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An investigation of deformations in stainless steel rings by heating Towfighi, Siyavash 2011

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  AN INVESTIGATION OF DEFORMATIONS IN STAINLESS STEEL RINGS BY HEATING  by Siyavash Towfighi  B.A.Sc., The University of British Columbia, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver) December 2011  © Siyavash Towfighi, 2011 ii  Abstract Screen cylinders are widely used in the field of pulp and paper for the removal of oversized contaminants. These screens may consist of several cylindrical partitions that are attached to each other by welding them to cylindrical rings. Since they are subject to various types of loading and placed in a corrosive environment, it is desirable to understand the changes that occur in them as a result of heating in a welding process. The focus of this study is to identify the cause and magnitude of deformations due to welding in the connecting rings of stainless steel AISI 304L. This is accomplished by investigating two potential mechanisms:  1) Microstructural changes such as phase transformations, and 2) Plastic deformation. To consider microstructural changes, a ferritescope is utilized for detecting the non-austenitic phases that may occur. Measurements taken from connecting rings prior to and after welding show that little phase transformation is caused in AISI 304L. In a second set of experiments, cylindrical specimens of AISI 304L are heated to elevated temperatures and compressed in the Gleeble 3500 machine. It is confirmed that heating and straining produce insignificant phase transformations in the rings. Material data is also obtained for elevated temperature stress strain curves at low strain rates that were not available. These results are used in a thermo- structural analysis of the rings for plastic deformation. A simplified constrained beam model is used to illustrate the possibility of compressive yielding in the heated region of the material. This analysis highlighted important aspects of the thermo-structural analysis including buckling/bending and the temperature profile. Experiments on square plates of AISI 304L confirmed some of these predictions. A finite element model created using LS-DYNA and C++ code predicted the deformations in square plate samples with reasonable accuracy. A simpler finite element methodology is then used as an extension of this work to examine plastic deformation in the AISI 304L rings after welding. About 440 simulations with varying parameters are performed with the results indicating that significant plastic deformation occurs during the welding procedure and suggestions are given for controlling this deformation.  iii  Table of Contents Abstract ......................................................................................................................................................... ii Table of Contents ......................................................................................................................................... iii List of Tables .................................................................................................................................................. v List of Figures ................................................................................................................................................ vi Abbreviations and Acronyms ....................................................................................................................... ix Acknowledgements ....................................................................................................................................... x Dedication..................................................................................................................................................... xi 1 Introduction .......................................................................................................................................... 1 2 Phase Transformation .......................................................................................................................... 7 2.1 Precipitate Formation ................................................................................................................... 7 2.2 Phase Transformation Due to Heating and Cooling ..................................................................... 7 2.3 Phase Transformation Due to Compression ................................................................................. 8 2.4 Experimental Verification for Phase Transformations ................................................................. 9 2.5 Conclusion .................................................................................................................................. 13 3 Plastic Deformation ............................................................................................................................ 14 3.1 Elevated Temperature Material Characteristics of Stainless Steel 304L .................................... 14 3.1.1 Introduction ........................................................................................................................ 14 3.1.2 Experimental Setup ............................................................................................................ 15 3.1.3 Results ................................................................................................................................ 18 3.1.4 Conclusion .......................................................................................................................... 25 3.2 Contractions in AISI 304L by Heating.......................................................................................... 26 3.2.1 Introduction ........................................................................................................................ 26 3.2.2 Mechanism of Deformation ............................................................................................... 27 3.2.3 Finite Element Model ......................................................................................................... 28 3.2.4 Experimental Setup ............................................................................................................ 30 iv  3.2.5 Results ................................................................................................................................ 31 3.2.6 Conclusion .......................................................................................................................... 35 4 Connecting Ring Case Study ............................................................................................................... 37 4.1 Stress State Finite Element Model ............................................................................................. 40 4.2 Ring Finite Element Model ......................................................................................................... 44 4.2.1 Buckling............................................................................................................................... 44 4.2.2 Geometrical Constraints ..................................................................................................... 47 4.2.3 Moving Heat Source ........................................................................................................... 50 4.2.4 Cooling Error Correction ..................................................................................................... 52 4.2.5 Numerical Convergence and Initial Results ........................................................................ 53 4.2.6 Parametric Study ................................................................................................................ 55 4.2.7 Conclusion .......................................................................................................................... 60 5 Conclusions ......................................................................................................................................... 61 5.1 Limitations .................................................................................................................................. 61 5.2 Contributions .............................................................................................................................. 63 5.3 Future Work ................................................................................................................................ 63 Bibliography ................................................................................................................................................ 64     v  List of Tables Table 2-1 - Ferritescope measurements ....................................................................................................... 9 Table 2-2 - Nomenclature for volume ratio equations ............................................................................... 11 Table 2-3 - Relations for the derivation of the initial and final volume ratio ............................................. 12 Table 3-1 - Chemical composition of sample specimens versus AISI standard for 304L Stainless Steel .... 17 Table 3-2 - Actual strain rates and maximum strains achieved for experiments ....................................... 18 Table 3-3 – AISI 304L properties and bilinear model parameters for up to a strain of 0.1 ........................ 22 Table 3-4 - The experimental maximum temperature  and calculated errors for the Gaussian model .... 34 Table 4-1 - Properties used in the model to test for convergence............................................................. 54 Table 4-2 – Results of various meshes with a constant time-step of 0.5s without springs ....................... 54 Table 4-3 - Results of various meshes with a constant time-step of 0.5s with springs .............................. 54 Table 4-4 - Results of various time-steps with a mesh of global edge length 3.00 without springs .......... 55     vi  List of Figures Figure 1-1 – From trees to logs to pulp and paper ....................................................................................... 1 Figure 1-2 - Micrographs of contaminants: (a) a shive seen within the cross-section of a piece of cardboard   (b) discrete fibre bundle. ........................................................................................................... 2 Figure 1-3 - Fine pulp fibres passing through the small slots from 0.10 to 0.35 mm................................... 3 Figure 1-4 – A schematic of a screen cylinder with slots (in grey) and a  rotor with multiple foils inside the cylinder (in blue) .................................................................................................................................... 3 Figure 1-5 – An example of a pulp screen cylinder ...................................................................................... 4 Figure 2-1 - Volume of a unit cell ............................................................................................................... 10 Figure 2-2 - Schematic diagram of the phase transformation in the ring .................................................. 11 Figure 3-1 - Test chamber of Gleeble 3500 without test sample ............................................................... 16 Figure 3-2 – Test specimen with protective layers of tantalum and graphite for each sample ................. 16 Figure 3-3 – Comparison of barreling sample and cylindrical compression .............................................. 17 Figure 3-4 - Experimental results for the true stress-strain curves of AISI304L up to a strain of 0.4 ........ 19 Figure 3-5 - Samples after heating and compression (see Table 1) ........................................................... 19 Figure 3-6 - Comparison of experimental results for the coefficient of expansion and elastic modulus [15; 16] ............................................................................................................................................................... 20 Figure 3-7 - Comparison of experimental results for 0.2% offset yield strength [9] .................................. 20 Figure 3-8 - Bilinear model (shown with the red and green line) versus actual curve at 293 K (20 °C) ..... 22 Figure 3-9 - Oscillations in the AISI 304L stress-strain curves at elevated temperatures .......................... 23 Figure 3-10 - Temperature profile of compression samples up to the point of cooling ............................ 24 Figure 3-11 - Ferritescope readings of samples after cooling .................................................................... 24 Figure 3-12 - Sample considered for monitoring thermal deformation..................................................... 27 Figure 3-13 –Modes of plastic deformation during thermal expansion .................................................... 27 Figure 3-14 – Mesh used for the finite element simulation ....................................................................... 28 Figure 3-15 - Material properties of AISI 304L stainless steel .................................................................... 29 Figure 3-16 - Convection coefficients for surfaces of a plate in air ............................................................ 30 Figure 3-17 - Experimental apparatus for testing samples ........................................................................ 31 Figure 3-18 – Upper plate with points and distances measured using the coordinate measuring machine32 Figure 3-19 - Temperature profiles of the samples for Thermocouple 1 ................................................... 