Dilatometry Study on a High-Chromium Cast Iron

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IN THE FIELD OF TECHNOLOGY DEGREE PROJECT

MATERIALS DESIGN AND ENGINEERING AND THE MAIN FIELD OF STUDY

MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2018,

Dilatometry Study on a High- Chromium Cast Iron

BENJAMIN SOLEM

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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Acknowledgement

This report is the result of a Master Thesis in Mechanical Engineering at KTH Royal Insti- tute of Technology, Stockholm, Sweden, performed between January and June 2018. The work has been conducted for Xylem Water Solutions, Sundbyberg, Sweden, based on a thesis proposal from the company.

I would like to acknowledge my gratitude towards all personnel in contact with this thesis at Xylem Sundbyberg and Emmaboda. Special thanks to Måns Bergsten, Stefan Ramström, and Keith Lothian for their support and guidance during this spring term, as well as for introduc- ing this complex and intriguing material to me.

Furthermore, I want to thank my supervisor at KTH, Prof. Bo Alfredsson, as well as lab technician Wenli Long at the Department of Materials Science and Engineering, for their help throughout the course of this project.

Sundbyberg, 2018-06-05 Benjamin Solem

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Abstract

High-chromium cast irons are used in certain applications where the demand on abrasion resistance is high. Such applications can be found in the milling industry and in pumps for transport of abrasive particles in liquid suspension. Soft annealed high-chromium cast iron containing 2.6 % C and 24.7 % Cr was supplied by Xylem Water Solutions, Sundbyberg, and investigated by dilatometry. The heat treatments were inspired by induction hardening procedures. The purpose of the investigation was to evaluate the effect of maximum temperature reached during heat treatment on the final length of the test specimen. The aim with this was to find the treatment yielding the maximum possible length which should be profitable to create desirable compressive stresses in the surface hardened area. The experimental results were used to create a finite element model in COMSOL Multiphysics accommodating for the maximum temperature, simulating the phase changes occurring in a geometry based on the experimental test specimen.

The experimental results did not reveal any clear correlation between the maximum temperature and the final length change. The hardness, however, increased with the increasing temperature in the treatment interval 900-1150 °C. The, by light optical microscopy, observed amount of secondary precipitated carbides decreased with increasing temperature. Martensite transformation was also affected; the transformation temperature decreased for increased treatment temperatures. From dilatometry it was also seen that the thermal strains were greatly affected by the direction of which the material was cut from the original cast material.

Samples taken perpendicular to the mainly investigated direction showed lower coefficients of thermal expansion and the final strain was clearly positive compared to the slightly nega- tive values found for the main direction. This phenomenon could possibly be explained by different macrostructures created during solidification of the melt causing anisotropy in the eutectic. The implementation in COMSOL by describing the phase transformation as ordinary differential equations did show partially good results in the simulation of thermal expansion. The difference in original material is noticeable in the dilatometry and the simulated martensite transformation deviates from the experimental results. The model needs to be validated against new intermediate test temperatures and the martensite transformation kinetics must be investigated further to yield better results to be able to combine the phase transformations with mechanical calculations.

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Sammanfattning

Högkromhaltiga vita gjutjärn används i vissa tillämpningar med höga krav på slitstyrka och nötningsmotstånd såsom stenkrossar och i pumputrustning lämpat för att förflytta partiklar i suspension. Mjukglödgat högkromhaltigt vitt gjutjärn med 2,6 % C och 24,7 % Cr tillhanda- hållet av Xylem Water Solutions, Sundbyberg, undersöktes med hjälp av dilatometri med värmebehandlingar motsvarande induktionshärdning. Detta utfördes med syftet att under- söka effekten av maximala temperaturen på slutgiltiga längdsförändringen av provet. Målet var att hitta den temperatur som gav maximal längdutvidgning då det bör resultera i mest gynnsamma spänningsförhållanden i ytskiktet följande en härdningscykel. Resultaten imple- menterades i en finit elementmodell i COMSOL Multiphysics för att simulera fasomvand- lingarna i en geometri matchande den experimentella med hänseende till den maximalt upp- nådda temperaturen.

Resultaten visade inte något tydligt samband mellan maximal temperatur och slutgiltig längd.

Dock ökade hårdheten påtagligt med ökad maximal temperatur samtidigt som den kvalitativt uppmätta mängden sekundärt utskilda karbider synliga i ljusoptiskt mikroskop minskade.

Martensitomvandlingstemperaturen sänktes vid ökad maximal temperatur. Tydligast resultat var att ett riktningsberoende fanns i det undersökta materialet då prover tagna vinkelrätt mot den huvudsakliga undersökningens riktning visade klart lägre termiska utvidgningskoefficien- ter och en markant ökning i längd jämfört med de andra provernas minimala krympning.

Detta kan möjligen förklaras av en delvis annorlunda riktning i eutektiket på grund av en anisotropisk stelningsriktning. Implementeringen i COMSOL med hjälp av beräkning av fasandelar uttryckta som ordinära differentialekvationer lyckades på ett rimligt vis återskapa den termiska utvidgningen. Skillnaden mellan olika ursprungsmaterial är dock märkbar i dila- tometriresultaten och den simulerade omvandlingen till martensit avviker markant från det experimentellt uppmätta beteendet. Modellen behöver valideras mot nya prover och marten- sitomvandlingen utvärderas noggrannare mot andra möjliga modeller för att ge bättre resultat i framtida simuleringar för att kunna implementeras i hållfasthetstekniska beräkningar.

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List of Figures

Figure 1 - Phase volume fraction of the alloy between 100 °C and liquidus. ... 7

Figure 2 - Phase volume fraction upon cooling of the melt. ... 8

Figure 3 - Mass fraction of all components, except Cr, in γ in the temperature range 850-1200 °C... 9

Figure 4 - Mass fraction of Cr in FCC in the temperature range 850-1200 °C. ... 9

Figure 5 - A Bähr dilatometer DIL 805A chamber with details marked with arrows. ... 10

Figure 6 - Schematic temperature history of the dilatometer samples A1-B3 ... 14

Figure 7 - Schematic idealization of the part where the specimens are cut from. ... 14

Figure 8 - Illustration of the cast macro structure of the thicker material with the notation of the directionality. ... 14

Figure 9 - The test cylinder as seen in the COMSOL mesh interface, corresponding to an eight part of the real specimen. The symmetry planes’ normal are shown by arrows. ... 20

Figure 10 - Time-Temperature curve for the samples heated to 1150°C. Curves with the same heating and cooling rates overlap. ... 21

Figure 11 - Temperature-Strain curve for the samples heated to 900 °C. ... 22

Figure 13 - Temperature-Strain curve for the slowly heated samples to 1000 °C. ... 23

Figure 17 - Time-Elongation curve for the samples heated to 925 °C. ... 25

Figure 20 - Mean final strain values for all heating schemes. Suffix L denotes the slower cooling rate and U the faster. B is the mean value for B1-B3... 27

Figure 21 - Temperature-Power curve (left axis) plotted on top of the Temperature- Strain curve (right axis) for A3 heated to 1150 °C. ... 28

Figure 22 - Temperature-Power curve (left axis) plotted on top of the Temperature- Strain curve (right axis) for P1 slowly heated to 1000 °C. ... 28

