state is then referred to as a quasi-stationary state, which

mathematically can be related to that (Pavel 2008).

(3.4)

Fig. 3.2 shows the quasi-stationary state, where the temperature will be at its peak just below

the heat source moving along x-axis at the velocity v. The temperature then decreases over

time and distance, which is shown by the isotherms.

Figure 3.2: 3-dimensional keyhole temperature distribution around a moving heat source

(Ion 1984, p.35).


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The next step in the process to find an analytical solution of the heat flow differential

equation are some basic assumptions made by Rosenthal (1946), who is often referred to

first in work about heat flow in welds in general (Kamala et al. 1993). The assumptions are

according to Ion (1984, p.33):

i. Heat is provided by a point heat source.

ii. Both latent heat of phase transformations and the fusion of the weld bead are

neglected, i.e. no energy is generated by material transformation.

iii. Thermal properties of the welded material are not dependent on temperature; this is

of course not true, but assumption made in original solution.

iv. Heat flow occurs only by conduction in the work piece; no heat losses through

surface.

v. Speed v will be constant; reasonable for automated welding processes.

Rosenthal (1946) presented the analytical solution to Eq. (3.4), by the use of complex

mathematical modelling and his assumptions. The work can be summarized in two main

equations (Eq. 3.5 and 3.6 resp.), the first describing 3-dimensional heat-flow, this from a

surface heat source, where heat is conducted radially through the material. The other

describes the 2-dimensional situation, where heat is only conducted laterally in the material.

Fig. 3.3 schematically show both these conditions. The heat flow will of course, despite

assumptions made by Rosenthal (1946), dissipate through the surfaces of the work piece,

although not shown in Fig. 3.3.

Figure 3.3: Heat-flow (orange) in thin-plate respectively thick-plate, i.e. 2D respectively 3D.


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( )

(

) {(

) } (3.5)

( )

(

) (

) (3.6)

For this step one need to notice that firstly the preheat/initial temperature T0 is added as the

initial temperature of the plates being welded and q is the power of the weld apparatus

(Darmadi et al. 2011). Secondly, K0 represents the Bessel-function of second kind and zero

order and while (Bass 1983, p.182). According to

Poorhaydari et al. (2005) simplifications of Eq. (3.5) and (3.6) were made by Easterling and

Ashby in order to produce thermal cycles for the HAZ, see Eq. (3.7) and (3.8).

( )

(

( )

) (3.7)

( )

( ) (

( )

) (3.8)

Further on in the project the focus will be on the 2-dimensional solution. There the

z-coordinate is ignored, and the thickness d of the plate is used directly as the solution

heavily depends on this parameter (Bass 1983, p.182).

3.2 Temperature-time profile in the HAZ

For the first task, the goal is to produce a temperature versus time plot. The problem faced

here is that the assumption made earlier in order to solve Eq. (3.4) was for a quasi-stationary

state, thus meaning the assumptions cannot be applied when seeking the temperature

distribution in a fixed point (Darmadi et al. 2011).

As this project is only looking at 2-dimensional cases, the heat only disperses laterally.

If one considers the point in the plane, x = 0, this gives: and ( ) .

Now r is defined as the lateral displacement in the plane, i.e. the distance from the point of

interest to the weld centre line. Then and ( ) , thus reaching Eq. (3.9).

( )

(

) (3.9)


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In order to reach the goal, thus producing a plot, which is principally shown in Fig. 3.4, the

exponential part of Eq. (3.8) must be considered mathematically. This was done by Ion

(1984, p.37) in order to obtain Eq. 3.9. Another addition is that if one considers a thin plate

solution, i.e. a 2-dimensional heat-flow, some of the input form the laser will escape through

the plate. Ion et al. (1996) defines the factor ( ), which is the absorptivity of the

welded plate of the beam in laser welding.

( )

(

) (3.10)

The factor Aq/(vd) (called absorbed energy per area unit) is broken out, as these parameters

are what may be controlled by the welding machine chosen, as well as producing the plots

with width versus Aq/(vd) (Darmadi et al. 2011).

Figure 3.4: Schematic plot of temperature versus time (Ion et al. 1984).

If the cooling curve is divided as in Fig. (3.4), it is easy to understand how Eq. (3.10)

actually works. The exponential part will control the rapid heating and when the time tends

towards infinity; this part tends to 1. The inverse part of Eq. (3.10) controls the cooling

phase of the curve (Ion 1984, p.38).

Equation (3.10) is highly sensitive to the radius r of the weld, because the radius is

sensitive to unpredictable variations. This knowledge is crucial if Eq. (3.10) should be used

practically (Poorhaydari et al. 2005). Profiles produced by Eq. (3.10) may be validated by

being plotted against a profile that has been arisen experimentally, work that has been done

by both Ion et al. (1984) and by Poorhaydari et al. (2005). Though their experimental

procedures differ somewhat, thin steel plates in which holes were drilled into the HAZ where

used in both experiments. Small thermocouples where placed in the holes, where the

temperature was then measured in order to plot the experimental curve that can be compared

with curves produced by the analytical solution, Eq. (3.10).


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3.2.1 Time constants derived from temperature-time profile

From the profile as in Fig. 3.4, where temperature is calculated from each separate time step,

two important time constants need to be described. These two quantities can be seen in Fig.

3.5. Firstly the constant is described as the time to reach peak temperature. The constant

is evaluated by differentiating Eq. 3.10 with respect to time and setting the resulting

differential to zero, thus obtaining Eq. (3.11) (Poorhaydari et al. 2005). Below, e is base of

the natural logarithm.

(

)

(3.11)

Secondly the cooling time; is stated. This refers to the severity of the quench through

phase transformations, i.e. the time taken to cool from 800 to 500 . The cooling time is

stated as the inverse part of Eq. (3.11), multiplied by a weight factor for the temperatures

that one seeks (Ion et al. 1984). It is important to note that, according to Ion et al. (1984), if

the peak temperature is below 900 , Eq. (3.12) may not be used.

(

)

(3.12)

where:

(

( )

( ) ) (3.13)

Figure 3.5: Typical temperature-time profile with time constants showed (Ion et al. 1984).

Eq. (3.11) and (3.12) are found to be reasonably suited for the estimation of the constants

cooling temperature and peak temperature, but according to Ion (1984, p.42) some

calibrations must be made to the welding equipment in order to get accurate answers. They

are though good for estimating the variations of and about a known value with a


13

change in welding conditions. Another thing to note is that the cooling time is totally

independent of the distance from the point heat source, at least when one looks at the HAZ.

This has been proven both numerically and experimentally, which supports the analytical

solution (Poorhaydari et al. 2005).

In Ch. 5 the deduction of why the cooling time, , is so important, is stated. This is

the real connection between the thermal modelling and the actual hardness simulations. The

fact that the volume fractions of phases are heavily dependent on the cooling rate is

mentioned in the early work done by Ion et al. (1984). This situation, that if the energy put

into the weld increases the cooling time, was experimentally confirmed by Poorhaydari et al.

(2005). In their work, they published Fig. 3.6. The shifting to the right in the figure shows

that the cooling time drastically increases as input energy increases. The slight difference

between the peaks of the thermal cycles (Input energy 1-3) in Fig. 3.6 is due to that the

thermocouples that measure temperature were placed at slight different places in order to

reach the same peak temperature, Tp (Poorhaydari et al. 2005).

Figure 3.6: Measured temperature profiles showing drastic increase in cooling time of weld HAZ

(Adapted from Poorhaydari et al. 2005).

3.3 Peak temperature-distance relationship

The meaning of the parameter Tp can be found in Fig. 3.5. This is the variation of peak

temperature with respect to distance from heat source. Eq. (3.14) is obtained using that the

condition at the peak is , thus differentiating Eq. (3.10) with respect to time.

(

)

(

) (3.14)

Both Poorhaydari et al. (2005) and Ion et al. (1984) pointed out that using Eq. (3.14) directly

versus radius from heat source would only show the variation of the peak temperature over

the HAZ. This is because this radius r will include some part of the molten weld pool, thus

not following the Rosenthal (1946) assumptions of a single-phase material.


14

3.4 Input energy – HAZ width relationship

The last goal of the thermal modelling in this project will be to look into the relation between

the input energy and width of the HAZ. This is interesting when studying the HAZ

geometry. This geometry will be examined later, thus finding the hardness and the

composition of the HAZ. The hardness will be plotted against the input energy, in order to

evaluate the hardness of the weld in a proper manner, by looking at the phase

transformations inside the HAZ (Ion et al. 1984).

The width of the HAZ can be calculated in two major ways, firstly Ion (2005, p.436)

states that: if the input energy to the HAZ is q’, then , where is the radius

of the molten weld pool. By substitution of this into Eq. (3.12) the same author states that:

(

)

[

( )

( )] (3.15)

Where Tm is the melting temperature and AC1 is the transformation temperature at the end of

the HAZ. When using Eq. (3.15), it must be noticed that some part of the actual weld bead

will be taken into the calculation. It is one of the assumptions that the thermal properties of

the material do not change during welding. This makes for a slight overestimation of the

HAZ-width, so the calculated value will be slightly greater than the experimental value (Ion

1984, p.39).

Tekriwal et al. (1988) also support this theory, which show that the HAZ will increase

somewhat by the transient heat from the weld, which makes the quasi-steady state somewhat

questionable. The second theory is stated by Poorhaydari et al. (2005), who state that the

width of the HAZ is found when calculating the radius-Tm and the radius-AC1 from Eq. (3.12),

then combining them:

(

)

[( )

( )( )] (3.16)

Which is the same as Eq. (3.15), which is used to be consistent with the rest of the project.

As comparison for results, Eq. (3.16) is important but not dealt with further. Eq. (3.17) and

(3.18) resp. may be used to calculate transition-temperatures as a function of composition

(Kamala et al. 1993):

(3.17)

(3.18)


15

Fig. 3.7 is taken from the study made by Poorhaydari et al. (2005). It shows both the 2 and

3-dimensional HAZ-widths, compared with actual experimental values evaluated by the

same. Important to note is that they plot the width against the heat input, not the energy

input, but the schematics of the plot remain the same with both. Fig. 3.7 provides a good

visual impression that the analytical solution gives computed values within the range of the

experimental values. This simulation has been performed numerically by Piekarska et al.

(2012) who deduced roughly the same conclusion. Their research on how the laser properties

affect the weld was conclusive about the thermal modelling in the sense that simulated

numerical values had the best fit, between the two analytical solutions.

Figure 3.7: HAZ-widths from three different solutions (Adapted from Poorhaydari et al. 2005)


16

3.5 Verification Rosenthal thermal modelling

Before going through what is produced by the GUI developed using the mathematical theory

about thermal modelling, remarks will be made about the validity of the modelling, therefore

discussing the validity of the graphs produced. Fig. 3.8 shows two curves derived by

experiments made with real welding apparatus versus the curves derived with thermal

modelling according to the functions and solutions, Eq. (3.10) and (3.12) compiled by

Rosenthal (1946).

Figure 3.8: Experimental versus thermal modelling values of temperature profiles (Ion et al. 1984).

The preface towards Fig. 3.8 was underlying experiments using a weld simulator; the whole

line represent the experimental curve, thus the thermal modelling is shown as the broken

line. Minor holes were drilled into the underside of a plate, in order to penetrate the HAZ,

thus gaining sufficient data to plot the profiles. Thermocouples were spot-welded inside the

holes, thus measuring the cooling time of the welding cycle (Ion 1984, p.20).

The validity of Rosenthal’s thermal modelling of a heat point source is sufficient enough, the

differences between the two curves are minor and more importantly, the reasons are known.

In section 3.1 the general solution to the Rosenthal equation was presented, and with it the

assumptions to get the solution (Ion 1984, p.33). As suggested by Ion (1984), these

assumptions play a crucial role in the error estimation, especially when stating that the

thermal properties of a material are constant.


17

This phenomena has been questioned by several authors, e.g. Goldak et al. (1986), who

incorporates not only one, but two of the original assumptions into their numerical model.

According to them, the original analysis made by Rosenthal (1946) cannot be extended to

incorporate the thermal properties due to their non-linear nature. To clarify what Goldak et

al. (1986) means by error estimation, Fig. 3.9 was derived to prove that there is a difference.