32 vii  Figure 3-20 – Comparison of thermocouple temperature readings with the Gaussian flux model for Sample 4 ..................................................................................................................................................... 33 Figure 3-21 - Deformations of the sample plates as illustrated in Figure 3-18 .......................................... 34 Figure 3-22 – Deformation of calibration sample after repeated heating cycles ...................................... 35 Figure 4-1 - Initial state of a small part of the ring prior to heating ........................................................... 37 Figure 4-2 - Free expansion of the outer part due to heating .................................................................... 38 Figure 4-3 - Stress state induced by compatibility during heating ............................................................. 38 Figure 4-4 - Free contraction of the outer element after plastic deformation and cooling ...................... 39 Figure 4-5 - Stress state induced by compatibility after heating ............................................................... 39 Figure 4-6 - Mesh generated in LS-DYNA for a ring with no slots .............................................................. 41 Figure 4-7 - Temperature profile for preliminary analysis ......................................................................... 41 Figure 4-8 - Y-Stress distribution after 20 minutes in the heated region (red nodes heated) ................... 42 Figure 4-9 – Plot of the stresses of the inner (S5) and outer (S271) elements .......................................... 43 Figure 4-10 - Vertical upward buckling of the ring ..................................................................................... 44 Figure 4-11 - Radial outward buckling of the ring ...................................................................................... 44 Figure 4-12 – Buckling behaviour for a 200mm long, 4mm thick rectangular sample .............................. 46 Figure 4-13 – Stress states induced by radial buckling versus heating the outer region ........................... 47 Figure 4-14 – Cylindrical strip used to identify the properties of spring elements .................................... 48 Figure 4-15 - Load applied to the top of the strip with respect to time..................................................... 48 Figure 4-16 – Scaled loading force and displacement relationship used for one spring element ............. 49 Figure 4-17 - Geometry of the 3D ring FEM model with 30 spring elements ............................................ 49 Figure 4-18 - Segmented approach for modeling a moving heat source ................................................... 50 Figure 4-19 - Linearized temperature profile for one segment using segmented temperature boundary conditions ................................................................................................................................................... 51 Figure 4-20 - A dip in the temperature profile prior to the approach of the heat source ......................... 52 Figure 4-21 - Node A may undergo further cooling due to the high temperature gradient caused by nodes B and C ............................................................................................................................................. 53 Figure 4-22 – The two parameters, D1 and D2, used to control the heated region on the ring ............... 56 Figure 4-23 - Maximum heating temperature results ................................................................................ 57 Figure 4-24 - Heating time results .............................................................................................................. 57 Figure 4-25 – D1 parameter results ............................................................................................................ 58 Figure 4-26 – D2 parameter results ............................................................................................................ 58 viii  Figure 4-27 – Constraint results ................................................................................................................. 59 Figure 4-28 - Contraction of the ring with D1 = 0 and D2 = 11mm ............................................................ 59   ix  Abbreviations and Acronyms BCC Body Centred Cubic BCT Body Centred Tetragonal CMM Coordinate Measuring Machine CPU Central Processing Unit FCC Face Centred Cubic FEM Finite Element Method UBC University of British Columbia      x  Acknowledgements Thanks to everyone that provided support and guidance with this research project. Without your help, progress would have been difficult. The support and supervision provided by Dr. Douglas P. Romilly and Dr. James Olson over the course of this thesis are appreciated. Amongst others that contributed their time are David Marechal, Fateh Fazeli, Markus Fengler, Dr. Daan Maijer, Rami Mansour, Ross McLeod and Nima Zirak. I also want to thank my family for being so supportive throughout my education. Thank you all for making this happen and best of luck in the future.  xi  Dedication   Dedicated to Khosrow Pouria and Batul Razavi   1  1 Introduction Pulp and paper and its associated suppliers are one of Canada’s largest industries accounting for 13% of exports from the country. The pulp and paper industry is focused on the process of converting either wood or recycled paper into pulp, and then forming the pulp fibres to produce paper. Figure 1-1(d) shows a magnified view of a sheet of paper where pulp fibres are in the range of 10 to 30 microns in diameter. These pulp fibres are contained inside the cells of a lignin matrix shown in the micrograph in Figure 1-1(c). The lignin matrix is actually a magnified cross sectional view of wood.  Figure 1-1 – From trees to logs to pulp and paper   2  There are numerous chemical and mechanical methods for the production of the fibres from the lignin matrix. This research is related to equipment used for the removal of oversized contaminants from  pulp after the “pulping” process, which may refer to either the production of fibres from the lignin matrix (i.e. wood) or from recycled paper products. Some examples of these oversized contaminants are fibre bundles, known as shives and plastic specks. Micrographs of some contaminants are shown Figure 1-2.  Figure 1-2 - Micrographs of contaminants: (a) a shive seen within the cross-section of a piece of cardboard (b) discrete fibre bundle. Since the quality of the paper decreases with the presence of contaminants, it is desirable to separate these oversized contaminants from the pulp fibres. This is typically done by passing the fibres through screen apertures which retain the oversized contaminants. This process is done while the fibres and contaminants are in a water-based suspension, with mass concentrations in the range of 1 to 5%.  3   Figure 1-3 - Fine pulp fibres passing through the small slots from 0.10 to 0.35 mm As the pulp (fibre) suspension flows through the slots, there is a tendency for the slots to become clogged with the fibres and for the flow to decrease or stop. To counter this effect, screening slots are commonly arranged as a screen cylinder with a rotor inside. The motion of the rotor creates turbulence and a pressure pulsation with a back flush which prevents material from accumulating in the slots [1].  Figure 1-4 – A schematic of a screen cylinder with slots (in grey) and a rotor with multiple foils inside the cylinder (in blue)  4  While there are various sizes and models of screen cylinders, a typical screen cylinder is shown in Figure 1-5. A common procedure for manufacturing these screens involves the assembly of different cylindrical partitions through a connecting ring to which both partitions are welded.  Figure 1-5 – An example of a pulp screen cylinder Since the rotor inside creates a cyclic pressure pulse on the screen, it is desirable to understand how the material characteristics of the connecting rings change due to this welding procedure. This is accomplished by exploring two potential avenues: 1) The possibility of a microstructural change (such as a phase transformation and dynamic recrystallization) caused by heating or straining, and 2) Plastic deformation caused by thermal loads in the ring and the corresponding residual stresses induced through welding. The objectives of the current research are thus outlined below. 1. Determine the elevated temperature material characteristics of stainless steel AISI 304L used to manufacture the rings; 2. Identify possible mechanisms by which the  welding procedure can alter the rings; 3. Use this knowledge to develop a model to predict the state of the rings after welding; and 4. Validate the model utilizing experimental methods within the laboratory. Weld Line  5  This thesis presents the results of an investigation into the welding process of screen cylinders in order to provide insight into its effects on the rings. The research performed to address each objective is organized into different chapters. Since the research topics are diverse, ranging from microstructural material characteristics to finite element modeling, the literature review for each topic has been incorporated into the corresponding chapter. Starting in Chapter 2, theoretical and experimental methods are employed to consider a phase transformation as the main cause of ring deformation. Equilibrium phase diagrams along with previous empirical data from the literature are presented to predict if a phase transformation occurs in heating. Ferritescope readings with proper interpretation are used to confirm the phases present in the ring before and after welding. The prospect of precipitation is also considered. The lack of a significant phase transformation from experiments implies there is little microstructural change in the material. This naturally leads to the possible deformation mechanism of plastic deformation considered in Chapter 3. A hypothesis is presented for how plastic deformation can occur with finite element models as quantitative support. It is proposed that the deformation of the ring is sensitive to its heating rate and temperature. To create an FEM model for the rings, elevated temperature stress-strain curves for stainless steel 304L were obtained experimentally using the Gleeble machine from Dynamic Systems Inc. After incorporating the material properties into the finite element model, small square plate samples were prepared and experimentally tested for model validation. The sample plates were heated with an oxy-acetylene torch reaching temperatures close to 1200 °C. Three thermocouples were used for monitoring the temperature of the samples at various locations. A CMM machine profiled the samples prior to and after heating to monitor deformations. A Gaussian flux distribution was utilized to model the torch with the power and size characteristics being determined from the experiments. This was done through C++ code which obtained the simulation temperature profiles and compared them with experimental thermocouple readings. About 280 simulations were analyzed to find the best match for each experiment. The results indicated compressive yielding which causes deformation and residual stresses. In Chapter 4, the finite element methodology is used to create a model for the rings. Unfortunately given the size of the rings, it was not possible to create a model with a mesh fine enough to capture the Gaussian flux distribution. A simplified segmented temperature boundary condition is assumed  6  to perform these simulations. Code written in C++ is utilized to automate the process allowing over 160 simulations to be analyzed. Several parameters such as the maximum temperature, heating rate, and the position of the weld line are varied. Chapter 5 is a summary of the work presented and its contributions. Limitations of the research are outlined and suggestions are given for future work.    7  2 Phase Transformation Stainless steel 304L is commonly used to build the connecting rings in pulp screen cylinders. This austenitic steel may have its properties altered depending on the manufacturing processes employed such as heating due to welding. Since deformation is observed in the rings after welding, it provides evidence for a change in material characteristics or the stress state of the ring. In general, materials expand during heating but are expected to return to their original size and shape after cooling. In order to provide insights into the cause of the deformation of the ring, the possibilities of precipitation and/or a phase transformation are explored. The following sections discuss these possibilities in more detail.  2.1 Precipitate Formation Heating steel is known to cause embrittlement or sensitization through precipitation. In stainless steel 304L, this process occurs by the formation of chromium carbides . As a result, a small region with a high density of chromium is created while the surrounding regions are depleted. Since chromium is essential for corrosion resistance, the depleted regions may corrode prematurely and serve as crack initiation sites. Furthermore, carbon is key to the formation of carbides which can aid in crack initiation. Using a stainless steel with lower carbon content is likely to help in preventing corrosion. This is indeed achieved in the screen being studied by using AISI 304L. The choice of AISI 304L for the screen cylinders is suitable as this stainless steel has a carbon of only 0.03% as opposed to 0.08% for AISI 304. For the connecting rings, higher corrosion resistance properties are expected as precipitate formation is not a significant cause for crack initiation.  