X-Ray Tomography in Material Science

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X-Ray Tomography in Material Science

0 HERMES Science Publications, Paris, 2000 HERMES Science Publications

8, quai du MarchC-Neuf 75004 Paris Serveur web : http://www.hermes-science.com

ISBN 2-7462-0115-1 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission, in writing, from the publisher.

Disclaimer While every effort has been made to check the accuracy of the information in this book, no responsability is assumed by Author or Publisher for any damage or injury to or loss of property or persons as a matter of product liability, negligence or otherwise, or from any use of materials, techniques, methods, instructions, or ideas contained herein.

X-Ray Tomography in Material Science

Jose Baruchel Jean-Yves Buffiere Eric Maire Paul Merle Gilles Peix

•cience

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Authors

ANDERSON P., Department of Biophysics in Relation to Dentistry, St Bartholomew's and The Royal London Scool of Medecine and Dentistry, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK BABOT D., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France BARUCHEL J., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France BELLET D., Laboratoire GPM2, INPG, BP 46, 38402 Saint-Martin-d'Heres BENOUALI A.-H., Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B-3001 Heverlee, Belgium BERNARD D., ICMCB, CNRS, 87 avenue du docteur Albert Schweitzer, 33608 Pessac, France BLANDIN J.-J., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France BOLLER E., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France BOUCHET S., Ecole des mines , ENSMP, 35 rue St Honore, 77300 Fontainebleau, France BRACCINI M., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France BUFFIERE J.-Y., GEMPPM INSA Lyon, 20 avenue Albert Einstein, 69621 Villeurbane Cedex, France CLOETENS P., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France DAVIS G., Department of Biophysics in Relation to Dentistry, St Bartholomew's and The Royal London Scool of Medecine and Dentistry, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK DEGISCHER H.P., Institute of Materials Science and Testing, Vienna University of Technology, Karlsplatz 13, A-1040 Wien

6

X-ray tomography in material science

DERBY B., Manchester Materials Science Centre, UMIST and the University of Manchester, Grosvenor Street, Manchester, Ml 7HS,UK DUVAUCHELLE P., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France ELLIOTT J., Department of Biophysics in Relation to Dentistry, St Bartholomew's and The Royal London Scool of Medecine and Dentistry, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK FOROUGHI B., Institute of Materials Science and Testing, Vienna University of Technology, Karlsplatz 13, A-1040 Wien FREUD N., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France FROYEN L., Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B-3001 Heverlee, Belgium GuiGAY J.-P., University of Antwerp, RUCA Groenenborgerlaan 171, B-2020 Antwerp, Belgium HEINTZ J.-M., ICMCB, CNRS, 87 avenue du docteur Albert Schweitzer, 33608 Pessac, France JOSSEROND C., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France JUSTICE I., Department of Materials, University of Oxford, Parks Rd, Oxford, OX1 3PH, UK KAFTANDJIAN V., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France KOTTAR A., Institute of Materials Science and Testing, Vienna University of Technology, Karlsplatz 13, A-1040 Wien LUDWIG W., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France MAIRE E., GEMPPM INSA Lyon, 20 avenue Albert Einstein, 69621 Villeurbane Cedex, France MARC A., LETI-CEA/Grenoble, 17 rue des martyrs, 38054 Grenoble Cedex 9, France MARTIN C.F., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France PEK G., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France PEYRIN F., CREATIS, INSA-Lyon, 69621 Villeurbane, France ROBERT-COUTANT C., LETI-CEA/Grenoble, 17 rue des martyrs, 38054 Grenoble Cedex 9, France SALVO L., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France SAVELLI S., GEMPPM INSA Lyon, 20 avenue Albert Einstein, 69621 Villeurbane Cedex, France SCHLENKER M., CNRS, Laboratoire Louis Neel, BP 166, F-38042 Grenoble, France SUERY M., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France

Authors

7

VAN DYCK D., University of Antwerp, RUCA Groenenborgerlaan 171, B-2020 Antwerp, Belgium VERRIER S., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France VIGNOLES G.-L., LCTS, CNRS-SNECMA-CEA, Universite Bordeaux 1, 3 allee La Boetie, F-33600 Pessac, France WEVERS M., Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B-3001 Heverlee, Belgium


Table of contents

Foreword

13

Chapitre 1. General principles

G. PEIX, P. DUVAUCHELLE, N. FREUD 1.1. Introduction 1.2. X and gamma-ray tomography: physical basis 1.3. Different scales, different applications 1.4. Quntitative tomography 1.5 Conclusion 1.6. References

15 15 16 20 23 26 26

Chapitre 2. Phase contrast tomography

P. CloETENS, W. LUDWIG, J.-P. GUIGAY, J. BARUCHEL, M. SCHLENKER, D. VANDYCK 2.1. Introduction 2.2. X-ray phase modulation 2.3. Phase sensitive imaging methods 2.4. Direct imaging 2.5. Quantitative imaging 2.6. Conclusion 2.7. References

29 29 30 32 38 38 42 43

Chapitre 3. Microtomography at a third generation syncrotron radiation facility

J. BARUCHEL, E. BOLLER, P. CLOETENS, W. LUDWIG, F. PEYRIN 3.1. Introduction 3.2. Syncrotron radiation and microtomography

45 45 46

10


3.3. Improvement in the signal to noise ratio in the 3D images 3.4. Improvement in the spatial resolution 3.5. Quantitative measurement (absorption case) 3.6. Present state of "local" tomography 3.7. Sample environment in microtomography 3.8. Phase Imaging 3.9. Other new approaches in microtomography 3.10. Conclusion 3.11. References

49 50 51 53 54 55 56 57 57

Chapitre 4. Introduction to reconstruction methods C. ROBERT-COUTANT, A. MARC

61

4.1. Introduction 4.2. Description of projection measurements 4.3. Backprojection 4.4. Projection-slice theorem 4.5. Fourier reconstruction methods 4.6. Filtering in Fourier methods 4.7. ART-type methods 4.8. Conclusion 4.9. References

61 62 65 66 67 69 70 74 74

Chapitre 5. Study of materials in the semi-solid state

S. VERREER, M. BRACCINI, C. JOSSEROND, L. SALVO, M. SUERY, W. LUDWIG, P. CLOETENS, J. BARUCHEL 5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

Introduction Experimental device and procedure Results on Al-Si alloys Results on Al-Cu alloys Conclusion and perspectives References

,...

77 77 79 80 85 86 87

Chapitre 6. Characterisation of void and reinforcement distributions by edge contrast

I. JUSTICE, B. DERBY, G. DAVIS, P. ANDERSON, J. ELLIOTT 6.1. Introduction 6.2. Dual energy X-ray microtomography 6.3. Experimental materials 6.4. Results and discussion 6.5. Conclusions 6.6. References

89 89 90 92 94 100 101

Table of contents

11

Chapitre 7. Characterisation of MMCp and cast Aluminium alloys

J.-Y. BUFFIERE, S. SAVELLI, E. MAIRE 7.1. 7.2. 7.3. 7.4. 7.5.

103

Introduction Experimental methods Results and discussion Conclusion References

103 104 107 112 113

Chapitre 8. X-ray tomography of Aluminium foams and Ti/SiC composites

E. MAIRE, J.-Y. BUFFIERE

115

8.1. General introduction 8.2. Aluminium foams 8.3. Titanium composites 8.4. General conclusion 8.5. References

115 116 121 124 125

Chapitre 9. Simulation tool for X-ray imaging techniques

P. DUVAUCHELLE, N. FREUD, V. KAFTANDJIAN, G. PEIX, D. BABOT 9.1. 9.2. 9.3. 9.4. 9.5. 9.6.

127

Introduction Background Simulation possibilities Simulation examples in tomography Conclusions and future directions References

127 128 129 132 135 136

Chapitre 10. Micro focus computed tomogrgraphy of Aluminium foams A.-H. BENAOULI, L. FROYEN, M. WEVERS 10.1. 10.2. 10.3. 10.4. 10.5. 10.6.

Introduction Production process of Aluminium foams Mechanics of foams Non-destructive investigation of Aluminium foams Conclusion References

139 139 140 142 144 151 152

Chapitre 11. 3D observation of grain boundary penetration in Al alloys W. LUDWIG, S. BOUCHET, D. BELLET, J.-Y. BUFFIERE

11.1. Introduction 11.2. Experimental set-up 11.3. Result 11.4. Conclusions 11.5. References

.*.

155

155 157 158 160 163

12


Chapitre 12. Determination of local mass density distribution

H.P. DESISCHER, A. KOTTAR, B. FOROUGHI 12.1. Introduction 12.2. Material 12.3. X-ray radiography 12.4. Result 12.5. Application of the mean local density distribution 12.6. References

165 165 166 166 168 172 175

Chapitre 13. Modelling porous materials evolution

D. BERNARD, G.-L. VIGNOLES, J.-M. HEINTZ 13.1. Introduction 13.2. Evolution of sandstone reservoir rocks by pressure solution 13.3. C-C 13.4. Ceramics sintering 13.5. Conclusions and forthcoming works 13.6. References

177 177 179 185 187 190 191

Chapitre 14. Study of damage during superplastic deformation

C.-F. MARTIN, J.-J. BLANDIN, L. SALVO, C. JOSSEROND, P. CLOETENS, E. BOLLER 14.1. Introduction to damage in superplasticity 14.2. Usual techniques of characterisation 14.3. Experimental procedure 14.4. X-ray microtomography results 14.5. Quantification of the coalescence process 14.6. Conclusions 14.7. References

193 193 197 198 199 200 203 204

Foreword

This book collects the texts of the lectures given during the Workshop on the application ofX-Ray tomography in material science which was organised by the Groupe d'Etudes de Metallurgie Physique et de Physique des Materiaux (GEMPPM) in Villeurbanne on October 28-29 1999. Researchers from several European universities, research centres and companies attended the lectures which were given by experts in both materials science and X-ray tomography. The workshop was subsidised by the INSA Lyon, the MMC Assess european network and the Region Rhone Alpes and we would like to acknowledge their support. The scope of this European workshop was to provide material scientists with a detailed presentation of X-Ray tomography techniques, including the latest developments, and to present recent applications of these techniques in the field of structural materials. The interest of material scientists in X ray tomography arises from two facts: 1) most structural materials are opaque, and 2) it is of very crucial importance to observe what occurs in the bulk of materials when they are subjected to a mechanical loading. The apparent contradiction between these two facts has been overcome by recent progress in X Ray tomography which has allowed 3D non destructive images of structural materials, with a resolution around 1 micron, to be achieved. Synchrotron radiation sources are necessary to record these very high resolution images. Moreover, the phase contrast images, easily obtainable with X ray sources emitting photons with a high spatial coherence, even permits the visualisation of features with weak attenuation differences. This technique is especially well adapted for studying metal matrix composites which are among the most promising structural materials and for which damage development under stress is of crucial importance. Within this framework, the workshop was divided into two parts. The first one included a global description of the technique itself, an introduction to the

14


reconstruction algorithms, and an overview of the new possibilities offered by synchrotron X ray sources with an emphasis on the phase contrast images. The second part was devoted to the presentation of some examples of the application of X-Ray tomography to investigating micro-heterogeneous structural materials. The use of synchrotron and laboratory X-Ray sources was illustrated. The workshop was a stimulating event which has given scientists with various backgrounds the opportunity to discuss and exchange ideas and experiences. We do hope that this book will bring useful information to material scientists looking for new characterisation methods in their research fields. The organisers, Jose Baruchel Staff Scientist, Group Leader ESRF Jean-Yves Buffiere Maitre de conferences INSA Lyon Eric Maire Charge de recherches INSA Lyon Paul Merle Professeur INSA Lyon Gilles Peix Maitre de conferences INSA Lyon

Chapitre 1

General principles

Among the different methods allowing to obtain, in a non-invasive way, the image of a slice of matter within a bulky object, X-ray transmission tomography is widely used in both the medical and the industrial fields. In the latter case, defect detection, dimensional inspection as well as local characterization are possible. Non destructive testing, process tomography and reverse engineering are thus feasible. A wide range of sizes can be 1 mm small inspected, starting from a sample, up to a whole rocket motor (several meters in diameter). The present paper describes the physical basis and give examples of some industrial applications. The main reconstruction artifacts are described.

