April 2018
A micro-CT investigation of density changes in pressboard due to compression
Johan Stjärnesund
Teknisk- naturvetenskaplig fakultet UTH-enheten
Besöksadress:
Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0
Postadress:
Box 536 751 21 Uppsala Telefon:
018 – 471 30 03 Telefax:
018 – 471 30 00 Hemsida:
http://www.teknat.uu.se/student
Abstract
A micro-CT investigation of density changes in pressboard due to compression
Johan Stjärnesund
Pressboard, a high-density cellulose-based product, is used both as load bearing structures and dielectric insulation in oil-filled power transformers. During trans- former operation, mechanical forces and vibrations are applied on the material. In particular, this investigation focuses on pressboard sheets placed between the turns of transformer windings, called the spacers, which during short circuit are subjected to high compressive forces of impulse nature. As a result of these forces, remaining deformations are created in the components. One step to reduce the negative consequences that come from the deformations is to understand how the fiber structure of the material changes by these forces, thus finding the week link.
Understanding these changes could lead to future modification of the material to better withstand short circuits.
To see the fiber structure and its changes in the material, pressboard has been investigated with a micro-CT at the Division of Applied Mechanics at Uppsala University. The scanned images have been reconstructed and analyzed in NRecon, CTAnalyser, and Matlab to investigate the density distribution changes and to identify the densification patterns. The study shows that pressboard initially has an inhomogeneous density distribution through the thickness and after mechanical indentations, the densification tends to begin in the more porous parts of the material. The project also included a prediction of the densification pattern, performed by finite element analysis (FEA) using a simplified material model. The results show that a material model with varying Young’s modulus through the thickness, based on a stiffness and porosity relationship, can produce similar densification patterns as in the experiments.
ISSN: 1401-5773, UPTEC Q 18010 Examinator: Åsa Kassman Rudolphi Ämnesgranskare: Kristofer Gamstedt
Handledare: Orlando Girlanda, Fredrik Sahlén och Reza Afshar
Undersökning av densitetsförändringar i pressboard
Johan Stjärnesund
Dagens ökande efterfrågan på elektrisk energi ökar även kraven på elsystemet, inklusive transformatorer och dess komponenter. Mellan kraftverk och elförbrukare finns transformatorer i flera olika storlekar. En transformator är en elektroteknisk komponent som används för att reglera elnätets spänning. Vid kraftverken höjs spänningen för att med så små förluster som möjligt kunna transportera elen långa sträckor. Vid fastigheter och fabriker sänks sedan spänningen gradvis till lämplig nivå för slutkund. Transformatorer är uppbyggda av två eller flera kopparspolar, lindade runt en gemensam järnkärna. Utöver koppar och järn utgör cellulosabaserade produkter en stor del av transformatorer. En sådan cellulosaprodukt är pressboard, som tack vare sina utmärkta isoleringsegenskaper och höga hållfasthet, används både som dielektrisk isolation och lastbärande struktur.
Tillverkningen av pressboard liknar vanlig papperstillverkningen. I en pappersmaskin sprutas oblekt fibermassa ut till stora ark. Arken pressas sedan under hög värme tills produkten är torr.
Den färdiga produkten är ett poröst och pappersliknande material. Den specifika komponenten som studien fokuserar på är en pressboard-bricka. Brickorna används mellan transformatorns kopparlindningar för att utgöra ett mellanrum där olja kan flöda och kyla. Den typen av pressboard som de här brickorna består av är 3 mm tjock och har en densitet mellan 1,1 och 1,2 g/cm3.
Trots materialets höga styrka så klara det inte av allt. Under drift utsätts transformatorns komponenter för flera mekaniska påfrestningar. Till exempel, under kortslutningar pressas kopparlindningarna ihop och brickorna emellan utsätts för väldigt höga och snabba tryckkrafter.
Dem här krafterna kan leda till kvarstående deformationer i materialet. Som ett steg mot att förhindra dessa deformationer påbörjades den här studien. Målet var att försöka förstå hur materialet förändrandes av krafterna, och se om man kunde förutspå dessa förändringar med hjälp av simuleringar.
I undersökningen användes en teknik kallad mikro-röntgentomografi. Det är en teknik som med hjälp av röntgen används för avbildning. Röntgentomografen tar flera röntgenbilder av ett prov ur olika vinklar. Med informationen från dessa bilder kan provet återskapas som en virtuell 3D- modell med en hög detaljnivå, för att sedan undersökas inuti.
För att skapa deformationer användes en förenklad provuppställning där proverna belastades med en sfärisk stålkula. Intrycken utfördes i tre olika belastningsfall för att frambringa tre olika stora deformationer. När proverna deformerats så skannades dem i mikro-röntgentomografen, för att sedan analyseras i olika mjukvaror. Med hjälp av röntgen-resultaten skapades sedan en förenklad materialmodell i simuleringsverktyget Ansys. Den här materialmodellen användes för att se om simuleringar med finita elementmetoden kan förutspå de experimentella resultaten.