2.2 Phase Transformation Due to Heating and Cooling Stainless steel 304L supplied from most manufacturers consists of mostly -austenite and very small traces of other phases such as 	-ferrite (about 1%) at room temperature [3] [4]. To assess the composition of different phases present after the welding process is finished, generally the temperature profile of the material is required. Since this profile was not available, it is assumed that the rings reach their melting temperature at 1400 °C. This temperature range allows for an assessment of the phase of the material through an equilibrium phase diagram [3]. By using a Fe-Cr- Ni ternary system at 70 wt% Fe with a sample composition of AISI 304L, it can readily be observed  8  that this steel maintains its austenitic phase until about 1300°C [5]. Higher temperatures may produce traces of 	-ferrite in the ring. This should be avoided if possible as the grain boundaries of the 	-ferrite phase serve as crack initiation sites and decrease the fatigue life of the material [4]. At much lower temperatures, the formation of martensite can become an issue. Martensite has a lower density than austenite and is generally more brittle. For a steel with a composition of 18%Cr - 8%Ni - 0.04%C, the transformation of austenite to martensite begins at approximately -100°C [5]. Since the composition of 304L is almost identical to this case (with the exception of 0.03% C composition), it is reasonable to assume martensite does not form at any stage as a result of heating. Finally it is remarked that the analysis presented above assumes equilibrium conditions. To perform an analysis that accounts for heating and cooling rates, an isothermal transformation or a time- temperature transformation (TTT) diagram is required [6]. The literature does not provide any results for such a diagram. This is likely due to the relative stability of AISI 304L in a wide temperature range. Based on the information available, it is quite likely that no significant phase changes in the ring occur due to temperature changes. Further experiments are carried out in Section 2.4 to prove this fact. 2.3 Phase Transformation Due to Compression Martensite formation in AISI 304L is also possible as a result of working the material. Rolling of the ring plate can produce such deformations. However, a significant amount of strain is required for martensite transformation to occur. At 22°C, a true strain of  = 0.2 results in the formation of approximately 3% martensite. This amount tends to be reduced as the temperature increases [5]. The welding process however does not produce very large strains. For example, a rough calculation with the linearized coefficient of thermal expansion of AISI 304L for a temperature change of 1000°C gives,  Δ = Δ = 19.8 × 10431000 = 2.67 2.1  ≈ Δ = 2.67 43 = 0.02  9  where  is the expansion coefficient,  is the perimeter of the ring, Δ is the perimeter change due to thermal loads, and Δ is the temperature change [7]. Hence, a phase transformation in the ring due to straining is not expected. 2.4 Experimental Verification for Phase Transformations As discussed previously, a phase transformation due to heating or straining is unlikely. To experimentally verify this claim, a ferritescope was used [8]. This device measures the magnetic permeability of a material. The particular model used for this experiment is the Ferritscope MP30E by Fischer Technologies Inc. The ferritescope works on the principle that martensite and ferrite are magnetic whereas austenite is non-ferromagnetic [9]. Since both the unassembled ring and an assembled screen cylinder were available, readings for existing phases in the material prior to and after heating were obtained. The measurements were taken at various locations with the results being in the range shown in Table 2-1. Table 2-1 - Ferritescope measurements Condition Reading (%) Prior to Heating 0.37 – 0.56 After Heating 0.21 – 0.29  It was assumed that the existence of any possible coating on the rings did not significantly alter the readings. This assumption was necessary for the assembled screen cylinder rings as polishing them would damage the equipment. These ferritescope readings do not differentiate between the magnetic phases present. The case-by-case analysis shown below considers two scenarios to provide insights for a possible phase transformation. Martensite to austenite: There may be some strain induced martensite in the rings due to unknown manufacturing methods. The percent volume of martensite present can be approximated by using a correlation factor to convert the readings from the ferritescope. For AISI 304, this factor is 1.7 [9]. Given that AISI 304 and 304L are very similar in composition, it is assumed this value can be applied to both steels. The rings are known to contract after welding although the cause of this deformation is unknown for AISI 304L. Should this contraction be caused by a phase transformation, knowing the percent  10  volume of martensite prior to and after heating, the diametric change of the ring can be approximated using the density ratio of martensite and austenite. The lattice parameters for the unit cells of " martensite and  austenite are given as a function of carbon content in reference [5]. For AISI 304L with a carbon composition of 0.03, these lattice parameters are, #$ = 3.%%& #'( = 2.86& The volume of the unit cell can be calculated by the formula ) = # as shown in Figure 2-1.  Figure 2-1 - Volume of a unit cell For  austenite and " martensite, the unit cell volume can be given respectively as, )$ *+,, = 3.%%& = 44.739& )'-*+,, = 2.86& = 23.394& Since " martensite has a BCT crystal structure (or BCC for low carbon stainless steels [6]) with two atoms per unit cell and  austenite has an FCC crystal structure with four atoms per unit cell [5], it can be deduced that two unit cells of " martensite produce one unit cell of  austenite [7]. It follows that,  ./- .0 = 12 34/-256612 40 2566 = 70 25667/-2566 = 88.9: .:8 = 0.9%6 2.2 where ;'- is the density of the martensite phase and ;$ is the density of the austenite phase. To proceed with deriving an expression for the volume change, the ring is modeled as a strip with a length equivalent to the perimeter of the ring. The transformation process is illustrated in Figure 2-2 with the corresponding volume of each phase shown.  11   Figure 2-2 - Schematic diagram of the phase transformation in the ring Using the nomenclature shown in Table 2-2 and the relations in Table 2-3, an expression for the final and initial volume ratio can be obtained. Table 2-2 - Nomenclature for volume ratio equations Nomenclature Description '(<  Mass of martensite before heating '(=  Mass of martensite after heating $< Mass of austenite before heating $= Mass of austenite after heating )'(<  Volume of martensite before heating )'(= Volume of martensite after heating )$< Volume of austenite before heating )$= Volume of austenite after heating < Correlated ferritescope reading before heating = Correlated ferritescope reading after heating )< Total volume before heating )= Total volume after heating   12  Table 2-3 - Relations for the derivation of the initial and final volume ratio Relation Description < = )'-< )$< + )'(< < is the percent volume of martensite before heating = = )'-= )$= + )'(=  = is the percent volume of martensite after heating ;'- = '-< )'-< = '- = )'-=  By the definition of density for martensite ;$ = $< )$< = $= )$=  By the definition of density for austenite '(< + $< = '(= + $= Conservation of mass  With some algebraic manipulation, ?)@ABC+ DEBFC. = 1 − ) = )< = 1 − < = H I I J;'-;$ + K 1 − << L ;'-;$ + M 1 − = = NO P P Q  (2.3) Plugging in the conservative ferritescope readings of < = 1.70.%6% = 0.009% and  = = 1.70.21 = 0.003%7 into (1) gives, ?)@ABC+ DEBFC. = 0.000261 = 0.026% Using the same analogy of modeling the ring as a strip, the percent volume change can be approximated by the diameter contraction observed as, ?)B*DSB, = 1 − ) = )< = 1 − ?< − Δ? ?< (2.4)  It is assumed there is no change in the width and thickness of the ring. The initial ring diameter is given as 432 mm with a minimum contraction of 1.8 mm. Numerically, Equation 2.4 reduces to,  ?)B*DSB, = 0.00416% = 0.417% 2.%  13  The actual contraction observed by heating is almost 16 times larger than the contraction predicted by a phase change. Thus a reasonable conclusion is that a transformation from martensite to austenite is not the main cause of the ring contraction. δ-ferrite to austenite: It is possible to have small amounts of δ-ferrite in AISI 304L, although this is not predicted by the equilibrium phase diagram [3]. In accordance to the ferritescope readings, the maximum percent volume change of δ-ferrite is 0.35%. If this portion was removed without being replaced by any austenite, it would still not be sufficient to account for the 0.417% volume change calculated by Equation 2.4. Thus, a transformation from δ-ferrite to austenite also cannot be the main cause of the ring contraction. 2.5 Conclusion After considering the possibilities of precipitation and a phase transformation due to heating or straining with equilibrium phase diagrams and experimental measurements with a ferritescope, no evidence to support this theory as the primary cause for the contraction of the ring is found. In the next chapter, plastic deformation is investigated as potentially being the cause for the deformation of the ring.    14  3 Plastic Deformation To consider how plastic deformation can occur in the ring, it is necessary to understand the mechanical behaviour of stainless steel 304L at elevated temperatures. Stress-strain curves at elevated temperatures are particularly important to create a finite element model. Since this data is not readily available in the literature, Section 3.1 presents experiments performed to determine both the mechanical and chemical behaviour of this material at elevated temperatures. In Section 3.2 these properties are incorporated into a finite element model for a stress analysis. Experiments are performed with an oxy-acetylene torch as the heat source to validate the model and physically show plastic deformation. After developing an understanding of the modes of deformation that need to be considered, similar principles are applied to the analysis of the ring to determine the cause of deformation. 3.1 Elevated Temperature Material Characteristics of Stainless Steel 304L This section presents newly-obtained mechanical response data for AISI 304L stainless steel at high temperatures.   This type of data is necessary for performing thermo-structural analysis of heating processes and was unavailable in the literature. Applications include manufacturing processes such as welding where previously assumptions about the material response had to be made due to a lack of published material data. The results presented here were obtained from compression tests using a Gleeble 3500 test machine for temperatures ranging from 293 K (20 °C) to 1500 K (1227 °C) loaded at low strain rates.  Elevated temperature stress-strain curves are provided with proposed coinciding bilinear response models.   The experimental results were verified by comparing the coefficient of expansion and modulus of elasticity with values available in the literature.  Dynamic recrystallization effects at high temperatures were observed and are expected to have negligible effects on the mechanical behavior for small strains. Ferritescope readings were obtained from the samples both before and after the heated compression tests to assess the possibility of phase transformations. 3.1.1 Introduction Stainless steel type AISI 304L is a widely used material for a variety of applications. The reduced carbon content of AISI 304L makes this stainless steel a preferred candidate material as it is more resistant to forming precipitates when exposed to elevated temperatures [6]. Knowledge of the elevated temperature mechanical properties of AISI 304L is a prerequisite for creating a model to predict dimensional changes due to thermo-mechanical processes, and thus is critical to this work.  15  While elevated temperature mechanical response data for similar steels (such as AISI 304) are available in the literature [2], elevated temperature stress-stain curves for AISI 304L suitable for simulating heating processes were previously unavailable. Furthermore, in previous numerical studies of welding processes, simplifying assumptions had to be made due to the unavailability of this material data. Zhu and Chao presented a welding study where a constant plastic modulus was assumed for a wide range of temperatures [3]. Although some recent studies have focused on stress-strain curves of AISI 304L to model forging processes [4], they provide limited data taken at high strain rates which unfortunately is not suitable for modeling thermal stresses in the current application. This is because thermal stresses caused by thermal expansion typically occur at very low strain rates. Therefore, an objective of the current work was to experimentally obtain the AISI 304L material response behavior at elevated temperatures in the range of 293 K (20 °C) to 1500 K (1227 °C). The tests were performed at low strain rates suitable for modeling thermal expansion due to heating. The Gleeble 3500 test machine was used for determining elevated temperature true stress-strain curves. This device was designed by Dynamic Systems Inc. specifically for performing elevated temperature compressive or tensile tests. An overview of the equipment, experiments and results are provided below. 3.1.2 Experimental Setup The Gleeble 3500 testing system consists of thermal, mechanical and digital control units. The thermal unit uses electricity to rapidly heat the test sample to the desired temperature. Thermocouples attached to the center of the sample provide measurements that are used as feedback to adjust or maintain the temperature of the sample. Two water cooling systems allow for rapid cooling of the sample (as may be desired in some applications) and also keep the machine components within a suitable operating temperature range. Figure 3-1 shows the test chamber that holds the sample in place.  