1.1. Introduction Tomography is referred to as the quantitative description of a slice of matter within a bulky object. Several methods are available, delivering specific images, depending on the selected physical excitation: - ultrasonics, - magnetic field (in the case of nuclear magnetic resonance imaging), - X and gamma-rays (y rays), - electric field (in the case of electrical impedance or capacitance tomography). In the field of industrial non-destructive testing (NOT), as well as in the field of materials characterization, X-ray or y-ray tomography is mostly used today. Tomography is a relatively "new" technique. The very first images were obtained in 1957 by Bartolomew and Casagrande [BAR 57]: they characterized the density of

16 X-ray tomography in material science particles of a fluidized bed, inside a steel-walled riser. The first medical images were performed by Hounsfield in 1972, and most industrial applications were developed much later, in the 1980's. This slow development can be explained by the huge amount of data to handle, and thus by the need for high speed and high memory computers. Industrial benefits of what is called computed tomography (CT) today are numerous. This is due to the wide range of potential applications, starting from the small sample, 1mm in size, dedicated to the characterization of advanced composite materials, and displayed in three dimensions with a one micrometer (urn) voxel size, up to the single slice image, across a 1 meter diameter riser, with a five centimeter pixel size. 1.2. X and gamma-ray tomography: physical basis 1.2.1. Different acquisition set-ups The simplest set-up consists in detecting the photons which are transmitted through the investigated object (Fig. 1.1): transmission tomography delivers a map of u, the linear attenuation coefficient, quantity which is in turn a function of p (the density) and Z (the atomic number).

Figure 1.1. X-ray transmission tomography

The clear separation between p and Z implies to perform either bi-energy tomography or scattered photons tomography [ZHU 95, DUV 98] (Fig. 1.2). This last technique is based on the clear differentiation between Compton and Rayleigh scattered photons. The ratio between those two measured quantities is purely proportional to Z and is not affected by the density. The third possibility is to detect photons emitted by the investigated object itself. Such is the case when gamma-ray sources are distributed inside a nuclear waste container, for instance. Emission tomography is thus performed [THI 99] (Fig. 1.3). An alternative is encountered when the distributed source is a positon emitter: the

General principles

17

local positon annihilation delivers pairs of 0.51 MeV annihilation photons which are detected outside. This is the PET technique, used in the medical field.

Figure 1.2. Scattered photons tomography

Figure 1.3. Emission tomography

1.2.2. X-ray transmission tomography The present paper will be focused on transmission tomography, which is widely used in both industrial and medical fields. It is based on the application of equation [1], known as the Beer-Lambert law, or attenuation law. Figure 1.4 describes the basic experimental set-up for transmission tomography inside a single slice.

N l=

Ar0exp[-

v(x,yi)dx]

[1]

path

Measuring the number N0 of photons emitted by the source and the number N, of photons transmitted throughout a single line across the sample allows to calculate the integral of ja along the considered path: [2] N

l

path

The term ((x,y) represents the value of the linear attenuation coefficient at the point (x,y). Repeating such a measurement along a sufficient number of straight lines within the same slice delivers the Radon transform of the object. Radon demonstrated in 1917 the possibility to find an inverse to that transform and thus to reconstruct the n(x,y) map of the slice [KAK 87].

18 X-ray tomography in material science As industrial tomography makes frequently use of an X-ray generator, we will focus our discussion on that kind of experimental set-up. Nevertheless, some comments will be made on gamma-ray tomography.

Figure 1.4. Physical basis of transmission tomography inside a slicef

1.2.3. The linear attenuation coefficient Transmission tomography delivers a map of (x,y), the linear attenuation coefficient, which is correlated to i) the photon energy E, ii) the density p and iii) the atomic number Z of the investigated material. Figure 1.5 displays the dependance between those quantities for carbon (Z=6) and iron (Z=26). It must be noticed that the quantity displayed on Fig. 1.5 is in fact the ratio /p, the mass attenuation coefficient.

Figure 1.5. Value of the mass attenuation coefficient for carbon and iron

Two main domains appear in Fig. 1.5. Below 200 keV, the photoelectric effect dominates and jj/p is sharply dependant on E and Z. Equation [3] is often used to describe this behaviour [ATT 68]:

General principles

19

[3]

where K is a constant. Such an equation implies that, for any given photon energy, is proportional to p and to Z4. Performing images in the photoelectric domain implies two main characteristics: - a comparison of p between two areas of the object (or between two objects) can be achieved only in the case when Z is constant (same atomic element or same composition), - a change in p between two areas can be cancelled by a change in Z in the opposite direction. It thus appears that a clear separation between Z and p can not be obtained, in the photoelectric domain, unless two tomographic images are performed, using two different energies. Within the Compton domain, above 200 keV, u can be considered as weakly dependant on Z and on photon energy. Tomography thus delivers an information which is nearly proportional to p. However, due to the higher photons energy, and hence to the lower value of u, the contrast within the object image is lower, as can be derived from Beer-Lambert law.

1.2.4. Different experimental set-ups In the field of industrial tomography, three different configurations are mainly encountered. They are displayed on figure 1.6.

Figure 1.6. Different experimental set-ups in the field of industrial tomography: a) first generation scanner, b fan-beam scanner, c) cone-beam scanner

20 X-ray tomography in material science Figure 1.6.a corresponds to the simplest experimental set-up. A single sensitive element is used and a rather long scanning time is needed, as the acquisition of a single linear "projection" needs a set of elementary translations. Successive projections are then acquired, corresponding to different value of the angle of rotation. A half turn is sufficient to reconstruct the image of a slice. Figure 1.6.b implies the use of a linear array. Acquisition is shorter, as a whole linear projection is acquired at a time. A complete turn is needed since the beam diverges. Figure 1.6.c makes the best use of the X-ray cone-beam; one turn of the object is needed. The Feldkamp algorithm [PEL 84] allows the direct 3D reconstruction of the whole object.

1.3. Different scales, different applications 1.3.1. Industrial tomography The main application in the field of X-ray tomography is Non-Destructive Testing (NOT) of manufactured components, i.e. detection of internal defects. Among other issues there are i) "reverse engineering", whose purpose is the geometrical inspection of a component, in such a way to assist the design, ii) local characterization of materials (density measurements, for instance) and Hi) process tomography, able to deliver some kind of control on a continuous manufacturing process. As industrial applications involve a broad range of sizes and a great variety of materials to be inspected, the corresponding devices may be very different. 1.3.1.1. Different photon sources Inspection of small components can be performed using a standard industrial Xray tube (160 kV for instance). Much attention must be paid to the stability of both the high-voltage and the anode current, because the consecutive projections must be acquired within constant conditions. A focus size within the range 1 to 3 mm is acceptable. Inspecting heavier components may require a 450kV tube, or even a linear accelerator. Two different high capacity scanners were constructed by the french Atomic Energy Commission (CEA-LETI, Grenoble). A 420 kV X-ray generator in the first case and an 8 MeV linear accelerator in the other case allow the complete inspection of a whole (empty) rocket motor, up to 2.3 meters in diametre, of a nuclear waste container or of a whole car engine. Gamma-ray sources can be used, in spite of the very low emitted photon flux. The Elf Research Centre (Solaize-France) uses a cesium 137 source with an activity up to 18 GBq (gigabecquerels). The high monochromatic energy (662 keV) delivered by the source allows to map the density of solid particles inside a fluidized

General principles

21

bed, through the steel wall of the riser (0.85 meter in diametre). A single source and a single detector (Nal) are used, thus constituting a first generation tomograph, as shown in Fig. 1.6a. The scan lasts 3 hours [BER 95]. The University of Bergen and the Norsk-Hydro Company built a static device using a set of five americium 241 sources (energy: 60 keV) distributed around a pipe [JOH 96]. A linear array comprising 17 semiconductor detectors is set opposite to each source, allowing a near real-time (0.1 second) imaging of the slice. The purpose is to visualize the liquid components (oil, water) apart from gas within a pipe. This application is an example of process tomography, i.e. fast imaging dedicated to the control of a manufacturing process. 1.3.1.2. Different families of detectors Four main families of detectors can be found: 1. gas ionisation detectors were used in the early medical scanners. They are still in use today in some industrial applications. Their main characteristic is their high dynamic range. Filled with gas having a high atomic number, they can be used even with high energies. Linear arrays are available. 2. image intensifiers (I.I.) are used in "desktop" scanners for industrial NDT of small components. Their low dynamic range and the inherent distortion of the image need some care. Significative 3D images can nevertheless be obtained. 3. scintillation detectors, composed of a fluorescent material (e.g. gadolinium oxysulphide Gd2O2S, or caesium iodide Csl) are nowadays widely used. Those detectors are of two kinds: i) the fluorescent material is directly coupled to an array of photodiodes [KAF 96] or of photomultipliers (in some cases the coupling is realized using tapered optic fibers), ii) the fluorescent material is spread on a screen, which is optically coupled to a CCD camera via a lens [CEN 99]. 4. arrays of semiconductors (e.g. CdTe or ZnCdTe), which allow a direct photon detection are promising. High energy applications are possible.