They state that the effect is so profound on the analytical solution that the numerical slution

is significantly better, especially for the 3-dimensional heat-flow case. The work-cost of

actually incorporating these non-linear thermal properties into the numerical analysis is

trivial according to Pavel (2008), stating that the numerical solution is the most satisfying

with appropriate data supplied.

Figure 3.9: Effect of thermal properties on a computed weld curve in steel,

(a) being variable properties, (b) being constant (Adapted from Goldak et al.1986).

Further discussed is the incorporation of latent heat developed by the weld bead. The most

difficult things to analyse are partly solid-state transformations, e.g. the austenite-pearlite

reaction, partly the transformation surface, in which the liquid-solid boundary is formed. The

latter is very difficult to incorporate exactly in the analytical solution, due to there being a

discontinuity in the thermal gradient of the boundary, when forming the moving boundary,

traveling along the weld-axis with the weld apparatus (Goldak et al. 1986). This phenomena

is also pointed out by Ion (1984) that states the following about the analytical solution; “It

cannot, however, describe the latent heat evolved from phase transformations and weld bead

solidification, although these phenomena do not affect the kinetics of grain growth, particle

coarsening etc. significantly.” (Ion et al. 1984, p. 92)


18

3.5.1 Rosenthal modelling for different materials

It was important to implement in the GUI easy access to a materials-library, in order to see if

the Rosenthal (1946) equations would work on other material in theory. The validity of

usage of these equations on different materials other than steel is not thoroughly developed,

but work carried out by Kou (1981) supports that the equations may be used, although Kou

(1981) prefers the usage of numerical modelling. In Fig. 3.10a the thermal cycle for a weld-

pass in 6061-aluminium is shown, thus providing the author with somewhat of verification.

The simulation cannot be reproduced, as Kou (1981) does not provide all the setup

parameters for reproduction of analytical simulation.

The difference provided for welding aluminium shows that there is not much of a

difference between numerical and analytical modelling, despite the age of the article by Kou

(1981), which supports that computer computations may have improved in later years. Fig.

3.10a also supports the singularity problem of analytical modelling, taken up in the results,

were temperature approaches infinity at small radii. Further, the weld pool calculations

made, seen in Fig. 3.10b, is not accurate, but provides sufficient correlation for interpretation

of results. Lastly, the latent heat of fusion-problem is also stated as an error-estimation, since

this is not implemented in the analytical solution to the heat-transfer solution provided by

Rosenthal (1946) (Kou 1981).

Figure 3.10a and b: Comparison of analytical versus numerical modelling for aluminium (Adapted from Kou 1981).

Kou (1981) also identifies the same observation in aluminium that Poorhaydari et al. (2005)

has pointed out in Fig. 3.6, that cooling time is highly dependent on the input/absorbed

energy, but also makes the note that aluminium specifically depends heavily on preheating.

Both the cooling time and buckling-effect in parent-material are drastically reduced when

preheating, but as a result the weld pool will extend, therefore making for more HAZ-region

and doubling solidification times of phase transformation.


19

4 Evolution of microstructure in the HAZ

Steels of different compositions will be the focus in this thesis, thus only hypo-eutectoid and

eutectoid steels are considered. By this limitation, phases of transformations are limited to

and simulation of hardness is easier. Also that alloying components may form undesired

precipitates of mixed compositions that are hard to control in the analytical solution (Ion et

al. 1984).

Figure 4.1: Sequence of steps for a technical solution in general and thesis specific.

Fig. 4.1 shows the evolution, which is often applicable for many technical problems

including this thesis. This procedure is emphasised, as the mathematical/thermal modelling

and solution to such a problem has been shown in the previous chapter that brings on the

physical interpretation and then discussion of results that could result in technical solutions.

4.1 Eutectoid transformation – pearlite, bainite or martensite formation

In order to further explain what will happen in the microstructure of the HAZ, one needs to

consider Fig. 4.2. The eutectoid composition will form at 0.78 wt% C.

Figure 4.2: Simplified phase diagram of low carbon steels, adapted from lecture1.

1 Ion, J. (2013). Lecture 6 – Phase transformations in steels. Phase transformations, MT622A. Malmö University.


20

What happens when the welding equipment passes a certain point in the base-material, the

HAZ microstructure will change depending on both the temperature and the cooling time,

. The hardness of any point chosen inside the HAZ can be calculated by using the rule

of mixture (Ion et al. 1984). This is done when knowing the correct volume fractions of

bainite, austenite, pearlite and martensite.

But before this happens the necessary calculations for composition of the microstructure

must be made for the chosen steel to arrive at volume fractions and knowing the hardness of

each constituent (Goldak et al. 2005, p.148). When the cooling occurs and the temperature

falls below the AC1 temperature, which is calculated by Eq. (3.18), the austenite ( ) will

become supersaturated with both the phases ferrite ( ) and cementite ( ) (see Eq. 4.1).

As a result of further cooling these two phases will make up either the microconstituent;

pearlite ( ) or the non-equilibrium phase; bainite ( ), a reaction highly

dependent on one important factor; cooling time (Porter et al. 2008, p.312).

(4.1)

4.1.1 Pearlite formation

The microconstituent pearlite is formed through diffusion of carbon in the austenite, growing

into the surroundings as a sheet-lamellae type microstructure. This is due to that the lamellae

are consisting of either cementite or ferrite that nucleates on the grain boundaries in the

austenite. According to Hawbolt et al. (1983) the mechanism of phase growth initiation is

totally random, either the ferrite or the cementite starts to grow along grain boundaries first.

The rate of transformation is deduced by the TTT-diagram, which will be constructed in

a later chapter in this report. The actual volume fraction is deduced by the rate of

undercooling, i.e. the cooling rate. Rate of formation is at its peak around 550 , around the

nose of the C-curve in the TTT-diagram that is shown in Fig. 4.3 (Porter et al. 2008, p.332).

Note that this curve is only schematic and the result will be different when different

compositions of steels and alloys are considered, which is discussed in section 4.2.

Figure 4.3: Schematic plot of a TTT-diagram for the formation from austenite (Adapted from Porter et al.2008, p.333).


21

4.1.2 Bainite formation

The second thing that may happen, if the cooling time is relatively short, is that bainite will

nucleate. The composition of bainite is the same as pearlite, although microstructurally they

are significantly different. It may seem like the two products are the same and form the same

C-shaped curve in the TTT-diagram (See Fig. 4.3a), but this product is more complicated to

derive and categorize, something that has been dealt with several times before (Hawbolt et

al. 1983).

At a relatively high temperature during cooling, around 350 - 550 , bainite will form a

needle-type structure, which is called upper bainite. According to Porter et al. (2008, p.334)

amongst others, deduction that microstructurally bainite is heavily dependent on the forming

temperature. While both constituents grow in roughly the same temperature span in the TTT-

diagram, the distinct difference lies within their crystallography and formation of the latter.

Upper bainite will form the characteristic needle-shape, as ferrite nucleates into the

surrounding austenite along the grain boundary. Whilst the undercooling, which is large at

this stage, is controlling the nucleation, these lath-needles thickens to such a degree that they

become supersaturated with carbon, which in turn builds up the cementite in the

microstructure of the HAZ (Porter et al. 2008, p.334-335).

Lower bainite will form at lower temperatures and higher undercooling below the A1

temperature, but a main temperature for the formation is hard to depict. This mainly depends

on the carbon content of the steel at which it forms in, thus being a highly complicated

transition development not stated in this report (Hawbolt et al. 1983)

In the end the difference of the two will not matter, as the fact that bainite is formed as

two different types of microstructure is disregarded in the hardness derivation later on. What

can be stated though is that bainite formation is a non-resolved issue, a dispute, initiated by

Ko et al. (1952), which is still on going. Basically bainite may be formed by a diffusion-

controlled process, or as a product of shear-transformation by surface relief (Porter et al.

2008, p.337-339).


22

4.1.3 Martensite formation

When passing the eutectoid point in the iron-carbon phase diagram, thus producing a low-

carbon-steel, one last non-equilibrium phase may be recognized, called martensite ( ). If the

cooling time for the passing of the weld is sufficiently rapid, the time for the eutectodial

diffusion-controlled decomposition process is not enough, resulting in a diffusionless

transformation of the austenite (Sourmail et al. 2005).

The transformation process is not fully understood, thus being a very complex

experiment to observe, due to the high speeds of formation that according to Porter et al.

(2008, p.397) may approach the speed of sound at roughly 800 - 1100 m/s. This

transformation procedure is deduced by, in carbon steels, that the austenite carbon ( )

will stay the same in the transition to martensite-carbon ( ).

Figure 4.4: Typical martensite structure, dark areas

represent high carbon content (Ion 2005).

When this supersaturated solid martensite is formed like in Fig. 4.4, the result in a weld is

high brittlement. Wang et al. (1993) state that the martensite together with the overheated

coarse-bainite will be the weakest point in hardness terms in the weld-HAZ. The simplest

explanation to this phenomenon that influence strength and toughness of martensitic steels,

for any austenitic grain size, the martensite will have a finer grain structure, and hence the

steel will be stronger but not necessarily ductile enough (Wang et al. 1993)

.


23

4.2 Transformation rates to TTT-diagrams – theoretical approach

The main objective of deriving the hardness of the HAZ may be done by using the analytical

approach of rule of mixture (Eq. 5.14), which incorporates the volume fractions of each

phase. These volume fractions may be deduced by a graphical approach of a TTT-diagram,

which in turn can be produced in theoretical approach introduced in this thesis.

The idea is by making general approximations to go from fraction transformation that is

made temperature dependent, to the TTT-diagram, a process presented in Fig. 4.5 This will

be presented as a thought, as it is recognised that the work involved to complete this theory

is too cumbersome to be completed in the time frame of this thesis.

Figure 4.5: Schematic transformation to TTT-diagram conversation (Adapted from Porter et al. 2008, p.285).

The theory starts by knowing from previous work by Lee et al. (1993) amongst others, that

the curve on a TTT-diagram is c-shaped when the transformation rate is controlled by

heterogeneous nucleation. Eq. (4.2) deduces the number of possible nucleation-sites (Porter

et al. 2008, p.257).

(

) (

) (4.2)

An approximation must be made, due to that Eq. (4.2) assumes spherical formation of nuclei

in the solid solution. This is not true for martensite, which may form as any shape, but it is a

necessary assumption to go further in theory (Sourmail et al. 2005). This estimation of

formation of grain size effect on the general theory could although be neglected, if one has

enough experimental data to support such an assumption.


24

The next step in the construction of necessary diagrams is to consider the equation, that

controls the volume fraction transformed at varying time and constant temperature, the

Avrami equation, see Eq. (4.3). This equation stated, as was developed to describe the

growth and nucleation rates involved in transformation of precipitates, but in this case used

to describe the phase transformations of the weld HAZ during specific cooling times

(Hawbolt et al. 1983).

( ), (4.3)

where is the volume fraction transformed and

( ) . (4.4)

The construction of Eq. (4.3) is so that the equation itself is not explicitly temperature-

dependent, which is the basic idea of this theoretical approach. Contained in Eq. (4.3) is c,

the parameter of which in turn is controlled by partly the nucleation rate described by Eq.

(4.2) and partly by the parameter that controls the nucleus development in three-dimensions,

i.e. describing how spheres are formed in the solid solution. The parameter n is temperature-

independent, instead controlled by the nature of the transformation, being between one and

four. The higher the number, the higher the degree of freedom to transform into, n = 4 being

close to a three-dimensional nucleation procedure in theory. Both k and n can be calculated

using a diagram as shown in Fig. 4.6 (Porter et al. 2008, p.287-288).

Figure 4.6: Eutectoid steel transformation versus logarithmic time plot for 675 2

2 Ion, J. (2013). Lecture 6 – Phase transformations in steels. Phase transformations, MT622A. Malmö University.


25

Next step includes a theoretical approach to the nucleation process, where one needs to

assume that both the energy barrier and activation barrier can be put into one

constant for all analysed temperatures, called . This assumption is the most cumbersome

part of all work, thus the nucleation formation of critical nucleus radii r* is temperature

dependent. Fig. 4.7 shows the problem of the Gibbs free energy barrier (Porter et al. 2008,

p.264).