16   Figure 3-1 - Test chamber of Gleeble 3500 without test sample As shown in Figure 3-2, a thin layer of tantalum (inner disks) and graphite (outer disks) are placed between every sample and the tungsten carbide pieces shown in Figure 3-1. The graphite layer serves as a lubricant and preserves the surfaces used to hold the specimens. The tantalum layer prevents any chemical diffusion between the test sample and graphite at high temperatures. This is necessary to preserve the properties of the sample. Once the sample is properly positioned, a servo- hydraulic system applies pressure to generate the stress-strain behavior of the material in compression.  Figure 3-2 – Test specimen with protective layers of tantalum and graphite for each sample The testing system requires cylindrical specimens with a diameter of 10±0.03 mm and a length of 15±0.03 mm. The AISI 304L cylindrical specimens tested in this work were machined from an annealed 11.11 mm (7/16”) diameter rod at low speed to prevent work hardening of the material. The ends were then ground to provide a surface finish of approximately 0.4 μm Ra (16 μ” AA). Table 3-1 shows the chemical composition of the samples based on a certificate provided by the material supplier. The standard chemical composition of AISI 304L from the ASM Metals Handbook [5] is also provided as a reference for comparison. Tungsten Carbide Pieces Space for Sample Thermocouple Tantalum Graphite  17   Table 3-1 - Chemical composition of sample specimens versus AISI standard for 304L Stainless Steel Source C Mn Si Cr Ni P S Experimental  Samples 0.019 1.740 0.297 18.140 8.030 0.031 0.028 ASM Metals Handbook 0.03 2.0 1.00 18.0-20.0 8.0-12.0 0.045 0.03  Figure 3-2 shows a finished test sample with the thermocouples spot-welded in the center. For tests reaching temperatures of 800 K (527 °C) and higher, S-type thermocouples made of platinum and 10% rhodium were used to prevent diffusion with the steel.  For temperatures lower than 800 K (527 °C), K-type thermocouples made of alumel and chromel were used. The thermocouples were spot welded with about 1 mm separation on the sample. A dial gauge was attached to the center of each sample to measure diametric changes. The presence of this device limited the maximum compressive strain that was achievable since very high strain values would cause the machine to crush the dial gauge. The diametric change was used to calculate the linear strain of the samples. This technique required the assumption that the volume of the material remained constant during plastic deformation and that it deforms as a cylinder. The linear strain was then estimated as,   = ln KVWVXL = ln M YX3 YW3 N 3.1 where Z< is the initial sample diameter, Z= is the final sample diameter, < is the initial sample length and = is the final sample length. At high temperatures, the samples can deform into non- cylindrical “barrel” shapes which would give poor data for the stress-strain curve. Figure 3-3 illustrates cylindrical compression versus barreling. To limit the potential effects of barreling, a maximum strain limit of 0.4 was used for all the results. Another limiting factor for the strain was that the load cell was limited to 70 kN.  As a result, some of the tests had strains lower than 0.4.  Figure 3-3 – Comparison of barreling sample and cylindrical compression Barreling Cylindrical Compression  18  The parameters specified for the tests using the Gleeble test system were: temperature, heating rate, strain rate and maximum strain. For the experimental test results presented here, a heating rate of 10K/s (10°C/s) was used with a target strain rate of 0.002/s to approximate the strain rate values for thermal expansion. 3.1.3 Results Table 3-2 shows the temperatures for the tests performed, along with the actual strain rates and maximum strains achieved in each test. The stress-strain curves for AISI 304L are shown in Figure 4 for various experimental test temperatures as examples of material response curves obtained. Figure 5 shows the final geometry of the sample after testing. Table 3-2 - Actual strain rates and maximum strains achieved for experiments Sample No. Temperature (K) Strain Rate (s -1 ) Maximum Strain 1 293 °K (20 °C) 0.00183 0.1131 2 600 °K (327 °C) 0.00196 0.1748 3 700 °K (427 °C) 0.00189 0.1803 4 800 °K (527 °C) 0.00184 0.2728 5 900 °K (627 °C) 0.00161 0.4569 6 1000 °K (727 °C) 0.00172 0.4967 7 1100 °K (827 °C) 0.00170 0.6144 8 1200 °K (927 °C) 0.00183 0.6891 9 1300 °K (1027 °C) 0.00263 0.7505 10 1400 °K (1127 °C) 0.00182 0.6963 11 1500 °K (1227 °C) 0.00197 0.6591   19   Figure 3-4 - Experimental results for the true stress-strain curves of AISI304L up to a strain of 0.4  Figure 3-5 - Samples after heating and compression (see Table 1) An approximate value for the modulus of elasticity was determined from the stress-strain curves. The coefficient of thermal expansion was also calculated using the changing diameter of the sample recorded at various temperatures during heating in the final test. Figure 3-6 and Figure 3-7 present a comparison between these experimental results and values available in the literature for the thermal expansion coefficient, elastic modulus and 0.2% offset yield strength [15; 16].   20   Figure 3-6 - Comparison of experimental results for the coefficient of expansion and elastic modulus [15; 16]  Figure 3-7 - Comparison of experimental results for 0.2% offset yield strength [9] -23 177 377 577 777 977 1,177 0 50 100 150 200 250 0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05 2.50E-05 3.00E-05 250 450 650 850 1050 1250 1450 Temperature (°C) E la st ic  M o d u lu s (G P a ) E x p a n si o n  C o e ff ic ie n t (K -1 ) Temperature (°K) Experimental expansion coefficient Literature expansion coefficient Experimental elastic modulus Literature elastic modulus -23 177 377 577 777 977 1,177 0 50 100 150 200 250 300 350 400 450 500 250 450 650 850 1050 1250 1450 Temperature (°C) Y ie ld  S tr e n g th  ( M P a ) Temperature (°K) Material Supplier Experimental Literature  21  The experimental data for the expansion coefficient compares reasonably well with values found in the literature.  The average error between the values was 13% which can be a result of variations in sample materials.  The modulus of elasticity was also determined from the stress-strain curves.  This can vary depending on which data points are used to calculate the slope. For example at 700 K (427 °C), by using two different data points occurring just 0.12 seconds apart, the values calculated for the modulus of elasticity vary from 195 to 245 GPa.  Since on average, the experiments match well with results available in the literature, it is reasonable to conclude reasonable agreement between the two. However, the results obtained experimentally for the yield strength and those available in the literature differ significantly.  According to the certificate provided by the material supplier, the yield strength of the sample at room temperature should be 455 MPa. This is close to the experimentally obtained result of 448 MPa but is significantly different from the value of 237 MPa obtained from Rho et. al.  [9] or 285 MPa as from Zhu and Chao [3].  Given the consistency between the experimental yield strength and the one provided by the material supplier, the material properties obtained experimentally seem to be more reliable. The differences in material properties observed between the experimental results and those available in the literature may be due to a variation in the composition of the samples or their initial state before the experimental tests. In order to provide a convenient method to input the material properties obtained in this research into the finite element modeling software, a bilinear model is proposed to approximate the stress- strain curves.  This approximation is constructed using the modulus of elasticity, the yield strength and the tangent modulus.  Since the strains in the thermo-structural simulations are expected to be small, a bilinear model accurately representing the stress-strain curve up to a strain of 0.1 was deemed sufficient.  Hence, the tangent modulus was calculated using the yield point and the point with a strain of 0.1. The equation below demonstrates this relation,  [D = \X.]\^_X.]_^  3.2 where ` a , cad are from the yield point,  <.e, c<.e are from the point with a strain = 0.1 and [D is the tangent modulus. Figure 3-8 shows a sample bilinear model created based on the stress-strain curve at 293 °K (20 °C) versus the actual stress-strain curve obtained.  22   Figure 3-8 - Bilinear model (shown with the red and green line) versus actual curve at 293 K (20 °C) The values for the elastic modulus, yield strength and tangent modulus for the bilinear approximation are shown in Table 3-3. The maximum error between the actual and bilinear yield strength is 14% with the average error being 8%. Table 3-3 – AISI 304L properties and bilinear model parameters for up to a strain of 0.1 Temperature Elastic Modulus (GPa) Actual Yield Strength (MPa) Bilinear Yield Strength (MPa) Bilinear Tangent Modulus (GPa) 293 °K (20 °C) 228 447.8 479.8 2.251 600 °K (327 °C) 208 320.1 363.7 1.418 700 °K (427 °C) 195 322.2 351.5 1.331 800 °K (527 °C) 163 294.2 326.9 1.150 900 °K (627 °C) 156 264.5 297.3 0.855 1000 °K (727 °C) 132 231.2 257.6 0.317 1100 °K (827 °C) 107 162.1 165.3 0.127 1200 °K (927 °C) 72 96.9 100.5 0.023 1300 °K (1027 °C) 45 55.3 57.1 0.048 1400 °K (1127 °C) 45 27.3 29.5 0.075 1500 °K (1227 °C) 16 12.5 13.3 0.054  Aside from uncertainty in experimental readings, phase transformations and grain recrystallization must also be considered in a thermo-structural analysis.  During the tests, the stress-strain response curves at temperatures above 1400 K (1127 °C) were found to experience oscillations with various peak points as shown in Figure 3-9. This may be due to nucleation and growth of grains during 0 100 200 300 400 500 600 700 800 0 0.02 0.04 0.06 0.08 0.1 0.12 S tr e ss  a t 2 9 3 K  ( M P a ) Strain Yield Line Tangent Line  23  deformation or dynamic recrystallization [10] and will affect the accuracy of predictions based on any linear-based models.  Figure 3-9 - Oscillations in the AISI 304L stress-strain curves at elevated temperatures Furthermore, the formation of non-austenitic phases as a result of heating and straining is plausible with effects on mechanical behavior. The formation of δ-ferrite due to heating has been shown to reduce fatigue life [9].  Straining and heating the specimens can also cause changes in the martensite content of the material [1].  The samples were not directly monitored during the compression tests for a phase transformation due to the presence of a vacuum in the test chamber which prohibited typical ferritescope use.  Instead, the ferritescope was used to check for any remnants of ferromagnetic phases after the samples were cooled to room temperature.  These measurements were taken every 2 mm along the length of the sample.  Correction factors for edge and curvature effects were applied in accordance to the Fischer Operator’s Manual for Ferritescope MP30E.  Polishing of the samples was not performed as it might have resulted in cold work of the sample surface. Instead, it was assumed that the outside layer formed after heating did not significantly alter the readings and the information was collected without polishing. Figure 3-10 shows the temperature profile for the samples. The maximum strain of each sample can be found in Table 3-2 with the average ferritescope readings for each sample shown in Figure 3-11. 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 S tr e ss  ( M P a ) Strain 1400°K (1127°C) 1500°K (1227°C)  24   Figure 3-10 - Temperature profile of compression samples up to the point of cooling  Figure 3-11 - Ferritescope readings of samples after cooling The ferritescope readings of the room temperature test specimen increased after compression to a strain = 0.113. Based on the work of Lacombe et. al. [1], a strain of 0.1 should produce approximately 2% martensite in a steel with a composition of 18% Cr and 8% Ni. Using a correlation factor of 1.7 for converting ferritescope readings to martensite [12], an increase of 1.17% = e.9 would be expected due to straining of the samples. This is very close to the experimentally observed increase of 1.35% (= 1.75% - 0.40%). Since strain induced martensite does not form easily at temperatures above 353 K (80 °C), a decrease in the ferritescope readings is expected as the test 0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 250 450 650 850 1050 1250 1450 1650 F e rr it e sc o p e  R e a d in g Maximum Sample Temperature (°K) Tested Samples Untested Sample  25  temperature increases as  shown in Figure 3-11 until  the temperature reaches approximately 900 K (627 °C) [1]. This approximate temperature for an inflection point on the curve is in close agreement with Herrera et. al. [11] who found that 823 K (550 °C) is the reversion temperature when strain- induced martensite disappears from the sample. The consequent rise in ferritescope readings at higher temperatures is likely due to the remnants of δ-ferrite or an oxide layer which may form at higher temperatures. Based on the oscillations of the stress-strain curves shown in Figure 3-9, a phase transformation or a recrystallization process may slightly affect the results of the compression experiments. However, these changes are negligible for small strains and for most thermo-structural simulations it is unlikely that they would significantly alter the outcome. 