Figure 23 - Elongation of A2 plotted on the left axis with smoothed and original derivative plotted on the right axis. Heating sequence is seen to the left and cooling on the right hand side. Arrows indicate direction of temperature change (black) and direction of differentiation (red). The dashed arrows connect the denotation of each transformation to the point seen marked on the derivative curve. ... 29

Figure 24 - Critical temperatures. ... 30

Figure 25 - 100x magnification untreated samples marked with direction and sample denotation. ... 31

Figure 26 - 1000x magnification Untreated samples ... 32

Figure 27 - 100x magnification P-samples. ... 33

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Figure 28 - 100x magnification A1-A6... 34

Figure 29 - 100x magnification A7-A12 ... 35

Figure 30 - 100x magnification B1-B3 ... 36

Figure 31 - 1000x magnification P-samples. ... 37

Figure 34 - 1000x magnification B1-B3 ... 40

Figure 35 - 1000x magnification for samples etched for shorter time. ... 41

Figure 36 - 1000x magnification of A10 and A12. The image is taken by cropping the original frame taken at 1000x magnification. ... 42

Figure 37 - Hardness Rockwell C measured for each specimen. ... 43

Figure 38 - Hardness Rockwell C averaged for each heat treatment separated by the cooling rate. ... 43

Figure 39 - Results from simulations of the Temperature-Strain behavior for the A and B samples. ... 44

Figure 40 - The average effective plastic strain in the cylinder during the simulated heat treatment. ... 44

Figure 41 - Temperature difference between the symmetry line at the surface and the core line, a) 925 °C, b) 1050 °C, c) 1150 °C. ... 45

Figure 42 - Results from simulations of the phase fractions apparent in the material during each temperature cycle. a) 925 °C, b) 1050 °C, c) 1150 °C. ... 46

Figure 43 - Final strain at 30 °C including the COMSOL model. ... 46

Figure 44 - Two indentations in the material subjected to hardness test. The distance from center to center is marked by arrows corresponding to the diameter of the top indentation. ... 50

Figure 45 - Schematic and exaggerated representation of carbide orientation in the different samples. ... 52

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List of Tables

Table 1 - Previously initiated samples in dilatometry. ... 13 Table 2 - List of the specimens tested in dilatometry with respective heat treatment,

direction. ... 15 Table 3 - The heat cycles tested in this dilatometry study. ... 15 Table 4 - Composition of the alloy (wt%) based on the smelt analysis of the batch

where A1-B3 are cut from. ... 15 Table 5 - Material data used in simulations. Data dependent on temperature and/or

phase fractions, is given a value for the initial microstructure at 20 °C. Direction

dependent values are marked with A or B. ... 18 Table 6 - Final strain for each specimen. ... 26 Table 7 - Critical temperatures observed by each heat treatment. Mean values for P-

samples, A- and B-samples, and all samples are presented in the bottom rows.

Subscript s indicates the start temperature of transformation and f the end (finish) temperature. K is the martensite transformation temperatures determined by the

conductor of the dilatometry. N/A indicates no available data. ... 29 Table 8 - Table of CTE found be the secant method. All B-samples are excluded in

rows starting with H, corresponding to the treatments described in Table 3. Instead, they are presented below starting with B. The ratio A/B is the ratio between the A- samples and B-samples. For heating, all A-samples are used and for the sections

occurring upon cooling, only the slowly cooled A-samples are accounted for. ... 30

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Nomenclature and Abbreviations

𝝐_𝒕𝒉 Thermal strain

Bulk

hardness Hardness of the bulk mass in contrast to micro hardness. Often measured in HRC.

CCT Continous Cooling

Transformation (Non- isothermal

Transformation Diagram)

CTE Coefficient of Thermal Expansion

dode Domain ODE physics node in COMSOL Eutectic The reaction where

melt is transformed into two solid phases (E.g. 𝐿 → 𝛾+Carbide)

F A compound of cold

phases present in the untreated material of this study

FE Finite Element

FEM Finite Element

Method flc1hs Continous first

derivative Heaviside function

flc2hs Continous second derivative Heaviside function

HCCI High-Chromium Cast Iron

HRC Hardness Rockwell C

ht Heat transfer node in COMSOL

HV Hardness Vickers

L Liquid phase

Liquidus The temperature and composition at which the liquid phase is in equilibrium with the multiphase phase area (L+solid)

LOM Light Optical

Microscopy

M Martensite, also used

as the final phase after dilatometry in this

study

M*C* Stoichiometric formula of carbides.

M=metallic phase (eg.

Fe,Cr), C=carbon.

Often seen as M7C3.

Mf Martensite end (finish) temperature

Micro-

hardness Hardness of specific component/phase, often measured in HV

Ms Martensite start

temperature

ODE Ordinary Differential Equation

Primary The first reaction at the liquidus

temperature (𝐿 → 𝛾 or 𝐿 →Carbide)

Secondary Mainly in connection with secondary precipitated carbides.

I.e. carbides formed from solid phases.

SEM Scanning Electronic

Microscopy solid Physics node in

COMSOL, for solid mechanics

Solidus The temperature and composition at which the solid phase is in equilibrium with the multiphase area (L+solid)

Tcf The final temperature for Curie

transformation Tcs The start temperature

for Curie transformation

TTT Time Temperature

Transformation (Isothermal Transformation Diagram)

Tγf The final temperature for 𝛾 formation Tγs The start temperature

for 𝛾 formation

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XRD X-Ray Diffraction

Xt Time derivative of

variable X

𝜶 Ferrite

𝜸 Austenite, also used as the possible

combination of austenite and other phases present in this study

𝚫𝝐_𝒊𝒋 Difference in thermal strain for phase i to j.

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Table of Content

1 Introduction ... 1

1.1 Xylem Water Solutions ... 1

1.2 Background ... 1

1.3 Project Outline ... 2

1.4 Needs and Limitations ... 2

2 Literature Survey ... 3

2.1 Heat Treatments ... 3

2.2 Carbides ... 5

2.3 Metallographic Evaluation ... 6

3 Theory ... 7

3.1 Metallurgical ... 7

3.2 Dilatometry ... 9

3.3 Hardness ... 11

3.4 Implementation of Data in FE-models ... 11

3.4.1 Heat Transfer and Phase Transformations ... 11

3.4.2 Thermal Strains ... 12

4 Method ... 13

4.1 Experimental ... 13

4.1.1 Specimen Description and Preparation ... 13

4.1.2 Dilatometry ... 15

4.1.3 Metallographic Evaluation ... 16

4.1.4 Hardness... 17

4.2 Simulations and Modelling ... 17

4.2.1 Material Data and Parameters ... 18

4.2.2 Heat Transfer and Phase transformations ... 19

4.2.3 Solid Mechanics ... 19

4.2.4 Geometry and Boundary and Initial Conditions ... 20

4.2.5 Study Configurations ... 20

5 Results... 21

5.2 Material Characterization ... 31

6 Discussion ... 47

6.2 Material Characterization ... 49

6.4 Future Research ... 52

7 Conclusions ... 54

8 Cited Works ... 55

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1 Introduction

This report is the result of a Master Thesis in Mechanical Engineering performed 2018 at KTH Royal Institute of Technology, Stockholm, Sweden. The work has been conducted for Xylem Water Solutions, Sundbyberg, Sweden, based on a thesis proposal from the company.