Figure 4.7: The two energy barriers that has to be crossed

for critical radius to be formed (Adapted from Porter et al. 2008, p.191)

By now knowing from Fig. 4.6 that at t = 102 s, 50 % of the austenite has transformed into

pearlite for the eutectoid composition and that NHET is the nucleation rate at a certain

specified time t could be translated into amount of volume % transformed we obtain Eq.

(4.5). The last fact is that in order for this to construct volume fraction transformed versus

time diagrams, it has to have the same predictability as Eq. (4.3):

(

) (4.5)

by assuming that .

By using Fig. 4.6 at 948 K (675 ), one can assume the following:

(

)

The n-value can be calculated from the Avrami-expression for steel or other materials. This

would be done when controlling the curve produced by the expression above, by obtaining: 1

< n < 4 (Porter et al. 2008, p.287).


26

The remainder of the above theory would be to combine this with Eq. (4.6), thus being able

to produce volume fraction transformed versus time diagrams for different temperatures,

thus making the equation temperature-dependent.

{ ( ) (

)

} (4.6)

Lee et al. (1993) have completed their work with experimental values compared to

predictions of their own model, which recognises the alloying of the steel and thereby they

can form an empirical model based on their approach to Eq. (4.3).

The results of such an operation would be to for a specific composition in a phase-

diagram to construct and render an almost complete TTT-diagram. Fig. 4.8 shows how the

TTT-diagram for the eutectoid-composition in the iron-carbon system looks like.

Figure 4.8: TTT-diagram for eutectoid steel at 0.78 wt% carbon (Adapted from Callister et al. 2011, p.686).


27

Following the schematic Fig. 4.5 which is a template for construction of TTT-diagrams, the

three blue curves in Fig. 4.9 represent different amounts of volume fraction transformed,

were the left most one is 1 % transformed, the broken line is 50 % transformed and the right

most one is 99 % transformed. The three temperatures in the bottom represent the

martensitic formation temperatures, which are stated as Eq. (4.7 – 4.9) (Ion 2005, p.534)

where the element symbols refer to concentration in wt %.

( ) (4.7)

( ) (4.8)

( ) (4.9)


28


29

5 Hardness in HAZ

After deducing the resulting phase transformational products, one needs to consider the

transformation of these numbers into usable applicable hardness notifications for chosen

welding and material-parameters. Contours of the constant hardness is represented and

calculated using the cooling time derived by Eq. (3.12) that is then used in the equations for

hardness calculations. Further calculations will make use of empirical equations based on the

derived chemical composition of the HAZ. Noticeable is that hardness above 350 HV is not

desirable (Ion et al. 1984). This is due to the fact that hardness is closely coupled to the

mechanical properties of the weld itself, thus reflecting the ability to withstand especially

dynamic loading cycles putting stresses upon the weld-area. The desired result is shown

schematically in Fig. 5.1, with the same basic structure (Ion 2005, p.535).

Figure 5.1: Typical microstructure - time diagram, schematic (Ion 2005).

Observation can be made in Fig 5.1 that the logarithmic cooling time scale is inversely

related to the applied energy from the welding apparatus, where the absorbed energy per area

unit is a linear scale along the x-axis contra the logarithmic scale of the cooling time

(Poorhaydari et al. 2005).

5.1 Analytical equations of phase volume fraction in low carbon steels

In order to arrive at a suitable diagram, the chosen carbon equivalent, see Eq. (5.1), which

must be derived for further calculations. All of the equations below are quoted from the

same source; Ion (2005), if none other is stated.


30

(5.1)

Where element symbols refer to the composition in wt %. This carbon equivalent is

important for Eq. (5.2) to (5.5), in order to be able to calculate the necessary critical cooling

times for diagram construction:

( ) (5.1)

( ) (5.3)

( ) (5.4)

{ (

) } (5.5)

Where is he cooling time for 50 % martensite formation in seconds, likewise

is

cooling time for 0 % ferrite formation, is cooling time for 0 % bainite formation and

the cooling time for 50 % bainite formation. Next the volume fractions are stated as Eq.

(5.6) to (5.8).

{ ( ) (

)

} (5.6)

{ ( ) (

)

} (5.7)

( ) (5.8)

Where is the volume fraction of martensite, volume fraction of bainite and is the

volume fraction for ferrite-pearlite mixture. Important to note is that these equations, when

used for evaluating the hardness in the HAZ, are only applicable for a carbon content in the

range of: 0.1 < CEq < 0.5 (wt%) (Ion et al. 1996). In accordance to the work by Lee et al.

(1993) they state that some empirical equations are applicable even up to CEq < 0.8 (wt%).


31

Other alloying element will also be a limitation, some thing that is emphasised in the

simulations of hardness in Ch. 7. Since hardness is heavily dependent on the martensite

volume, experiments have shown that a peak temperature around 1400 will shift the

hardness distribution, thus the shape of the broken line in Fig. 5.1 (Goldak et al. 2005,

p.149). Where Eq. (5.2 – 5.7) may not be used, due to the limitation of carbon content, other

equations may be applied to get a more appropriate result, see Eq. (5.9) and (5.11) (Ion et al.

1984).

(5.9)

(5.10)

(5.11)

5.2 Hardness calculations by rule of mixtures

The last steps in order to reach the complete diagram are the hardness calculations, where

the rule of mixture is applied on. Noteworthy is that Eq. (5.12-5.14) can be stated different

from one work to another, because of the relation between the carbon content and hardness

of the individual phases. Derivation of these are made from well-fitted experimental data,

and thus for plane carbon steels stated as Eq. (5.12 – 5.15) (Ion et al. 1996):

(HV) (5.12)

(HV) (5.13)

(HV) (5.14)

(HV) (5.15)

Where Hm is the hardness of martensite in the HAZ in Vickers hardness (HV), Hb is the

hardness of bainite and Hfp is the hardness of the ferrite-pearlite mixture. Hmax is then the

simulated maximum hardness of the HAZ. Eq. (5.15) is the one that will be experimentally

verified, thus answering the main objective. The experimental data by Ion et al. (1996) is to

be plotted as the dotted line in Fig. 5.1 and used for validation simulations in Ch. 7.


32

Eq. (5.12 – 5.14) are relevant to plain carbon steels from which they are calculated.

Eq. (5.9 – 5.11) must be used for these alloyed steels, to calculate the hardness of each phase

is also different, not just the carbon-equivalent CEq (See Eq. 5.16 – 5.18) but other alloying

elements are included. The cooling rate V’ may be calculated according to Goldak et al.

(2005, p.149) by Eq. (5.19), where Eq. (3.12) is included:

(5.16)

( ) (5.17)

( ) (5.18)

(

) (

) (5.19)


33

6 Results and discussion – thermal modelling

This chapter will be describing the thermal modelling produced by the mathematical

procedure explained in Ch. 3. A MATLAB® Graphical User Interface (GUI) developed by

the author produces the plotted functions, which represent the results in this chapter.

Ch. 7 describes data obtained using the empirical equations stated in Ch. 5, where results

will be produced in volume fraction diagrams and estimated hardness of chosen simulation,

also represented in diagrams.

Lastly, Ch. 8 will carry out the hardness estimation using a simpler graphical estimation

using TTT-diagrams, where results will be compared to those of Ch. 7.

6.1 Thermal modelling simulations

6.1.1 MATLAB® implemented GUI for thermal simulation

In order to easily derive the graphs, a GUI was developed for representation, although in this

project the plots will take into the figures one-by-one. This section will just briefly show the

GUI and how it looks like when used accurately. The code for this GUI may be found in

App. A and the front facia of the program can be seen in Fig. 6.1.

Figure 6.1: Basic MATLAB® GUI starting screen, with plotted functions.


34

All the equations used in the background of this GUI are stated in the mathematical

modelling from Ch. 3. The equations are gathered from Ion (2005) and Ion et al. (1984) with

one exception for the critical thickness equation. This is taken from the work done by

Poorhaydari et al. (2005), which is according to them used as a boundary to determine when

the criteria for through-welding, 3-dimensional heat flow are attained.

6.1.2 Graphical simulations of thermal modelling

The first run with the thermal modelling was performed based on data described by

Kannatey-Asibu (2009) who in his book, see p. 239-245, makes temperature modelling,

along with the modified Bessel-function first seen in Eq. (3.6), in order to calculate the

temperature in a specific point in the HAZ. To validate the GUI-implementation; the

analytical simulation by Kannatey-Asibu (2009, p.239-245) will be replicated and presented

in Fig. 6.2. The following conditions were used in Eq. (3.10) (Simulation 6.1):

Power input, q = 6 kW

Plate thickness, d = 2.5 mm

Welding speed, v = 50 mm/s

Initial temperature, T0 = 298 K

Absorptivity, A = 0.7 (70%)

Density, = 7870 kg/m3

Specific heat capacity, = 452 J/kg K

Thermal conductivity, k = 73 W/m K

Radius, r =3.2 mm

Table 6.1: Comparison of calculation of replicated analytical model.

Calculation method Temperature (K) Mean relative error

Kannatey-Asibu 2009 (p.239-245) 985.4 1.34 %

Ion 2005 (p.532-535) 972.2

Figure 6.2: Temperature profile obtained by simulation 6.1with peak temperature at 972.2 K.


35

Following that the tolerance is somewhere about 0.15 mm (Ion et al. 1984) and the plate is

2.5 mm, thus the tolerance would be 6 %, the experiment replication of 1.34 % mean relative

error calculated in Table 6.1 is acceptably lower than the tolerance of manufacturing

thickness limitations. Next simulation will consist of the properties that are stated as pre-set

for the start-up facia of the GUI. The material data is taken from Ion (2005) whilst welding

parameters are chosen in conjunction with the supervisor. Following conditions were used in

Eq. (3.1) and (3.14) (Simulation 6.2):

Power input, q = 4 kW

Plate thickness, d = 5 mm

Welding speed, v= 20 mm/s

Initial temperature, T0 = 298


Density, = 7790 kg/m3

Specific heat capacity, = 560 J/kg K

Thermal conductivity, k = 32.5 W/m K

Distance, r =2.3 mm

Figure 6.3: Temperature versus time profile for pre-set values in GUI from simulation 6.2.

Figure 6.4: Radius versus peak temperature for pre-set values in GUI from simulation 6.2.


36

Figure 6.5: Input energy versus HAZ-width plot for pre-set values in GUI from simulation 6.2.

Table 6.2: Calculated properties by GUI pre-set values.

Absorbed energy (J/mm2) ( ) HAZ-width (mm) Tp (K) dC

56.00 4.871 2.150 1648 10.39

Fig. 6.3 - 6.5 and Table 6.2 is interpreting what the GUI produces with the help of the

equations used. Caution should be taken when considering the HAZ-width. This value will

be a slight overestimation due to the original assumptions; this will be discussed later on.

When using the GUI with these equations, it is very important that the radius to the point

of interest is specified correctly. Eq. (3.10) is especially sensitive, as the radius will be

squared in the exponential part, the part that controls the heating of the HAZ. This is

explained in Fig. 3.4. In Fig. 6.6 different radii have been implemented for the same energy

input to show the previous statement and it visualises the sensitivity of the radius to the heat-

source. The problem using these equations is when the radius will be very small, thus the

answer will be incorrect as shown by Eq. (6.1) (Simulation 6.3):

( ) (

) (6.1)

Table 6.3: Results for simulation 6.3

Radii (mm) TP (K) Absorbed input (J/mm2) ( )

1.20

1.60

2.00

2.40

2.80

3.20

3.60

2265

1770

1484

1286

1145

1039

957

42.78 2.842


37

Figure 6.6: Plot showing sensitivity to radii shifting versus time.

This phenomenon can be shown by plotting peak temperature versus radius from heat-source

to end HAZ that has been done for the different radii (See Table 6.3) in Fig. 6.7, which

shows that TP rises to infinity approaching the heat-source. This fundamental problem in

Eq. (3.10) has been analysed by Darmadi et al. (2011), whom compared the analytical values

with the numerical to get the best fit.

Figure 6.7: TP versus radius plot for actual energy input.


38

Next simulation will replicate the experimental procedure of Poorhaydari et al. (2005) by

shifting the absorbed energy to obtain results to confirm that the analytical equations work as

the experiment. The parameters that are not changed are the same as in simulation 6.2 and

results are presented in Fig. 6.8. Following parameters were changed in Eq. (3.10) and (3.14)

(Simulation 6.4):


Figure 6.8: Simulation of shifting input energy for cooling time derivation.