3.1.4 Conclusion The elevated temperature experiments performed with the Gleeble 3500 testing system  have provided material response curve data and estimates for the elastic modulus, yield strength and tangent modulus of AISI 304L stainless steel . These elevated temperature mechanical properties are a critical input to future thermo-structural finite element simulations. The low strain rates used make the material response curves suitable for applications where deformation is caused purely by heating and constraining a structure.  Specific material properties (such as yield strength) were found to be significantly different from the data available in the literature.  Reasonable agreement was observed for the elastic modulus and coefficient of thermal expansion.  Ferritescope measurements were also taken of the samples before and after heating to account for possible phase transformations.  The readings were in good agreement with other sources in the literature showing that very limited phase transformation occurred during the compression testing and may be neglected for reasonable mechanical analysis.  Slight effects from dynamic recrystallization of the grains were also observed at elevated temperatures.  For small strains with little plastic deformation, these effects are expected to be insignificant as the grains are not compressed enough to recrystallize and cause changes in the mechanical behavior.  However, for applications requiring larger deformations, dynamic recrystallization at high temperatures may become significant.  After consideration of all the possible material changes discussed above, the bilinear models proposed for AISI 304L can pave the way for more accurate predictions in thermo-structural finite element studies in the future.   26  3.2 Contractions in AISI 304L by Heating This section presents a study of contractions in AISI 304L stainless steel as a result of heating. These experiments are similar to a welding process and are intended to examine plastic deformation caused by thermal stresses. A simplified beam model is used to show two possible modes of permanent deformation in the heated region. These predictions are verified by a three-dimensional nonlinear thermo-structural finite element model to predict the magnitude of contraction in the sample slots. A testing facility is also constructed to physically demonstrate slot contractions and to validate the finite element models. Since a phase transformation and dynamic re-crystallization were found to be insignificant for small strains, as discussed in Section 3.1, they are neglected in the modeling work. Heat treated samples tested using an oxy-acetylene torch show slot contractions for all cases. Higher maximum temperature was found to result in greater slot contraction. Temperature profiles generated from a Gaussian flux model and employed in the finite element model to represent the torch heat input matched well with thermocouples readings from the experiments. The finite element predictions of the slot contractions were found to be in reasonable agreement with experimental results. Repeated heating cycles were observed to increase the contraction to roughly 1 mm in a 12 mm slot. The magnitude of this deformation was altered by changes in the temperature profile generated from the heat source and by reducing the number of heating cycles. 3.2.1 Introduction The main focus of welding studies is to reduce the dimensional changes of the pieces being welded. The aim of this study is to explore some important analytical aspects that can aid with this task and to develop confidence in the finite element methods and inputs employed in this research work. Hence sample plates of AISI 304L are considered to examine thermally induced contractions in a slot as shown in Figure 3-12.  27   Figure 3-12 - Sample considered for monitoring thermal deformation The following sections present an analysis of the mechanisms involved in deformation of this sample through heating. Results from three dimensional nonlinear thermo-structural finite element models are given for different cases and a testing platform is used to experimentally show slot contraction and validate the finite element results. 3.2.2 Mechanism of Deformation Thermal stresses have been known to cause plastic deformation in welding. It is important to understand the structural aspects of this process so that the desired level of deformation can be achieved. For simplicity one may suppose that there is a very high temperature gradient between the heated region and its surrounding regions. The mechanical properties of the heated region are then much lower than its cooler counterparts. This region can be modeled as a beam surrounded by rigid walls to gain insights into its deformation. As the heated region expands, it can undergo either pure compression or buckling and bending as illustrated in Figure 3-13.  Figure 3-13 –Modes of plastic deformation during thermal expansion Based on this simplified model, the following observations can be made: • High temperature gradients are important for inducing differences in mechanical properties • Differentiation in mechanical properties and thermal expansion can cause plastic deformation Pure Compression Bending/Buckling Heated region 12mm slot  28  • Dimensional changes (i.e. contractions) may be induced by compressive yielding during thermal expansion • Since higher compressive stresses are reached in pure compression, avoiding bending and buckling should result in greater contraction Some of these principles have been shown previously in other applications. For example several studies have demonstrated plastic radial shrinkage of welded pipe-flange joints [1; 4; 5]. Temperature fields have also been known to significantly affect mechanical properties as well as residual stresses during welding [1; 6]. These principles are utilized to investigate the contraction of slots using an oxy-acetylene torch through finite element modeling and experimental validation. These square plates are designed to have a height and thickness that is roughly the same as the rings. The heated region is chosen to be small to minimize buckling and provide for larger contractions which are easier to measure. 3.2.3 Finite Element Model Any finite element model for this system needs to be capable of fully capturing the deformational effects caused by bending, buckling and high temperature gradients. This is accomplished by performing a nonlinear transient thermo-structural simulation using LS-DYNA. To account for bending or buckling action, a three-dimensional model was created as two-dimensional models may be inadequate [19; 22]. The mesh consists of 55,612 tetragonal elements and 12,404 nodes with finer resolution in the heated region as shown in Figure 3-14.  Figure 3-14 – Mesh used for the finite element simulation Thermocouple 1 Constrained nodes Thermocouple 3 Region of flux input Thermocouple 2  29  The elevated temperature properties used for stainless steel AISI 304L are illustrated in Figure 3-15 with the experimentally obtained properties incorporated in [10; 13; 14; 21]. When there is a high temperature gradient on an element, there can be excessive convection cooling on the cold nodes. This is due to the fact that surface cooling is calculated using the average temperature of the nodes on the surface. As a result, room temperature nodes may experience cooling caused by hot nodes in the vicinity. To partially correct this problem, the structural properties at room temperature were extended to -30 °C with the exception of thermal expansion that was deliberately set to zero so the deformation of the material is not affected by erroneous cooling.  Figure 3-15 - Material properties of AISI 304L stainless steel Convection and radiation cooling boundary conditions were incorporated into the model. The convection coefficients for a plate surface in air with different orientations are presented in Figure 3-16  assuming an ambient temperature of 27 °C (300 °K) [14]. Surfaces that were not completely vertical or horizontal were assumed to be vertical for convection. The small fixtures that hold the plate in place were assumed to be far away from the heat source and neglected for conduction cooling. Radiation cooling was applied using an emissivity of 0.17 [8]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -30 170 370 570 770 970 1170 M a te ri a l P ro p e rt y  V a lu e Temperature (°C) Specific Heat x 1.4E-3 (J/kg °K) Conductivity x 2.5E-2 (W/m °K) Thermal Expansion x 4.0E+4 (1/°K) Poisson's Ratio Elastic Modulus x 5.0E-3 (GPa) Yield Strength x 3.3E-3 (MPa) Tangent Modulus x 4.0E-1 (GPa)  30   Figure 3-16 - Convection coefficients for surfaces of a plate in air A Gaussian flux distribution proposed by Pavelic et. al. was implemented for the thermal input of the torch [14]. Goldak et. al. suggest it to be an accurate model for a preheat torch [15]. The equation for this flux distribution can be written as:  fg, h = ij*3 kllX 3/*3kaaX3/*3 3.3 where n(Watts) is the energy input,  (meters) is the characteristic radius of flux distribution, g< (meters) and h< (meters) are the coordinates for the center of the torch. C++ code was written to read the LS-DYNA keyword file and apply the proper flux value at the surface of each element exposed to the torch. This method readily lends itself to parametric studies as new models can be setup by running the subroutine with different parameters. The sample plate was geometrically fixed using a small nut and bolt. The nodes close to the screw were fully constrained in the finite element model as illustrated in Figure 4. Creep was neglected as previous welding studies have indicated it is not a cause for significant deformations [17]. To ensure the finite element method reached convergence in spatial dimensions and time, the scenario with the highest heat input was computed with finer mesh and smaller time steps. The time step size was 0.1 seconds for both thermal and structural calculations. 3.2.4 Experimental Setup A small scale testing facility was constructed for experimental validation using an oxy-acetylene torch as shown in Figure 3-17. 0 5 10 15 20 25 -30 470 970C o n v e ct io n  C o e ff ic ie n t (W /m 2 °C ) Temperature (°C) Top Surface Bottom Surface Vertical Surface  31   Figure 3-17 - Experimental apparatus for testing samples A linear bearing ensures the flame maintains the same direction relative to the plates in all tests. The distance of the flame from the plate can be adjusted to vary the heating parameters. Previous experiments have demonstrated that by changing the flow of oxygen and acetylene alone, the heat flux of the torch can range from 2380 kW/m 2  to 3920 kW/m 2  giving flame temperatures of 1800 °C to 2700 °C [16]. The heat input into the samples is varied using these two methods. The torch was adjusted to provide a neutral flame for the experiments. The samples were manufactured from 4.76 mm (3/16”) plates which were reduced in thickness to 4 mm by milling on a numerically controlled (NC) machine. The profiles of the samples were cut using a water-jet cutter. The holes for the thermocouples were cut on a milling machine. To ensure residual stresses caused by manufacturing did not affect the samples, they were heat treated in a vacuum oven by annealing to 1065 °C (1950 °F) and cooled in accordance to the ASM guidelines for stress relieving AISI 304L stainless steel [12]. A coordinate measuring machine (CMM) was used to measure the profile of the samples at various points with an accuracy of 2.54 μm (1/10,000”) before and after the experiments.  The temperature of the samples was monitored using three K-type thermocouples placed 10 mm apart from each other as illustrated in Figure 3-14 and Figure 3-17. 3.2.5 Results The deformations in the plate were monitored before and after heating to measure contractions of the slot, changes in the thickness and height of the heated region. Figure 3-18 illustrates the points and distances measured with the coordinate measuring machine. Sample Thermocouples Linear bearing Torch  32   Figure 3-18 – Upper plate with points and distances measured using the coordinate measuring machine The readings from Thermocouple 1 for the first 90 seconds of each test are shown in Figure 3-19. Overall, six sample plates were tested.  Figure 3-19 - Temperature profiles of the samples for Thermocouple 1 To determine the power input and characteristic radius of the Gaussian flux torch model corresponding to the thermocouple readings, a series of simulations were performed with varied parameters to find the closest match. Code written in C++ evaluated the square error function between the experimental and LS-DYNA temperature profiles in a similar manner to regression problems. Since the maximum temperature reached is critical, Thermocouple 1 and Thermocouple 2 were given greater emphasis in calculating the error as shown in Equation 3.4, -100 100 300 500 700 900 1100 1300 0 20 40 60 80 T h e rm o co u p le  1  ( °C ) Time (s) Sample 1 (S6) Sample 2 (S4) Sample 3 (S2) Sample 4 (S7) Sample 5 (S8) Sample 6 (S9) Thermocouple 3 Thermocouple 2 Thermocouple 1  33   [ = o `p],qrst p],uvw t d3xe.y`p3,qrst p3,uvwt d 3x`pz,qrst pz,uvwt d 3 { {|}e  3.4 where [ is the square error function,  is the number of data points, ~,€|  is experimental temperature reading from thermocouple ‚ and ~,ƒ„…|  is the LS-DYNA temperature reading from a node corresponding to thermocouple ‚. The first 150 seconds of the experiment were considered for the error calculation as they were deemed to affect the deformations of the slot more than the cooling phase. A finite element model with a coarse mesh of 13,574 elements and 3567 nodes was used for this task. From over 280 simulations, the optimum values for power input, characteristic radius, torch position and heating time were selected and re-evaluated in a model with a fine mesh consisting of 55,612 elements and 12,404 nodes. A comparison of the resulting temperature profile for Sample 4 is shown in Figure 3-20.  