1.1 Xylem Water Solutions

Xylem Water Solutions (Xylem) is a multinational company with 12 500 coworkers in 150 counties, working with water solutions. The company works to help people using water more efficiently and have been part of the founders of Stockholm Water Prize which is one of the ways the company works to increase people’s consciousness about the importance of the available water resources. [1] All brands in the Xylem group work within a wide area connected to water solutions. The products and services are used within dewatering, water and wastewater treatment, water and wastewater transport, applied water systems and analytics.

[2]. Flygt is a Xylem brand with a large part of its production in Emmaboda, Sweden, where they have been producing pumps since the 1930’s [3].

1.2 Background

Some submersible wastewater pumps, drainage pumps and slurry pumps are used to transport abrasive particles suspended in liquid. To handle this abrasive environment the products are sometimes made of abrasion resistant cast irons [4, 5]. One group of these irons is the High-Chromium Cast Iron (HCCI) consisting of a high fraction of chromium carbides in an austenitic or martensitic matrix. HCCI is a wide spectrum of cast irons with high alloying content of Chromium and sometimes Molybdenum, that can be sufficiently hard while withstanding a certain degree of corrosive environments [4, 5, 6]. However, they are also intimately connected with a somewhat difficult choice of composition, conditions in the casting, and casting geometry that may discourage foundries from using this material [6].

Not only is it difficult to cast a product free from cracks, another possible reason for the material being seldom used outside of the milling and pump industry is the brittleness. If the product needs increased hardness to reach a higher level of abrasion resistance it is possible to harden the goods, e.g. by fully harden or by induction hardening of a specific surface which is standard practice for many applications such as for gears and shafts. To get compressive stresses in the surface region that reduces the risk of crack initiation and propagation it is preferable with a volume expansion after a temperature cycle corresponding to hardening [7]. As for this class of materials though, it has been shown in dilatometry studies that shrinkage may occur when subjected to certain temperature cycles [8, 9].

To obtain data regarding the thermal strain behavior, dilatometry can be utilized where the linear expansion of the material is determined. This method is also relevant today when the trend in materials science and mechanical engineering points towards calculation-based research and engineering. This can be seen in the calculation-based material science software such as ThermoCalc, and the many finite element method (FEM) software available. Howev- er, it is always necessary to possess mechanical and material data for the problem to perform simulations and dilatometry makes it is possible to get data regarding the expansion of materials and revealing critical temperatures for transformations [7, 10, 11, 12, 13, 14].

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1.3 Project Outline

The purpose of this study was to investigate how HCCI may behave during temperature cycling in terms of elongation change after a temperature cycle corresponding to typical induction hardening procedures. The sensitivity to temperatures and cooling rates on the final dilatometric change were evaluated to find out if it is possible to maximize the volumetric change by the choice of holding temperature. This should be advantageous in terms of resid- ual stresses in the surface of the hardened part after cycling.

Many works aim at the destabilization and/or the destabilization followed by subcritical heat treatment of as-cast material. Some of the destabilization processes are usually applied for the hardening of the material and the combination for annealing for machining. It is, however, not easy to find the destabilization-subcritical treatment followed by another destabilization treatment/hardening step in the literature, especially for shorter time sequences as in induction hardening. This study contributes with material data for this sequence of treatments and a proposal of implementation of this data in a simplified FE-model in the commercial software, COMSOL Multiphysics (COMSOL), simulating the temperature change in a sample geometry. The result from the simulation will show the phase distribution during a heat cycle corresponding to those used in the experiments and be a first step in making it possible to simulate hardening of products made of HCCI.

A hypothesis regarding the final length contraction is presented. An increased amount of precipitated carbides in the hardening step could be favorable in the aspect of giving the maximum volume expansion after the thermal cycle. A higher fraction of carbides leads to lower C and Cr content in the austenite (𝛾) phase. This might lead to lower hardness but higher martensite start temperature (Ms) and lower combined coefficient of thermal expansion (CTE). The earlier transformation upon cooling should yield a higher fraction of martensite, and the CTE for previously for martensite is lowest of all phases [8]. Due to its behavior as the strengthening phase in the composite, more carbide should lead to less elastic and plastic deformation of the bulk material martensite transformation. Hence, the final expansion should be maximized. The exact opposite was considered as well, since higher C content will lead to harder martensite and a more expanding lattice, causing the actual expansion of the martensite phase to be maximized.

In this report, fractions in percentage are shown by the symbol % regardless of the meaning of the fraction. Furthermore, if not explicitly stated differently, volume fractions are used for describing phases and mass fractions are used for describing alloying contents.

1.4 Needs and Limitations

The following limitations and needs were considered upon choosing the material and method used. The experiments needed to yield information such as the thermal strain behavior, temperature dependence of critical temperatures for phase changes, heat capacity, thermal conductivity, modulus of elasticity, yield stress and flow/hardening function, and fracture stress.

Limiting factors were mainly the time frame of 20 weeks, financial aspects, and available equipment and material. Therefore, only critical temperatures and thermal expansion were evaluated, and only light optical microscopy (LOM) and hardness measurements were used for the material characterization. Also, it was necessary to use material supplied by Xylem and work specifically with this previously heat treated material which makes it more difficult to apply results to a broader class of materials but might give indications of aspects needed to evaluate further in a more experimental manner.

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2 Literature Survey

This chapter will highlight the existing knowledge regarding the investigated material and its production. HCCI is a relatively wide group of metals with a narrow field of use and are of- ten found in abrasion resistant applications [15, 16]. Zhou et al. [17] and Bedolla-Jacuinde et al. [18] describe HCCI as a material with 11-30 % Cr and 1.8-3.6 % C. The material investi- gated in this study falls within the Swedish standard SS-EN 12513:2011 [19], in literature often referred to the previous SIS 140466:1971 [20]. Nearest American standard, ASTM A532, is Class III Type A [20]. This is a material with 23%-30% Cr and 2.0%-3.3% C [20].

HCCI have been reported both as hypoeutectic, eutectic and hypereutectic alloys where the primary crystals are austenite dendrites for hypoeutectic alloys and large M7C3 carbides for most hypereutectic materials [18, 21, 22, 23]. The following equation (1) has been proposed to determine the content of alloying elements in the Fe-Cr-C binary eutectic structure (𝛾+M7C3) [24]

(% C) + 0.0474 × (% Cr) = 4.3 (1)

If the left-hand side equals less than 4.3 the composition is hypoeutectic [24].

The HCCI materials are known to be brittle and show little or no yield in tension at low temperatures. Even in compression tests, some of the materials of this class have shown little or no plasticity prior to fracture, especially for the eutectic and hypereutectic alloys [25]. The metals may also show anisotropic behavior, especially following cold working such as rolling and extrusion [26].

2.1 Heat Treatments

There are several reasons for subjecting a metal to heat treatments after casting. In this section the heat treatments of HCCI found in literature are presented. Commonly, the procedures reported regarding HCCI are called destabilization and subcritical heat treatment, often performed in that order. While evaluating several materials in the HCCI class, Sare and Ar- nold [27] found that “each of the four commercial grades of material studied must be treated as separate entities, and that there are no generic trends that apply to all alloys”, especially in connection with abrasion resistance. In addition, Tabrett and Sare [28] reports that there are significant differences in microstructural changes and properties for the Cr15Mo3 and Cr27 white iron subjected to equal heat treatments. Since the material investigated falls within the higher range of chromium content, the otherwise common Cr17 are not described in detail.