De experimentella resultaten visar att pressboard från början har en ojämn densitetsfördelning.
Pressboard har en lägre densitet i mitten av tjockleken. Densiteten är också lägre allra längst ut på ytorna, detta på grund ojämnheter. Den typen av densitetsfördelning har tidigare visats i tunnare pappersprodukter. Vid deformation tenderar lågdensitets-områdena i mitten att förtätas först och mest. När densiteten har övergått till en mer homogen fördelning, skapar ytterligare deformation en förtätning nära belastningspunkterna. Simuleringsresultaten visade kvalitativt liknande förtätningsmönster som de experimentella resultaten, där den största förtätningen sker i mitten av materialets tjocklek. Dock behövs en mer utvecklad modell för att kvantitativt återskapa de verkliga resultaten.
Examensarbete 30 hp på civilingenjörsprogrammet Teknisk fysik med materialvetenskap
Uppsala universitet, 27 april 2018
Acknowledgements
This thesis is the result of a collaboration between ABB Corporate Research, Västerås and the Division of Applied Mechanics, Department of Engineering Sciences at Uppsala University.
I would like to thank the following people:
Kristofer Gamstedt, subject reader at the Division of Applied Mechanics, Uppsala University Orlando Girlanda and Fredrik Sahlén, supervisors at ABB Corporate Research
Reza Afshar, supervisor at the Division of Applied Mechanics, Uppsala University Carl Saxén, Master Thesis Student at the Division of Applied Mechanics, Uppsala University
For showing me how the μCT works
Everybody else at the Division of Applied Mechanics, Uppsala University For welcoming and supporting me during the project
Contents
1 Introduction... 1
2 Material ... 3
3 Experimental methods ... 5
3.1 Micro computed tomography (μCT) . . . 5
3.2 Mechanical loading . . . 7
3.3 Image analysis . . . 7
4 Simulation method...10
4.1 Finite Element Analysis (FEA) . . . 10
4.2 Calculation of the relationship between stiffness and porosity . . . 11
5 Results and discussion ...14
5.1 Density profiles . . . 14
5.2 Density maps . . . 17
5.3 Densification maps . . . 18
5.4 Finite Element Analysis . . . 20
6 Conclusions ...22
7 Suggestions of Future Work ...23
8 References ...24
9 Appendix ...25
9.1 Appendix A . . . 25
9.2 Appendix B . . . 27
9.3 Appendix C . . . 30
9.4 Appendix D . . . 31
9.5 Appendix E . . . 34
1 Introduction
With increasing demand for electrical energy, the requirements on the electrical system, including transformers and its components, also increase. Between power plant generators and electricity consumers in real estate and factories there are a variety of transformers in different sizes. A large power transformer can be seen in Figure 1. The transformer’s job is first to increase the voltage to transport the electricity over long distances, then gradually reduce it to an appropriate level for the consumer. A transformer consists of two or more copper coils winding around a common iron core [1]. In addition to copper and iron, cellulose-based products form a large part of oil-filled transformers, see Figure 2. An example of a cellulose- based product is pressboard, which thanks to its excellent insulating properties and high strength, is used both as load-bearing structure and dielectric insulation in oil-filled transformers [2].
Figure 1: Large power transformer next to a Volvo XC60 for comparison (ABB).
Figure 2: Internal components of a power transformer wrapped in cellulose-based products [3].
1 Introduction 27 april 2018
The insulation material is an important part, and the transformer’s life expectancy depends on the reliability of the insulation. The specific component this study focuses on is a pressboard spacer, see Figure 3a. The spacers are used to provide a gap between the copper windings so that the oil can flow and cool in a right way [2]. Figure 3b shows the spacers mounted between the windings. During transformer operation, the spacers are affected by several of mechanical forces and vibration [3]. For example, during a short circuit the copper windings are pressed together, and the spacers between them are subject to high compressive forces of impulse nature. As a result of these forces, remaining deformations are created in the components. One step to reduce the harmful effects is to understand how the fiber structure of the material changes by these forces, thus finding a week link. Understanding these changes could in future lead to modification of the material to better withstand short circuits.
Figure 3: Pressboard spacers (a). Spacers between the copper windings (b) (ABB).
The present thesis investigates the mechanisms responsible for permanent deformations in pressboard. Structural changes in pressboard are affected by a number of factors. This paper focuses on the density distribution through the thickness of the material and its impact due to mechanical stress. To investigate this a technology called micro computed tomography (μCT) was used. In addition to the image analysis, a finite element (FE) model has been created to see if results from simulations can be compared to the experimental ones, in order to predict the behavior of pressboard.
To understand how the material behaves in compression and see the possibility to show similar behavior via finite element analysis (FEA), the project is supposed to answer the following questions:
• Can a μCT be used to measure the density of the material?
• How is the density distribution in pressboard?
• How is the density distribution affected by mechanical stress?
• Where does the densification occur?