Figure 3-20 – Comparison of thermocouple temperature readings with the Gaussian flux model for Sample 4 The values for the square error function, [, calculated for each experiment and the matching finite element simulation are shown in Table 3-4.    0 200 400 600 800 1000 1200 -100 100 300 500 700 T e m p e ra tu re  ( °C ) Time (s) Th1 Simulation Th1 Experiment Th2 Simulation Th2 Experiment Th3 Simulation Th3 Experiment  34  Table 3-4 - The experimental maximum temperature and calculated errors for the Gaussian model Sample Maximum Temperature (°C) Regression Error (Eq. 3.4) 1 760 164.41 2 863 273.18 3 889 207.66 4 1027 458.05 5 1079 584.42 6 1133 997.97  The magnitude of expansions and contractions in the plate denoted by positive and negative values respectively are shown in Figure 3-21 for the experiments and the LS-DYNA finite element model.  Figure 3-21 - Deformations of the sample plates as illustrated in Figure 3-18 Increasing contraction in the slot is observed with increasing maximum temperature although there are discontinuities in this trend. These differences may be caused by variations in the thermal input (i.e.  the torch) or they may be due to variations in the composition of the material of the samples. An expansion in thickness and height is seen and expected since compression in one direction should result in an expansion in other directions. Sample 4 and 5 were found to have the same Gaussian flux parameters as their thermocouple temperature profiles were similar. The predicted deformations are thus identical. An extra sample was tested with 14 heating cycles and measured -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 700 800 900 1000 1100 1200 D e fo rm a ti o n  ( m m ) Maximum Temperature (°C) Experimental Slot LS-DYNA Slot Experimental Thickness LS-DYNA Thickness Experimental Height LS-DYNA Height  35  with the CMM machine.  As this sample underwent repeated heating cycles further deformations occurred. Figure 3-22 shows the deformation of its slot which underwent a 0.94 mm contraction.  Figure 3-22 – Deformation of calibration sample after repeated heating cycles This high level of dimensional change is likely due to the fact that the repeated heating cycles cause a hysteresis loop to form on the stress-strain curve of the material and each cycle increases plastic deformation. Hence, depending on the application and the geometry of the part, repeated weld cycles will increase plastic deformation. 3.2.6 Conclusion The primary objective of this section was to examine the mechanisms of deformation and highlight how they can change dimensions in 304L stainless steel material due to localized heating. Developing confidence and validation of the finite element modeling approach for quantifying dimensional changes was a secondary objective of this combined experimental and analytical work. The dimensional changes were first predicted by applying a simplified stress analysis using a constrained beam model for the heated region which identified important deformation modes such as pure compression versus buckling or bending. Results from this model highlighted important characteristics of thermal input that can increase plastic deformation, such as higher temperatures and temperature gradients. A small testing facility physically validated compression in the heated region and showed that higher temperatures produce greater dimensional changes. The thermal input from the torch was incorporated into the computational model as a Gaussian flux distribution. The temperature profiles from the experiments and LS-DYNA simulations were in reasonable Compressed drilled hole Bulging due to compression 11mm slot  36  agreement as shown by the regression errors in Table 3-4. The magnitude of the deformations measured experimentally for the samples were also in reasonable agreement with the simulations; the average difference between two was 0.031, 0.027 and 0.040 mm for the slot, thickness and height dimensions respectively. Some possible sources for the disagreement may be the lack of complete convergence in the model since further refinement of the time-stepping parameters and the mesh result in unfeasible computations. Variation between the material properties of the samples of AISI 304L can also induce error in the calculations, not to mention the errors between the temperature profiles with the Gaussian flux distribution. Despite all these possibilities, the results obtained are realistic and the trends generally agree with expectations. The experimental results presented demonstrate that thermal deformation can cause significant compressive yielding in 304L stainless steel components and provide insights into how these deformations can be controlled.   37  4 Connecting Ring Case Study In the previous chapter, plastic deformation was identified as the main alteration that can happen in AISI 304L as a result of heating and a basic stress analysis with a beam model demonstrated some important principles for predicting dimensional changes. The same methodology can be applied to other 304L components such as the welded ring. Consider a small piece of a ring divided into two elements: one inside the ring (inner element) and one outside (outer element). Suppose that each element has a constant temperature distribution and the outer one is heated up during welding.  Figure 4-1 - Initial state of a small part of the ring prior to heating By increasing the temperature of the outer element, it will have a tendency to expand as shown in Figure 4-2. Heated Outer Element Inner Element  38   Figure 4-2 - Free expansion of the outer part due to heating However, geometric compatibility must hold between different parts. The ring will resist this expansion and cause compressive stresses the outer element. The inner element will also resist the expansion as it is attached to the outer element resulting in more compressive stresses on it. The combination of these compressive stresses can cause the outer element to yield. The stress states of both elements are shown in Figure 4-3.  Figure 4-3 - Stress state induced by compatibility during heating Now suppose the outer element has undergone compressive yielding in the hoop direction. Since the outer element is in compression, it would shrink in height and widen its width. After returning to room temperature, it will tend to be shorter and wider in terms of its original dimensions as shown in Figure 4-4.  39   Figure 4-4 - Free contraction of the outer element after plastic deformation and cooling  As discussed previously, geometric compatibility must hold again. Thus the outer element must to go into tension to expand while the inner element will go into compression to shrink as illustrated by Figure 4-5.  Figure 4-5 - Stress state induced by compatibility after heating Although the inner and outer elements will reach a geometrically compatible position, the overall length of the section will contract while its thickness expands. This analysis assumes no buckling or bending will occur in any direction. The driving cause of the deformation is the geometrical incompatibility caused by the temperature difference between the elements. If the outer element is being heated at a low rate, there is increased heat flow to the inner element. This decreases the temperature difference and the amount of the resulting deformation. The ring should therefore show greater contraction with high rates of heating which is unfortunately the case in welding.  40  Since stainless steel AISI 304 is commonly used in industrial applications for welding, previous work is available regarding their thermo-structural analysis [10; 14; 18; 26]. Malik et. al. performed finite simulations on thin walled pipes during welding[14]. In their work the diametric shrinkage of stainless steel 304 cylinders after welding was predicted and experimentally observed. This study serves as a good indication that plastic deformation is in fact the main cause of deformation in the AISI 304L rings. There are differences in the geometry and material properties with previous welding studies which limits their applicability to the connecting rings of screen cylinders however. To provide some quantitative evidence in support of the hypothesis presented above, a preliminary finite element model for the stress states of the inner and outer elements is presented in Section 4.1. The results shown are from the software package LS-DYNA. The underlying principles of the program for performing thermo-structural simulations can be found in the LS-DYNA Theory Manual [15]. The finite element methodology used is identical to the one in Section 3.2. More detailed descriptions are presented in the following section. 4.1 Stress State Finite Element Model A 2D model can be an efficient method to demonstrate the stress state of the ring elements when reaching high temperatures. This is possible by making the following assumptions: • The local temperature of the ring is constant throughout its thickness • The 4mm ring plate is properly modeled using plane stress elements • Deformations occur only in the hoop and radial directions of the ring The mesh generated in LS-DYNA for the ring is shown Figure 4-6 with the heated nodes.  41   Figure 4-6 - Mesh generated in LS-DYNA for a ring with no slots The temperature profile in Figure 4-7 was assumed in this preliminary model.  Figure 4-7 - Temperature profile for preliminary analysis The residual stresses in the ring can only be determined when the ring has completely cooled to room temperature to avoid the stresses caused by the expansion due to thermal effects. The simulation was continued until the heated region returned to a temperature of 32.4°C which is 0 100 200 300 400 500 600 700 800 900 0 50 100 150 200 T e m p e ra tu re  ( °C ) Time (s) Heated Nodes  42  roughly equal to the defined room temperature of 30°C. It is reasonable to assume that the stresses are not significantly altered by further cooling. In explicit integration schemes, to maintain computational stability, the smallest element size is used by LS-DYNA to determine the time-step. For the ring above, a time-step in the order of 10 -7 s was calculated requiring 8 hours of computation time for a heating process spanning 140 seconds. Such calculations are not practical. Hence implicit time-stepping methods in LS-DYNA were used. By implicit time-stepping, the 8 hour computation was reduced to a minute. The two methods were in reasonable agreement. Figure 4-8 illustrates the normal stresses in y-direction while Figure 4-9 plots them numerically and compares them to the hypothesis presented.  Figure 4-8 - Y-Stress distribution after 20 minutes in the heated region (red nodes heated)  Heated Nodes Elements Used to Monitor Stress  43   Figure 4-9 – Plot of the stresses of the inner (S5) and outer (S271) elements The results in Figure 4-9 are in agreement with the hypothesis presented regarding the deformation of the ring by plastic deformation. They show each element changing stress states as predicted. A more detailed simulation of the welding process is thus considered to eliminate some of the simplifying assumptions made in this section. These include: • Two dimensional model does not allow for buckling or bending in all directions • Coarse mesh may produce inaccurate results computationally Each of these issues is addressed in Section 4.2.       44  4.2 Ring Finite Element Model Prior to creating a finite element model of the ring, it is useful to determine which assumptions can be made to reduce the complexity of the problem. Preliminary calculations can give good insights for this task. In the following sections, analysis is presented regarding buckling, moving boundary conditions, high temperature gradient numerical errors and convergence. 4.2.1 Buckling The plastic deformation hypothesis presented in Section 3.2 is a simplified two dimensional treatment that neglects the possibility of buckling of the rings. In reality there are two distinct modes by which buckling can occur: vertically and radially. Both modes are illustrated in Figure 4-10 and Figure 4-11.  Figure 4-10 - Vertical upward buckling of the ring   Figure 4-11 - Radial outward buckling of the ring  45  A simple calculation can be done to illustrate the possibility of buckling purely by thermal loads for a rectangular beam. The idea is to determine the thermal stress caused by thermal expansion on a sample fixed at both ends. Using the thermal expansion coefficients provided in Section 3.2.3, the expansion of the rectangular sample of AISI 304L is,  Δ =  p Δ 4.1 where Δ is the change in length,  is the length at the initial temperature, p is the coefficient of expansion and Δ is the change in temperature. Assuming the sample is fixed by rigid walls on both ends, this thermal expansion can be used to evaluate the compressive strain on the sample as,   = ln KVx†VV L 4.2 The bilinear stress strain curve approximation gives the corresponding stress for the given strain. This stress can be compared with the Euler buckling stress to check whether buckling occurs as a result of heating. The Euler beam buckling formula can approximate whether buckling occurs before plastic deformation,  ‡ = j3ˆ‰ŠV3 4.3 where ‡ is the critical load, [ is the elastic modulus, ‹ is the moment of inertia,  is the unsupported length of the beam, Œ is the effective length factor (Œ = 0.