For hypoeutectic HCCI the solidification results in primary solidified 𝛾-dendrites followed by a eutectic formed by 𝛾 and M7C3 carbides (𝐿 → 𝛾+M7C3) [16, 29, 30]. The eutectic can take different forms based on the interaction with the dendrites and is either formed as a “flower- like structure” [24] starting from a point in-between the dendrites, or as lamellar form [24, 30]. M7C3 carbides have been found to grow as rods or sometimes as plates in the eutectic, oriented in the columnar direction of solidification [31]. Fredrikson and Remeaus [16] investigated a Cr26 alloy with 2.54 %C and 2.77 %C respectively. The as-cast structure consisted of primary 𝛾-dendrites and eutectic (𝛾+M7C3) with a thin liner of martensite around the carbides [16].

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4 The solidified austenite contains an excessive amount of C and Cr and is therefore in industry destabilized at temperatures around 930-1060 °C to precipitate carbides and deplete the matrix on allowing elements allowing for transformation products to form upon cooling or treatment at lower, subcritical temperatures [15, 32]. The carbides formed upon the eutectic reaction have shown to be left unaffected by most heat treatments [15]. Depending on alloying content and temperature of isothermal heat treatments either or both M7C3 and M23C6 may form as secondary carbides [16, 30]. Different theories are presented for the temperature ranges. For the Cr26 alloy investigated in [16] the M23C6 carbides are assumed to form upon the eutectoid reaction forming a ferrite (𝛼)+carbide cluster (𝛾 → 𝛼+carbide) at lower temperatures (<775 °C) while the M7C3 carbides are precipitated at higher temperatures (~975

°C). In [33] several different HCCI were investigated and the common conclusion was that the isothermal treatment resulted in c-shaped curves with the fastest decomposition at 950- 1000 °C and 650-700 °C. For the higher temperature range, M7C3 was formed above the temperature where 𝛼 is no longer stable (often denoted A3) and between the eutectoid temperature, below which 𝛾 is no longer stable (often denoted A1), and A3 a combination of M23C6, 𝛼, and 𝛾 were formed [33]. For the lower temperature range, subcritical treatment below A1, 𝛾 formed 𝛼 and M3C carbides. This precipitation and depletion of alloying elements of the austenite matrix causes the Ms temperature to rise close to the carbides, hence forming allowing martensite to form subsequent quenching [15, 16, 33]. The martensite is often formed as a liner around the carbides while austenite is retained further away from the carbides due to higher carbon content away from the carbides as reported in [16, 21]. The

“magnitude of martensitic transformation” defined as the expansion due to martensite formation is discussed in [30]. It was found that higher temperatures and greater precipitation lead to a greater magnitude.

The maximum hardness of different heat treatments is often seen as a c-curve of the destabilization temperature with a maximum found at around between approximately 980-1050 °C depending on the investigated alloy [17, 30, 33, 34]. Reference [16] report that the hardness of the material is determined by the hardness of the martensite and the content of martensite in the matrix. They write that the hardness of the martensite is strongly dependent on the carbon content in the martensite and the solubility of carbon in austenite is increasing with increasing temperature; however, the Ms temperature is decreased and the formation of martensite might be suppressed. Similar reasoning regarding the maximum hardness of the matrix is pointed out in the extensive study performed by Maratray and Usseglio-Nanot [30] as well. The authors found that the decrease in hardness at low temperatures is explained by the lower hardness of the martensite due to the carbon content and at very high temperatures by the increased amount of retained austenite. Maratray and Poulalion [29] connected this behavior with the retained austenite content and found that around 20 % retained austenite was found at the maximum hardness. Sare and Arnold [27] showed that the higher austenitization temperature will lead to increased fracture toughness and in the case of Cr27 the higher austenitization temperature (1100-1150 °C) resulted in the lowest abrasive wear as well, even though the latter relation was subtler. The abrasion resistance did, however, depend strongly of the tempering temperature; even though the microstructure was reportedly similar. The relation with retained austenite showed that the optimal value is about 30% for the wear resistance.

Destabilization at 1100 °C is reported to redissolve secondary carbides precipitated at lower temperatures while they remained stable at 1000 °C [28, 35]. Lai et al. [36] presents a con-

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5 stantly decreasing amount of carbides with temperature (950-1100 °C) and Bedolla et al. [18]

also shows that previously precipitated carbides may dissolve at higher temperatures while further precipitation is detected for lower destabilization temperatures (900-1000 °C).

There are several proposed procedures described in the literature to decrease hardness and retained austenite and thus increase the machining properties. Even though austenite is relatively soft; it is sometimes unstable and therefore possible to transform to martensite upon certain isothermal conditions, such as increased strain upon deformation [37, 38]. In [32]

either subcritical (between 690 °C and 705 °C) or full annealing are recommended for HCCI;

the latter, performed in the range 955-1010°C followed by a longer holding time at 760 °C is recommended for alloys with particularly good hardenability. The same treatment is also recommended in [15] where the austenite is described to be destabilized at the higher temperature range followed by a formation of a ferrite-carbide structure at the lower range.

Maratray [33] writes that this annealing can be performed either by isothermal holding between A1 and A3 or by destabilization above A1 followed by slow cooling to about 650 °C and that the most easily machined structure is the ferrite-carbide structure consisting of sphe- rodized carbides of possibly M23C6. A heat treatment of a Cr27 alloy at 1000 °C followed by a subcritical heat treatment at 550 °C or 600 °C showed a great decrease in retained austenite and hardness and exhibited much higher wear losses than the as-cast alloy [28].

No experimental studies investigating the hardening treatment following soft annealing of the higher range of chromium rich HCCI have been found in this literature survey.

2.2 Carbides

In the case of the high-alloyed cast irons, the carbide content may be very high. Equation (2) proposed by Maratray [33] is used in the literature to estimate the amount of carbides in HCCI and has shown good correspondence to both calculations performed in ThermoCalc as well as experimental results [39, 40].

% carbides = 12.33 × (% C) + 0.55 × (% Cr) − 15.2 (2)

Liu et al. [25] also showed that the macrostructure caused by the casting of these materials has impact on the strength and yield stress. They saw that the materials’ stress-strain relationship was affected by the directionality of the material, especially for the slightly hypereutectic alloys and at elevated temperatures. They also found that the eutectic structures grew in the direction of solidification using a setup to control the solidification direction. The same di- rectionality in the morphology of the carbides was found in a study performed by Lu et al.

[24]. In that study the unidirectional solidification was investigated and showed a clear fi- brous network of eutectic carbides growing in the direction of solidification and the HCCI was treated as a fiber composite. It was seen that the strength was drastically increased in the direction of the carbide network. Tabrett and Sare [31] investigated the columnar macrostructure formed during solidification in a regularly cast material. They found that the mainly columnar structure of eutectic carbides in a Cr27 alloy had a clear impact on the fracture toughness and that fractures often occurred along the carbide-matrix interface. The growth as plates or bars of the eutectic hexagonal M7C3 is reported in [18] as well. By aligning the columnar structure, and the longitudinal axis of the carbides, perpendicular to the crack plane the fracture toughness increased [31].