1.3.2. Microtomography Considering advanced materials characterization, the need of 3D images with a very high resolution (a few um) obtained through a non invasive method is growing. Figures 1.7 and 1.8 show two specific examples of 3D tomographic images performed with two different scanners, conceived and built in our laboratory [KAF 96] [CEN 99]. Such 3D images are then used by the researchers for the modelisation of the mechanical properties of materials, within finite elements models computations. For such applications micro-focus X-ray tubes, with a focus size in the range 5 - 1 0 micrometers, are used. A very low focus size allows to set the investigated object directly at the window of the tube. A geometrical magnification

22 X-ray tomography in material science can thus be obtained. Figure 1.9 shows that the magnification can be easily modified. A limit exists to the magnification: the geometrical unsharpness [HAL 92] must be kept lower than p, the size of the sensitive element of the detector (sampling step). In practice, this upper boundary to the magnification Gg can be computed according to equation [4], where represents the size of the focus: [4]

Figure 1.7. 3D rendered view of a tomographic image of a composite material with 400 yon glass balls inside an organic matrix. (Herve Lebail; Laboratory GEMPPM). The voxel size is set to 42 jjm

Figure 1.8. 3D rendered view of a tomographic image of an aluminium foam (density 0.06) (Eric Maire; Laboratory GEMPPM). The voxel size is 150 pm. The size of the sample is 3cm

Figure 1.9. According to the location of the investigated object between the focus and the screen, different geometrical magnifications are attained

General principles

23

Designing and building such a kind of scanner implies some care in at least three domains: - the low photon flux delivered by the micro-focus X-ray tube results in long exposure times; the camera must therefore deliver a very low noise, - the choice of the photon energy is important: low energy photons deliver images with an higher contrast, but also with an higher relative noise, - the accuracy of the mechanical setting must be better than the expected image resolution. Today, the most powerful tool involves the use of synchrotron radiation. The European Synchrotron Radiation Facility (ESRF-Grenoble) delivers a huge X-ray flux and thus allows very short exposure times. A complete scan can be acquired within a few minutes, with a spatial resolution down to 1 um. On beam-line ID 19, the source is located far from the working hutch (145 meters), thus delivering photons with a high spatial coherence. This property of the X-ray flux generates diffraction features which underline the edges within the sample, and thus highlighting sharp defects. Such a phenomenon, the so-called "phase contrast" [CLO 97], allows very small defects to be detected. As the beam is non-diverging, the resolution is set by the detector itself. Transparent luminescent screens are used, with a 5 jam sensitive layer of an yttrium-aluminium (YAG) or lutetium-aluminium (LUAG) garnet, epitaxially grown on a YAG monocrystal, 170 jim in thickness; they allows a high resolution (1 fim) and a 4% to 8% efficiency for 14 keV photons.

1.4. Quantitative tomography As mentionned earlier, tomography offers many possibilities. If the goal is just defect detection, the selected resolution must therefore be adjusted to the size of the details to be observed. Much attention must also be paid to the noise of the camera or, more precisely, to its dynamic range [CEN 99]. When the inspection's issue is the determination of the accurate size of some internal feature, or the local characterization of materials (density measurement for instance), then an increased attention must be paid to the reconstruction artifacts. They create artificial patterns inside the reconstructed slice (streak artifacts), or they locally modify the pixels values (cupping effect), and hence the quantitative result [ISO 99] [SCH 90]. In the following lines, we will describe the main physical mechanisms leading to erroneous reconstructions, as well as the shape of the corresponding artifact in the reconstructed image. - Beam hardening As an X-ray tube delivers a polychromatic spectrum, differential attenuation of photons within the investigated object leads to the rapid attenuation of the lowest

24 X-ray tomography in material science

energy photons, and hence to the gradual increase of the mean energy along the path. The reconstruction algorithm uses, for the reconstruction of any single point, experimental data corresponding to individual rays impinging the point of interest, but coming from different orientations. The corresponding information therefore corresponds to different attenuations, and hence different energies, and different values of ji. Two kinds of artifacts are generated by beam-hardening: i) cupping effect and ii) streaks. Cupping effect corresponds to measured values of \JL which are corrupted, thus preventing the measurement of the "true" density. As the measured values, inside an homogeneous sample, are lower at the center than at the edges, the name of cupping effect is generally used to describe this artifact. Projections can be corrected by acquiring an image of a step-wedge, made of the same material, in such a way to correlate the mesured attenuation to the true material thickness. Streaks artifact correspond to abnormal values along lines which correspond, inside the object, to high attenuation. Beam hardening artifacts can be avoided when using some filter, i.e. a metallic foil, directly set at the window of the X-ray tube and intended to pre-harden the spectrum [KAF 96]. Figure 1.10 displays an example of streaks inside the tomographic image of a set of six samples surrounded by air (Fig l.lO.a); the streaks are suppressed by the use of a copper filter, 0.1 mm in thickness (Fig. l.lO.b).

Figure 1.10. The reconstructed slice (l.lO.a) is corrupted by streaks due to beamhardening (l.lO.a). Filtration with a foil of copper, (0.1 mm) nearly suppresses the streaks (l.lO.b). The high voltage used for both images is 100 kV

Beam hardening is also avoided when using a monochromatic y-ray source. But it must be kept in mind that y-ray sources deliver a very low photon flux (typically one hundredth of the flux delivered by a tube). Tomography using synchrotron

General principles

25

radiation does not generates artifacts because a monochromator is always used, thanks to the huge X-ray flux. - Detector saturation To obtain a reconstruction which is free of defect, the signal delivered by every cell of the detector must be strictly proportional to the photon flux. Thus high values (approaching the upper limit of the digitization range) as well as low values (approaching the noise level) of the flux must be avoided. Streaks artifacts, similar to those obtained in the case of beam-hardening, are generated along lines which correspond to high attenuation. - Aliasing High (spatial) frequencies are encountered in the signal corresponding to every projection. They are due to the steep edges which are eventually present in the object. As the detector samples the signal (all along the projection) with a non-zero step, high frequencies corrupt the data, within the Fourier domain. Streaks are generated [KAK 87]. On figure 1.11, aliasing is visible at the corners of the objects. - Scattered photons Photons scattered by the sample or by its environment deliver a wrong information which leads to cupping effect. Collimation can improve the reconstructed image.

Figure 1.11. Aliasing at the corners

Figure 1.12. Ring artifacts

-111 corrected detector The signal delivered by every sensitive cell of the detector must be linearly spread between the offset level (corresponding to the absence of photons) and the gain level (corresponding to the non-attenuated flux). A bad correction of one cell will generate, in the reconstructed image a "ring artifact", i.e. the image of a ring,

26 X-ray tomography in material science

centered on the pixel corresponding to the location of the rotation axis. On figure 1.12 a great number of concentric rings are visible. - Spatial distortion of the detector Distortions of the projections, due for instance to the camera (e.g. distortions due to the lens) deliver artifacts which can be corrected by software. - Centering error The reconstruction requires the knowledge of the location of the projection of the center of rotation within the detector. Distortions are generated when the reference to the centre is erroneous.

1.5. Conclusions X and y-ray tomography allow a great number of potential applications. The measured quantity is in fact the linear attenuation coefficient \i, and not directly the density. A careful choice of the photons energy and the selection of a detector with a high dynamic range allows to lessen the noise to a reasonable level. Coefficient \JL can be estimated with an accuracy slightly better than 1%.

1.6. References [ATT 68] Anrx F.H.R., ROESCH W.C., Radiation Dosimetry, Academic Press, 1968. [BAR 57] BARTHOLOMEW R.N., CASAGRANDE, R.M., "Measuring solids concentration in fluidized systems by gamma-ray absorption", Industrial and Engineering Chemistry, vol. 49, n. 3, p. 428-431, 1957. [BER 95] BERNARD J.R., Frontiers in Industrial Process Tomography, Engineering Foundation, Ed. DM SCOTT& RA WILLIAMS, New-York, p. 197, 1995. [CEN 99] CENDRE, E. et al., "Conception of a high resolution X-ray computed tomography device; Application to damage initiation imaging inside materials", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 362-369, 1999. [CLO 97] CLOETENS P., PATEYRON-SALOME M., BUFFIERE J.-Y., PEK G., BARUCHEL J., PEYRIN F., SCHLENKER M., "Observation of microstructure and damage in materials by phase sensitive radiography and tomography", J. Appl. Phys., vol. 81, n. 9, p. 5878-5886, 1997. [DUV 98] DUVAUCHELLE P., Tomographie par diffusion Rayleigh et Compton avec un rayonnement synchrotron: Application a la pathologic cerebrale, these de doctoral, universite de Grenoble 1, 1998. [PEL 84] FELDKAMP L.A., DAVIS L.C., KRESS J.W., "Practical cone-beam algorithm", J. Opt. Soc., vol. 1, n. 6, p. 612-619, 1984.

General principles

27

[HAL 92] HALMSHAW R., "The effect of focal spot size in industrial radiography",flrif/s/i Journal of NOT, vol. 34, n. 8, p. 389-394, 1992. [HAR 99] HARTEVELD W.K. et al. "A fast active differencial capacitance transducer for electrical capacitance tomography", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 571-574, 1999. [ISO 99] iso/TC 135/SC 5 , ISO document "NDT Radiation methods- Computed tomography", Part I: Principles; Part II: Examination Practices, 1999. [JOH 96] JOHANSEN G.A, FR0YSTEIN T., HJERTAKER B.T., OLSEN O., "A dual

sensor flow imaging tomographic system", Meas. Sci. Techn., vol. 7, n. 3, p. 297-307, 1996. [KAF 96] KAFTANDJIAN V., PEDC G., BABOT D., PEYRIN F., "High resolution X-ray computed tomography using a solid-state linear detector", Journal of X-ray Science and Technology, vol. 6, p. 94-106, 1996. [KAK 87] KAK A.C., SLANEY M., Principles of Computerized Tomographic Imaging, IEEE Press, 1987. [PIN 99] PlNHEIRO P.A.T. et al., "Developments of 3-D Reconstruction Algorithms for ERT", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 563-570, 1999. [SCH 90], SCHNEBERK D.J., AZEVEDO S.G., MARTZ H.E., SKEATE M.F., "Sources of error in industrial tomographic reconstruction", Materials Evaluation, vol. 48, p. 609-617, 1990. [THI 99] THIERRY R. et al., "Simultaneous Compensation for Attenuation, Scatter and Detector Response for 2D-Emission Tomography on Nuclear Waste within Reduced Data", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 542-551, 1999. [ZHU 95] ZHU P., PEIX G., BABOT D., MULLER J., "In-line density measurement system using X-ray Compton scattering", NDT & E International, vol. 28, n. 1, p. 3-7, 1995.


Chapitre 2

Phase contrast tomography

Hard X-ray radiography and tomography are common techniques for medical and industrial imaging. They normally rely on absorption contrast. However, the refractive index for X-rays is slightly different from unity and an X-ray beam is modulated in its optical phase after passing through a sample. The coherence of third generation synchrotron radiation beams makes a simple form of phase-contrast imaging, based on simple propagation, possible. Phase imaging can be used either in a qualitative way, mainly useful for edge-detection, or in a quantitative way, involving numerical retrieval of the phase from images recorded at different distances from the sample.

2.1. Introduction The phase of an X-ray beam transmitted by an object is shifted due to the interaction with the electrons in the material. Imaging using phase contrast as opposed to attenuation contrast is a powerful method for the investigation of light materials but also to distinguish, in absorbing samples, phases with very similar X-ray attenuation but different electron densities. Phase contrast imaging was pioneered in the early seventies by Ando and Hosoya [AND 72], who obtained images of bone tissues and of a slice of granite using a Bonse-Hart type interferometer [BON 65]. This technique developed into a quantitative three-dimensional imaging technique. Because of the limited quality of available lenses, elaborate forms of phase contrast imaging such as Zernike phase-contrast [ZER 35] or off-axis holography [LEI 62] are presently ruled out for hard X-rays. Three methods of phase sensitive imaging exist: the interferometric technique [MOM 95, BEC 97], the Schlieren technique [FOR 80, ING 95] and the propagation technique [SNI 95, CLO 96]. They are compared in section 3. The main advantages of the method used in this work, the propagation technique, are the extreme simplicity of the set-up and the better spatial resolution.

30


This technique was mostly used up to now in the so-called 'edge-detection regime' to image directly the discontinuities in refractive index in the object. It is however possible to fully exploit the quantitative information entangled in the Fresnel diffraction patterns towards high resolution quantitative phase tomography. The 'holotomographic' reconstruction is performed in two steps: first the optical phase of the wave exiting the sample is retrieved numerically from images recorded at different distances from the sample. The refractive index distribution is then reconstructed from a large number of phase maps using a classical tomographic algorithm. Results of quantitative phase tomography on samples of interest to materials science are discussed.