% for fixed boundary conditions) [26]. The moment of inertia ‹ = ee ℎ where  and ℎ are the base and height of the rectangle respectively. The Euler buckling stress, c, is given as,  c = j 3ˆA3 eŠV3 4.4 For a sample with a length of 200mm and a height of 4mm, the thermal stress, yield strength and Euler buckling stress are shown with respect to temperature in Figure 4-12. It can be observed that buckling is indeed an important consideration as thermal stresses exceed the Euler buckling stress from 327°C to 727°C.  46   Figure 4-12 – Buckling behaviour for a 200mm long, 4mm thick rectangular sample The 200 mm rectangular sample analyzed above is significantly different from the actual geometry of the ring. However, this calculation illustrates the importance of a three dimensional model which accounts for buckling. Prior thermo-structural studies have also confirmed that two dimensional models may not be sufficient [14]. As a result, to simulate a weld moving around the ring a three- dimensional model is more appropriate. The radial buckling of the ring is also significant in the analysis. This is due to the fact that when the ring bends outwards its outer region experiences tension and its inner region experiences compression. This stress state is contrary to the one created by heating the outer region as illustrated in Figure 4-13 during peak temperatures. Furthermore buckling allows the ring to maintain its expanded length during heating causing less compressive yielding. 0 100 200 300 400 500 600 0 200 400 600 800 1000 1200 1400 S tr e ss  ( M P a ) Temperature (°C) Bilinear yield strength Euler buckling stress Bilinear thermal stress  47   Figure 4-13 – Stress states induced by radial buckling versus heating the outer region To account for cooler parts of the cylinder preventing the radial buckling of the ring, geometrical constraints need to be considered. 4.2.2 Geometrical Constraints The ring of the screen cylinder being analysed is roughly 45 cm in diameter. An attempt was made to generate a fine 3D mesh for this ring to be solved in LS-DYNA. However, the size of the matrices to be solved led to failures in the computer software. To overcome these computational challenges, simplifications were made to the model. It is desirable to have the finest mesh possible on the ring so that temperature gradients can be captured for the welding process; yet any attached cylindrical parts of the cylinder screen to the ring which reduce buckling should also be at least partially accounted for. To approximate the effects of such restraints without adding a significant number of new solid elements, discrete nonlinear elastic spring elements were used. The force displacement behaviour of these spring elements was determined by performing a thermo-structural finite element simulation of a longitudinal strip of a cylindrical pipe as illustrated in Figure 4-14. Thermal Stress Radial Buckling Compressive Stress Tensile Stress  48   Figure 4-14 – Cylindrical strip used to identify the properties of spring elements To account for the thermal effects of the welding process on the strip, the top end was fixed at a temperature of 700°C with the remaining surfaces under cooling by convection and radiation. The load shown in Figure 4-15 was applied at the top of the strip.  Figure 4-15 - Load applied to the top of the strip with respect to time  -50 0 50 100 150 200 250 300 350 0 50 100 150 200 250 F o rc e  ( N ) Time (s)  49   Figure 4-16 – Scaled loading force and displacement relationship used for one spring element Since adding thirty spring elements manually can be difficult, a program was written in C++ which read the model mesh and added the spring elements accordingly. The LS-DYNA model with 30 springs is show in Figure 4-17.  Figure 4-17 - Geometry of the 3D ring FEM model with 30 spring elements 0 1000 2000 3000 -0.001 0.001 0.003 0.005 0.007 F o rc e  ( N ) Displacement (m)  50  Although many approximations have been made in the proposed model, they are done so out of necessity as alternatives are computationally unfeasible. One resolution to this problem is by using adaptive re-meshing so that as the weld moves along the ring, the heated region is refined while other regions coarsen. However, the mesh being used was generated in MD-Patran as 3D mesh generation is not supported in LS-DYNA. The structural limitations of this approach are discussed in more detail in Section 5.1. 4.2.3 Moving Heat Source There are flux models available in the literature to capture the heating effects produced by welding. The most widely accepted is the double ellipsoidal heat source model originally presented by Goldak et al. [24] A similar model was implemented in Section 3.2.3. Such models require a fine mesh to capture the flux distribution properly and are not ideal for the mesh being used for the rings. As a result temperature boundary conditions are applied to the nodes instead. Since the welding temperatures obtained in the manufacturing process of cylinders are also unknown, this research is limited to providing conceptual insights that can be used when welding parts to the rings. A trend- like analysis is sought by varying parameters in the temperature profile such as the maximum temperature and the rate of heat input while trying to minimize possible deformations in the ring. As in the case of the square plate samples, code was written in C++ to modify the LS-DYNA keyword files and set the proper temperature for the desired nodes. The heat source was modeled by dividing the ring into cylindrical segments as shown in Figure 4-18.  Figure 4-18 - Segmented approach for modeling a moving heat source  51  The segments are heated in order to simulate a moving weld. By observing the temperature profiles generated by the previous simulations, an approximate linearized temperature profile was created for setting the temperature boundary conditions. The parameters used to study the effects of heating on the ring were heating time (t_heat) and maximum temperature (T_max) as shown in Figure 4-19. The heating time is the time it takes for the segment to reach its maximum temperature. Constant slopes for cooling were assumed with the cooling time solved for algebraically.  Figure 4-19 - Linearized temperature profile for one segment using segmented temperature boundary conditions  The heating of the segments were coordinated such that once a segment reached its maximum temperature, the next segment would begin to heat up. The temperature profiles were applied only to the top surface of the ring. The welding speed for the model above can be approximated as,   = ‘K 3’ “ L D”5•–  4.%  52  where  is the speed of the welder, — is the radius of the ring, ˜ is the number of segments used for the boundary conditions, and ™A+BD is the heating time for one segment. Typical values used for ™A+BD ranged from 2s to 8s giving a welding speed range of 1.4 mm/s to 5.6 mm/s for 120 segmentations. The small values for heating time imply high gradients for the temperature in the ring. Finite element simulations with high temperature gradients are prone to specific numerical errors caused in cooling. This is discussed in the next section. 4.2.4 Cooling Error Correction During some of the simulations of the ring, it was observed that for brief periods, the minimum global temperature of the model went below room temperature. The ring should not cool to temperatures lower than its environment. This happened for nodes on the bottom surface of the ring below the heated region. Figure 4-20 shows the temperature profile for one of these nodes.  Figure 4-20 - A dip in the temperature profile prior to the approach of the heat source  Although the source code for LS-DYNA is not available, the following hypothesis is presented to explain this behaviour. Consider the triangular surface shown below with two high temperature  53  nodes (B and C) and one room temperature node (A). This configuration occurs just before the welder reaches node A.  Figure 4-21 - Node A may undergo further cooling due to the high temperature gradient caused by nodes B and C Since convection and radiation are surface boundary conditions, LS-DYNA needs the surface temperature to calculate the heat transfer out of the ring. The surface temperature is likely to be obtained by averaging the temperature of the three nodes shown above. This implies that although node A is at room temperature, it will still undergo significant cooling because the average temperature of the surface is elevated by the two neighbouring nodes. It is possible to reduce this error by refining the mesh further; however, this would increase the computation time. The other method is by extending the room temperature material properties of stainless steel 304L to temperatures below zero. This way, the excessively low temperatures calculated do not influence the structural behaviour of the ring. 4.2.5 Numerical Convergence and Initial Results To check numerical convergence, three different meshes were generated with varying global edge length parameters for the mesh generator. All three meshes were setup and run in LS-DYNA with the properties shown in Table 4-1.    A B C Room Temperature Node High Temperature Node  54  Table 4-1 - Properties used in the model to test for convergence Property Value Maximum Temperature 1200°C Heating Time 5 s Heated Region 11 mm (61%) Heating Segments 120 Welder Velocity 2.25 mm/s  The computation time on a quad core Intel processor at 2800 Mhz was recorded along with the horizontal diametric contraction of the ring. Table 4-2 – Results of various meshes with a constant time-step of 0.5s without springs Global Edge Length Total Nodes Total Elements Computation Time Diametric Contraction (mm) 3.60 7076 22026 2hr 26min 1.110 3.00 10701 36644 5hr 10min 0.853 2.30 17456 63494 9hr 13min 0.847  Given that the contraction of the ring diameter did not change significantly after using a global edge length of 3.00 to generate the mesh, it was assumed that the model had achieved reasonable convergence. Refining the mesh further increased the computational time significantly for the last case with the global edge length of 2.30. The same computations were performed on the models with 30 spring elements added as constraints for the ring. Table 4-3 - Results of various meshes with a constant time-step of 0.5s with springs Global Edge Length Total Nodes Total Elements Computation Time Diametric Contraction (mm) 3.60 7076 22026 2hr 35min 2.259 3.00 10701 36644 4hr 39min 2.190 2.30 17456 63494 9hr 53min 2.177   55  Incorporating the spring elements as constraints for the strips caused a two and a half fold increase in the contraction of the ring. This is likely due to the fact that buckling occurs to a much lesser extent as discussed in Section 4.2.1. To check numerical convergence for time-stepping, three different time-steps were tested with the same mesh. The results for the model without springs are shown in Table 4-4. Table 4-4 - Results of various time-steps with a mesh of global edge length 3.00 without springs Time-step (s) Total Nodes Total Elements Computation Time Diametric Contraction (mm) 1.00 10701 36644 2hr 43min 0.848 0.50 10701 36644 5hr 10min 0.853 0.25 10701 36644 9hr 25min 0.848  Since there was no significant difference in the results, it is likely that the spatial numerical errors are greater than the time-stepping numerical errors. The automatic time-step control feature in LS- DYNA was used to reduce computation time. 4.2.6 Parametric Study To consider methods to decrease the deformation of the ring, a parametric study was performed. Based on the analysis for plastic deformation presented in Section 3, the maximum temperature of the ring is important as it affects the expansion of the material. The heating rate is also important due its role in the temperature gradient achieved. These parameters were incorporated into the study by modifying the temperature profile of the segments accordingly as shown in Figure 4-19. Other parameters included were the position of the heat affected zone. This was done by varying the width of the ring heated and also the location of the weld line.  56   Figure 4-22 – The two parameters, D1 and D2, used to control the heated region on the ring Code written in C++ automated the generation of boundary conditions for different combinations of the parameters. Once the simulations were performed in LS-DYNA, the code also measured changes in the distance between specified nodes to determine deformations. Over 160 simulations were performed in 22 days of computation, generating 215 gigabytes of data. Each simulation was a combination of different values for the parameters chosen shown below. • Maximum Temperature (800°C, 900°C, 1000°C, 1100°C, 1200°C) • Heating Time (2s, 5s, 8s) • D1 (0 mm, 5 mm) • D2 (7 mm, 11 mm, 15 mm) • Buckling constraints versus no constraints To analyze the data, a MATLAB program was written to fix all except for one parameter and record its effect on ring contraction. The analysis worked by fixing four of five parameters and varying the fifth. The value for the varied parameter which resulted in the least deformation in the ring was recorded. The total number of times a value was recorded is the “score” for that parameter. This process was repeated for all five parameters.  The results are shown from Figure 4-23 to Figure 4-27. The value causing the least diametric deformation is highlighted in green.  57   Figure 4-23 - Maximum heating temperature results The results for the maximum temperature imply that in the majority of circumstances, decreasing the temperature while maintaining all other parameters reduces the ring deformation. Figure 4-24 implies similar results for increasing the heating time.  Figure 4-24 - Heating time results The parameters D1 and D2 are used to vary the weld line position and also the width of the heated region. The computational results suggest that the smaller the heated region is, the less deformation occurs. 