Table 6.4: Results of shifting input energy simulation

Laser power (W) Absorbed energy (J/mm2) ( )

1500

3000

4500

6000

13.13

26.25

39.38

52.50

0.27

1.07

2.41

4.28

Results derived experimentally by Poorhaydari et al. (2005) concluded that the cooling time

is very much dependent on the energy absorbed in the welding process. Noteworthy is the

cooling times them selves. With a welding apparatus of 1500 W power-output and the

parameters chosen for the simulation, an extremely quick cooling time is derived. This time

would give a significant martensite volume-fraction formation in the HAZ, and when

weighting with the rule of mixture (Eq. 5.14), the hardness would be high compared to the

parent-material.


39

Figure 6.9: Calculated values from simulation 6.4 with interpolated fitted curve-showing relationship.

Observation in Fig. 6.9 can be made that a point has been plotted at an absorbed energy

lower than that of the first point in Table 6.4, which is only as an interpolation-point, but can

not be discussed theoretically as this point cross the critical thickness which in turn is

heavily controlled by the factor input energy per unit length (q/v). This observation has been

pointed out for aluminium as well, where this factor has a profound effect on the thermal

cycle theoretically, numerically and experimentally (Kou 1981).

Fig. 6.9 is important as the analytical solution shows that increasing the power more will

eventually tend to stabilise the amount of absorbed energy, thus when absorbed energy

approaches 200 J/mm2 the extra power input will only theoretically increase the cooling

time. Following Eq. (6.2) this implies that there is a fundamental calculation problem that

the analytical solution is thought to be depending on the fact of a constant absorption-factor

(A). This observation may imply that A is not constant, but follows a theoretically non-

linear/linear behaviour, that is dependent on either or both material properties and laser

properties.

(

)

(6.2)

The last diagram produced in simulation 6.4 is Fig. 6.10 produced by Eq. (3.14), where TP

has been plotted against r. This is done in order to deduce if the analytical equations will

follow the fundamental thermodynamic principle that the heat would dissipate faster at a

smaller amount of absorbed energy. According to the simulation done by Darmadi et al.

(2011) they confirm this by numerical principles, but it works the same with analytical

calculations. The biggest difference would be that their numerical implementation uses

thermal properties of non-linear nature, which is difficult to implement in the analytical

solution, see section 3.5.


40

Figure 6.10: Results for TP versus r, from simulation 6.4.

Two more simulations will be made before discussion of the general thoughts on the

analytical solution to the thermal modelling problem. This is done to get total understanding

of what happens when changing of parameters in order to understand the hardness-problems

of the HAZ when cooling times are to fast.


41

Simulation 6.5 will handle how the thickness affects both and Tp. This will give a

picture of how these two parameters will vary with plate thickness in order to get some

comparison for discussion. All calculations will be below the critical thickness between 2

and 3-dimensional heat-flow. The following conditions were used in Eq. (3.10) and (3.14), if

changed; otherwise see simulation 6.2 (Simulation 6.5):

Power input q = 4.5 kW

Welding speed v = 30 mm/s

Distance, r =2 mm

Figure 6.11: T versus t plot for thickness-variation-simulation.

Table 6.5: Results of simulation 6.5

Thickness (mm) Absorbed energy (J/mm2) Peak Temperature (K) ( )

1.50

2.00

2.50

3.00

4.00

70.00

52.50

42.00

35.00

26.25

2239

1754

1463

1269

1026

7.61

4.28

2.74

1.90

1.07

The results, seen in Fig. 6.11, does follow the predictions, as more energy is absorbed, the

peak will be higher, thus the cooling time will be extended, but as a result peak temperature

will be significantly higher. At a plate thickness of 1.5 mm, the peak temperature, seen in

Table 6.5, would most likely be so high it would warp the base-plate, as this is a very high

temperature that will conduct to the rest of the base-plate’s extent. This can be explained in

Fig. 6.12, which shows the predictions of a higher temperature further out in the peripheral

material boundaries.


42

Figure 6.12: Results for TP versus r, from simulation 6.5.

Identification needs to bee done with the fact that simulation 6.5 is done with a constant

energy input (q/v) of 150 J/mm. The difference in this and the absorbed energy is important

to recognise, as the absorbed energy (Aq/vd) will differ with varying thickness. The amount

of absorbed energy will be higher at smaller thickness due to the base-plate’s ability to be

easily penetrated by the output power of the welding apparatus. But the cost of this is the

risk of distortion in the base-plate and the enlarged HAZ, which follows that if the absorbed

energy is greater, the width will be greater, see Eq. (3.15).

Figure 6.13: Results plot for simulation 6.5.


43

For better understanding of simulation 6.5, Fig. 6.13 shows the diagram plotted for thickness

versus both peak temperature and cooling time. This is done to see how the two parameters

follow the thickness variation mathematically.

The interpolated curves of the simulation results are plotted as seen in Fig. 6.13, which

shows that for each input-energy, there will be an intersection at which the cooling time will

dissipate faster than the peak temperature. The problem will be to plot closer to d = 0,

were both the thickness would be to small, but also that both curves would most probably

tend towards infinity.

Last simulation will show the implemented function of material choice in the GUI that

will give a picture of how the temperature profile will depend heavily on material/thermal

properties. Fig. 6.14 shows the temperature profile with the following conditions in Eq.

(3.10) are used (Simulation 6.6):

Power input q = 5 kW


Distance, r =2 mm


Figure 6.14: T versus t plot for different materials.

Table 6.6: Results of simulation 6.6

Material Density

(kg/m3)

Thermal

conductivity

(W/mK)

Heat

capacity

(J/kgmK)

Thermal

diffusivity

(mm2/s)

Peak

temperature

(K)

( )

Low-carbon

steel

Titanium

Gold

AISI304-steel

Nickel

7790

4500

19300

7870

8900

32.5

23.0

296

25.5

72.0

560

523

132

450

560

7.45

9.77

116

7.20

14.4

1106

1797

1638

1294

1006

1.32

3.46

0.25

2.07

0.52


44

Fig. 6.14 show that when changing the material for the same amount of input energy,

differences according to thermal properties will occur. By analysing Table 6.6 we can

deduce some key-factors in the changing thermal cycles. Observation of how cooling time is

dependent on thermal conductivity is well shown by the difference in low-carbon steel and

nickel, which both have the same specific heat capacity. Nickel has about double the value

of thermal conductivity compared to low-carbon steel, but the cooling time is 60 % faster.

This is also applicable for gold, which has a very high conductivity, thus the cooling time

will be very rapid, as the material properties state that heat dissipates faster. Observation can

be made that gold should have, in relation to the very high conductivity, an extremely rapid

cooling time. But as in Eq. (3.10) the cooling is governed by the inverse part, the high

density and heat capacity will extend this time. Heating the gold plate will be very quick,

due to the very high thermal diffusivity, thus the heat can travel very fast in the material, and

as this is controlling the inverse factor in the heating exponential part of Eq. (3.10).

Further observation can be made about Eq. (3.14), where the peak temperature for both

gold and titanium can be compared. This equation is governed by density and heat capacity

( ) in the inverse; therefore the peak temperature will correlate somewhat. The difference

in the materials and their curves will be the calculation of cooling time, where gold’s high

thermal conductivity will control the inverse part of Eq. (3.12), thus decreasing the cooling

time compared to that of titanium.


45

6.2 Discussion of thermal modelling

The general analysis using the Rosenthal (1946) derived equations may be used for easier

application in the industry, but it is not without its drawbacks. In the above simulations using

the equations implemented in the MATLAB® GUI, discussion will follow.

In section 3.5 the problems using the original equations are highlighted. The basic

problem has been dealt with by Darmadi et al. (2011). They state that the analytical

modelling is much weaker than the numerical analysis in the sense that assumptions made by

Rosenthal (1946) in the analytical solution to the heat transfer equation are the biggest

errors. In the numerical solution these errors are not included as this numerical model may

incorporate things as the non-linear thermal properties or that the conductive heat inside the

parent material may dissipate into the plate surroundings.

Firstly the thermal properties are dependent on temperature, a property that is very hard

to implement in the analytical solution because of its non-linear nature. The analytical

solution would then have to be solved for every step of change in thermal properties as they

change for each in temperature. That would be unreasonable, compared to the solution by

numerical analysis. Further it was necessary to state that the heat from the heat source would

only dissipate through conduction in the parent-material, i.e. no heat dissipates through

surface. This fact is the one were least work has been done by others recently. This is mostly

likely due to that the fact that transformation take place inside the microstructure of the HAZ

or in the surroundings of the welded area, not outside in the base-plate-surroundings.

The largest problem is that the latent heat developed in the weld-bead is not

implemented in Rosenthal solutions. Both plots of Fig. 6.4 and 6.5 are therefore

overestimating their results a small amount. Plotting peak temperature versus radius will

only be an estimation of the calculation for the two parameters. The plots show the principle

dissipation of temperature through the parent-plate, but result should be handled with care

and analysed further if were to be used in actual material analysis. The schematics of such a

plot are although confirmed by Fig. 6.6 were the thermal cycles are plotted and the peak

temperatures will form the base for Fig. 6.7, showing that the schematic plot of peak

temperature against radius is correlating although the model is not considering the latent heat

development. The curve in Fig. 6.6 showing thermal cycle for a radius of r = 1.2 mm should

be treated as only schematic, as a peak temperature of approximately 2250 K is unreasonable

and would distort the base-plate.

This renders the next topic of the Rosenthal (1946) analytical modelling that Equation

3.10 houses a singularity problem at small radii, explained in Eq. (6.1). This can not be

overrun, as the radii is evaluated in square by the exponential part controlling the heating of

the base-plate, which makes the analytical solution useless at small radii with constant

absorbed energy. Thought should be taken when simulating with small radii and nearing the

melting temperature of the parent-material, because after that point the errors will be too big

to validate the simulation analytically. This problem is only erasable in a numerical model,

which may handle singularities by evaluating several points around the analysis point

chosen, although it is not easy to implement.


46

This work has been handled by both Darmadi et al. (2011) and Kou (1981), where both use a

weighting method of combining the numerical and analytical results for a complete analysis,

which is observed in Fig. 6.15. This shows how schematically one can handle the singularity

problem of the analytical solution, by combining with other approaches, thus providing a full

picture of the model.

Figure 6.15: Temperature profile very close to the weld line, longitudinal to weld (Adapted from Darmadi 2011)

Following observation made during simulation 6.4, when increasing the amount of input

power of the welding apparatus, the increase in absorbed energy will make the cooling time

eventually tend toward infinity. Meaning that at some point, a miniscule increase in power

input will significantly increase the cooling time. This is believed to depend on the constant

absorptivity set to A = 0.7 by Ion (2005). Caution should be taken, because very absorbed

energy-value is not applicable in practice, as there is a limit to how much power could be put

into the base-material plate. In theory if the thickness d is decreased for a specified constant

power input q, the absorbed energy would increase, but then as the parent-material plate-

thickness decreases, so does the ability to cope with high power inputs. This means that Fig.

6.9 will be a theoretical approach to the problem, but although makes for a discussion on

weather it is applicable to state that absorptivity A is a constant, linear or a non-linear

relation to the power input (q/v). This work would be very extensive and is not covered in

this thesis. Another thought would be to form easy and practically applicable curves for

deducing the amount of input energy for specific thicknesses and material (thermal)

properties, thus implementing the absorptivity A from an experimental approach.

Simulation 6.5 induced a change in thickness for a constant input energy. From Fig. 6.11

one can see that as the thickness d increases, the peak temperature will decrease in

accordance, as well as decreasing the cooling time significantly. The higher absorbed energy

at thinner plates is due to the construction of Eq. (3.10), where thickness d is controlling the

denominator of the absorbed energy. In theory this will of course increase at thinner plates,

in practical manners it has the meaning that the heat generated by the weld will be higher


47

further out relative to the thickness, something that is graphically illustrated by Fig. 6.12.