2.2. X-ray phase modulation The interaction of a wave with matter affects its amplitude and phase. This can formally be described by the complex refractive index n of the medium. Because its value is nearly unity, it is usually written for X-rays as n = l - < J + i/?

[1]

A plane monochromatic wave propagating along the z-axis in vacuum is of the form exp(i^ L z) with A the X-ray wavelength. In a material with refractive index n this becomes exp(m^ L z). The refractive index decrement 6 results in a phase variation compared to propagation in vacuum. The imaginary part J3 determines the attenuation of the wave. The X-ray intensity is the squared modulus of the wave and the absorption index (3 is simply proportional to the linear absorption coefficient p.

,=

f>

[2]

The absorption index has a complex energy and composition dependence. It varies abruptly near the characteristic edges of the elements. The refractive index decrement 6 on the other hand is primarily due to Thomson scattering and has a much simpler dependency on the energy and the material characteristics. S is essentially proportional to the electron density in the material. Generally, it can be expressed as

where the sum extends over all atoms p, with atomic number Zp, in the volume V, rc = 2.8 fm is the classical electron radius, and f'p is the real part of the wavelengthdependent dispersion correction, significant near absorption edges, to the atomic scattering factor. If the composition of the material is known in terms of mass fractions qp, the following equivalent expressions can be used

Phase contrast tomography

31

[5]

with NA Avogadro's number and Ap the mass number. 6P and pp are respectively the refractive index decrement and mass density of the pure species. If the dispersion correction fp can be neglected, 6 is proportional to the electron density pe, i.e. S = r c A 2 p e /(27r). The ratios ZP/AP appearing in Equation 4 are similar for many atomic species (« 1/2), and 6 is thus to a good approximation determined by the mass density p of the material [GUI 94]

[6] Both 6 and ft are small, typically 10~5 - 10~6 and 10~8 - 10~9 respectively for light materials, indicating the power of phase sensitive imaging compared to the absorption. Figure 1 shows the ratio S /ft, a figure of merit for phase effects compared to attenuation effects, as a function of the X-ray energy E for aluminium. The energy range includes soft X-rays and hard X-rays. In the soft X-ray range, more precisely in the 'water window' where soft X-ray microscopes usually operate, a gain exists but it is relatively modest. On the other hand in the hard X-ray range (energies above 6 keV) this ratio increases with energy to huge values (up to 1000). Practically, if one selects for example an X-ray energy of 25 keV to be able to cross a thick aluminium sample, a hole in this metal should have a diameter of at least 20 /zm to produce 1 %

Figure 2.1. Ratio S//3 of the refractive ondex decrement and the absorption index as a function of the X-ray energy for the element aluminiu. This is a figure of merit for phase effects compared to attenuation effects

32


absorption contrast. Using the effect on the phase, the minimum detectable hole is reduced to about 0.05 yum. X-rays are adapted for imaging of thick samples thanks to their low absorption at high energies. If it is possible to visualise the phase of the transmitted wave, the sensitivity and spatial resolution remain good. For inhomogeneous samples the wave at the exit of the sample will be modulated in both phase and attenuation. Propagation inside the sample itself can usually be neglected and it is possible to project the object onto a single plane perpendicular to the propagation direction. The transmission function T(x, y) gives the ratio of the transmitted and the incident amplitudes. It can be compared to exp(— f n(x, y, z)dz) that gives the ratio of the transmitted and the incident intensities according to LambertBeer's law. This transmission function corresponds to the projection of the refractive index distribution through T(x,y) = A(x,y)eirtx>ri

[7]

with the amplitude A(x,y) = e-W*'*)

and

B(x,y} - y j 0(x,y,z)dz

[8]

(p(x, y) = Y / [1 - (15x15x15 voxels) for the voxels along the planes shown in (a)

The iso-surface presentations of the limits of mean local mass density reveal as well the 3D arrangement of the corresponding hard or soft regions across the sample Fig I2.5b shows a V-like interconnection of hard regions across the sample which will increase the compression resistance in the long direction. The soft regions of that sample - see Fig. 12.5c, d - are oriented rather parallel to the direction of compression suggesting, that they will not be identical with deformation bands [KRI yybj. Consequently the spatial distribution of hard and soft regions has to be considered too as an additional quality criterion. High resolution CT reveals details in the cellular structure as shown in Fig 126 The 3D iso-surface representation of the structure shows cell walls down to about 100 Mm thickness as well as tiny shrinkage pores in cell wall nodes. Such CT data can be used to study deformation mechanisms by computing 3D-displacements of structural features [FOR 98].

Determination of local mass density distribution

171

Figure 12.5. Iso-surfaces of mean local mass densities in one AISU2-ALULIGHT test sample (a = b = 22 mm, c = 30 mm) of average density pm= 0.5 g/cm3: a) hard regions of mean local densities p > 1.67-pm in averaging volumes of 2x2x2 mm3, b) hard region of p > 1.33-pn, within 6x6x6 mm3, c) soft regions of p < 0.67- pm within 6x6x6 mm3, d) same density limit as c) but within 7 x 7 x 10mm3, i.e. 1/3 of the edge lengths of the sample

172


Figure 12.6. Iso-surface at a level of 50% of pAI computed from high resolution 3D technical CT data (voxelsize: 40x40x 40 fjm3). The volume of 8x8x8 mm3 within a casting alloy ALL/LIGHT sample (same as in Fig. 5) shows a few cells, where the roughness of the cell walls, defects in cell walls, shrinkage pores in nodes and an ensemble of neighbouring cells can be seen

12.5. Application of the mean local density distribution X-ray transmission images of cellular samples are ambiguous in detecting big pores. Any conventional X-ray transmission system including medical CT is suitable for identification of density variations of cellular metallic parts of regular shape, based on 2D mean local density maps. 3D X-ray computed tomograms of even moderate resolution provide the basis for the calculation of 3D mean local mass densities to identify soft and hard regions and their spatial arrangement within the component. However the experimental results show, that the arrangement of these regions has to be considered when describing the mechanical behavior of foams (macroscopic anisotropic behavior). Fig. 12.7 shows the measured compressive stress-strain curves and the density distribution of two samples viewed from perpendicular directions. The density distribution is derived from medical CT measurements. The 2D local mean densities were calculated by averaging the CT data over columns having the thickness of the sample (20 mm) and a cross-section of 5 x 5 mm. The largest extension of the columns is oriented in y-direction (z-direction) for the xz-mapping Cry-mapping). The yield strength shows a variation of about 20%, although the average density of samples was fixed at 480 kg/m3. This high variation of yield stress can be explained by the density distribution. The sample A is characterized by the lower yield stress and shows a soft zone of low density, oriented nearly perpendicular to the loading direction; sample B has a higher strength and has a hard zone of high density


173

parallel to the loading direction. Density variation limits can be chosen as quality criteria, but have to be combined with a reasonable choice of averaging volume.

Figure 12.7. Compressive stress-strain curves of Alulight (cast alloy) samples, both having an average density of 0.48 g/cm3 but different local density distribution as shown by mappings of these samples in xz and xy planes

Figure 12.8. The sample A: a) The cellular structure along the observed surface recorded by a digital camera at 2% overall strain. Deformation zones are marked by the white lines; b) distribution of the corresponding calculated equivalent plastic strain (PEEQ) on this surface

The assessment of the spatial arrangement of such hard and soft regions in mechanically loaded components can also be used for the meso-mechanical simulation of heterogeneous materials. The results of this simulation for foamed aluminium are presented, by demonstrating the calculated elastic-plastic behavior of a sample. A 3D density mapping which was calculated with an averaging volume of 5 x 5 x 5 mm3 was used in this finite element simulation. The detailed modeling has

174


been described in [KRI]. Fig. 12.8b shows the distribution of the equivalent plastic strain [HIB 98] occurring on the sample's surface, which was observed in the experiment. The calculated maximum strain is in the same position as observed in the experiment by optical recording (Fig. 12.8a). The capability of the 3D model enables to follow the forming of deformation bands in the interior of the sample too. The calculated 3D plastic strain field for this sample is given in Fig. 12.9. Four stages, showing the growth of deformation bands, are depicted in this figure by indicating the regions having more than 1% equivalent plastic strain.

Figure 12.9. Simulated 3D propagation of plastic regions in the interior of the sample A at a) 0.7%; b) 1.0%; c) 1.23% and d) 1.5% overall strain. Direction x is the compression direction

Acknowledgements The authors gratefully acknowledge: the provision of ALULIGHT samples by Leichtmetall-Kompetenzzentrum Ranshofen (A) and Slovak Academy of Science, Bratislava (SK); the admission to use CT at the Department of Radiology, Division of Osteoradiology, University of Vienna (A) and the Ferderal Institute for Materials Research and Testing, Berlin (D). The work was funded by the Austrian Ministry of Science and Transport.


175

12.6. References [ASH 99] ASHOLT P., "Aluminium Foam Produced by the Melt Foaming Route Process Properties and Applications", MetFoam99 a>, p. 133-140, 1999. [BAM 97] "Computertomographie", Leaflet, Federal Institute for Materials Research and Testing, Berlin (Germany), 1997. [BAU 99] BAUMGARTNER F., GERS H., "Industrialisation of P/M foaming process", MetFoam99 a>, p. 73-78, 1999. [COR 99] CORNELIS E., KOTTAR A., SASOV A., VAN DYCK D., "Desktop X-ray micro-tomography for studies of metal foams", MetFoam99 a), p. 233-240, 1999. [COP 94] COPLEY D., EBERHARD A., MOHR A., "Computed Tomography Part I : Introduction and Industrial Applications", Journal of Materials, vol. 46, no 1, p. 14-26, 1994. [DAX 99] DAXNER T., BOHM H.J., RAMMERSTORFER F.G., "Influence of microand meso-topological properties on the crash-worthiness of aluminium foams", MetFoam99 a>, p. 283-288, 1999. [DEG] DEGISCHER H.P., DOKTOR M., PRADER P., "Assessment of metal matrix composites for innovations - a Thematic Network within the 4th EUframework", Euromat 99 (to be published). [DEG 99] DEGISCHER H.P., KOTTAR A., "On the Non-Destructive Testing of Metal Foams", MetFoam99 a), p. 213-220, 1999. [EVA 99] EVANS A.G., HUTCHINSON J.W., "Mutifunctionality of Cellular Metal Systems", MetFoam99 a>, p. 45-56, 1999. [FOR 98] FOROUGHI B., "Study of cellular deformation of Al-Foam under Compressive Loading", Junior Euromat, 1998. [GIB 97] GIBSON L.J., ASHBY M.F., Cellular Solids : Structure and Properties, 2nd Ed., Cambridge University Press, 1997. [GRO 99] GROTE F., SCHIEVENBUSCH A., "Characterization of cast and compressed foam structures by combined 2D-3D analysis", MetFoam99 a), p. 227-232, 1999. [HIB 98] HIBBIT, KARLSSON and SORENSON INC., HKS ABAQUS/Standard user manual, Version 5.8, 1998. [HOP 99] HOPLER E., SCHORGHUBER F., SIMANCIK F., "Foamed aluminium cores for aluminium castings", MetFoam99 a>, p. 79-82, 1999. [KRE 99] KRETZ R., HOMBERGSMEIER E., EIPPER K., "Manufacturing and testing of aluminium foam structural parts for passenger cars demonstrated by example of a rear intermediate panel", MetFoam99 a>, p. 23-28, 1999. [KRI] KRISZT B., FOROUGHI B., KOTTAR A., DEGISCHER H.P., "Mechanical Behavior of Aluminium Foam Under Uniaxial Compression ", Euromat 99 (to be published). [KRI 99a] KRISZT B., KOTTAR A., DEGISCHER H.P., "Strukturanalyse von geschaumten Aluminium mittels Computertomographie", Metalle/Werkstoffwoche 98, Symposium 8, Ed.: R. Kopp, Bd. 6, p. 687-692, Wiley-VCH, 1999.