Among other features, the extent of secondary carbide precipitation was investigated in an extensive study by Maratray and Usseglio-Nanot [30]. Their results for destabilization at 800-

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6 1000 °C showed that for samples with a high amount of carbides the extent of the isothermal carbide formation at destabilization does not alter Ms; however, the temperature at which the precipitation is performed does, as described in the previous section.

2.3 Metallographic Evaluation

The structures of the investigated materials in literature are often evaluated by LOM and scanning electron microscopy (SEM). These methods yield a mostly qualitative understand- ing of the material structures. For the metallography it is necessary to prepare samples according to a regular metallographic procedure by polishing and often etching. A chemical etchant attacks different compositions or different crystalline directions of grains at different rates, causing irregularities in the surface that is detectable in the LOM and not only depending on the relative reflectivity of the phases in the polished state. [41, 42] The choice of etchant, however, has been done differently throughout different studies. 4% Nital used in [17], Viella’s etchant used in [18, 35, 43], and aqua regia with inhibitor [44] are some choices for etching similar materials, mostly with a chromium content of 14-30 %.

SEM is used in addition to LOM to resolve fine structures and can be used in combination with deep etching to evaluate the network of carbides formed. Since SEM in general gives higher resolution and magnification it is possible to distinguish certain structures from each other which otherwise can be difficult by the magnifications used in LOM [30]. Another commonly used experiment in addition to LOM is X-Ray diffraction (XRD). This method is used to detect phases present in the investigated material. The amount of austenite has been found difficult or impossible to quantitatively determine by metallographic evaluation [29].

However, both these methods are time consuming and costly compared to LOM that is often easily obtainable and user friendly.

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7

3 Theory

In this chapter theory behind the metallurgical processes are described, as well as the dilatometry, hardness measurements, and FEM regarding this research area.

3.1 Metallurgical

A primary study performed by the author reveals some thermodynamic relations based on computational thermodynamics. The software ThermoCalc was utilized to create diagrams of relevant aspects in the material handling based on equilibrium equations in the database TCFE7 for the material composition corresponding to that of the samples in dilatometry.

Figure 1 show the volume fraction of phases thermodynamically stable in the material for the temperature range 100-1500 °C. It consists mainly of austenite, 𝛾, and M7C3 carbides at the higher temperatures and the austenite is transformed to ferrite, 𝛼, and carbides at the eutectoid temperature around 800 °C. The transformations around the melting point are seen in detail in Figure 2. Fredriksson and Remaeus [16] present isothermal sections of the Fe-Cr-C phase diagram from Bungardt [45] that shows the increased solubility of Cr and C in 𝛾 with increasing temperatures and that high chromium content may cause the carbide M23C6 to form as well as M7C3. The solubility of alloying elements in 𝛾 is further described by Figure 3 and 4Figure 4, showing the temperature dependent equilibrium mass fraction of alloying elements in 𝛾 for the investigated material based on calculations in ThermoCalc.

Figure 1 - Phase volume fraction of the alloy between 100 °C and liquidus.

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8

Figure 2 - Phase volume fraction upon cooling of the melt.

An estimation of the phase fraction found in treated samples can be done by several methods. Point counting of the revealed microstructures after preparation by applying a grid to the microstructure either in the ocular, or in post processing after taking a photograph of the cross section, is one of the most commonly used methods. The lowest possible magnification giving sufficient level of detail is to be used and the grid size is adjusted for the microstructure to be evaluated. The points are counted at the grid intersections where half a point is counted when the intersection coincides with a phase boundary. The volumetric phase amount is then calculated by dividing the points with the total amount of grid intersections.

[46]

Not visible in the Figure 1 and 2 is the Curie temperature (Tc), below which the 𝛼-ferrite is ferromagnetic [47]. For unalloyed iron, this temperature is around 770 °C [47] (1043 K) and a proposed relationship (3) seen in [48] is insensitive to carbon, but atomic fraction of manga- nese (𝑥Mn) is considered

𝑇_𝑐= 1042 K − 𝑥_Mn× 1500 K (3)

Belyakova et al. [49], however, report that the same author in a previously published article found the Curie temperature depending on chromium by drastically lowering Tc with increasing Cr content. They present a decrease in transformation temperature from 750 °C to 538 °C when increasing the Cr content from 13.6 % to 33.5 %. The effect of the magnetic transformation on the dilatometry is pronounced when looking at the power needed to keep a constant heating rate over the temperature interval as seen in reports regarding hypoeutec- tiod steels [48, 50].

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9

Figure 3 - Mass fraction of all components, except Cr, in γ in the temperature range 850-1200 °C.

‘

Figure 4 - Mass fraction of Cr in FCC in the temperature range 850-1200 °C.

3.2 Dilatometry

There are reportedly several possible methods to determine phase transformation temperatures and some yield additional information such as phase volume fractions. Dilatometry is one of these methods [51]. It is used to detect length changes of a material during heating and cooling and through the operationalization of connecting phases with different densities makes it possible to observe critical points of phase transformations and the phase fractions [48, 50, 52]. The Curie transformation is also detectable due to the change in power of the inductor and an additional observed strain [53, 54, 55].

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10 The dilatometer seen in Figure 5 show the main parts in the measurement chamber. The linear length change (𝛥𝐿) of the specimen is recorded for the changing temperature (𝛥𝑇) by (4)

Δ𝐿 = 𝐿 − 𝐿₀= 𝐿₀𝛼_𝐿Δ𝑇 (4)

for the length 𝐿 and original length 𝐿0 [11]. 𝛼𝐿 is the CTE and is defined by the differentiation in equation (5)

𝛼_𝐿= ¹

𝐿₀ 𝑑𝐿

𝑑𝑇 (5)

and is a function of temperature [11]. Other definitions exist as well, such as the secant expansion defined by the difference in temperature and length change in two points, divided by the initial length. For materials showing a crystalline anisotropy 𝛼𝐿 vary with the direction but for an isotropic material the volumetric coefficient of expansion due to temperature changes are approximately three times 𝛼_𝐿. [11]

Figure 5 - A Bähr dilatometer DIL 805A chamber with details marked with arrows.

If the dilatation of pure iron is performed under experimental circumstances, it is possible to determine the phase fraction from the lever rule method between two phases. For complex systems such as iron alloyed with carbon, however, this is not applicable for several reasons.

One is due to the carbon diffusion that will cause the remaining austenite to change in composition and therefore in volume [48, 50]. HCCI further complicates interpretations of dilatation since the system is more complex at temperatures where austenite is the only stable phase in steels. Especially for dilatation after certain heat treatments, the initial material of ferrite-pearlite [50] will contain more phases, such as eutectic carbides and secondary precipitated carbides and austenite will not be the only phase present at start of cooling from higher temperatures as discussed in chapter 2.

The determination of critical temperatures is mostly used for determination of Ms, but the same practice applies to other transformations as well. The use of differentiating the dilatation curve has been used for showing subtle changes in the dilatometer curve that else could not be visualized [56, 57]. A more statistically correct method for martensite transformation determination in steels is the offset method presented by Yang and Bhadeshia [14] which

Induction Coil

Quartz Push Rod Sample

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11 utilizes the lattice expansion of the phases and a fixed limit of offset from the expected dilatation.