• Is it possible to predict the densification with finite element analysis?
a b
2 Material
The material subject of the present investigation is pressboard, a high-density paper material produced similar to regular paper making. Unbleached wood fibers are mixed with water to form a pulp that is then sprayed out on a rotating wire. When this wire passes a forming roll, the pulp is transferred over from the wire to the forming roll until the desired thickness is reached. The pulp is then cut off and removed from the roll as a 3x4 meter sheet and consist of about 70 % water. Figure 4 shows the paper machine. The sheets are then hung up before they will be placed in an oven where each sheet is hot-pressed individually. Figure 5 shows the final product, pressboard. The plates that the sheets are pressed between are provided with a steel mesh to improve the drying process. This mesh creates remaining marks in the material, so- called wired marks. These marks can be polished to some extent and result in calibrated pressboard [2]. Figure 6 shows SEM-pictures where the wire marks and the dense thickness can be seen.
Depending on type, pressboard has a density of 0.9 to 1.3 g/cm3. In this project 3 mm thick calibrated high-density pressboard (HDPB) was used, which has a density between 1.1 and 1.2 g/cm3.
Figure 4: The paper machine with the forming roll in the front and the wire behind (ABB Figeholm).
2 Material 27 april 2018
Figure 5: Pressboard in different thicknesses (ABB Figeholm).
Figure 6: (a) Pressboard surface with wire marks. (b) Cross section in the thickness direction of pressboard, (K. Wei and F Sahlén, ABB).
a b
0
3 Experimental methods
3.1 Micro computed tomography and reconstruction
Micro computed tomography (μCT) is a method using X-rays for imaging. It is a non-invasive method used to study the internal structure of different materials. The μCT can investigate several types of materials, including organic materials which are challenging when other types of testing machines are used. In a desktop μCT, a cone-shaped X-ray beam is generated from the source through the sample to the detector, unlike a synchrotron μCT where the source produces parallel X-rays. The advantage of a desktop μCT is that the sample can be enlarged only by moving it closer to the X-ray source, so no lenses are needed. The contrasts of the image depend on how much the scanned material absorbs the X-ray, known as the attenuation level. The intensity of the X-ray is different along the distance it travels through the sample and is characterized by the Beer-Lamberts law:
𝐼 = 𝐼0𝑒−𝜇𝑥 (1)
where 𝐼0 is the output intensity, x the distance traveled by the x-ray in the sample and 𝜇 is the attenuation coefficient. The μCT takes several projections of the object that simultaneously rotates around its axis [4]. The projections of the sample are then reconstructed via a cone-beam algorithm, and new 2D images of the cross-section are created. The schematic picture of the procedure is presented in Figure 7. In different software, these cross-sections can then be analyzed or constructed as virtual 3D models.
Figure 7: Schematic picture of the μCT scanning procedure.
The μCT used in this project is a Bruker Skyscan 1172 desktop style. Figure 8 shows the μCT machine with an open chamber and a sample mounted inside. First of all, a background image must be scanned, it is then subtracted from the final image and darkens areas with lower intensity. The background image is scanned without samples, and the average intensity should be between 50 – 60 %. Then, a bright-field and a dark-field image are taken for flat field correction. After this, the sample is placed in front of the source, and a new image is taken. The new image should have a minimum intensity of 20 – 30 % [5]. If the intensity at any stage is not within the given frames, the procedure should be repeated but now with increased or decreased exposure time and voltage. With rotational symmetry samples, image quality is improved because the X-rays travels through a constant mass regardless of the rotation angle [5]. Therefore, the material was cut into cylindrical samples with a diameter of 20 mm.
3 Experimental methods 27 april 2018
Figure 8: The μCT machine standing on a table (a). Closer look at the chamber (b) including one sample (1), the detector (2) and the X-ray source (3).
Table 1 shows the scanning parameters. Parameters such as voltage, current and exposure time were tested for the intensities in the images to fall within recommended limits. The number of pixels and pixel size were chosen based on the size of the sample and what was to be investigated. In this case, when the focus was on density and density distribution, a high level of details was not required, and larger pixels could then be selected to reduce scan times. Since desktop style μCT-scanner creates a polychromatic spectrum of X-rays, low energy X-rays will absorb in the outer parts of the sample and give an incorrect picture of the density of the sample.
This phenomenon is called Beam hardening artifact. To reduce this artifact, an aluminum filter was placed in front of the X-ray source. The filter absorbed or blocked the lowest energy X- rays [6].
Table 1: Scanning parameters used in the μCT Parameter [unit]
Voltage [kV] 80
Current [μA] 125
Exposure time [ms] 850 Pixel size [μm] 10.60 Number of pixels 2000 x 2000
Filter Al 0.5 mm
The reconstruction from the sample projections to transverse cross-sections was made in the software NRecon v.1.7.0.4. Table 2 shows the reconstruction parameters. A high degree of smoothing was chosen to reduce noise and get a clearer picture of the density distribution through the sample. The smoothing was done with Gaussian kernel level 10. Ring artifacts were reduced by Ring Artifact Correction level 20. Despite the aluminum filter in the μCT, some beam hardening artifacts still appeared. The remaining beam hardening artifacts were removed with 35 % Beam Hardening Correction.
