Temperature profiles and hardness
estimation of laser welded heat affected
zone in low carbon steel
Axel Lundberg
One year Master's Degree
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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
i
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
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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-1K-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-1m-1K-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
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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... 32 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
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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
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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.
2 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 low-carbon equivalent will be at maximum CEq = 0.78 wt %. The analysis is only about the HAZ, so what happens in the weld
3 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).
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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
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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.
10 ( ) ( ) {( ) } (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).
( ) ( )
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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)
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.
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)
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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.
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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
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.
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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).
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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)
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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.
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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
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).
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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-phase-diagram. Fig. 4.8 shows how the TTT-diagram for the eutectoid-composition in the iron-carbon system looks like.
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)
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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)
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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.
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
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 Absorptivity, A = 0.7 (70%) 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.
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.