This Figure is only used as a graphical tool to show that in a thicker plate, the heat will have

more difficulty to travel through the material, thus the curve in Fig. 6.12 shows a flatter

curve for a thinner plate relative to a thick plate. Fig. 6.13 implements the results from the

simulation as two curves plotted in the same space to see the correlation between the two

calculated variables; Tp and . They both follow a non-linear trace, although the

interpolated curves are not true, as they should tend towards infinity when thickness d goes

towards 0 mm. This error is traced to both the calculations made with Rosenthal equations,

that includes the error from original assumptions, but also that the simulations is only done

for smallest thickness d = 1.5 mm as smaller values would give a non-realistic value, only

applicable in theory.

Last simulation, 6, deduces that the thermal modelling is applicable for other material

than steel, where validation made by Kou (1981) stands as reference. The equations used

follow what the predictions of the thermal properties suggest, but caution should be taken

when using them in practice, not only as theoretical results.

Other errors that must be noticed is that calculations with a peak temperature below

900 should not be used according to Ion (2005) due to their non-valid nature when

calculated. If the width of the HAZ is calculated using Eq. (3.15), the temperature at end of

HAZ TR must not be to close a value to the melting temperature Tm, see Eq. (3.17) and

(3.18), as this will give an even more faulty calculation of the width, as this already includes

a part of the molten weld-pool according to original assumptions of thermal modelling, see

section 3.1.


48


49

7 Results and discussion – empirical hardness estimation

7.1 Empirical hardness estimation using calculated volume fractions

This chapter will handle the hardness calculations by the means of the empirical

calculations, using the equations derived by Ion (2005). The simulations will use a number

of cooling times deduced by the, in Ch. 6, validated the thermal modelling simulations.

These numbers will be plugged into the GUI developed, which in turn gives an estimation of

the hardness for the simulation chosen. The representation will be in graphical form, where

results will be compared and discussed compared with results deduced experimentally, from

sources chosen in conjunction with the supervisor.

7.1.1 MATLAB® implemented GUI for hardness simulation

In order to easily deduce the diagrams and results used in the chapter, a MATLAB® GUI

was developed, where alloys and cooling times can be put in, thus producing the results. The

main face of the GUI may be seen in Fig. 7.1 and the code may be found in App. B.

Figure 7.1: MATLAB® GUI used for hardness estimation and simulation of curves.

Equations working in the background of the GUI are taken from Ch. 5 and come from both

Ion (2005) (Eq. 5.1-5.8 and 5.12-5.15) and Ion et al. (1984) (Eq. 5.6-5.11 and 5.15-5.19).

Both have their own limitations, which will be brought up in the discussion.


50

7.1.2 Graphical results of hardness simulations

First simulation in this results chapter of hardness will consist of a simulations where the

carbon equivalent CEq is altered to see how the curves react. By using Eq. (5.1 -5.5), Ion et

al. (1996) has noted that the limitation of this model is a carbon content of about C = 0.18

wt%, this will be observed in the discussion. Fig. 7.2 shows how altering only the carbon

content affects the curves, whilst in Fig. 7.3 and 7.4, manganese, resp. silicon-content will be

altered as these are the two alloys used by Eq. (5.5). Following conditions were used

(Simulation 7.1):






Average density, = 7790 kg/m3

Average specific heat capacity, = 560 J/kg K


Melting temperature, Tm = 1804

Figure 7.2: Altering the carbon contents to get different carbon equivalent.

Table 7.1: Results of both thermal and hardness modelling for simulation 7.1.

Carbon content

(wt%)

Carbon equivalent

(wt%)

Absorbed

energy (J/mm2)

Cooling time (s) Hardness

(HV)

0.10 0.10

27.2 1.15

213

0.14 0.14 255

0.18 0.18 330


51

Fig. 7.3 shows how altering the manganese content will affect both carbon equivalent and

hardness. The limits for using this model are again adapted from Ion et al. (1996) where the

maximum empirical manganese content is 1.4 wt%. The variables derived from thermal

modelling will be the same here as used in Table 7.1.

Figure 7.3: Altering the manganese content, thus carbon equivalent to derive hardness.

Table 7.2: Results of altering manganese content.

Carbon content

(wt%)

Manganese content

(wt%)

Carbon equivalent

(wt%) Hardness (HV)

0.12

0.40 0.153 276

0.80 0.187 342

1.20 0.220 389

Table 7.2 and 7.3 show that the hardness will increase rather much when introducing alloys

into the low carbon steel. The last curve, where the carbon equivalent approaches a high

value is hard to deduce, as there actually is an empirical limitation to the model.

In order to get comparison for the hardness versus carbon equivalent, the silicon also

needs to be altered. Lastly, these will be combined in Fig. 7.5 where both the manganese and

the silicon will be altered, then the results of all these altering’s will be visualised in Fig. 7.6

showing how the hardness develops when alloys are introduced in different ways, seen in

Table 7.4. Note that these results are only schematic, as these alloying combinations are not

actual steel types used in the industry.


52

Figure 7.4: Altering the silicon content, thus the carbon equivalent to derive hardness.

Table 7.3: Results of altering silicon content.

Carbon content

(wt%)

Silicon content

(wt%)

Carbon equivalent

(wt%) Hardness (HV)

0.12

0.18 0.128 242

0.36 0.135 249

0.54 0.143 258

Now by altering carbon, manganese and silicon, Fig. 7.5 shows what happens with the

hardness at this stage. This is done in order to get validation that the model actually follows

the equation for carbon equivalent stated by Ion (2005).


53

Figure 7.5: Altering either alloy or carbon to get results of hardness.

Table 7.4: Results of altering all the alloying components in steel using equations from Ion (2005)

Carbon content

(wt%)

Manganese

content (wt%)

Silicon content

(wt%)

Carbon equivalent

(wt%)

Hardness

(HV)

0.10 0.40 0.18 0.141 266

0.14 0.40 0.18 0.181 342

0.10 0.80 0.18 0.174 330

0.10 0.40 0.36 0.148 280

The results from Table 7.4 can be seen in Fig. 7.6, where the overall impression is that if one

increases the carbon content or the alloying composition, the carbon equivalent will increase

following Eq. (5.1) thus increasing the hardness of the weld HAZ. This is due to that when

the alloys raise the carbon equivalent, the curve of volume fractions will shift to the right,

thus the composition of the HAZ will get an increased martensite volume fraction.

According to Eq. (5.1), the increase of carbon content has the most significant effect on the

hardness. This happens when increasing the carbon equivalent makes for both shifting

curves and by that the martensite hardness will increase. As the martensite is the hardest

phase measured in HV, it also affects the overall hardness the most. Note should be taken,

that the distance between curves are not equal, as the x-axis is logarithmic.

A problem has been noted that Eq. (5.1 – 5.5) and (5.12 – 5.14), taken from Ion (2005)

has a limit of around CEq = 0.24 wt% as the empirical model developed only has values

below this limit. A comparison is done with Eq. (5.9 – 5.11) and (5.16 - 5.18) taken from Ion

et al. (1984), that might work for hypo-eutectic steels of higher carbon equivalent. Test for

hyper-eutectic will not be included in this thesis, therefore max CEq = 0.78, the eutectic

composition.


54

Figure 7.6: Results of simulation 7.1 showing typical trending of hardness.

Fig. 7.6 shows that the trend of all the lines is that increasing carbon equivalent will result in

increased hardness, which follows the proportions of Eq. (5.1). Observation is made that

changing the carbon content will shift the curves the most between each step of content

increase, but by altering the manganese content will result in the biggest shift in hardness,

although the increase in manganese also results in the biggest increase of carbon content

compared to silicon.


55

Simulations 7.2 will use the model by Ion et al. (1984) for comparison. Note that manganese

and carbon will alter the carbon equivalent, as silicon is not included in Eq. (5.9), but it is

included in Eq. (5.16 – 5.18). Simulation 7.2 can be seen in Fig. 7.7 and results in Table 7.5

(Simulation 7.2):

Figure 7.7: Simulation 7.2 using model by Ion et al. (1984).

Table 7.5: Results of altering all the alloying components in steel using model by Ion et al. (1984).

Carbon content

(wt%)

Manganese

content (wt%)

Silicon content

(wt%)

Carbon equivalent

(wt%)

Hardness

(HV)

0.10 0.40 0.18 0.167 277

0.14 0.40 0.18 0.207 303

0.10 0.80 0.18 0.233 317

0.10 0.40 0.36 0.167 313

The problem when using the model derived by Ion et al. (1984) on modern low carbon steels

becomes apparent by the results above. As these steels often is consisting of carbon,

manganese and silicon, with additional alloys of minute content of around wt% < 0.05, the

three mentioned, will be those who affects the hardness most significantly as the model by

Ion (2005, p. 534) is only dependent on manganese and silicon in the calculation.(Sokolov et

al. 2011).

As silicon is not added into Eq. (5.9), it does not affect the carbon equivalent, but in turn

it affects Eq. (5.16 – 5.18), thus it in turn affects the hardness seen in Table 7.5. There one

sees that row one and row four has the same carbon equivalent but row four has a 13 %

higher hardness in the HAZ. This may be misguiding when using, but Simulation 7.3 shows

that the model by Ion et al. (1984) can be used for high carbon content, such as eutectic

steel. Same conditions as in simulation 7.1 are used, where parameters used may be seen in

Table 7.1.


56

Fig. 7.8 shows the curves produced for steels with higher carbon content and the results are

seen in Table 7.6, simulation 7.3:

Figure 7.8: Results of simulation 7.3 using higher carbon equivalent and model by Ion et al. (1984).

Table 7.6: Results of simulation 7.3 altering the alloying components in steel using model by Ion et al. (1984).

Carbon content

(wt%)

Manganese

content (wt%)

Silicon content

(wt%)

Carbon equivalent

(wt%)

Hardness

(HV)

0.78 0 0 0.78

0.78

0.78

0.78

887

0.70 0.48 0 816

0.78 0 0.36 897

0.70 0.48 0.36 826

Simulation 7.3 reveals a serious problem with the

empirical model. Such a high hardness, of above

800 HV, would only work as a theoretical value,

not applicable in practice. This suggests that

neither of the empirical models tested may not be

applied to carbon steel with a very high carbon

equivalent. Fig. 7.9 shows how the hardness will

deviate, but at values above the limit of 350 HV

(Sokolov et al. 2011) one may deduce that a

reasonable maximum carbon equivalent of

around CEq = 0.28 is to be set.

Figure 7.9: Results of simulation 7.3.


57

Before simulating what happens when altering the cooling time to derive the hardness on

different time steps, an experiment by Sokolov et al. (2011) will be replicated and compared.

They derived the hardness for thick pieces of s355 constructional steel by experiment. Using

very powerful laser-apparatus they achieved penetration welding, i.e. 2-dimensional heat

flow, even in 20 – 25 mm thick pieces. The composition of s355 steel is taken from their

work, while the thermal properties are adapted from Piekarska et al. (2012). Following

conditions were used, with results in Fig. 7.10 and Table 7.7 (Simulation 7.4):

Power input, q = 20 - 25 kW








Figure 7.10: Results of simulation 7.4 using parameters from Sokolov et al. (2011).

Table 7.7: Results of simulation 7.4 with difference between experimental and empirical method.

Laser-

power

(kW)

Absorbed

energy

(J/mm2)

Cooling

time (s)

Carbon

equivalent

(wt%)

Hardness

simulation

(HV)

Hardness

Sokolov et

al. (2011)

(HV)

Relative

mean error

(%)

20 17.50 0.45 0.225

429.1 432.7 0.83

25 21.88 0.66 421.3 425.5 0.91


58

Simulation 7.4 shows that the empirical approach using equations by Ion (2005) works well,

considering a less than one percent difference. It may be a coincidence for these results only.

It should be noted that the thermal simulation made to deduce the cooling time warned for a

precise cross over the limit between 2 and 3-dimensional heat flow, but this was neglected to

show the results. In the experiment by Sokolov et al. (2011) they deduced by sections of the

weld that by using the laser-power stated, a 2-dimensional heat flow is induced. It must be

implied that the composition of the s355 steel may vary and was by meaning kept below

CEq = 0.235, thus being in the range where equations by Ion (2005) may be used, therefore

verifying that they may be used for modern low carbon steels.

The fifth simulation in this chapter will be using different cooling times, i.e. altering the

welding speed, in order to arrive at the graphical representation of the broken line in Fig. 5.1.