3.3 Hardness

There are several hardness measurement methods available. In common is that the indentation depth/width is connected to the hardness value through specific formulas when tests can be systematically performed. In the Rockwell test the hardness is determined by the difference in indentation depth of two different loads and in many testing machines the value is given on a display after indentation [46]. Different critical distances are given in the literature [46, 58]. However, the most conservative guideline states that the distance from the edge of the sample should be no less than 1 mm and the center-to-center distance of two indentations should exceed 2 mm [58]. Rockwell C (HRC) is measured by a Brale penetrator first indented by 10-kgf and then by 150-kgf load [46, 58].

3.4 Implementation of Data in FE-models

This chapter will only briefly explain some theoretical foundations for describing the heat transfer, phase transformations, and stress and strain relationships in a FEM-simulations related to the studied phenomena of dilatation and induction hardening. A thorough explana- tion of the theory and equations used for describing the latent heat and the phase transformation can be found in [59] and, among others, [60] and [61] are works using FEM software to model phase changes in steel containing various information of the theory behind the modelling.

3.4.1 Heat Transfer and Phase Transformations

Austenite transformation is often described by the formula proposed by Leblond and De- veaux [62] that consider the influence of time and temperature. Martensite fraction (M) is, however, described frequently by the exponential K-M equation (6) [60, 61, 62, 63] based on austenite fraction at Ms, 𝛾0, positive kinetics parameter 𝛽, and temperature 𝑇, formed by Koistinen and Marburger [64].

𝑀 = 𝛾₀(1 − exp(−𝛽(𝑀𝑠 − 𝑇))) (6)

The equations can be implemented into FEM software as (6) and the following ordinary differential equation (7) [59, 63]. Here the time derivative (subscript t) of austenite (𝛾) and martensite are considered. 𝜏(𝑇) is a kinetics parameter, 𝑇(𝑡) is the temperature as a function of time, 𝛾𝑒𝑞 is a linear function from 0 to 1 describing the maximum fraction of austenite, 𝑇𝛾𝑠

and Ms are the start temperature of respective transformation found in practice, and 𝐻 is the Heaviside step function.

𝛾_𝑡(𝑡) = ¹

𝜏(𝑇)max{[𝛾_𝑒𝑞(𝑇(𝑡)) − 𝛾(𝑡)], 0} 𝐻[𝑇(𝑡) − 𝑇𝛾_𝑠] − 𝑀_𝑡 (7) Neglecting radiation and mechanical effects on the thermal properties the energy equation is described as (8) [59, 65]

𝜌(𝑇)𝐶𝑝(𝑇)^𝜕𝑇

𝜕𝑡− ∇ ∙ (−𝑘(𝑇)∇𝑇) = 𝐹1 (8)

where 𝜌, 𝐶_𝑝, and 𝑘, denotes the temperature dependent variables density, heat capacity at constant pressure and conductivity respectively. The heat consumed or released during phase transformations is described as a heat source, 𝐹1, by (9)-(11) [59]

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12 𝐹₁= 𝐹₁₁(𝐻(−𝑇_𝑡)) + 𝐹₁₂ (9)

𝐹₁₁= 𝜌(𝑇)𝐿_𝑚𝑀_𝑡 (10)

𝐹₁₂= −𝜌(𝑇)𝐿_𝛾_𝜏(𝑇)¹ max{[𝛾_𝑒𝑞(𝑇(𝑡)) − 𝛾(𝑡)], 0} 𝐻[𝑇(𝑡) − 𝑇𝛾_𝑠] (11) 𝐿_𝛾 and 𝐿_𝑚 are positive values for the latent heat, hence heat is released upon martensite transformation and consumed for austenite transformation from the original cold phases.

3.4.2 Thermal Strains

Instead of the usual heat expansion 𝜖𝑡ℎ(𝑇) = 𝛼_𝐿(𝑇 − 𝑇_𝑟𝑒𝑓) equation (12) can be implemented in COMSOL [66]

𝜖_𝑡ℎ= ∑ 𝜖_𝑖(𝑇) 𝑓_𝑖 (12)

where 𝑓_𝑖 is the volume fraction, and 𝜖_𝑖 are temperature dependent functions describing the strain of the phases 𝑖. The CTE for each phase is accurately described by the lattice parameters and carbon content as well as temperature dependence [60, 61]. A secant thermal expansion has been used as well to describe the expansion and contraction due to temperature changes [65].

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13

4 Method

This chapter will highlight the choice of method and describe the setup for the experimental part of this study. Lastly, the procedure of implementing experimental data in the FEM- simulations in COMSOL is presented.

4.1 Experimental

The main section of this study was based on experimental results. Dilatometry was the choice of method to evaluate the volumetric change after temperature cycling. There are other methods available to determine critical temperatures during cycling for implementation in simulations such as cooling curve analysis [51]. However, since the volumetric change was the focus of this study the dilatometry could yield more relevant data. A metallographic evaluation and hardness measurements is usually evaluated in connection to dilatometry [10, 67].

By evaluating the specimens by several methods, it is easier to rule out alternative explana- tions to the behavior found in dilatometry.

4.1.1 Specimen Description and Preparation

The specimens in this report refer to previously initiated samples in dilatometry described in Table 1, as well as the specimens subjected to dilatometry experiments designed by the author, see Table 2. The heating rate, holding time, cooling rate as well as the destabilization temperature are reported as different heat treatments further described in Table 3.

Table 1 - Previously initiated samples in dilatometry.

Sample Heat treatment Direction

P1 H1000LL Iso

P2 H1000LL Iso

P3 H1000U Iso

P4 H1000U Iso

P5 H1000L Iso

P6 H900U Iso

P7 H900U Iso

P8 H900L Iso

Pobh Untreated Iso

Pobh is untreated material from the same section as P1-P8. All specimens in Table 2 were cut from material with the composition in Table 4, based on the analysis taken on the melt before casting. The samples in Table 1 are of the same material standard, falling in the limits of ASTM Class III cast irons; however, no charge analysis is available. The temperature history was available for A1-B3 and is schematically presented in Figure 6. Samples cut from the other main material have followed a similar standardized scheme.

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14

Figure 6 - Schematic temperature history of the dilatometer samples A1-B3

Figure 7 and 8 show the direction of which the specimen was cut from the manufactured part. The original part had a slight curvature not depicted in the illustration. P1-P8 are taken from a product with a thinner section and were only cut in the Iso-direction.

Figure 7 - Schematic idealization of the part where the specimens are cut from.

Figure 8 - Illustration of the cast macro structure of the thicker material with the notation of the direc- tionality.

A1-B3 were cut out from a bigger casting with a thickness of 20 mm. A typical casting macro structure is seen in Figure 8. The samples were carefully cut in a Beuhler Abrasimet 2 with coolant. For A1-B3, eight square rods of approximately 6×6×50 mm were cut out of the main material.

Iso Ani

Depth

A ni

Iso

Ani

Iso

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15

Table 2 - List of the specimens tested in dilatometry with respective heat treatment, direction.