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[KRI 99b] KRISZT B., FOROUGHI B., FAURE K., DEGISCHER H.P., "Deformation behavior of aluminium foam under uniaxial compression (a case study)", MetFoam99 a), p. 241-246, 1999. [MEP 96] "Alulight", Leaflet, Mepura Ges.m.b.H., Ranshofen/Austria, 1996. [MIY 99] MIYOSHI T., ITOH M., AKIYAMA S., KITAHARA A., "Aluminuim Foam, 'ALPORAS': The Production Process, Properties and Applications", MetFoam99 a>, p. 125-132, 1999. [SEE 99] SEELIGER H.-W., "Application Strategies for Aluminum-Foam-Sandwich Parts (AFS)", MetFoam99 a>, p. 29-36, 1999. [SIM 99] SIMANCIK F., MINARIKOVA N., CULAK S., KOVACIK J., "Effect of foaming paramters on the pore size", MetFoam99 a\ p. 105-108, 1999. a)

Metal foams and porous metal structures, International conference, 14th-16th June 1999, Bremen (Germany), Ed.: J. Banhart, M.F. Ashby, N.A. Fleck

Chapitre 13

Modelling of porous materials evolution

By providing 3D images of the micro-geometry, synchrotron micro-tomography is offering huge possibilities to porous materials evolution modelling. Through three examples, reservoir rock diagenesis, carbon-carbon composite densification and ceramics sintering, this text illustrates those possibilities and puts into evidence the need of a strong theoretical framework. The volume averaging method is succinctly presented in the case of pressure solution in sandstone reservoirs. The fundamental concept of Representative Elementary Volume is introduced. Various problems specific of the considered materials are described and solutions to be used during the acquisition or in post-processing are outlined.

13.1. Introduction Understanding the evolution with time of natural or artificial porous materials is essential for many applications. In this paper, three examples that are presently studied within the research group CM3D* from Bordeaux, will be considered: - mineral diagenesis of reservoir rocks for oil exploration and production, - carbon/carbon (C/C) composites elaboration by vapour-phase densification of carbon-fibre preforms for thermostructural materials production, - sintering of advanced ceramics with controlled porosity distribution for mechanical properties enhancement (strength and toughness). In porous media physics the concept of "change of scale" is fundamental. The common objective of all the methods used to perform the change of scale (volume averaging, homogenisation, etc...) is to move from the local scale (the pore scale for the cases considered here) to a larger one where the porous material behaves as an

178


equivalent continuous material characterised by effective properties. In this paper we will only make use of the volume averaging method. At the local scale everything is known; the equations and the associated boundary conditions governing the physical phenomena, the values of the physical parameters appearing in those equations and the geometry. Applying the change of scale operator to the problem defined at the local scale will make the equations of the large scale physics emerge. The effective properties appearing in those equations can be calculated from solutions of differential problems (eventually integrodifferential) stated at the local scale: the closure problems. Furthermore some methods, like volume averaging, provide conditions that must be verified to insure that the change of scale is possible. Depending on the treated problem, those conditions might be more or less difficult to verify, as illustrated by the three examples examined here. Mineral diagenesis of reservoir rocks is an evolution process for which the above-cited conditions are easily verified. Indeed, it is a very slow process that takes place within very large domains that are rather homogeneous. In this case the change of scale puts into evidence an evolution governed by physical phenomena at the pore scale but having a kinetics directed by fluxes at the large scale (fluid and mater fluxes). For C/C composite elaboration, the change of scale might be difficult if the size of the sample is too small with respect to the characteristic dimensions of the textile architecture or if the characteristic time of the chemical reactions responsible for the densification is too short. Practically, this last case is avoided because it produces reaction fronts and heterogeneities that drastically reduce the mechanical properties. For large enough samples, the change of scale is possible and all the remarks made for mineral diagenesis are valid. For small samples, direct simulation is still possible but one cannot define effective properties intrinsically related to the porous material. In particular, the equivalent properties would be affected by the boundary conditions applied at the "large" scale. In the case of sintering, particles are very small and the conditions for the change of scale are generally easily verified. Evolution, i.e. sintering, is caused by diffusion within the solid phase. This diffusion is governed by the local curvature of the interface and the kinetics is controlled by large-scale parameters, like temperature for instance. The corresponding coupling is weaker than for mineral diagenesis because those parameters are constant at the local scale (it is not the case for the local fluid flow for instance). Because of the differences pointed out above, the modelling strategy will slightly differ for each of these three examples. Nevertheless, a good characterisation of the 3D micro-geometry is always essential. For this purpose, computed microtomography (CMT) is a very well adapted tool, as it will be shown in the following paragraphs. In the first one, the volume averaging method is succinctly introduced


179

through its application to an important mechanism of mineral diagenesis: pressure solution. The usefulness of CMT will appear clearly in this example and will be reinforced by the two ensuing paragraphs where preliminary but encouraging results will be given for C/C composites and ceramics sintering. 13.2. Evolution of sandstone reservoir rocks by pressure solution Rock deformation by pressure-solution is due to the variation of the chemical potential with stress; in some conditions, the stress concentration at the grain-tograin contacts (Figure 13.1) induces an accelerated dissolution of minerals in these zones and a precipitation of those minerals at interfaces under low stress. This microscopic mechanism is the origin of one of the three most important deformation modes of earth crust. More practically, research of a better understanding of this mechanism is justified by the fact that in deep hydrocarbon exploration (more than 3 km), compaction and cementation of the quartz matrix is the main cause of error in sandstone hydrocarbons reservoir quality evaluation. In the literature there are two main theoretical models for pressure solution: - the free-face pressure solution model adopted in this work, where dissolution occurs at the periphery of the contacts. This process leads to a reduction of the contact area, plastic deformation and finally collapse of this zone. - the water-film diffusion model where dissolution is supposed to occur within the grain-to-grain contacts. In this model, the existence of a thin water film able to support a high normal effective stress is necessary. Dissolved species are transported from the film to pore space by diffusion. 13.2.1. Local equations and volume averaging The pore scale configuration is pictured on Figure 13.2. The two phases, the fluid P and the solid a, are in contact at Apa their interface. A dissolved compound a is moving within the fluid phase by diffusion (to simplify the demonstration, convection is not considered here). The compound a can react with the solid phase at the interface. Equations [1] and [2] govern the transport at the local scale. —- = V.(DVCfl)

in the p phase

- npo . (DVCa) = k (Cfl - C*)

on the interface A po

[1] [2]

where Ca is the concentration of the compound a in the p phase, D its molecular diffusivity, t the time, k the reaction rate coefficient and C* the equilibrium concentration. C* is a function of pressure, temperature and stress.

180


Figure 13.1. Tangential stress concentration in a simple spheres packing: all the zones having a stress value greater than 20% of the maximum are near the contacts

Figure 13.2. Local scale geometry and notations used in the text

The intrinsic phase average of the concentration is defined by: [3]

where V is the Representative Elementary Volume. Applying this operator to equations [1] and [2] and using the spatial averaging theorems [WHI 99], the following equation is obtained:

where GRAY's decomposition (equation [5]) of the local concentration has been used to distinguish the smooth part, the averaged concentration linked to transport at the large scale, from the spatial deviation produced by the reaction at the interface and by the interface itself.

In a similar way the equilibrium concentration is decomposed into two terms:


181

Assuming that k is constant and introducing the specific surface av defined by:

equation [4] can now be written:

[8]

Equation [8] is still containing microscopic terms that have to be eliminated. Following the same way as [WHI 99], it can be inferred that the spatial deviation of the concentration can be represented by the following expression:

where f is a vector and s a scalar. Both are solutions of partial-differential problems, called closure problems, which have to be solved at the local scale:

f mainly takes into account the effects of the micro-geometry on diffusion and s takes into account the previous effects plus the effects of the chemical reactions on the interface. The problems [10] are similar to the closure problems classically obtained for diffusion-reaction in porous media, the specific effects of local stress concentration being carried by £*• Indeed, this term is not an unknown for transport equations. It is a local property of the interface that has to be calculated by solving an almost decoupled local mechanical problem. The transport equation at the large scale can now be written:

182


where the effective diffusivity is given by [WHI 99]:

[12]

The local flux of matter can be expressed by equation [13]. A new version of the model, taking into account the existence of water films at the grain-to-grain contacts, is under development. When the new expression of the local flux will be available, it will be possible to compare the evolutions predicted by both models for the same starting geometry. In parallel, both models will be confronted to real data (§ 13.2.3) looking for arguments supporting a choice between them. 13.2.2. REV size of a Fontainebleau sandstone sample The concept of Representative Elementary Volume (REV) is central in the change of scale theory for porous media. However there is no explicit formula for the REV size ro, the main indication being that it has to be large enough compared to the local scale characteristic lengths (lp, la) and small enough compared to the largescale characteristic length (L): [14]

Using tomography data acquired at the ESRF (ID 19) for ELF-EP, the 3D microgeometry shown on Figures 13.3 and 13.4 has been reconstructed. It is the central zone of cylindrical sample (diameter of 6 mm). The voxels are cubic (edges of 10 jam) and the complete data set comprises 256 x 256 x 256 voxels. To estimate the REV size of this Fontainebleau sandstone sample for different physical properties, the following numerical experiments have been performed: - cubic volumes of different sizes have been extracted from the central region of the complete data set; - those volumes being considered as REV, three physical properties have been computed: - porosity values from the number of voxels belonging to the fluid phase, - effective diffusivity values using equation [12] after resolution of problem [10],


183

-permeability values after resolution of a closure problem obtained by volume averaging of Stokes equations [BER 95]; - the results have been plotted as functions of the volume size (Figure 13.5).

Figure 13.3. Visualisation of the solid phase of a Fontainebleau sandstone sample

Figure 13.4. Visualisation of the fluid phase of the same sample

Figure 13.5. Evolutions of porosity, effective diffusivity and permeability with the size of the computational cell

Figure 13.5 is clearly demonstrating that it is possible to obtain, using CMT, 3D images of porous samples large enough to attain the REV size for various physical properties. 10 times the average grain diameter can be considered as a good

184


approximation of the size of a cubic REV for simple granular porous media. The subsequent and complementary question is now: what is the required size for the voxel in order to characterise correctly the fluid-solid interface? The importance of this aspect is obvious when dealing with more complex porous media and with coupled phenomena like in the following example.