Table 2: Reconstruction parameters used in NRecon Parameter
Smoothing (Gaussian) 10 Ring Artifact Correction 20 Beam Hardening Correction 35 %
1
2 3
a b
3.2 Mechanical loading
To induce localized high loads in the material, the samples were loaded by indentation between two plates in a tensile machine, where a 6 mm in diameter spherical steel ball was attached to the upper plate. The spherical indenter was chosen for a simplified test setup. Displacement was measured by three strain gauges. See Figure 9 for the setup. The indentation was performed in three different load cases, where each sample was loaded with a single indentation where the maximum force was 615, 1450 or 2850 N. These forces correspond to the mean contact pressure of 90, 120 and 150 MPa, according to the Hertzian contact stress [7]. The purpose of the three stresses was to create remaining deformations with stresses of around 80-120 MPa, which is the maximum allowed stress on spacers during short circuit [8]. However, the load is applied in a different way, not by an indenter as in this case.
Figure 9: Sketch of the set up used for mechanical loading of the samples.
3.3 Image analysis
The image analyzes were performed in different software depending on what was being investigated. In this study, mainly CT Analyser V.1.16.4.1, Fiji and Matlab were used. Some images have also been produced via CT Vox V.3.2.0, which provides good 3D images.
The reconstructed images from the μCT are grayscale pictures with each pixel having a value between 0 and 255, where 0 is black and 255 is white. The denser the material, the lighter the image becomes. To show the results as density, the grayscale was translated into density by assuming a linear relationship between them. With the measured density of a sample and the average gray value from the μCT for each sample, the connection was set as shown in Figure 10. The reconstructed 3D models were cut into new vertical cross-section, as shown in Figure 11, to investigate the structure through the thickness. Needle marks were created as reference points to find the loading points both before and after indentation. The markings were about half a millimeter deep and can be seen in the circle in Figure 11.
3 Experimental methods 27 april 2018
Figure 10. Example of the linear relationship between gray value and density for one of the samples.
Figure 11: Vertical cross section for a loaded sample. In the circle shows a needle mark used to navigate through the sample.
Visualizing the results and changes in unprocessed grayscale images can be difficult. Therefore, the results from the image analyzes are shown as either line profiles or contour maps. In Fiji, a line could be drawn through the sample (as shown in Figure 12) to get the profiles of the grayscale versus distance through the thickness.
Figure 12: The line as the density profile was measured through the thickness, before (a) and after (b) loading.
a b
The contour maps were created in Matlab. The images were first smoothed with ImGaussfilt level 16 to reduce noise. ImGaussfilt applies a filter that makes the images blurrier but with more homogeneous grayscale distributions, which also means that the transition between specimen and air became unclear. As an edge-preserving action, pixels in the image below the gray value of 50 have been replaced by 0. Despite the edge-preserving action, the values near the edges should be considered with some reservation, due to the high level of smoothing. The level of contours was set to 8. A balance between smoothing, edged-preserving and number of contour levels were necessary to clearly show the results without reducing too much details.
With less smoothing and more contour levels, the maps would be more accurate, but it would then be difficult to see the trends visually.
To produce the densification maps, the scanned cross sections were standardized, and the densification is shown according to:
𝐷(𝜒´) =𝜌´(𝜒)𝜌(𝜒) (2)
where 𝐷(𝜒´) is the densification ratio, 𝜌(𝜒) is the density of pixel 𝜒 in the undeformed state and 𝜌´(𝜒) is the density of pixel 𝜒 in the deformed state. The Matlab codes for the density and densification maps can be seen in Appendix A.
27 april 2018
4 Simulation method
4.1 Simplified Material Model of Pressboard
The finite element analysis (FEA) was performed in Ansys 18.2. The aim of the FEA was to see if the simulations could produce the same densification results as the experimental ones.
Developing a material model that would in all respects be comparable with such a complex material as pressboard would require extensive and time-consuming material testing. In this study, several simplifying assumptions have been made. Pressboard is an orthotropic and porous material with an elastic stiffness that increases during compression due to densification [9]. In the present investigation, the first approach was to use an isotropic, homogeneous and linear elastic model. The argument for using an isotropic model was that the simulation solely focused on load and deformation in the thickness direction. The model has been provided with a varying Young’s modulus through the thickness, in an attempt to create similar densification as in the experiments. This varying module is based on an exponential stiffness and porosity relationship, see Figure 13. The y-axis in Figure 13 shows E/Es where E is Young’s modulus of the porous material and Es is Young’s modulus of the cell wall material, in this study the pressboard specimen respectively the cellulose fiber. The Young’s modulus of the fiber was assumed to be 25 GPa, according to [10]. The geometry was 2D axisymmetric. As in experiments, the specimen was set to be 3 mm thick and with a radius of 10 mm and the indenter spherical with a radius of 3 mm. See the deformed geometry in Figure 14.