Fictional simulations will be used to schematically show the broken line, these will be

marked. The parameters and compositions used for the thermal modelling and the hardness

estimations are taken from Ion et al. (1996) for comparison (Simulation 7.5):



Melting temperature, TM = 1801



Thermal conductivity, k = 30 W/m K

1st Steel type: Fe37B, CEq = 0.171 wt%

2nd

Steel type: HSE, CEq = 0.238 wt%

Figure 7.11: Results for comparison with Fig. 5.1 using data from Ion et al. (1996).


59

Table 7.8: Results of shifting welding simulation 7.5.

Welding speed

(mm/s)

Absorbed energy

(J/mm2)

Cooling time (s) Hardness

Fe37B (HV)

Hardness

HSE (HV)

f* ---- 0.001 383.2 417.4

70.00 6.33 0.055 382.9 417.2

25.00 17.73 0.431 369.8 415.8

20.00 22.17 0.675 352.0 413.5

17.50 25.33 0.881 335.7 411.0

16.00 27.71 1.054 320.6 408.4

14.50 30.57 1.283 302.0 404.1

13.00 34.10 1.597 280.3 397.2

11.50 38.55 2.041 260.4 385.8

10.00 44.33 2.690 249.7 366.6

7.50 59.11 4.798 245.7 305.0

3.50 122.70 22.030 208.9 267.6

f* ---- 100 183.1 249.3

f* ---- 1000 175.8 240.5 *Fictive simulation, theoretical value, only for schematic plotting and calculations purpose.

The results of hardness simulation can be seen in Table 7.8, where the results are plotted

against one another in Fig. 7.11. What can be deduces by this simulation is that the

schematic plot seen in Fig. 5.1 is correct, where the broken line is here represented by the

points plotted. The difference between the two steels can clearly be seen in the table, where

the shifting of the curves for both bainite and martensite to the right for HSE, due to the

higher carbon equivalent, makes for a higher hardness value further in the plot. HSE steel

simulation shows that the cooling curve for martensite is shifted much to the right compared

to the Fe37B, thus the hardness will be higher, due to higher martensite volume fraction in

the HAZ, which is validated by the results. Also that the hardness should be higher, due to

higher carbon equivalent in HSE compares Fe37B, which has been shown by earlier

simulations, see Fig. 7.6.


60

Before the final simulation when comparing experimental and simulated results, simulation

7.6 will bring forth a statement from section 2.2, to test how much the preheat temperature

affects weld-hardness. Conditions used in this simulation are the same as in simulation 7.5

(except for welding speed v), but by altering the preheat temperature one would be able to

decrease the hardness, which will of course affect Eq. (3.13) heavily, thus also affecting the

cooling time of Eq. (3.12). Simulation 7.6:

Table 7.9: Results of simulation 7.6, altering preheat temperature T0.

Preheat temperature

( ) Cooling time (s)

Hardness Fe37B

(HV)

Hardness HSE

(HV)

273 (0) 0.59 359 409

298 (25) 0.68 352 406

323 (50) 0.77 345 405

373 (100) 1.02 323 400

423 (150) 1.41 292 391

473 (200) 2.03 261 371

523 (250) 3.09 248 332

573 (300) 5.12 245 278

623 (350) 9.69 238 256

673 (400) 22.86 206 250

723 (450) 95.55* 183 242

*Simulation should be seen as a purely theoretical value.

Figure 7.12: Results of simulation 7.6 made graphical for easy interpretation.


61

Simulation 7.6, together with results in Fig. 7.12 and Table 7.9, validates that the preheat

temperature has a dramatic effect on the hardness. For steel Fe37B, one sees that only a

slight preheat of 50 - 100 will give the weld much better values of hardness, actually

bringing it below the limit of 350 HV (Sokolov et al. 2011). Simulations also confirm that

steels with higher carbon content will favour when preheated, as the hardness goes below the

limit eventually.

Last simulation will compare the results of Ion et al. (1996) derived experimentally, with

those brought forth by the equations in Ion (2005). Table 7.10 shows the properties of the

steels, while composition may be found in Ion et al. (1996).

Table 7.10: Physical properties of steels investigated (Ion et al. 1996).

Steels

Property Fe37B Fe52D HSD HSE

Density (kg/m3) 7860 7860 7860 7860

Thermal conductivity (W/mK) 30 30 30 30

Specific heat capacity (J/kgK) 680 680 680 680

AC1 Temperature (K) 994 994 989 994

Melting temperature (K) 1801 1798 1802 1799

Carbon equivalent (wt%) 0.171 0.205 0.162 0.232


62

The parameters used will be given in Table 7.10, taken directly from Ion et al. (1996).

Values of hardness in Table 7.11 are set up so the value deduced experimentally is inside

brackets, whereas the simulated value, deduced by this thesis is outside the brackets.

(Simulation 7.7):

Table 7.11: simulated results versus experimental values in parenthesis of hardness.

HAZ Hardness (HV)

Plate thickness

(mm)

Power

(kW)

Speed

(mm/s)

Applied energy

(J/mm2)

Fe37B Fe52D HSD HSE

4.0

2.5

10.00 63 245

(234)

254

(248)

291

(363)

292

(339)

23.33 27 326

(340)

368

(348)

404

(413)

409

(407)

3.8

35.00 27 345

(328)

376

(368)

402

(378)

408

(413)

40.00 23 318

(283)

386

(413)

408

(395)

412

(413)

6.0

3.8

6.67 94 233

(210)

221

(210)

256

(283)

259

(293)

10.00 63 245

(222)

254

(245)

278

(339)

299

(334)

13.33 47 232

(214)

282

(269)

341

(373)

354

(378)

5.0

18.33 47 225

(212)

290

(286)

349

(363)

362

(358)

23.33 36 262

(242)

340

(325)

385

(368)

393

(413)

8.0 5.0

8.33 75 235

(218)

253

(242)

279

(321)

274

(293)

10.00 63 245

(215)

254

(237)

301

(325)

292

(312)

Mean relative

error 10 % 5 % 9 % 7 %

Discussion of results may be found in the next part below.


63

7.2 Discussion of empirical hardness simulation

Simulation 7.1 was done in order to see if the implemented model reacted according to what

is expected, that is an increase in hardness when increasing the carbon equivalent in the

observed steels. Fig. 7.2- 7.4 shows the expected results, Fig. 7.2 shows a large space

between curves, thus a great variation in hardness values is observed, reflected also in

Fig. 7.4, where small increments gives less of increase in hardness. Caution should be taken,

thus the model used, Ion (2005), is an empirical model, which has its limits. The model will

not work above a carbon equivalent of about CEq = 0.24, where especially Eq. (5.5) will

show clear singularity problems due to its empirically derived nature as it is interpolated

from experimental data. If one instead want to use the equations by Ion et al. (1984), the

problem facing is that most modern low-carbon steels will incorporate silicon as a major

alloying component, thus this element not being taken into Eq. (5.9). This problem is

reflected well by Fig. 7.7 in simulation 7.2, where the results are considered inconclusive.

The model also uses an estimated cooling rate V’ in Eqs 5.16 – 5.18, which is hard to

deduce, thus rendering the usage of this model unpractical for modern use. The advantage is

that it may be used at higher levels of carbon equivalent than Ion (2005), but it has its limits

here as well. Simulation 7.3 shows how unrealistic a simulation using a very high carbon

equivalent and the model by Ion et al. (1984) is, when reaching hardness levels above 800

HV, values only theoretically achievable. Ion et al. (1996), Sokolov et al. (2011) amongst

others point out acceptable levels of hardness as 350 HV.

Simulations 5 thus show how the carbon equivalent will affect the hardness drastically.

HSE-steel with a high carbon equivalent has a much higher hardness at same cooling times

as Fe37B, where acceptable hardness levels is achieved approximately four times as fast, at

0.675 s for Fe37B against 2.690 s for HSE. This verifies the necessity of appropriate

hardness versus cooling time-diagrams; although caution should be shown to what empirical

model was used to deduce the hardness levels.

All of the above simulations have been implemented using cooling times derived by the

analytical solution to the thermal/mathematical modelling, thus it brings its own errors

along. Validity of thermal modelling has been deduced as appropriate, but thermal modelling

deducing cooling times using numerical analysis will be superior if more accurate values is

needed. In turn this would result in an accurate thermal cycle, but the empirical model for

deriving the hardness is still very approximate, with an error margin of about 10 %

according to Ion et al. (1996). These error margins are deduced by the fact that the carbon

equivalent is only deduced by three components of the alloy, thus rejecting the remaining

ones.

Using the latter stated error-margin, simulation 7.7 shows acceptable mean relative error

of 10 % or less for the four steels simulated. One should although be cautious, where the

experimental values, as well as simulated results may vary in accordance. Simulation may

vary due to steps of thermal modelling into hardness empirical model, giving errors along

the way. Notable is that the relative mean error for each material varies, where Fe37B has

eight out of 11 simulations above 10 % relative error, whilst Fe52D has none. This may be

down to the composition, where Fe52D has higher carbon content, thus a relatively higher

carbon equivalent. Therefore the errors deduced in both HSE and HDE-steels may be traced


64

to their alloying composition and carbon content, as they both show that four out of 11

simulations have 10 % relative error or more.

The later statement may also be the reason that simulation 7.4 shows a small relative

error of less than 1 %. Note should be taken that the hardness value taken from Sokolov et

al. (2011) is an average hardness in the HAZ, not a point-hardness. Nevertheless such a

small relative error reflects the validity of the model deduced and presented by Ion (2005).

Plotting the two data-arrays show no trends of hardness errors deduced by applied energy,

where distribution of errors a significantly equal.

Simulation 7.6 is reflecting what was brought up was has been stated in section 2.2,

where Table 7.9 and Fig. 7.12 significantly validates that preheating the steel prior to

welding will affect the hardness drastically. Although preheating is a practically

cumbersome process, one might argue that it may give hardness levels below the limit of

350 HV (Sokolov et al. 2011) even a slight preheat. For steels with higher carbon equivalent

a higher preheat is needed, but this might also favour, as these steel already are hard to bring

below the limit thanks to their composition.

The simulations, seen in Table 7.11, have been repeated in App. C, for the equations by

Ion et al. (1984). If one follows the same mean relative error procedure as for the equations

by Ion (2005), it may be deduced that they are not applicable for the steels tested as all of the

simulations ended up showing mean relative error of 20 % or more. Many of the simulations

that failed the most may be traced to Eq. (5.16 – 5.18), where one has to specify all the

alloying components to be calculated in the carbon equivalent. This makes for a somewhat

faulty simulations, where the answers may not be as accurate as one had hoped. Eq. (5.19)

for the cooling rate is a big factor in the error; as it decreases the hardness in Eq. (5.16 -5.18)

already at relatively fast cooling times. Therefore many of the simulations in App. C are way

of, often lower than the experimental.


65

8 Results and discussion – graphical hardness estimation

This part will conclude if the theory in Ch.4 may be used for estimation of hardness in the

HAZ and serves only as a theoretical approach to if the analysis made graphically is

sustainable against a much more elaborate empirical analysis, such as the on presented in

Ch. 7. The analysis will utilize an already created TTT-diagram for a hypo-eutectoid steel

with a carbon equivalent of CEq = 0.45 wt%. Due to this factor, only the equations by Ion et

al. (1984) may be applied to the problem.

The analysis is based on the cooling times printed in Figure 8.1, where the times have

been adapted form simulation 7.7, Ch. 7. This approach is very crude, thus predicted results

using the empirical hardness approach uses an adapted composition of modern low carbon

steels (Sokolov et al. 2011).

Figure 8.1: TTT-diagram of hypo-eutectoid steel of CEQ = 0.45 wt% with simulated cooling times

(Adapted from Callister et al. 2011, p.699).


66

Results of the simulation comparison using the two methods is presented in Table 8.1, where

B = bainite, M = martensite and P = pearlite.

Table 8.1: Results of the empirical versus the graphical simulation.