13.2.3. An example of pressure solution evolution from the ELLON field Among the numerous difficulties encountered when studying mineral diagenesis of reservoir rocks, two are rather specific: - phenomena occur at very different scales (from the pore scale -100 urn- to the basin scale -500 km) and are strongly coupled, - the initial state (-100 My) is not directly accessible. Having that in mind, the Ellon field (Alwyn area, North Sea) can be considered as exceptional. Indeed, in an early stage of its evolution this sandstone reservoir has been drastically and quickly modified by two diagenetic events [POT 97]: first a general and almost complete calcite cementation (porosity initially around 40% and about 3% after cementing) and secondly a localised dissolution of the calcite cement leading to an heterogeneous formation composed of two domains separated by sharp fronts (Figure 13.6). During the subsequent evolution, the cemented zone remained unchanged (Figure 13.7) and the uncemented zone has been modified by pressuresolution and quartz over-growths (Figures 13.8-9) giving an actual porosity of about 20%. Those processes mainly occurred at the pore level with small coupling with

Figure 13.6. Sandstone sample from the ELLON field showing the interface between the cemented (C) and the non-cemented (NC) zones

Figure 13.7. Reconstructed section (512x512 pixels of 6.5 um) of a sample at the C-NC interface. Black zones are pores, dark grey zones are quartz grains and light grey zones correspond to calcite


Figure 13.8. Reconstructed section (512x512 pixels of 6.5 urn) of a NC sample. Black zones are pores and dark grey zones are quartz grains

Figure 13.9. Reconstructed (700x700 pixels of 1.8 urn) of sample. Quartz overgrows are visible. Pores are partly filled with

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section a NC clearly clay

larger scales (no water flow) and the initial state has been frozen by calcite cementation; it is why the Ellon field is an exceptional case to study. Examining Figures 13.7 and 13.8, it is evident that this sandstone is more complex than Fontainebleau sandstone. Porosity is larger but spatial distribution of this porosity is completely different. Both were rather similar quartz grains packing at the deposit (Figure 13.7) but the diagenetic mechanisms that took place have been different. The combination of classical CMT (Figures 13.7, 13.8) and local CMT (Figure 13.9) seems to be a promising solution to problems where more than one scale are relevant.

13.3. C-C Carbon/carbon composites are well known high-performance materials for thermostructural applications, such as rocket nozzles or aeroplane brakes, and their market is in appreciable extension. They are usually produced either by impregnation of a preform made of carbon fibres by pitches or mesophases or by vapour-phase densification of the same preform [NAS 99]. Chemical Vapour Infiltration (CVI) is a variant of the Chemical Vapour Deposition (CVD) process involving the cracking of gaseous species (precursors) which lead to the deposition of a solid phase on a hot substrate by heterogeneous reaction. For instance, a mixture of hydrocarbons and hydrogen is used to obtain a pyrocarbon deposit. The gaseous species are transported inside the preform by viscous flow or by diffusion.

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In isobaric-CVI (I-CVI) low pressures are required in order to avoid diffusional limitations and premature pore plugging. As high temperatures are required for the heterogeneous reaction, one has to deal with transport by Knudsen diffusion (also called Klinkenberg effect or slip flow), in addition to ordinary diffusion. The role of porous medium transport is fundamental for the monitoring and optimisation of the CVI process, and it may be easily understood that it depends in a rather complex way on the preform geometry. One of the possible ways to investigate precisely the properties of interest either for direct use (mechanical, thermal) or for processing (permeability, diffusivity, and specific surface...) is to obtain accurate data concerning the 3D architecture of real fibrous preforms at various stages of densification. For this, CMT is a powerful tool having a resolution compatible with the size of an isolated single carbon fibre (7-8 jam diameter). Furthermore, the 3D character of the method yields essential information about the connectivity of the porous and solid phases. It is possible, on the basis of the reconstructed 3D images where the interface between porous and solid phases is visible, to compute the essential geometrical properties [LEE 98] and effective transport properties (permeability [BER 95], diffusivity and Knudsen diffusivity [VIG 95] as well as mercury penetration curves for the porous phase [HAZ 95], thermal and electrical conductivity [QUI 93], stiffness tensor... [POU 96]). It is also possible to perform on such images simulations of the evolution of the microstructure under some constraints, such as matrix deposition or infiltration, and chemical or mechanical degradation. Images were collected at the ID 19 beam line of ESRF (European Synchrotron Radiation Facility). The outstanding quality of the secondary beam allows to image samples at resolutions as good as 0.8 jam. Here, a resolution of 1.8 urn has been chosen because of the wide size of the representative elementary volume (REV). Accordingly, the studied samples measured less than 2 mm in diameter. Even though, it is not claimed that 2-mm width images do contain a C/C composite REV. The problem that arises specifically when imaging C/C composites is linked to the low absorption coefficient of C at the frequencies of use, together with the important coherence of the beam, and the order of magnitude of the resolution. In such conditions, the X-ray beam reveals itself much more sensitive to phase shifts than to intensity absorption [CLO 96]. When the detector plane is placed roughly at one centimetre behind the sample, the image obtained after reconstruction displays an enhancement of the void-solid interface under the form of a bright-dark double band, while the interior of both phases are of approximately the same mean grey level (Figure 13.10). The double band is the result of an interference between the coherent rays passing close to the interface, through both phases, one of which (carbon) induces a phase shift.


Figure 13.10. Image extracted from one slice of a reconstructed 3D image

of a C/C partially densified preform obtained by synchrotron X-ray CMT

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Figure 13.11. 3-D rendering of the treated sample. The circles focus on rounded fibre

tips

If phase contrast allows the immediate production of human-understandable images, it is not suited for subsequent computations. Nevertheless, the important information is contained inside the phase-contrast image, since for most of the interface, the bright layer is always on the same side of the interface (e.g. the fluid side) and the dark layer lies always on the other. A variety of methods may be designed to answer to the following question: for any point (voxel) of the discretized image, what is the phase it lies in? An image treatment sequence has been designed to extract pertinent information from C/C tomographs displaying a phase contrast structure enhancing the void/solid interface. After having applied the algorithm, the image is fully binary. However, some closed porosity remains, which is removed using a classical percolation algorithm. It is also possible to use a mask and recover the exact grey-level values of the pixels that touch the interface, in order to use subsequently accurate surface tessellation procedures. Phase contrast images, after this treatment, are now suited for subsequent physico-chemical computations at pore-scale. As an example, the image treatment suite has been applied to a 200x200x50 voxels sub-sample of a tomography taken at 2 urn spatial resolution (Figure 13.10). The result is shown on Figure 13.11.

13.4. Ceramics sintering Sintering is known as the process allowing the transformation of a powder into a compact material presenting at least some mechanical properties. The widest used model of solid state sintering considers the ceramic as ideal packing of facetted

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grains with its associated porosity situated on the grain edges [COB 61]. In this representation the process is completely defined by two parameters: the grain size and the density. In real systems, grain size distribution and initial packing inhomogeneities lead to internal stress gradients originating differential densification phenomena (constrained densification). Pore evolution should be considered and analysed independently of density and grain size changes. Sintering of a powder deposited on a dense and rigid substrate (Figure 13.12) of the same chemical nature, is an interesting model to investigate constrained sintering at the macroscopic scale.

Figure 13.12. Schematic of (a) a powder sintered without constraints (free sintering) (b) a powder layer sintered on a rigid substrate (constrained sintering)

A new phenomenological model of solid state sintering has been developed. In this model, pore size is explicitly considered [LET 94]. This parameter can be determined by image analysis of polished sections permitting a correct description of free and constrained sintering. If macroscopic shrinkage behaviour is correctly obtained, development of localised microstructural phenomena cannot be taken into account in this framework. For instance, large pore defects present in the compact can evolve and lead to de-sintering phenomena [LAN 89, HEI 94]. So, improvement of sintering modelling needs more detailed analyses of densifying phenomena, especially at a lower scale. It requires a more accurate description of porosity and CMT associated to 3D reconstruction techniques looked as the up to date technique to investigate pore structures in real ceramic samples at the appropriate scale (grain scale). Experiments were carried out on a well-known ceramic material, i.e. A12O3. The powder was a pure alumina powder (Baikowski DF 1200) and the average grain size was estimated to be 4 urn. A reference sample (initial state) was obtained by cold pressing this powder and then by sintering it at low temperature (1300°C, 1 min.) in order to get a minimum of cohesion without any significant change of the microstructure. The role of a constraint on the sintering and on the development of the porosity was investigated by comparing free sintered samples (powder naturally sintered) to constrained samples (powder layer deposited on a sapphire, i.e. dense alumina, substrate and sintered). The sintering temperature was fixed to 1600°C and


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the sintering times were 15 min, 30 min and 60 min. Due to the large initial grain size, relative density of the free samples remained low and was 62.5 %, 67 % and 68 % respectively. The constrained samples were assumed to exhibit even lower relative densities [LET 94].

Figure 13.13. 2D sections of alumina ceramics samples sintered during 30 min. at 1600°C (image size: 512x512 pixels, pixel size: 0.9 /jm2). Free sintering on left and constrained sintering on right (on left is the sapphire substrate)

Figure 13.14. 60 min. free sintered ceramic. Formation of a denser shell around a large pore is observed while no change in density can be noticed along the crack. The first effect may be attributed to differential sintering (localised constrained sintering phenomenon) as it has been predicted and observed in 2D within ceramic and composites [LAN 96]. Image processing analysis tools will be used to obtain 3D information on the micro structure around the pore to relate them to previous studies

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X-ray computerised micro-tomography of these alumina ceramics were performed on ID 19 beamline. All specimens were scanned at the two lowest resolutions (0.8 jam and 1.8 jam), using a monochromatic beam (17.5 keV) obtained using a bent multilayer device (strong decrease of the acquisition time). A control section was numerically reconstructed for each sample. Unfortunately, as it can be seen on Figure 13.13, phase contrast artefacts related to the low grain size (large number of solid-gas interfaces) superimposed themselves on the grey level image, preventing an easy separation of the two phases (pore/matter) at the micrometer scale. Nevertheless, numerical analysis of the 3D signal is planned to extract average density fields close or far from the substrate. Furthermore, some samples reveal interesting features justifying an deeper exploration: for example, a free sintered specimen (60 min.) shows two types of internal defects producing different microstructural modifications (Figure 13.14). 13.5. Conclusions and forthcoming works The three examples presented in this paper illustrate the huge possibilities offered by CMT to porous materials evolution modelling. Recent developments (phase contrast imaging, local tomography) increased notably the quality of the data that can be obtained and extended the domain of application to new fields. CMT is opening very promising perspectives but one must keep in mind that, because of the amount of data that is generated, an efficient use of this tool requires consequent intellectual and material investments. CM3D* is orienting its activities towards those goals: volume averaging is providing the theoretical framework for data exploitation, new codes are developed for data treatment and physical properties computation, new computing facilities will be available in near future (next camera will generate images of 2048 x 2048 pixels!!!). If data acquisition has been rather simple for rocks (§ 13.2), it has not been the case for C/C composites and ceramics (§ 13.3 and 13.4). More elaborated acquisition procedures are required there. Development and testing of those procedures will be possible only if the collaboration between users and beamline staff continues.

Acknowledgement All the micro-tomography data has been acquired at the ESRF (European Synchrotron Radiation Facility, Grenoble, France) on the ID 19 beamline. 3D * CM3D is a thematic research group from Bordeaux (Caracterisation et Modelisation 3D de 1'evolution des milieux poreux reels. Contact: [email protected]).


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reconstructions have been performed using VOLUMIC, a code developed by CREATIS (UMR5515, CNRS-INSA Lyon).