Figure 13: Example of a stiffness and porosity relationship in a porous material [11].
Figure 14: A quarter of the indenter and the deformed specimen.
4.2 Calculation of the relationship between stiffness and porosity
With the experimental results over the density distribution from the line profiles and the knowledge of the density of cellulose of 1.5 g/cm3 [10], the porosity through the thickness could be obtained by
𝑐 = 1 −𝜌𝜌
𝑓 (3)
where 𝑐 is the porosity, 𝜌 the density of the specimen and 𝜌𝑓 the density of cellulose.
The samples Young’s modulus after indentation were calculated from the unloading curves in the force vs. displacement graphs, according to the Oliver and Pharr-method [12]. Figure 15 shows the graph of one sample, including markings for the loading curve (a), unloading curve (b) and the permanent indentation depth (hp). The displacements were obtained as a mean of the three strain gauges. The nomenclature in Figure 15 and the calculations follows the indentation manual [13]. The graphs of the other five samples can be seen in Appendix B.
Figure 15: Indentation graph of sample 7. (a) is the loading curve, (b) is the unloading curve, and hp is the permanent indentation depth.
a
b
h
p4 Simulation method 27 april 2018
The calculations were performed according to the following steps [12, 13]:
𝑆 = 𝑚 ∙ 𝐹𝑚𝑎𝑥∙ (ℎ𝑚𝑎𝑥 − ℎ𝑝)−1 (4.1)
ℎ𝑟 = ℎ𝑚𝑎𝑥−𝐹𝑚𝑎𝑥𝑆 (4.2)
ℎ𝑐 = ℎ𝑚𝑎𝑥 − 𝜀 ∙ (ℎ𝑚𝑎𝑥− ℎ𝑟) (4.3)
𝐴𝑝 = 𝜋 ∙ 2 ∙ 𝑅 ∙ ℎ𝑐 (4.4)
𝑆 = 𝑚 ∙ 𝐹𝑚𝑎𝑥∙ (ℎ𝑚𝑎𝑥 − ℎ𝑝)−1 (4.5)
𝐸𝑟 = √𝜋∙𝑆
2∙𝛽∙√𝐴𝑝,ℎ𝑐 (4.5)
𝐸𝑝𝑏 = 1−𝜐𝑝𝑏2
1 𝐸𝑟 − 1−𝜐𝑖2
𝐸𝑖
(4.6)
where
S slope of the unloading curve at Fmax
Fmax maximum indenter load (force) hmax maximum penetration depth hp permanent indentation depth
hr the depth when the tangent of the unloading curve at Fmax crosses the y-axis hc the depth of contact of the indenter with the sample at Fmax
Ap projected contact area R radius of indenter
Er combined Young’s modulus
Epb Young’s modulus of the pressboard sample Ei Young’s modulus of the indenter
pb Poisson’s ratio of pressboard
i Poisson’s ratio of the indenter m, , geometric constants
Geometric and material constant were set as:
R = 3 mm Ei = 210 GPa
pb = 0 [9]
i = 0.3 m = 3.4
= 0.75
= 1
These calculations gave three different Young’s moduli, one from each load case. The residual porosities in the material under the load points were calculated by taking the mean value of Eq.
3 through the samples, also here one remaining porosity from each load case was obtained. The initial porosity and Young’s modulus in the out-of-plane direction for the specimen were set to 20 % and 235 MPa [9]. With these four modules and porosities, the connection between the pressboard stiffness and porosity could be calculated, as seen in Figure 16. The red line is a two terms exponential trend line.
Figure 16: The calculated relationship between stiffness and porosity in pressboard.
With the knowledge of the porosity through the thickness of the samples, and the relationship between pressboard stiffness and porosity, the varying Young’s modulus used in the FEA could be calculated. Figure 17 shows the Young’s modulus through the thickness of a pressboard sample, according to these calculations. The red line is a second order polynomial trend line, without regard to the porous surfaces. A loop in Ansys then defined the Young’s modulus for each element, according to the equation for the trend line in Figure 16. The loop used in Ansys can be seen in Appendix C.
Figure 17: The varying Young's modulus through the thickness.
27 april 2018
5 Results and discussion
5.1 Density profiles
The results of the indentation test are presented as density vs. through-thickness position graphs shown in Figures 18 through 21. The dotted lines are obtained from the scanning results, and the solid ones are polynomial trend lines of order 6 to more clearly show the results. The y-axis shows density and the x-axis the vertical position from 0 to -3 mm, where 0 is the top surface and -3 the bottom.
The density profiles of the samples before loading follow all the same patterns, see Figure 18 for the profile of sample 3. The profiles of the other five samples can be seen in Appendix D.
First of all, lower density at both surfaces can be observed. There are two reasonable explanations for it, the first is the rough surface due to the wire marks from manufacturing, and the other is that the images are smoothed which makes the edges less clear. It can also be seen from the profile that the density tends to drop in the middle of the test piece. Thinner machine- made paper products have shown this type of density distribution before [10].