Cooling

time (s)

Volume fractions,

graphically (%)

Hardness,

graphically

(HV)

Volume fractions,

empirically (%)

Hardness,

empirically

(HV)

0.77 100M 379 97M, 2B, 1P 371

1.25 60M, 40B 268 93M, 5B, 2P 360

7.25 10M, 30B, 60P 159 8M, 39B, 53P 183

Mean relative error: 43 %

Discussion of graphical approach to hardness estimation

The results of the graphical results may be deemed very inconclusive. Many problems may

be deduced, firstly that the equation implemented by Ion et al. (1984) in Ch. 7 was

concluded as inconvenient at use for higher carbon equivalent. Thus other equations may be

derived or located in order to deduce a plausible answer to the graphical approach. Secondly,

the graphical implementation itself is of vague nature, where the direct approach, if

implemented in a GUI, is of much more reasonable nature and use. If the theory in Ch. 4

were to be successfully implemented in other work thus producing TTT-diagrams for

various carbon equivalents, it would still be a cumbersome process to graphically deduce the

hardness, as computing capability is of today in abundance for such simple calculations.

Conclusion can be drawn that a relative mean error of 43 % is not near acceptable, thus

rendering this approach, at the moment, very inconvenient.


67

9 Conclusion

This final part of the report gives some concluding remarks, mainly referring to the question

given in section 1.3. Also, potential improvement areas of the analysis are brought forth.

9.1 Conclusions

The first question to be answered were if the Rosenthal model was valid and if it solutions

may be implemented into a quick and easy to use program, in order to derive quick

calculations. This basic theory about thermal cycles and thermal modelling has been dealt

with in Ch.3, where the verification has been dealt with in Ch. 3.5. Although this is an

analysis of the analytical solution to the heat transfer problem, it provides a very quick way

to derive the thermal cycles needed for further analysis, together with purposely-calculated

parameters. The analytical solution was deemed valid for the usage in this thesis, although it

is not without flaws. Many of the problems may be traced to the original solution proposed

to the differential equation of heat transfer of quasi-stationary state. Using these analytical

solutions plugged into the GUI is although very easy, but this tool require, to a larger extent,

critical evaluation of the results calculated. But the entire thermal cycle may be brought forth

with ease; it gives one great choice in the analysis of both thermal analyses and hardness

evaluation in the HAZ. However, it again requires wise choices of the user to get results

applicable in practical situations. The equations of thermal modelling have been employed

by other authors on different materials, where the validity has been stated as statistically

satisfying, but not without flaws. There are advantages of doing the thermal cycle-analysis

by numerical means,

Next, a program was brought forth for the simulation of the hardness of the HAZ, and

how the phases were distributed in terms of volume fractions. This is a rather straightforward

suite, where one only has to plug in few variables in order to get results. Again though

caution by the user has to be made, reasonable thinking about the derived results must be

made. The program may work for a higher carbon equivalent, but the results may be deemed

inconclusive at further analysis, which may be connected to the empirical model used in the

background. Although the main interest of the program was fulfilled, easy interpretation and

usage. Using computers of current standard, solving problems in any of the GUI:s takes no

longer than a few seconds, but basic understanding of the equations used must be meet, as

well as understanding the phase transformation shortly brought forth in Ch. 4. Also in Ch. 4,

a theory about making a fast way to produce diagrams that could be used for graphical

interpretation of hardness simulation. This was unfortunately put aside as the workload was

deemed to cumbersome for the timeframe of this thesis.

Secondly, the question about if one may use the empirical model brought forth by Ion

(2005). The work was done on a basis of simulating and comparing to other experimental

work, but also comparing to results of other empirical models, such as equations by Ion et al.

(1984).


68

One may say that comparing the values to experiments of certain age is not relevant, but the

conclusion is that the accuracy of the simulation is valid for both low carbon steels of some

age and that of which is used today. When comparing the model to work done in 2011 the

validity is significantly accurate, thus confirming the experimental results within a

statistically valid region. The limitations will although be a great drawback, as the carbon

equivalent of more modern steel-types are approaching the maximum of the empirical

model, thus there is argument for developing a new set of equations. The model by Ion

(2005) is also sparse in considering the alloying components, but it although works very well

in predicting the hardness inside the HAZ. Even though the model is predicting accurate, any

engineer working with the topic, should still be cautious and evaluate the simulation due to

singularity related issues in thermal modelling, which may be reflected in the hardness

simulation.

Whether or not, the objective has been to deal with the task in an easy manner, thus

significantly reducing the time to incorporate the analytical solution, by fitting them into

GUI:s, which is a fast and graphically simple way of dealing with the simulations needed for

further interpretations. But experience in the field and some discretising of the results are

needed in order to make them valid for the tasks at were it would be brought forth about.

Even the simplest computations require the engineer to highlight reasoning and the errors of

the analyses should be weighed in for significance of the simulation if to be used for other

than purely theoretical purpose. Of course the theoretical values are of importance, but when

using results for a practical manner, caution is advised and more advanced models for

calculation may be better at hand.

Important conclusions in thesis:

The thermal analysis using mathematical modelling by the solutions proposed firstly

by Rosenthal (1946) is applicable for theoretical and practical applications, if the

engineers using them are familiar with the potential problems of the analytical

solution.

The solutions may also be applied to other material than carbon-steels, with some

restrictions to material that has thermal properties varying significantly with

temperature, as solution to the heat-equation are based on assumptions of these.

The equations by Ion (2005) of heat transfer are applicable with the stated absorption

for laser welding and HAZ-property simulation.

When simulating the hardness of the HAZ of a weld, the equations by Ion et al.

(1984) may not be applied for valid results when using compositions of more modern

carbon steels due to its approximation of the carbon equivalent, see Eq. (5.9).

The Equations by Ion (2005) are valid for modern low-carbon steels, although

reasoning and understanding of the hardness simulation-results of the HAZ produced

must be made and scrutinised by engineer with basic understanding of the area.

Testing against experimental values show significant correlation of the values

derived simulated.

Preheating may favour the hardness levels in low-carbon steels, although being a

cumbersome process practically achievable.


69

9.2 Future work – Possible improvements

Developing two GUI:s and understanding all the things involved in the process of hardness

in the HAZ was time consuming. There is simply not enough time to dive into the topics that

one finds really noticeable in the discussions, as many of the thoughts that arrive are too

cumbersome for the time space at hand. Below is a list with different areas where future

work could be focused:

The thermal modelling by Rosenthal needs updating in several fields to incorporate

some of the assumptions made, see section 3.1, in order to compete better with the

numerical solutions made by many authors. The most urgent is the latent heat

developed in the weld bead that may cause further phase transformation in the HAZ.

If found that enveloping both the thermal properties of the material and that there are

surface losses through conduction to the surroundings into the analytical model it

would be considered much more accurate. At this point though, implementing the

non-linear thermal properties of any material would be a very cumbersome process

for the analytical solution.

More work should be done in finding a empirical model that spans greater carbon

equivalents, or defining different models for different spans of composition. This

would of course demand a project of great proportions to cover many different types

of steel.

Research done in the field of TTT-diagrams and finding the hardness by graphical

interpretation could be of interest, mostly for practical usage in the industry. But it

might be too cumbersome, and therefore more forfeitable to develop a program

relying entirely on the empirical models developed.

More attention should be attained towards what preheating the steel prior to welding

actually does. Is the results really so significant or is it just a product of the equation

used for deriving the cooling time? One might argue that it would make significant

difference in the industry if this is the case when dealing with unwanted hardness.


70


71

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and thermal stress analysis. Optics and laser technology, 42(5): 760-768.


73

Appendix A: MATLAB® GUI-code for temperature profiles function varargout = NewWeld(varargin) gui_Singleton = 1; gui_State = struct('gui_Name', mfilename, ... 'gui_Singleton', gui_Singleton, ... 'gui_OpeningFcn', @NewWeld_OpeningFcn, ... 'gui_OutputFcn', @NewWeld_OutputFcn, ... 'gui_LayoutFcn', [] , ... 'gui_Callback', []); if nargin && ischar(varargin{1}) gui_State.gui_Callback = str2func(varargin{1}); end if nargout [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:}); else gui_mainfcn(gui_State, varargin{:}); end function NewWeld_OpeningFcn(hObject, eventdata, handles, varargin) handles.output = hObject; guidata(hObject, handles); function varargout = NewWeld_OutputFcn(hObject, eventdata, handles) varargout{1} = handles.output; %-----------------Listbox of material choice----------------- function listbox1_Callback(hObject, eventdata, handles) Type_of_material=get(handles.listbox1,'Value'); if Type_of_material==1; set(handles.Tm,'String',1810); set(handles.roe,'String',7790); set(handles.lamda,'String',32.5); set(handles.c,'String',560); set(handles.Tr,'String',996); elseif Type_of_material==2; set(handles.Tm,'String',1773); set(handles.roe,'String',7870); set(handles.lamda,'String',25.5); set(handles.c,'String',450); set(handles.Tr,'String',694); elseif Type_of_material==3; set(handles.Tm,'String',1950); set(handles.roe,'String',4500); set(handles.lamda,'String',23); set(handles.c,'String',523); set(handles.Tr,'String','No value'); elseif Type_of_material==4; set(handles.Tm,'String',1726); set(handles.roe,'String',8900); set(handles.lamda,'String',72); set(handles.c,'String',560); set(handles.Tr,'String','No value'); elseif Type_of_material==5; set(handles.Tm,'String',932); set(handles.roe,'String',2704); set(handles.lamda,'String',238); set(handles.c,'String',1000); set(handles.Tr,'String','No value'); elseif Type_of_material==6; set(handles.Tm,'String',1356); set(handles.roe,'String',8930); set(handles.lamda,'String',375); set(handles.c,'String',471); set(handles.Tr,'String','No value'); elseif Type_of_material==7; set(handles.Tm,'String',693); set(handles.roe,'String',7140); set(handles.lamda,'String',111); set(handles.c,'String',420); set(handles.Tr,'String','No value'); elseif Type_of_material==8; set(handles.Tm,'String',1340); set(handles.roe,'String',19300); set(handles.lamda,'String',296); set(handles.c,'String',132); set(handles.Tr,'String','No value'); end


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function listbox1_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function d_Callback(hObject, eventdata, handles) function d_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Tm_Callback(hObject, eventdata, handles) function Tm_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function roe_Callback(hObject, eventdata, handles) function roe_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function lamda_Callback(hObject, eventdata, handles) function lamda_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function c_Callback(hObject, eventdata, handles) function c_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function text12_DeleteFcn(hObject, eventdata, handles) function v_Callback(hObject, eventdata, handles) function v_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function A_Callback(hObject, eventdata, handles) function A_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function q_Callback(hObject, eventdata, handles) function q_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function r_Callback(hObject, eventdata, handles) function r_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),


75

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function T0_Callback(hObject, eventdata, handles) function T0_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Tr_Callback(hObject, eventdata, handles) function Tr_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end % --- Executes on button press in plot. function plot_Callback(hObject, eventdata, handles) % hObject handle to plot (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB® % handles structure with handles and user data (see GUIDATA) %-------------------Taking in from input------------------------ d=str2num(get(handles.d,'String')); A=str2num(get(handles.A,'String')); v=str2num(get(handles.v,'String')); q=str2num(get(handles.q,'String')); r=str2num(get(handles.r,'String')); T0=str2num(get(handles.T0,'String')); Tm=str2num(get(handles.Tm,'String'));

Tr=str2num(get(handles.Tr,'String')); roe=str2num(get(handles.roe,'String'));

lamda=str2num(get(handles.lamda,'String')); c=str2num(get(handles.c,'String')); t0=str2num(get(handles.t0,'String')); t_end=str2num(get(handles.t_end,'String')); %-------------------Basic calculations------------------------ d=d./1000; v=v./1000; r=r./1000; in_en=((A.*q)./(v.*d)); roe_c=roe.*c; a=lamda./roe_c; t=t0:0.0005:t_end; in_en_plot=linspace(0,100,500); %---------------------Setting stop!--------------------------- if A>1 errordlg('Absorptivity cannot be more than 100%!') return end if t0>t_end errordlg('Check time settings...') return end if v>0.030 errordlg('!Welding speed very high, >30mm/s!') end if in_en>120.*1e6 errordlg('!Absorbed energy seems to high! >120 J/mm^2') return end if r<0 || r>2.*d; errordlg('Weld radius is either <0 or >2d!') return end set(handles.output_in_en,'String',in_en./1e6); set(handles.a_out,'String',a.*1e6); %-------------------Calculation AXES-1------------------------ T=T0+in_en*(1./sqrt(4.*pi.*roe_c.*lamda.*t)).*exp(-((r.^2)./(4.*a.*t))); [Tp,num]=max(T); axes(handles.axes1) plot(t,T,'r',t(num),Tp,'ko'), grid on xlabel('Time (s)'), ylabel('Temperature (K)') title('Plot 1: Temp. vs. Time') legend('Temperature profile','Current Tp') axis([t0 t_end min(T) Tp+50]) if Tp<900