13.6. References [BER 95] BERNARD D., "Using the Volume Averaging Technique to Perform the First Change of Scale for Natural Random Porous Media", Adv. Methods for Groundwater Pollution Control, Gambolati & Verri Eds., p. 9-24, Springer Verlag, New York, 1995. [CLO 96] CLOETENS P., BARRETT R., BARUCHEL J., GUIGAY J., SCHLENKER M., "Phase Objects in Synchrotron Radiation Hard-X-ray Imaging", J. Phys. D : Appl. Phys., vol 29, p. 133-146, 1996. [COB 61] COBLE R.L., "Sintering Crystalline Solids. I. Intermediate and Final State Diffusion Models", J. Appl. Phys., vol. 32, p. 787-792, 1961. [HEI 94] HEINTZ J.M., SUDRE O., LANGE F.F., "Instability of Polycrystalline Bridges than Span Cracks in Powder Films Densified on a Substrate", J. Am. Ceram. Soc., vol. 77 [3], p. 787-91, 1994. [LAN 89] LANGE F.F., "Powder Processing Science and Technology for Increased reliability", J. Am. Ceram. Soc., vol. 72 [1], p. 3-16, 1989. [LEE 98] LEE S-B., STOCK S.R., BUTTS M.D., STARR T.L., BREUNIG T.M., KINNEY J.H., "Pore Ggeometry in Woven Fiber Structures: 0°/90° Plain-Weave Cloth Lay-up Preform", J. Mater. Res., vol 13(5), p. 1209-1217, 1998. [LET 94] LETULLIER P., Ph.D. thesis, n°1247, University Bordeaux I, 1994. [NAS 99] NASLAIN R., "Key Engineering Materials", CSJ Series - Publications of the Ceramic Society of Japan vol. 164-165, Switzerland, Trans Tech Pub., p. 38, 1999. [HAZ 95], HAZLETT R.D., "Simulation of Capillary-Dominated Displacements in Microtomographic Images of Reservoir Rocks", Transport in Porous Media, vol 20(1-2), p. 21, 1995. [QUI 93] QUINTARD M., WHITAKER S., "Transport in Ordered and Disordered Porous Media: Volume-Averaged Equations, Closure Problems, and Comparison with Experiment", Chem. Eng. Sci., vol 48, p. 2537, 1993. [POT 97] POTDEVIN J-L., HASSOUTA L., "Bilan de matiere des processus d'illitisation et de surcroissance de quartz dans un reservoir petrolier du champ d'Ellon (zone Alwyn, Mer du Nord)", Bull. Soc. Geol. France, vol 168 (2), p. 219-229, 1997. [POU 96] POUTET J., MANZONI D., HAGE-CHEHADE F., JACQUIN C.G., BOUTECA M.J., THOVERT J-F., ADLER P.M., "The Effective Mechanical Properties of Reconstructed Porous Media", Int. J. Rock. Mech. Min. Sci. & Geomech. Abstr., vol 33(4), p. 409-415, 1996. [VIG 95] VIGNOLES G.L., "Modelling Binary, Knudsen, and Transition Regime Diffusion inside Complex Porous Media", J. de Physique IV, vol 65(1), p. 159166, 1995.

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[WHI 99] WHITAKER S., The Method of Volume Averaging : Theory and Applications of Transport in Porous Media, Theories and Applications of Transport in Porous Media, V. 13, Kluwer Acad. Pub., Dordrecht, 1999.

Chapitre 14

Study of damage during superplastic deformation

Superplastic deformation of some industrial alloys can induce damage, leading to premature fracture. This damage by cavitation is generally divided in three main steps: nucleation of the cavities, their growth and finally coalescence between cavities. Up to now, this latter point has been poorly documented due to a difficulty to get reliable data with conventional techniques of 2D characterisation of damage. In this work, high resolution X-ray micro-tomography is used as a technique of quantification of the population of cavities due to the ability to obtain 3D information and a particular attention is given to the coalescence process. In the case of a superplastically deformed Al-Mg alloy, it is shown that coalescence occurs in a large strain interval and that just before fracture, most cavities are connected together. A parameter is proposed to quantify the coalescence process.

14.1. Introduction to damage in superplasticity Superplasticity is frequently defined as the ability for a polycrystalline material deformed at high temperature, to reach elongations to fracture at least one order larger than those obtained in conventional plasticity. Elongations larger than 1000 % may be obtained in the case of metallic alloys. A sample of aluminium alloy deformed in superplastic tensile conditions is presented in figure 14.1. Superplasticity requires both specific experimental (temperature and strain rate) and microstructural conditions. It is associated to the predominance of grain boundary sliding (GBS) as the main mechanism of deformation. The movement of grains during superplastic deformation is illustrated by figure 14.2, which displays a

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SEM micrograph of a Pb-Sn alloy of which a marker line was drawn on the polished surface before testing [DUP 97]. After deformation, the marker line inside the grains is not significantly modified whereas gaps are detected through the grain boundaries, which indicates that movements of grains are predominant.

Figure 14.1. Example of an aluminium alloy deformed in superplastic conditions

In consequence, superplastic properties are promoted by a reduction of the mean grain size of the material. For metallic alloys, grain sizes of about 10 microns are generally needed to exhibit superplasticity at strain rates compatible with industrial processes. From a rheological point of view, superplastic deformation occurs under very moderate flow stresses (typically less than 10 MPa) and is associated with a large plastic stability resulting from the high value of the strain rate sensitivity parameter m, deduced from the conventional viscoplastic law between the flow stress a and the strain rate e :

m a = Ke

[1]

Superplastic forming (SPF) is to day an industrial forming process to produce components with complex shapes. It concerns particularly titanium and aluminium and major applications have been developed in the aeronautical industry. However, in the case of single-phase materials, like aluminium alloys, superplastic deformation induces damage through the microstructure, leading to premature fracture but also to a reduction of service properties of the alloy after SPF. Such damage can be observed on figure 14.3, which shows a SEM micrograph of a superplastically deformed aluminium-magnesium alloy [LAR 98]. To day, industrial SPF overcomes this difficulty by forming components under superimposed pressure, which inhibits the damage process but increases the cost and limits the maximum size of the components to be shaped. In consequence, a way for the promotion of SPF in the future, is the ability to superplastically form aluminium alloys under atmospheric pressure. Previous works have demonstrated that some


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benefits in terms on damage sensitivity can already be obtained by appropriate heat treatments before testing [BLA 96]. Nevertheless, a better understanding of the damage process in superplastic conditions appears as a key parameter for the future development of SPF.

Figure 14.2. SEM micrograph showing the predominance ofGBS in superplasticity

Figure 14.3. Example of strain-induced cavitation when an aluminium alloy is superplastically deformed

The damage process induced by superplastic deformation is usually divided in three main steps: nucleation of the cavities, growth and coalescence leading to fracture. Cavity nucleation is attributed to microcracking or vacancy agglomeration and is mainly located at triple junctions or near intergranular particles through the alloy [RAJ 77], as a result of stress concentrations generated by GBS. Cavity growth is generally interpreted thanks to models initially developed for materials deforming in creep conditions [RIC 69, HAN 76, CHO 86]. For small cavities, diffusion is expected to contribute predominantly to cavity growth whereas for larger cavities, it is considered that growth is controlled by plastic deformation of the matrix surrounding the cavity. In this case, the variation with strain 8 of the volume V of a cavity is given by [HAN 76]: V = V0 exp(r|ge)

[2]

with V0 a constant and r|g the cavity growth parameter. Some expressions of the parameter r)g have been proposed in the past, in particular as a function of the rheology of the matrix [PIL 85]. However, it must be underlined that, since these models of cavity growth were initially developed for alloys deforming in creep conditions, they do not take into account GBS, although it is the predominant mechanism of deformation in superplasticity.

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Despite its crucial link with fracture, the coalescence process between growing cavities has been poorly documented [STO 83, STO 84]. Very limited experimental data have been reported and few models were proposed. Moreover, these models are generally based on very strong hypotheses about the spatial distribution of the cavities through the microstructure and the criteria of coalescence. In particular, they assume that the cavities are randomly distributed through the microstructure and that all the cavities have a spherical shape and the same diameter. Lastly, coalescence is supposed to occur only when cavities impinge. It is expected that such assumptions are not experimentally satisfied. As already mentionned, many cavities are preferentially nucleated near grain boundary particles, which are not randomly distributed through the microstrucutre. Moreover, the shapes of large cavities after superplastic deformation can be very irregular, as it is illustrated by figure 14.4, which shows an optical micrograph of the fracture zone for a superplastically deformed aluminium alloy. This irregularity of the shape of the cavities is attributed to GBS, which is the main mechanism of deformation in superplasticity. Experimental data about strain-induced damage in superplastic conditions are frequently quantified from the variation with strain of the cavity volume fraction Cv. Figure 14.5 displays such a variation of Cv with s in the case of superplastic deformation of Al-Mg alloy. The cavity volume fraction continuously increases with strain and after a period of apparent incubation in which the level of cavitation remains limited (less than 1 %), a sharp increase is obtained and the cavity volume fraction, when fracture occurs, is generally high in the case of superplastic alloys, typically more than 10 %.

Figure 4. Optical micrograph of the fracture zone after superplastic deformation of an Al-Li alloy

Figure 14.5. Variation with strain of cavity volume fraction Cv during superplastic deformation of an Al-Mg alloy


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The variation with strain of Cy is frequently rationalised according to: Cv = Cvo exp(Tiapps)

[3]

where CVo is a constant and r\app the apparent parameter of cavitation sensitivity. The logarithm of experimental values of Cv is plotted as a function of strain and in most cases, a straight line can be roughly obtained in a relatively large strain interval, which allows the measurement of a slope r|app. The corresponding value of r)app is then compared to those deduced from the cavity growth models. Indeed, these models predict also an exponential variation of the volume of the cavity with strain under two main assumptions : firstly, growth is controlled by plastic deformation of the matrix which surrounds the cavity; secondly, the number of cavities is roughly constant in the corresponding strain interval. The differences between such predictions and r)app are frequently discussed in detail and sometimes interpreted in terms of continuous nucleation of cavities during deformation. Despite the fact that these assumptions (in particular the constancy of the number of cavities per unit volume in a large strain interval) appear very questionable, it is difficult to draw conclusions with data deduced from conventional 2D techniques of characterisation.

14.2. Usual techniques of characterisation Two techniques of characterisation are generally used to quantify strain-induced cavitation in superplasticity: variation of relative density of the alloy and quantitative metallography from polished sections. Both techniques have some notable disadvantages. Density variation measurements are relatively easy to perform and allow detection of low cavity volume fraction, typically less than 0.01 %, but the measurements appear doubtful for large cavity volume fraction. Indeed, for such levels of cavitation, some cavities may connect with the outer surfaces of the specimen and artificially modify the results, leading to an apparent slackening of the cavitation increase with strain. Moreover, some precautions must be taken to systematically check the possible variation of density of the alloy during an heat treatment similar to that undergone by the sample during high temperature deformation [VAR 89]. Finally, this technique is a global one since only Cv can be measured whereas no data about the population of the cavities (number, size...) are available.