Figure 18: Pressboard density versus distance from top surface for sample 3.
The density profiles in Figure 19, 20 and 21 show the comparison before and after load for one sample in each load case. The comparison profiles of the other three samples can be seen in Appendix D.
In the 90 MPa load case (Figure 19), the profile shows a more homogeneous density distribution than before deformation. This change could be explained by the fact that the more porous low- density area in the middle is mechanically weaker and therefore more prone to deformation.
The next load case, 120 MPa (Figure 20), shows again that the low-density area in the middle is gone, but also an increased density from the indentation side. The highest load case, 150 MPa
(Figure 21), shows a curve similar to the previous one but with even more increased density from the indentation side. The density is now undoubtedly highest on the deformation side.
Figure 19: Density versus distance before (blue) and after (red) mechanical load on sample 1.
Figure 20: Density versus distance before (blue) and after (red) mechanical load on sample 4.
90 MPa
120 MPa
5 Results and discussion 27 april 2018
Figure 21: Density versus distance before (blue) and after (red) mechanical load on sample 5.
The analysis of the profiles indicates that the translation from grayscale to density and the use of the μCT method to investigate density of a material is working well.
The comparative density profiles for pressboard before and after deformations also indicate two mechanisms that occur in the material under load. Figure 22 schematically shows the mechanisms. The y-axis corresponds to density and the x-axis the vertical position from top surface to the bottom of the sample. The first mechanism (1) can be seen in Figure 22a and shows that the center of the material’s thickness and more porous part appears to deform and densify first. The second mechanism (2) can be seen in Figure 22b and shows that when the density has become more homogeneous throughout the sample, further deformation has led to densification at the indentation side. After unloading, the density tends to be highest at the indentation side and becomes lower through the thickness, Figure 22c.
Figure 22: Schematic figure of how the density through the thickness is changed by deformation. First the low-density middle is densified (a). When the density has become homogeneous, further deformation has led to densification on the indentation side (b). The
residual density after unloading tends to be highest at the load side (c).
150 MPa
a b c
5.2 Density maps
The following figures show contour maps of the density in the material after deformation, for one sample in each load case. The maps of the other three samples can be seen in Appendix E1.
Unlike the density profiles that only showed the density directly below the loading point, these maps show the density over a wider area of the vertical cross-section. The y-axis shows the vertical position from 0 to -3 mm and the x-axis the radial position from -3.5 to 3.5 mm. The loads have taken place in position (0, 0). Density is given as g/cm3102.
It is difficult to draw any conclusions from the density maps in the lower load cases, but at higher loads, the different density areas are more clearly visible. Figure 24 shows how the area with highest density extends almost throughout the sample. The area is quite narrow and placed below the center of the load point. Figure 25 clearly shows a triangular area near the load point where density is highest.
Figure 23: Density map of sample 1 after load.
Figure 24: Density map of sample 3 after load.
90 MPa
120 MPa
5 Results and discussion 27 april 2018
Figure 25. Density map of sample 5 after load.
5.3 Densification maps
The following figures show contour maps of the densification in the material after deformation, one sample from each load case. The densification maps for the other three samples can be seen in Appendix E2. The y-axis shows the vertical position from 0 to -3 mm and the x-axis the radial position from -3.5 to 3.5 mm. The applied force is placed in position (0, 0). The densification ratio is given as D102.
Figures 26, 27 and 28 show that the highest densification tends to occur straight under the indentation and approximately in the middle of the test piece, for all load cases. It also shows how the densification seems to be limited to the area under the indentation, as very little or no densification at all occurs beyond the deformed surfaces. This indicates that within these load cases, the material is deformed only in the vertical direction.
Figure 26: Densification map of sample 1.
150 MPa
90 MPa
Figure 27: Densification map of sample 3.
Figure 28: Densification map of sample 5.
The line profiles have shown how the material is initially more porous in the middle of the thickness. The densification maps and the line profiles after load indicate that the material densifies more in the middle. These observations can probably be explained by the fact that the porous low-density areas make the material weaker. In the highest load cases, both the line profiles and density maps show how the material has the highest residual density at the indentation side. It is possible that the component with a higher and more homogeneous density through the thickness would show a different densification pattern.
120 MPa
150 MPa
5 Results and discussion 27 april 2018
5.4 Finite element analysis
The following figures show the densification maps from FEA, compared with the densification maps from experiments. Figure 29 shows the comparison with 90 MPa load case, Figure 30 with 120 MPa load case, and Figure 31 with 150 MPa load case. The left part of the figures shows the experimental densification maps, as shown in previous section, but the difference here is that the contours are now filled to match the maps from the FEA. To the right, the densification from the simulations is shown. The densification ratio in the FEA results is obtained from the Gibson and Ashby strain-density relationship for porous material [14] as follows:
𝐷𝐹𝐸𝐴(𝜒´) =𝜌𝜌
0= 𝜀+11 (5)
where 𝜌0 is the density before deformation, 𝜌 is the density after deformation and 𝜀 is the transverse strain. The densification ratio in following figures is given as D102.