Specimen ID Heat treatment Direction

A1 H925L Iso

A7 H925L Iso

B1 H925L Ani

A10 H925U Iso

A4 H925U Iso

A2 H1050L Iso

A8 H1050L Iso

B2 H1050L Ani

A11 H1050U Iso

A5 H1050U Iso

A3 H1150L Iso

A9 H1150L Iso

B3 H1150L Ani

A12 H1150U Iso

A6 H1150U Iso

H2 Untreated Depth

Hi1 Untreated Iso

Tö Untreated Ani (mid-section)

Tu Untreated Ani (close to surface)

Vy1 Untreated Iso

The choice of original rod for each sample and trial order were made randomly by using a Matlab random generator. The square rods were then machined into cylindrical pieces of 10 mm length and 4 mm diameter at the Swedish research institute Swerea KIMAB, Stockholm.

Table 3 - The heat cycles tested in this dilatometry study.

Denotation Heating rate [°𝐂/𝐬]

End temperature [°𝐂]

Holding Time [𝐬]

Cooling rate [°𝐂/𝐬]

H900L 100 900 1 20

H900U 100 900 1 100

H1000LL 0.5 1000 1 10

H1000L 100 1000 1 20

H1000U 100 1000 1 100

H925L 50 925 3 10

H925U 50 925 3 50

H1050L 50 1050 3 10

H1050U 50 1050 3 50

H1150L 50 1150 3 10

H1150U 50 1150 3 50

Table 4 - Composition of the alloy (wt%) based on the smelt analysis of the batch where A1-B3 are cut from.

Alloy C Si Mn P S Cr Ni Mo Cu Fe

HCCI 2.60 0.72 0.41 0.016 0.018 24.70 0.19 0.10 0.01 71.2

4.1.2 Dilatometry

The dilatometry was used for different reasons. One was to evaluate the thermal expansion and compression of the material during an arbitrary temperature cycle. It can also be used to

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16 detect phase changes in the material. This is otherwise not directly observable, but the feature is linked to a sudden volumetric change that is detectable by the dilatometer. This is to say that the feature of phase changes is operationalized by observing the length change of the material during heating and/or cooling. Hence, the method was used in this study.

The dilatometer used was a “Bähr dilatometer DIL 805A” at the research institute Swerea KIMAB, Stockholm, Sweden. The instrument is shown previously in Figure 5. The sample was held in the inductor by quartz push rods. The apparent length change (including the possible expansion of the quartz rods) of the specimen was measured simultaneously as the temperature is measured and controlled by a spot-welded thermocouple of type S, Pt and Pt+10 % Rh. The dilatometry was performed in two sessions. The first tests were, as men- tioned above, planned and initiated outside of this study. Therefore, the following description of the experimental randomization setup is valid for A1-B3 only. The results, however, are reported together but with the distinct difference in denotation of the samples.

During the experiment, all samples were given two different numbers; one for the identifica- tion for the position at the original material (works as specimen marker) and one randomized for the experiment order. Hence, the experimental setup was randomized twice. The randomization was done to allot the sample to a specific procedure and the second to determine the sample order of the trials. This was to reduce the risk of influence of systematical errors on the results. See Table 3 for the different heat treatment schemes and Table 2 for each specimen denotation.

The recorded data in dilatometry was time, temperature, length change, and nominal temperature. Additionally, it was possible to get the power used for the inductor. The strain was found by dividing the recorded length change by the initial length of each specimen, given in mm with two decimals. To more easily determine the phase transformations the derivative of the experimental Temperature-Elongation data was calculated. Since fitted to experimental data, the derivative values and the power values were then smoothed by “local regression using weighted linear least squares and a 2nd degree polynomial model” [68] called “loess” in Matlab before plotting in figures. In the case of noisy in-data the dilatation data were

smoothed as well. The raw data on cooling for the A and B-samples were smoothed by the span 0.025 for slowly cooled (suffix L) and 0.05 for the higher cooling rates (suffix U). The derivatives were evaluated both as unsmoothed and smoothed using a span of 0.05 at heating and five times the span of the raw data smoothing during cooling.

Using the Temperature-Strain data the (secant) CTE is determined for all samples. This procedure was divided into two sections. Regression points for all sections of the dilatometer curve were graphically chosen. Following this, a simplified curve was considered to evaluate data for simulations for fewer sections. In the latter evaluation, only A- and B-samples were considered.

4.1.3 Metallographic Evaluation

The metallographic evaluation included material previously tested in dilatometry (P3-P8) as well as the samples prepared and designed by the author (A1-B3) and the samples without heat treatment. The material from dilatometry was received as cylindrical pieces of 10 mm length and 4 mm diameter. Common praxis for sample preparation was used, including casting in thermosetting Bakelite (Struer’s PolyFast), wet grinding (240P, 320P, 600P, 1200P), and finally polishing using 3 µm diamond paste. Etching was performed in V2A etchant (100 ml H2O + 100 ml HCl + 10 ml HNO3 + 0.3 ml Nonidet P-40) at around 50-55 °C for 4-12

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17 seconds. The same procedure was repeated for the samples A1-B3 after dilatometry and for the specimens without heat treatment. All samples are inspected in an optical microscopy (Nikon Optiphot 150S) with resulting magnification ranging from 50x-1000x. A Nikon D500 DSLR + Nikkor 50mm f/1.8 D was mounted to enable photographs to be taken. The amount of primary phase was estimated at some samples by point counting photographs of the structures with a 7×11 grid.

4.1.4 Hardness

Following the microscopy evaluation, the samples were polished as described in the previous section and then tested in a hardness tester Otto Wolpert Werke of type Dia testor 2Rc with test load 150 kpf as is practice for Rockwell C measurements. The measurements were directly read of the scale on the machine with a precision of 1 HRC. A calibration sequence where the stability of obtained hardness values for calibration plates of HRC 59.2 and HRC 30.2 was performed. The hardness of the samples was then estimated with one decimal and then rounded to the closest integer before averaging over the specimens and heat treatments.

First, a measurement was taken in the center of the cylinders to accompany to the standard distances; however, at least one additional indentation was made between the center and edge. For the untreated samples that had a much larger cross section, several measurements were taken in rows from edge to edge.

4.2 Simulations and Modelling

Data from the dilatometry study were combined with data found in the literature to create a model of a material exhibiting the similar heat treatment response as the investigated HCCI.

The simulations were conducted in COMSOL Multiphysics 5.3a which is a commercially available FEM-software. The model was constructed using three different nodes, Heat Transfer in Solids (ht), Domain ODEs and DAEs (dode), Solid Mechanics (solid) as well as the Multiphysics node. This node connects the temperature between ht and solid and enables calculations of thermal strains. As a rule, most functions given by data in tables were interpolated with piecewise cubic functions to avoid discontinuities in the derivative otherwise found in the data points.

The analysis was a transient study in three dimensions. A demonstration model of a different problem involving induction heating and phase transformations provided by COMSOL Mul- tiphysics, based on the works of [59, 69, 70, 71, 72], was used for some material data as well as a reference for the implementation of the ordinary differential equations used to describe the martensite phase evolution. The data were in some cases modified to better match data found for HCCI and adjusted to better fit the dilatometry results. In this case, the dilatometry curves were assumed to show the correct behavior of the material as a bulk mass, even though it might deviate from the demonstration model and/or findings regarding other simulations of low alloyed steels. This is due to the difficulty of estimating the material behavior and constituents and therefor makes it diverge from normal calculation models that are often based on calculations of each individual phase.