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By quantitative metallography, such data about the population of cavities can be obtained but it requires to polish the surfaces to observe, which can modify the apparent size and shape of the cavities, particularly in the case of aluminium alloys. Moreover, as already mentioned, irregular shapes of cavities can be detected in superplastically deformed alloys, as shown in figure 14.4. In such conditions, the study of cavity coalescence from two-dimensional data appears very hazardous. From these remarks, it appears very interesting to use a technique of threedimensional (3D) characterisation of the population of cavities in a superplastically deformed alloy. In consequence, high resolution X-ray micro-tomography seems to be a very promising technique since it can provide three-dimensional images of the bulk of materials [HIR 95, BUF 99].

14.3. Experimental procedure Tomography experiments were carried out at the ID 19 beamline of the European Synchrotron Radiation Facility (ESRF). ID 19 is devoted to highresolution imaging. The beam energy was 17.5 keV. The samples were set on a goniometer allowing a precise positioning of the sample. A scan of the samples consisting of the recording of 800 two-dimensional radiographs was performed during a 180° rotation around the vertical axis. Those radiographs were recorded on a 1024 x 1024 CCD camera developed at ESRF [LAB 96]. The average exposure time for a radiograph was 0.3 s and the whole scan lasted about 10 minutes. The pixel size of the camera was 2x2 (irn2. The detector was set 3 mm behind the sample. For each sample, the investigated volume was approximately 0.6 x 0.6 x 0.6 mm3, knowing that for large strains, the final thickness of the sheet after SPF in industrial conditions, may be less than 1 mm. In a first step of characterisation, only cavities with a volume larger than 10 voxels were taken into account. It is the reason why the interpretation of the results deduced from X-ray micro-tomography were focused on the coalescence process, in order to deal with relatively large cavities. The studied material was an aluminium-magnesium alloy (Al - 4.2Mg - 0.7Mn 0.2Fe - 0.1 Cr, wt %). The alloy was provided in the form of sheets of 2.5-mm thickness. The mean grain size was about 10 urn. The superplastic properties of this alloy have been investigated in detail [MAR 99] and the cavitation behaviour in superplastic conditions was studied by deforming the alloy at 525°C and 10"4 s"1 at different strains. In these conditions of deformation, the elongation to fracture was about 400 %. Density variation measurements were performed on a micro-weighing machine allowing detection of relative density variation close to 0.005 % and limited quantitative metallography on polished sections were also carried out on SEM micrographs.


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14.4. X-ray micro-tomography results Figure 14.6 shows a 2D section of the superplastically deformed alloy, deduced from X-ray micro-tomography data. Well-contrasted sections are obtained, resulting from the difference of X-ray attenuation between the cavities (in dark) and the aluminium alloy.

Figure 6. 2D section showing good contrast between cavities and the aluminium alloy.

Figure 7. Comparison of the variations with strain of Cv between X-ray microtomography and density variation measurements

Figure 14.7 compares the variations with strain of the cavity volume fractions deduced from density variation measurements and from X-ray micro-tomography. A good correlation between the results obtained by these two independent techniques is found. It confirms the validity of X-ray micro-tomography as a fruitful technique of quantification of strain-induced cavitation in superplastic alloys. Moreover, this technique allows to get information about the population of cavities as illustrated by figure 14.8, which displays the reconstructed image of the spatial distribution of the cavities after an elongation of about 170 %. This condition corresponds to a mean cavity volume fraction of about 1 %. For this elongation, most cavities are isolated through the microstructure, although some connections between cavities can be detected.

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Figure 14.8. Reconstructed image of the population of cavities after an elongation of about 170% (Cv »1 %)

14.5. Quantification of the coalescence process As already mentioned, the increase with strain of the cavity volume fraction is frequently described according relation [3]. A value of the parameter r|app is thus estimated and compared to those predicted by the cavity growth models. In the case of the investigated alloy, a value of r|app close to 5 was obtained which is agreement with previous works (IWA 91, FRI 96). Nevertheless, this value is significantly larger than the values predicted by cavity growth models (typically close to 2 in this case). This difference is generally attributed to continuous nucleation of cavities during strain, resulting in an apparent increase of the value of r\. To confirm this interpretation, the variation with strain of the number of cavities nA per unit area is frequently determined from SEM micrographs. This work was carried out in the case of the studied alloy [LAR 98] and an increase of nA with strain was obtained in the investigated strain interval. From the 3D data obtained in tomography, it is also possible to estimate the variation with strain of the apparent number of cavities par unit area for a given family of planes (as it is shown in figure 14.6). This estimation has been performed in the case of the investigated alloy and an increase of this apparent number of cavities per unit area is obtained whatever the planes of observation. Consequently, these results have confirmed those deduced from SEM micrographs, even if, for a given strain, the values of nA may depend on the technique of characterisation: nA is lower in the case of the tomography results, which can be attributed to the fact that small cavities (i.e. equivalent diameter smaller than 5 urn) have not be taken into account in the treatment of the X-ray tomography data.


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However, as already mentioned, when strain is increased, the cavities become very irregular and consequently, the interpretation of the associated variation of nA may be delicate. It appears more reliable to study the variation with strain of the number of cavities nv per unit volume. Figure 14.9 shows the variation with elongation of the number nv of detected cavities per unit volume. Between elongations from 200 % to more than 400 %, a continuous decrease of nv is obtained, which points out the extent of the coalescence process. The data obtained for an elongation close to 150 % has to be considered with caution since it must be kept in mind that only cavities with a volume larger than 10 voxels were taken into account in the procedure of counting. It means that an apparent increase of ny with strain can be partially attributed to cavity growth.

Figure 14.9. Variation with elongation of the number of cavities per mm

From the results presented in figure 14.9, it can be concluded that the effects of strain on the variation of nv and nA (deduced from SEM observations or from X-ray micro-tomography data) are contrary. These differences between the dependencies on strain of nv and nA confirm the fact that a the usual interpretation of the cavitation processes from 2D characterisations is very hazardous. The variation with strain of the number of cavities per unit volume is a first approach to quantify the coalescence process. However, this parameter gives only limited indications about the mechanism of coalescence. A way to get additional data is to follow the variation with testing conditions of the largest cavity in the investigated volume, since it may give indication about the extent of connection between cavities through the microstructure. In this view, figure 14.10 displays a 3D observation of the largest cavity obtained after an elongation of about 400 % for which Cv is about 14.5 %. It can be seen in figure 10 that the largest cavity admits a very irregular shape and extends beyond the studied volume. Moreover, this cavity

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corresponds to about 75 % of the total cavity volume fraction, which indicates that most cavities have connected together. From this conclusion, a coalescence parameter CP can be defined, according to: CP =

volume of the largest cavity — -x 100 total volume of cavities

Figure 14.10. Reconstructed image of the largest cavity through the microstructure after an elongation of about 400 %

Figure 14.11 shows the variation with strain of the coalescence parameter CP. The value of this parameter remains limited up to a strain of about 1.2 and then sharply increases. This strain corresponds, for the associated conditions of deformation (525°C and 10"4 s"1) to a value of Cv close to 5 %. These results confirm the importance of the coalescence process during superplastic deformation since they point out that coalescence occurs in a large domain of strain. It must be kept in mind that between 8 « 1.2 (i.e. an elongation « 230 %) and fracture, which is obtained after an elongation « 400 %), the mean cavity volume fraction Cv increases roughly from 5 % to 15 %. Consequently, it indicates that coalescence takes place not only in a large strain interval but also in a large domain of Cv.


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Moreover, from figures 14.9 and 14.11, it seems that two domains of coalescence may be identified. In a first step (for strains between 1.0 and 1.3), a significant reduction of nv is obtained whereas the value of the coalescence parameter remains low. It means that a large number of cavities coalesce but that the volume of the largest cavity through the volume remains comparable to the mean cavity volume of the cavities. It is expected that this step of coalescence is associated to a nonuniform spatial distribution of cavities. Indeed, as already mentioned, cavity nucleation takes place preferentially near intergranular particles which are not uniformly dispersed in the alloy, as a result of the thermomechanical process undergone by the material. Indeed, previous work has shown that, as a result of the thermomechanical process undergone by the material, some stringers of second phase particles are present in the studied alloy [LAR 98]. In a second step (when 8 > 1.3), the value of CP increases sharply. It indicates that a large fraction of cavities connect together, leading to one cavity which concentrates the coalescence process.

Figure 14.11. Variation with strain of the coalescence parameter .CP

Complementary data are nevertheless required before any definite conclusion about the relevance of these two steps of the coalescence process.

14.6. Conclusions High resolution X-ray micro-tomography appears as a very promising technique to characterise cavitation induced by superplastic deformation of industrial alloys. It confirms the difficulty to interpret experimental information deduced from conventional 2D quantitative metallography. In the case of an Al-Mg alloy, the coalescence process has been preferentially investigated since it is concerned with large cavities.

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Despite a quite limited number of experiments, significant results have been obtained. Coalescence occurs in a large strain interval, which means that the frequently used assumption that the number of cavities remains roughly constant during deformation is not valid. Just before fracture, most cavities are connected together. A parameter CP was proposed to quantify the degree of connection of the cavities in the superplastically deformed alloy and the variation with strain of CP suggests a coalescence process in two steps. Additional experiments are however needed to establish a reliable quantitative correlation between CP and the experimental conditions of testing of superplastic alloys.

14.7. References [BLA 96] BLANDIN J.J., HONG B., VARLOTEAUX A., SUERY M., L'ESPERANCE G., Acta Mater., vol. 44, p. 2317-2326, 1996. [BUF99JBUFFIERE J.Y., MAIRE E., CLOETENS P., LORMAND G., FOUGERES R., Acta Mater., vol. 47, p. 1613, 1999. [CHO 86] CHOKSHI A.H., J. Mater. Sc., vol. 21, p. 2073, 1986. [DUP 97]DUPUY L., DEA INP Grenoble, 1997. [FRI 96] FRIEDMAN P.A., GHOSH A.K., Metall. Mater. Trans., vol. 27A, p. 3827, 1996. [HAN 76] HANCOCK J.W., Metal Sc., vol. 10, p. 319, 1976. [HIR 95] HlRANO T., USAMI K., TANAKA Y., MASUDA C., J. Mater. Res., vol. 10, p.

381, 1995. [rwA91]IWASAKi H., HIGASHI K., TANIMURA S., KOMATUBARA T., HAYAMI S., Proc. of Int. Conf. on Superplasticity of Advanced Materials (ICSAM), p. 447, 1991. [LAB 96]LABICHE C., SEGURA-PUCHADES J., VAN BRUSSEL D., MOY J.P., ESRF

Newsletter, vol. 25, p. 42, 1996. [LAR 98]LARIVIERE D., DEA INP Grenoble, 1998. [MAR 99] MARTIN C.F., Thesis INP Grenoble, 1999. [PIL 85] PILLING J., RIDLEY N., Res. Mechanica, vol. 23, p. 31, 1985. [RAJ 77] RAJ R., Acta Metall., vol. 25, p. 995, 1977. [RIC 69] RICE J.R., TRACEY D.M., J. Mech. Phys. Solids, vol. 17, p. 201, 1969. [STO 83] STOWELLMJ., Metal. Sc., vol. 17, p. 1, 1983. [STO 84] STOWELL M.J., LIVESEY D.W., RIDLEY N., Acta Metall., vol. 32, p. 35, 1984. [VAR 89] VARLOTEAUX A., BLANDIN J.J., SUERY M., Mater. Sc. Tech., vol. 5, p. 1109, 1989.

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