The FEA results show in all load cases that the largest densification occurs in the middle of the test piece, similar to the experimental results. However, the values of the densification differ significantly between the results. Figure 29 shows that the maximum densification from FEA is 1.19, and 1.58 from the experiment. In Figure 30, it can be seen that the maximum densifications are almost equal in experiment and simulation, 1.48 and 1.42 respectively. In the highest load case, the difference is significant. Figure 31 shows that the maximum densification from the experiment is 1.56, and 2.19 from FEA. The maximum densifications in experimental results are all around 1.50, regardless of load case. However, the densification in the simulations becomes higher with increased deformation. The differences are partly due to the assumption to simulate with a linear elastic material model. In addition, a reasonable thought is that the densification of the material in the experiments is saturated at a certain level, when the porosity is absent or very small. The simulations do not take this saturation into account.
Thus, with the aforementioned simplifying assumptions and the varying Young’s modulus, the model may produce qualitatively similar densification patterns as the experimental results.
Figure 29: Densification map: experimental result of sample 1 to the left and FEA result to the right.
90 MPa
Exp. FEA
Figure 30: Densification map experimental result of sample 3 to the left and FEA result to the right.
Figure 31: Densification map: experimental result of sample 5 to the left and FEA result to the right.
120 MPa
150 MPa
Exp.
Exp.
FEA
FEA
27 april 2018
Conclusions
• The μCT method is well suited to investigate density changes in pressboard.
• A density gradient through the thickness of the unloaded specimen was detected.
• For load cases 90, 120 and 150 MPa the density increases near the load point.
• Densification tends to be larger in the middle of the thickness where density is initially lower.
• An FE model with varying Young’s modulus has been developed and show similar densification patterns as experiments.
Suggestions of Future Work
• Investigate very small samples with a Synchrotron μCT to achieve higher resolution, in an attempt to track fibers and find a weak link to the deformations on a smaller scale.
• Similar study, but oil-impregnated pressboard is loaded by copper windings for a more real-life case.
• Develop a more accurate material model for FE modeling of pressboard with Usermat.
• Simulate to optimize material density profiles and control the manufacturing to achieve suitable structure.
• Use the μCT method as a quality control procedure.
27 april 2018
References
[1] S.V. Kulkarni, S.A. Khaparde, Transformer Engineering: Design, Technology, and Diagnostics, Second Edition, CRC Press, 2012.
[2] H. P. Moser, Transformerboard, Scientia Electria, 1979.
[3] G. Bertagnolli, The ABB approach to short-circuit duty of power transformers. Zurich, Switzerland: ABB Management Services, 2007.
[4] E. L. Ritman, Micro-Computed Tomography - Current Status and Developments, Annu. Rev. Biomed. Eng. 6, 185–208, 2004.
[5] Bruker, Method notes, SkyScan 1172 micro-CT: scanner operations, 2008.
[6] M. F. V. Tarplee, N. Corps, Acquiring optimal quality X-ray μCT scans, 2008.
[7] K. L. Johnson, Contact mechanics, Cambridge University press, 1985.
[8] IEC Standard 60076-5:2006, Power transformers – Ability to withstand short circuit, 2006.
[9] D. D. Tjahjanto, O. Girlanda, S. Östlund, Anisotropic viscoelastic–viscoplastic continuum model for high-density cellulose-based materials in Journal of the Mechanics and Physics of Solids. https://doi.org/10.1016/j.jmps.2015.07.002, 2015.
[10] C. A. Bronkhorst, K. A. Bennett, Deformation and Failure Behavior in Handbook of Physical Testing of Paper Vol 1, R. E. Mark, J. Borch, 356-363. Washington: CRC Press, 2002.
[11] L. Nielsen Fuglsang, Strength and Stiffness of porous materials in Journal of the American Ceramic Society vol. 73 no. 9, 1990.
[12] A. C. Fischer-Cripps, Nanoindentation, New York: Springer, 2002.
[13] CSM Instruments, Indentation Software User’s manual, 2008.
[14] L. J. Gibson, M. F. Ashby, Cellular solid – Structure and properties, Cambridge University press, UK, 1997.
Appendix
8.1 Appendix A
Appendix A contains the Matlab codes used to create the density maps (A1) and densification maps (A2).
A1: Matlab code to create density map.