When discussing both the experimental and simulated dilatometer curve, the different sections of the curve will be denoted as in regular steels, i.e. cold phases (𝐹), austenite (𝛾), and martensite (𝑀), even though the material may consist of different amount of carbides and phases as well at all steps. The original material is assumed to be stress free and consist of one single phase, 𝐹, which in practice is the composite of materials existing in the dilatometry samples before heat cycling. The CTE were assumed to be constant until the phase trans-

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18 formation at around 880-960 °C, denoted 𝐹 → 𝛾. The CTE was then determined by a linear combination of the present phases until the transformed material is going through the 𝛾 → 𝑀 transformation upon cooling below 400-200 °C where the CTE of 𝑀 was added.

4.2.1 Material Data and Parameters

Data used in this simulation were found for different materials and were accepted in absence of more precise data for the HCCI in question. Material parameters are presented in Table 5.

Temperature dependent values for the thermal conductivity, 𝑘, Young’s modulus, 𝐸, and heat capacity at constant pressure, 𝐶𝑝, were obtained from [49] while the temperature dependent tangent modulus, 𝐸_tan, and the nonlinear combination of phase and temperature dependent initial yield stress (𝜎0) accepted as found and combined in [63]. Values for the initial temperature and microstructure are presented in Table 5, and graphs of the material data are available in Appendix 1. The latent heat of fusion for both the modelled phase changes was set according to reference [59]. Density, 𝜌, and Poisson’s ratio, 𝜈, are set as con- stants shown in Table 5.

Table 5 - Material data used in simulations. Data dependent on temperature and/or phase fractions, is given a value for the initial microstructure at 20 °C. Direction dependent values are marked with A or B.

Parameter Value Comment

𝒌 13.9 [W/(m∙K)] Non-linear, temperature de-

pendent

𝑬 213 [GPa] Non-linear, temperature de-

pendent

𝑪𝒑 0.45 [kJ/(kg∙K)] Non-linear, temperature de-

pendent

𝑬_𝐭𝐚𝐧 6.579 [GPa] Non-linear, temperature de-

pendent 𝑳_𝜸 82 [J/g]

𝑳_𝑴 82 [J/g]

𝝈_𝟎 320 [MPa] Non-linear, phase and tempera-

ture dependent.

𝝆 7600 [kg/m³]

𝝂 0.28 [1]

𝜶_𝑳,𝑴 A: (-13×Tmax/280+369/7)e-6 [1/K]

B: (-5.27×Tmax/14010+301.71/7)e-6 [1/K] Secant coefficient for A and B, function of maximum temperature in °C.

𝜶_𝑳,𝑭 A: 14.66 [1/K] B: 12.32 [1/K] Secant coefficient, constant 𝜶_𝑳,𝜸 A: 20.53 [1/K] B: 17.48 [1/K] Secant coefficient, constant 𝚫𝝐_𝑭𝜸 A: 1.06e-2 [1] B: 0.87e-2 [1] Different for A and B samples 𝚫𝝐_𝑭𝑴 A: 6.81e-4 [1] B: -1.47e-2 [1] Different for A and B samples

𝜷 A:

0.035 [1/s], 0.035 [1/s], 0.015 [1/s]

B: 0.050 [1/s], -- --

Linear interpolation of maximum temperature (Tmax) in

°C. Values given for Tmax: 925, 1050, 1150 °C.

𝑴𝒔 A:

400 [°C], 320 [°C], 220 [°C]

B: 410 [°C]

-- --

Linear interpolation of maximum temperature (Tmax) in

°C. Values given for Tmax: 925, 1050, 1150 °C.

𝑻𝜸_𝒔 887 [°C]

𝒁^𝑹 0.31 [1]

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19 4.2.2 Heat Transfer and Phase transformations

The heat transfer was purely based on conduction; no convection or radiation was accounted for. The phase transformations were described by equation (6)-(7) found in chapter 3.4.1 and implemented in the dode node as equation (13), (14), and (15). Equation (16) was implemented as a regular variable.

𝛾_𝑡(𝑡) = ¹

𝜏(𝑇)max{[𝛾_𝑒𝑞(𝑇(𝑡)) − 𝛾(𝑡)], 0} 𝑓𝑙𝑐2ℎ𝑠(𝑇 − 𝑇𝛾_𝑠− 10,10) − 𝑀_𝑡 (13)

𝑇_max = 𝑇_𝑡× (𝑇_𝑡> 0) × (𝑇 > 𝑇_max) (14)

𝛾₀= 𝛾_0𝑡× (𝛾_0𝑡> 0) × (𝛾 > 𝛾_0𝑡) (15)

𝑀(𝑡) = 𝛾₀(1 − exp(−𝛽 max( 𝑀𝑠 − 𝑇, 0))) (16)

𝛾_𝑒𝑞 is a linear function from 0 to 1 defining the equilibrium content of austenite defined from the dilatometer curve. 𝜏 is rapidly decreasing with temperature and the values found in reference [62, 63] were adjusted to better fit the material used in this study. The Heaviside function was described by the COMSOL-function, 𝑓𝑙𝑐2ℎ𝑠(𝑥, 𝑏), which has a continuous second derivative and is equal to 1 when 𝑥 ≥ 𝑏 and 0 when 𝑥 ≤ −𝑏. Hence, 2𝑏 is the span over which the step occurs. Wide spans are used for the temperature range to avoid numeri- cal errors. 𝛽 and Ms were linearly interpolated between approximated values based on dilatometry, shown in Table 5. The final 𝜏 varied nonlinearly with the temperature (𝑇) and the maximum reached temperature (𝑇_max), respectively. Graphs of these variables are available in Appendix 1.

The heat consumed or released during phase transformations were defined as in 3.4.1 except for an additional factor (1 − 𝑍^𝑅) where 𝑍^𝑅 was set to 0.31 to accommodate for the carbides estimated by the equation (2) and Figure 1.

4.2.3 Solid Mechanics

The solid mechanics node was utilized to enable the Multiphysics node “Thermal expansion”

to define the isotropic thermal expansion (17) by modifying a proposed equation in [63].

𝜖_𝑡ℎ= 𝐹(𝑡)[𝛼_𝐹(𝑇 − 𝑇_ref)] + 𝛾(𝑡)[𝛼_𝛾(𝑇 − 𝑇_ref) − (1 − 𝑍^𝑅)𝛥𝜖_𝐹𝛾] + 𝑀(𝑡)[𝛼_𝑀(𝑇 −

𝑇_ref) − (1 − 𝑍^𝑅)𝛥𝜖_𝐹𝑀] (17)

where the factors 𝛥𝜖_𝑖𝑗 were determined as the difference in compaction between the cold phase, 𝐹(𝑡), and the current phase, 𝑀(𝑡) and 𝛾(𝑡), for A-samples set to 1.06e-2 as found in [63] and 6.81e-4 based on the dilatometry results. Values for B-samples, also found in Table 5, were adjusted to fit the dilatometry results. The reference temperature, 𝑇ref, was set to room temperature (20 °C). The expansion coefficients were based on the dilatometry results and are described in Table 5.

The influence of plasticity on the dilatation was evaluated by applying a large plastic strain model based on von Mises yield function and a linear isotropic hardening model in the Plas- ticity sub node in solid. The yield level, 𝜎_𝑌, is based on the isotropic tangent modulus, 𝐸_iso, and plastic strains 𝜖pl, and described by equation (18) [73].