76

errordlg('OBS: Peak-temp below 900K may give faulty calculations!') end %---------------------Delta-t---------------------- theta2=(1./((773-T0).^2))-(1./((1073-T0).^2)); delta_t=(in_en.^2).*(1./(4.*pi.*lamda.*roe_c)).*theta2; set(handles.D_t,'String',delta_t); set(handles.peak_temp,'String',Tp); %-------------------Calculation AXES-2------------------------ axes(handles.axes2) %----------------Peak-T vs. distance--------------- r_plot=0.0005:0.00005:(0.09+r); Tp1=T0+(in_en.*sqrt(2./(pi.*exp(1))).*(1./(2.*roe_c.*r_plot))); Tp_ny=T0+in_en.*sqrt(2./(pi.*exp(1))).*(1./(2.*roe_c.*r)); plot(r_plot,Tp1,'b',r,Tp_ny,'r.'), axis([0 0.03 0 3000]), grid on xlabel('Weld radius (m)'), ylabel('Peak temperature (K)') title('Weld radius vs. Temp'), legend('Curve','Current radius') %-------------------Calculation AXES-3------------------------ axes(handles.axes3); Type_of_material=get(handles.listbox1,'Value'); %------------------CALCULATIONS-------------------- if Type_of_material==1 || Type_of_material==2 in_en=in_en/1e6; theta1=(1/roe_c).*(1./(Tr-T0)-1./(Tp-T0)); w=(in_en_plot.*sqrt(1./(2.*pi.*exp(1))).*theta1)*1e9; wny=(in_en.*sqrt(1/(2.*pi.*exp(1))).*theta1)*1e9; set(handles.w_calc,'String',wny); plot(in_en_plot,w,'g',in_en,wny,'k.'), grid on xlabel('Absorbed energy Aq/vd (J/mm^2)'), ylabel('HAZ-width (mm)') title('Absorbed energy vs. HAZ-width'), legend('Curve','Current HAZ-

width') else axes(handles.axes3) imshow('Warnigforgraph.png') end %------------------CRITICAL THICKNESS-------------------- d1=((1./(773-T0))+(1./(1073-T0))); d2=((A.*q)./(2.*roe_c.*v)); d_c=sqrt(d1.*d2); set(handles.d_crit,'String',d_c.*1000) if d>d_c errordlg('Warning, critical thickness between 2D to 3D heat-flow

exceeded!!!') return end function output_in_en_Callback(hObject, eventdata, handles) function output_in_en_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Resetbutton_Callback(hObject, eventdata, handles) %-------------------Resetting plots-------------------- axes(handles.axes1) cla reset; axes(handles.axes2) cla reset; axes(handles.axes3) cla reset; %-------------------Resetting values-------------------- set(handles.Tm,'String',1810); set(handles.roe,'String',7790); set(handles.lamda,'String',32.5); set(handles.c,'String',560); set(handles.Tr,'String',996); set(handles.listbox1,'Value',1) %-------------------Resetting outputs------------------- set(handles.D_t,'String',0); set(handles.peak_temp,'String',0); set(handles.output_in_en,'String',0); set(handles.w_calc,'String',0); set(handles.a_out,'String',0); set(handles.d_crit,'String',0) %-------------------Resetting other------------------- set(handles.T0,'String',298); set(handles.d,'String',5);


77

set(handles.A,'String',0.7); set(handles.q,'String',4000); set(handles.v,'String',10); set(handles.r,'String',2.3); set(handles.t0,'String',0); set(handles.t_end,'String',10); function t0_Callback(hObject, eventdata, handles) function t0_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function t_end_Callback(hObject, eventdata, handles) function t_end_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function a_out_Callback(hObject, eventdata, handles) function a_out_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function D_t_Callback(hObject, eventdata, handles) function D_t_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function peak_temp_Callback(hObject, eventdata, handles) function peak_temp_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function w_calc_Callback(hObject, eventdata, handles) function w_calc_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Exitbutton_Callback(hObject, eventdata, handles) close all function pushbutton7_Callback(hObject, eventdata, handles) Infoweldaxel function d_crit_Callback(hObject, eventdata, handles) function d_crit_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end


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79

Appendix B: MATLAB® GUI-code for hardness estimation function varargout = newceq(varargin) gui_Singleton = 1; gui_State = struct('gui_Name', mfilename, ... 'gui_Singleton', gui_Singleton, ... 'gui_OpeningFcn', @newceq_OpeningFcn, ... 'gui_OutputFcn', @newceq_OutputFcn, ... 'gui_LayoutFcn', [] , ... 'gui_Callback', []); if nargin && ischar(varargin{1}) gui_State.gui_Callback = str2func(varargin{1}); end if nargout [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:}); else gui_mainfcn(gui_State, varargin{:}); end function newceq_OpeningFcn(hObject, eventdata, handles, varargin) handles.output = hObject; guidata(hObject, handles); function varargout = newceq_OutputFcn(hObject, eventdata, handles) varargout{1} = handles.output; function C_Callback(hObject, eventdata, handles) function C_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Mn_Callback(hObject, eventdata, handles) function Mn_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Si_Callback(hObject, eventdata, handles) function Si_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Cr_Callback(hObject, eventdata, handles) function Cr_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Mo_Callback(hObject, eventdata, handles) function Mo_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Cu_Callback(hObject, eventdata, handles) function Cu_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor'))


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set(hObject,'BackgroundColor','white'); end function V_Callback(hObject, eventdata, handles) function V_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Ni_Callback(hObject, eventdata, handles) function Ni_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function edit9_Callback(hObject, eventdata, handles) function edit9_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function delta_t_Callback(hObject, eventdata, handles) function delta_t_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Vm1_Callback(hObject, eventdata, handles) function Vm1_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Vb_Callback(hObject, eventdata, handles) function Vb_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Vfp_Callback(hObject, eventdata, handles) function Vfp_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Hm_Callback(hObject, eventdata, handles) function Hm_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Hb_Callback(hObject, eventdata, handles) function Hb_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Hfp_Callback(hObject, eventdata, handles)


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function Hfp_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Hmax_Callback(hObject, eventdata, handles) function Hmax_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end % --- Executes on button press in pushbutton1. function pushbutton1_Callback(hObject, eventdata, handles) C=str2num(get(handles.C,'String')); Mn=str2num(get(handles.Mn,'String')); Si=str2num(get(handles.Si,'String'));

Cr=str2num(get(handles.Cr,'String')); Cu=str2num(get(handles.Cu,'String'));

Mo=str2num(get(handles.Mo,'String')); V=str2num(get(handles.V,'String')); Ni=str2num(get(handles.Ni,'String')); Delta_t=str2num(get(handles.delta_t,'String')); %----------------IF-sats for Ceq----------------- if C>0.2315 Ceq=C+(Mn/6)+((Mo+Cr+V)/5)+((Cu+Ni)/15); %Ion 1984 V_prim=(300./Delta_t)*(1./3600); delta_t50m=exp((6.022*Ceq)-1.376); delta_t50b=exp((5.456*Ceq)-0.512); Hm=127+949*C+27*Si+11*Mn+8*Ni+16*Cr+21*V_prim; Hb=-323+185*C+330*Si+153*Mn+65*Ni+144*Cr+191*Mo... +(89+53*C-55*Si-22*Mn-10*Ni-20*Cr-33*Mo).*V_prim Hfp=42+223*C+53*Si+30*Mn+12.6*Ni+7*Cr+19*Mo+... (10-19*Si+4*Ni+8*Cr+130*V).*V_prim Vm=exp(log(0.5)*((Delta_t/delta_t50m).^2)); Vb=exp(log(0.5)*((Delta_t/delta_t50b).^2))-Vm; Vfp=1-(Vm+Vb); Hmax=Vm*Hm+Vb*Hb+Vfp*Hfp; else Ceq=C+(Mn/12)+(Si/24); %Ion 2005 delta_t50m=exp((17.724*Ceq)-2.926); delta_t0f=exp((19.954*Ceq)-3.944); %Max ceq=0.175!!!! delta_t0b=exp((16.929*Ceq)+1.453); delta_t50b=exp(log((delta_t0b*delta_t0f)/2)); Hm=295+515*Ceq; Hb=223+147*Ceq; Hfp=140+139*Ceq; Vm=exp(log(0.5)*((Delta_t/delta_t50m).^2)); Vb=exp(log(0.5)*((Delta_t/delta_t50b).^2))-Vm; Vfp=1-(Vm+Vb); Hmax=Vm*Hm+Vb*Hb+Vfp*Hfp; end set(handles.Vm,'String',Vm*100); set(handles.Vb,'String',Vb*100); set(handles.Vfp,'String',Vfp*100); set(handles.Hm,'String',Hm); set(handles.Hb,'String',Hb); set(handles.Hfp,'String',Hfp); set(handles.Hmax,'String',Hmax); set(handles.Ceq,'String',Ceq); delta_t_plot=0.01:0.01:10000; Vm_plot=exp(log(0.5).*((delta_t_plot./delta_t50m).^2)); Vb_plot=exp(log(0.5).*((delta_t_plot./delta_t50b).^2)); %---------------PLOTTING----------------- axes(handles.axes1) cla reset; semilogx(delta_t_plot,Vm_plot*100,'k',delta_t_plot,Vb_plot*100,'r') grid on,hold on line([Delta_t,Delta_t],[0.00001,100],'LineStyle','--','LineWidth',1) legend('Martensite curve','Bainite curve','Chosen \Deltat_{8-5}') xlabel('Cooling time (s)','FontSize',14),ylabel('Volume fraction

(%)','FontSize',14) set(gca,'FontSize',14); axis([0.01 10000 0 100]) set(gca, 'XTicklabel', {0.01 0.1 1 10 100 1000 10000}) plot(delta_t50m,50,'r*',delta_t50b,50,'b*'),hold on


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if Ceq<0.1749 plot(delta_t0f,100,'b*',delta_t0b,0,'b*'), hold on end function Vm_Callback(hObject, eventdata, handles) function Vm_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end % --- Executes on button press in pushbutton2. function pushbutton2_Callback(hObject, eventdata, handles) close all % --- Executes on button press in pushbutton3. function pushbutton3_Callback(hObject, eventdata, handles) close(gcbf) newceq function Ceq_Callback(hObject, eventdata, handles) function Ceq_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),

get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end function Delta_t_Callback(hObject, eventdata, handles) function Delta_t_CreateFcn(hObject, eventdata, handles) if ispc && isequal(get(hObject,'BackgroundColor'),



get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end


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Appendix C: Table for hardness simulation comparison

Table C.1: Hardness simulation using model by Ion et al. (1984).

HAZ Hardness (HV)

Plate thickness

(mm)

Power

(kW)

Speed

(mm/s)

Applied energy

(J/mm2)

Fe37B Fe52D HSD HSE

4.0

2.5

10.00 63 107

(234)

128

(248)

350

(363) 324 (339)

23.33 27 303

(340)

320

(348)

273

(413) 297 (407)

3.8

35.00 27 316

(328)

322

(368)

275

(378) 298 (413)

40.00 23 293

(283)

305

(413)

262

(395) 287 (413)

6.0

3.8

6.67 94 98

(210)

105

(210)

134

(283) 130 (293)

10.00 63 105

(222)

125

(245)

341

(339) 315 (334)

13.33 47 215

(214)

264

(269)

401

(373) 400 (378)

5.0

18.33 47 239

(212)

286

(286)

393

(363) 395 (358)

23.33 36 337

(242)

347

(325)

326

(368) 343 (413)

8.0 5.0

8.33 75 98

(218)

106

(242)

220

(321) 200 (293)

10.00 63 107

(215)

128

(237)

350

(325) 324 (312)

Relative mean

error 31 % 28 % 21 % 20 %

Table C.1 replicates what was done in simulation 7.7, purely for the purpose of comparison

in the results and conclusions-.part. It can be concluded that the errors are significantly too

large for this to be an experiment that can be validated and therefore useful in the simulation

of HAZ hardness.

temperature profiles and hardness estimation of laser welded

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