0 5 10 15 20 25 30 800 900 1000 1100 1200 S co re Temperature (°C) 0 5 10 15 20 25 30 35 40 45 2 5 8 S co re Heating Rate (s)  58   Figure 4-25 – D1 parameter results  Figure 4-26 – D2 parameter results The data was also analyzed to determine whether the constraints provided by the spring elements altered the ring deformation. It was found that if the springs were absent to allow radial buckling, deformations occurred to a much lesser extent. This is reflected by the score of the cases with springs versus the cases without springs in Figure 4-27. 0 2 4 6 8 10 12 0 5 S co re D1 (mm) 0 5 10 15 20 25 30 7 11 15 S co re D2 (mm)  59   Figure 4-27 – Constraint results Figure 4-28 provides the magnitude of the diametric contraction from the simulations at the point where the weld starts. These values are only meant to provide for conceptual insights as the ring model has not been validated.  Figure 4-28 - Contraction of the ring with D1 = 0, D2 = 11mm and spring elements present 0 10 20 30 40 50 60 70 80 Springs No Springs S co re Constraints  60  The results indicate that to minimize deformations caused by welding, it is desirable to decrease the heating rate. Decreasing the area heated by weld is also likely to decrease deformations. 4.2.7 Conclusion Based on the numerical analysis, several trends can be deduced to decrease the ring deformation: • Increasing the heating time • Decreasing the heat affect zone • Removing constraints to allow radial buckling of the ring Although further experimental work is required to validate these results, it is expected that their incorporation into the welding process would aid with manufacturing the screen cylinders.   61  5 Conclusions This thesis presented an analysis of the welding process of screen cylinders to identify the mechanism of deformation in the ring during heating. The prospect of a phase transformation for stainless steel AISI 304L was considered and experimentally tested by using a ferritescope. Compression experiments were performed on the Gleeble 3500 machine that provided insight into other material characteristics such as the stress-strain curves at high temperatures. It was found that phase transformations and dynamic re-crystallization are not the main cause of the deformation in the rings. The next deformation mechanism considered was plastic deformation caused by thermal stresses. Modeling the heated region as a constrained beam led to the hypothesis that compressive yielding in the hoop direction can induce deformations. Furthermore, this deformation was expected to be sensitive to the temperature distribution. Square plate samples of AISI 304L were experimented on using an oxy-acetylene torch to test this hypothesis. Thermocouples and a coordinate measuring machine were chosen to monitor the samples. MD Patran and LS-Prepost were utilized to generate a 3D mesh of the square plates and a finite element model was constructed for LS-DYNA. Code written in C++ modified the models to produce the Gaussian flux boundary conditions for simulating the torch and compared the resulting temperature profiles to experimental readings. Both experimental and numerical results validated compressive yielding as the main mechanism of contraction. The slot contraction was indeed shown be sensitive to the temperature profile of the sample during heating. These same principles were applied to the ring and verified in a parametric study. To decrease deformations in the ring caused by thermal loads, it is recommended to decrease the heating rate and the area exposed to the weld. If possible allowing the ring to buckle or bend radially is also expected to be beneficial. 5.1 Limitations In this study several assumptions have been made that require discussion. While some have been mentioned in previous chapters, this section presents a list for interested readers. Section 3.1: Elevated Temperature Material Characteristics of Stainless Steel 304L • Oxidation: The cylindrical samples were tested in a vacuum to prevent reactions such as oxidation caused by air. During the welding process, the presence of air may have small effects in the material properties. Stainless steel AISI 304L does not oxidize easily and these effects are ignored.  62  • Bilinear Stress-Strain Curve: Bilinear models are an approximation of the actual stress strain curves. Since LS-DYNA does not support full stress strain curve input for thermo-structural simulations, this assumption was necessary. The bilinear models were chosen for strains less than 0.1 which are suitable for deformations caused by thermal loads yet allow for a more accurate bilinear representation. Section3.2: Contractions in AISI 304L by Heating • Torch Model: The finite element model assumes the torch provides a circular heating region. This is not completely true in all cases as the gases from the torch reflect off the plate. This backlash was more significant for the samples reaching higher temperatures as the torch was placed closer to them. It was neglected as it would require much more sophisticated thermo-fluids simulations. • Computational Results: While the trend of deformations for the square plates such as the slot contraction agreed with computational results, there were significant differences in some cases. Possible causes for discrepancy are discussed in Section 3.2.6. Section 4.2: Ring Finite Element Model • Temperature Boundary Conditions: Since the flux boundary conditions for welding simulations require a very fine mesh, temperature boundary conditions were used instead. It is acknowledged that these boundary conditions are not ideal for capturing the temperature distribution of the rings and further experimental validation is necessary. They are simply used to perform a trend like analysis to give insights into the welding process. • Welding Pieces: The ring is only one piece of the parts being welded together. The other piece was not included as it would significantly increase computational time. • Geometric Constraints: As the weld moves along the ring, the cylindrical partitions attached to it can provide resistance to its expansion. The effect of this constraint was only partially accounted for using spring elements for two reasons. The first is that the spring elements are not attached to each other and the resulting stresses in the hoop direction do not transfer in between them. The second is that they are created to model a cylindrical strip heated at the top to a temperature of 700 °C. Once the force displacement curve is input, the temperature of the spring elements is constant and is not altered by the moving weld.  63  This creates symmetry in the temperature distribution of the springs in the ring which would not occur in reality as the weld moves around. 5.2 Contributions The contributions made in this thesis are relevant to the welding process of screen cylinders as well as other applications using similar materials. Material properties of stainless steel AISI 304L that were not available in the literature were determined experimentally and can be used in future simulations. Phase transformations were found be negligible as the material reaches high temperatures. Further analysis using a simplified constrained beam model showed pure compression and bending/buckling as two possible modes of deformation which may occur in welding or any heating application. These concepts illustrate the importance of the high temperatures and high temperature gradients in the rings. These are important design considerations that can be incorporated into welding and other manufacturing processes. Numerical simulations for the square plate samples demonstrated that the Gaussian flux distribution is a good approximation for a still torch. For thermo-structural simulations with high temperature gradients, it was also shown that specific nodes can be cooled excessively and it was proposed that if mesh refinement is not an option, the material properties for the room temperature can be extended to cooler temperatures. 5.3 Future Work This research provided some of the conceptual framework needed to analyze the welding process of pulp screens. There were some limitations due to computational costs which lead to a trend analysis of the welding parameters. These limitations are likely to be overcome in the future as the computational power of personal computers increases. Based on the analytical insights provided, it would be suitable to go beyond a trend analysis and create a finite element model of the rings with a full double ellipsoid flux distribution and the other welding pieces. Experimental validation of the ring model would be a good way to compare the results with the temperature boundary conditions to see if the simplifications made were justified. Complete models of the ring welding procedure can also be used to redesign this manufacturing process. By doing this, many of the limitations addressed in Section 5.1 can be resolved to provide more accurate results.   64  Bibliography 1. Numerical Simulation and Experimental Measurement of Pressure Pulses Produced by a Pulp Screen Foil Rotor. Feng, Mei, et al. s.l. : ASME, March 2005, Journal of Fluids Engineering, Vol. 127, pp. 347-357. 2. Fracture and the Formation of Sigma Phase, M23C6, and Austenite from Delta-Ferrite in an AISI 304L Stainless Steel. Tseng, C. C., et al. June 1994, Metallurgical and Materials Transactions A, Vol. 25A, pp. 1147-1158. 3. The effect of δ-ferrite on fatigue cracks in 304L steels. Rho, Byung Sup, Hong, Uk Hyun and Nam, Woo Soo. 2000, International Journal of Fatigue 22, pp. 683–690. 4. Lacombe, P, Baroux, B and Beranger, G. Stainless Steels. 1993. 5. Callister, W D. Materials Science and Engineering: An Introduction (7th edition). s.l. : Quebecor Versailles, 2007. 6. Budinski, Kenneth G. and Budinski, Michael K. Engineering Materials: Properties and Selection. 9th. s.l. : Pearson Education Inc., 2010. 7. Comparison of different methods for measuring strain induced α'-martensite content in austenitic steels. Talonen, J., Aspegren, P. and Hanninen, H. 2004, Materials Science and Technology, pp. 1506-1512. 8. American Welding Society. ANSI/AWS A4.2-97: Standard Procedures for Calibrating Magnetic Instruments to Measure the Delta Ferrite Content of Austenitic and Duplex Austenitic-Ferritic Stainless Steel Weld Metal. 1997. 9. Stress–strain curves for stainless steel at elevated temperatures. Chen, Ju and Young, Ben. 2006, Engineering Structures, pp. 229-239. 10. Numerical simulation of transient temperature and residual stresses in friction stir welding of 304L stainless steel. Zhu, X. K. and Chao, Y. J. 2004, Journal of Materials Processing Technology, Vol. 146, pp. 263-272.  65  11. High-rate Characterization of 304L Stainless Steel at Elevated Temperatures for Recrystallization Investigation. Song, B., et al. 2010, Experimental Mechanics, Vol. 50, pp. 553–560. 12. ASM International. Metals Handbook - Properties and Selection: Irons, Steels and High Performance Alloys. 10th ed. 1990. p. 843. Vol. 1. 13. Kim, Choong S. Thermophysical Properties of Stainless Steels (ANL-75-55). Argonne, Illinois : Argonne National Laboratory, 1975. 14. Residual stress and plastic strain in AISI 304L stainless steel/titanium friction welds. Kim, Y. C., Fuji, A. and North, T. H. 1995, Materials Science and Technology, Vol. 11, pp. 383-388. 15. Recrystallization in type 304L stainless steel during friction stirring. Sato, Yutaka S., Nelson, Tracy W. and Sterling, Colin J. 2005, Acta Materialia, Vol. 53, pp. 637–645. 16. Microstructural refinement during annealing of plastically deformed austenitic stainless steels. Herrera, C., Plaut, R. L. and Padilha, A. F. 2007, Materials Science Forum, Vol. 550, pp. 423-428. 17. Analysis of circumferentially welded thin-walled cylinders to investigate the effects of varying clamping conditions. Malik, A. M., et al. 2008, Journal of Engineering Manufacture, Vol. 222, pp. 901-914. Part B. 18. Experimental and Numerical Study of Multi-Pass Welding Process of Pipe-Flange Joints. Troive, L., Nasstrom, M. and Jonsson, M. s.l. : ASME, August 1998, Journal of Pressure Vessel Technology, Vol. 120, pp. 244-251. 19. Numerical investigation of residual stresses and distortions due to multi-pass welding in a pipe- flange joint. Abid, M. and Qarni, M. J. 4, 11 2010, Journal of Process Mechanical Engineering, Vol. 224, pp. 253-267. 20. Finite element prediction of thermal stresses and deformations in layered manufacturing of metallic parts. Mughal, M. P., Fawad, H. and Mofti, R. 1, 2006, Acta Mechanica, Vol. 183, pp. 61-79. 21. Finite Element Modeling and Validation of Residual Stresses in 304L Girth Welds. Dike, J. J., Ortega, A. R. and Cadden, C. H. Pine Mountain : Sandia National Laboratories, 1998, 5th International Conference on Trends in Welding Research, pp. Dike 1-5. SAND98-8641C. 22. Holman, J. P. Heat Transfer. 9th. New York : McGraw-Hill, 2002. pp. 322-334.  66  23. Experimental and Computed Temperature Histories in Gas Tungsten-Arc Welding of Thin Plates. Pavelic, V., et al. 7, 7 1969, Welding Research Supplement, Vol. 48, pp. 295-305. 24. A New Finite Element Model for Welding Heat Sources. Goldak, J., Chakravarti, A. and Bibby, M. 2, 6 1984, Metallurgical Transactions B, Vol. 15B, pp. 299-305. 25. Ablation behaviors of ultra-high temperature ceramic composites. Tang, Sufang, et al. 2007, Materials Science and Engineering A, Vol. 465, pp. 1-7. 26. Thermomechanical Analysis of the Welding Process Using the Finite Element Method. Friedman, E. s.l. : ASME, August 1975, Journal of Pressure Vessel Technology, pp. 206-213. 27. Hallquist, John O. LS-DYNA Theory Manual. Livermore Software Technology Corp. [Online] 2006. [Cited: 6 26, 2010.] http://www.lstc.com/pdf/ls-dyna_theory_manual_2006.pdf. 28. Beer, Ferdinand P., Johnston, E. Russell Jr. and DeWolf, John T. Mechanics of Materials. 3rd. New York : McGraw-Hill, 2001. p. 611. 29. ASM International. Metals Handbook - Properties and Selection: Irons, Steels and High Performance Alloys. 10th ed. 1990. p. 843. Vol. 1.   

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