% Density map --- clc;
clear all;
close all;
% Load image --- SA=imread('Sample1.bmp'); % Deformed
% Rotate the image ---
SAT = imrotate(SA,-0.3,'bilinear','crop');
% Cut out the deformed area ---
SATC = imcrop(SAT,[580 39 680 275]);
% Smooth the image ---
blurr = imgaussfilt(SATC,16);
figure,imshow(blurr);
% Edge preserving ---
blurr(blurr < 50) = 0;
% Convert GV to Density (g/cm^3 x 100) ---
blurrd = double(blurr);
mean = mean2(blurrd);
k = mean/1.2;
blurrd = (blurrd/k)*100;
blurr = uint8(blurrd);
% Make the image to a contour plot --- [C,h] = contour(blurr,8,'LineWidth',2);
colormap(jet);
% Plot settings --- set(gca,'ydir','reverse','FontSize',20);
c = colorbar;
axis equal;
box off;
ylabel('Vertical position (mm)');
xlabel('Radial position (mm)');
xticks([56.8 151.2 245.6 340 434.4 528.8 623.2 717.6]) xticklabels({'- 3','- 2','- 1','0','1','2','3','4'}) yticks([0 47.2 94.4 141.6 188.8 236])
yticklabels({'0','- 0.5','- 1','- 1.5','- 2','- 2.5'}) caxis([50 155]);
clabel(C,'manual');
c.Label.String = 'Density (g/cm^3 x 10^2)';
% ---
Appendix 27 april 2018
A2: Matlab code to create densification map.
% Densification map --- clc;
clear all;
close all;
% Load images ---
S=imread('Sample1.bmp'); % Undeformed SA=imread('Sample1.bmp'); % Deformed
% Rotate the images ---
ST = imrotate(S,0,'bilinear','crop');
SAT = imrotate(SA,-0.3,'bilinear','crop');
% Cut the images ---
STC = imcrop(ST,[895 60 880 270]);
SATC = imcrop(SAT,[700 38 880 270]);
% Normalize the images to show the densification - STC = double(STC);
SATC = double(SATC);
Dens = (SATC./STC)*100;
Dens8 = uint8(Dens);
% Smoothing ---
blurr = imgaussfilt(Dens8,16);
figure,imshow(blurr);
% Edge preserving and noise reduction ---
blurr(blurr < 50) = 0;
blurr(blurr > 0 & blurr < 100) = 100;
% Make the image to a contour plot --- [C,h] = contour(blurr,8,'LineWidth',2);
colormap(jet);
% Plot settings --- set(gca,'ydir','reverse','FontSize',20);
c = colorbar;
axis equal;
box off;
ylabel('Vertical position (mm)');
xlabel('Radial position (mm)');
xticks([31.2 125.6 220 314.4 408.8]) xticklabels({'-2','- 1','0','1','2'}) yticks([0 47.2 94.4 141.6 188.8 236])
yticklabels({'0','- 0.5','- 1','- 1.5','- 2','- 2.5'}) caxis([80 160]);
clabel(C,'manual');
c.Label.String = 'Densification x 10^2';
% ---
8.2 Appendix B
Appendix B contains force-displacement graphs from indentation tests. These graphs were used to calculate the Young’s modulus of pressboard.
Figure B 1: Indentation graph of sample 1.
Figure B 2: Indentation graph of sample 2.
Appendix 27 april 2018
Figure B 3: Indentation graph of sample 3.
Figure B 4: Indentation graph of sample 4.
Figure B 5: Indentation graph of sample 5.
Figure B 6: Indentation graph of sample 6.
Appendix 27 april 2018
8.3 Appendix C
Appendix C contains the loop used to define the varying modulus in Ansys.
!---
! Varying Young’s modulus
*get,elementcount,elem,,count ! Calculate element centroid
*get,maxmp,mat,0,num,max ! Calculate modulus at given location mat_ref = maxmp
*do,i,1,elementcount
*get,depth,elem,i,cent,y
value = 810*(depth)*(depth)+2470*(depth)+2400 mat_ref = mat_ref+1
MP,EX,mat_ref,value ! Defining the new material MP,PRXY,mat_ref,0
emodif,i,mat,mat_ref ! Assign the new material to the current element
*enddo
!---
8.4 Appendix D
Appendix D contains all density profiles before and after deformation.
Figure D 1: Density versus distance before (blue) and after (red) load on sample 1.
Figure D 2: Density versus distance before (blue) and after (red) load on sample 2.
Appendix 27 april 2018
Figure D 3: Density versus distance before (blue) and after (red) load on sample 3.
Figure D 4: Density versus distance before (blue) and after (red) load on sample 4.
Figure D 5: Density versus distance before (blue) and after (red) load on sample 5.
Figure D 6: Density versus distance before (blue) and after (red) load on sample 6.
Appendix 27 april 2018
8.5 Appendix E
Appendix E contains contour maps of the density (E1) and densification (E2) for all samples.
E1
Figure E1 1: Density map of sample 1 after load.
Figure E1 2: Density map of sample 2 after load.
Figure E1 3: Density map of sample 3 after load.
Figure E1 4:Density map of sample 4 after load.
Appendix 27 april 2018
Figure E1 5: Density map of sample 5 after load
.
Figure E1 6: Density map of sample 6 after load.
E2
Figure E2 1: Densification map of sample 1.
Figure E2 2: Densification map of sample 2.
Figure E2 3: Densification map of sample 3.
Appendix 27 april 2018
Figure E2 4: Densification map of sample 4.
Figure E2 5: Densification map of sample 5.
Figure E2 6: Densification map of sample 6.
