1 Department of Earth and Planetary Science, University of California, Berkeley, CA 94720,USA2 Department of MTM, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
Received 17 February 2004Published 5 July 2004Online at stacks.iop.org/RoPP/67/1367doi:10.1088/0034-4885/67/8/R02
Abstract
A large number of polycrystalline materials, both manmade and natural, display preferredorientation of crystallites. Such alignment has a profound effect on anisotropy of physicalproperties. Preferred orientation or texture forms during growth or deformation and ismodified during recrystallization or phase transformations and theories exist to predict itsorigin. Different methods are applied to characterize orientation patterns and determinethe orientation distribution, most of them relying on diffraction. Conventionally x-ray pole-figure goniometers are used. More recently single orientation measurements are performedwith electron microscopes, both SEM and TEM. For special applications, particularly textureanalysis at non-ambient conditions, neutron diffraction and synchrotron x-rays have distinctadvantages. The review emphasizes such new possibilities.
A second section surveys important texture types in a variety of materials with emphasison technologically important systems and in rocks that contribute to anisotropy in the earth. Inthe former group are metals, structural ceramics and thin films. Seismic anisotropy is presentin the crust (mainly due to phyllosilicate alignment), the upper mantle (olivine), the lowermantle (perovskite and magnesiowuestite) and the inner core (ε-iron) and due to alignment byplastic deformation. There is new interest in the texturing of biological materials such as bonesand shells. Preferred orientation is not restricted to inorganic substances but is also present inpolymers that are not discussed in this review.
Page1. Introduction 13702. Measurements of textures 1370
2.1. Overview 13702.2. X-ray pole-figure goniometer 13712.3. Synchrotron x-rays 13712.4. Neutron diffraction 13742.5. Transmission electron microscope 13742.6. Scanning electron microscope 13752.7. Comparison of methods 1375
3. Data analysis 13763.1. Orientation distributions and texture representations 13763.2. From pole figures to ODF 13783.3. Use of diffraction spectra 13783.4. Statistical considerations of single orientation measurements 13793.5. From textures to elastic anisotropy 1380
6.2. Thin films and coatings 13966.2.1. Silicon and diamond 13966.2.2. Nitride, carbide and oxide coatings 13976.2.3. Epitaxial films 1398
7. Textures in minerals and rocks 13997.1. Calcite (CaCO3) 14007.2. Quartz (SiO2) 14027.3. Olivine (Mg2SiO4) 14047.4. Sheet silicates 1406
Texture and anisotropy 1369
7.5. Ice (H2O) 14077.6. Halite (NaCl) and periclase (MgO) 14087.7. Polymineralic rocks 14097.8. Cement minerals 14117.9. Earth structure 14127.10. Textures as indicators of strain history 14137.11. Anisotropy in the deep earth 1415
8. Textures in mineralized biological materials 14198.1. Nacre of mollusc shells (aragonite) 14198.2. Bones (apatite) 1419
9. Conclusions 1421References 1422
1370 H-R Wenk and P Van Houtte
1. Introduction
Preferred orientation of crystallites (or texture) is an intrinsic feature of metals, ceramics,polymers and rocks and has an influence on physical properties such as strength, electricalconductivity, piezoelectricity, magnetic susceptibility, light refraction and wave propagation,particularly in the anisotropy of these properties. The directional characteristics of manypolycrystalline materials were first recognized not in metals but in rocks and were describedas ‘texture’ (d’Halloy 1833). In the 20th century texture research was largely pursuedby metallurgists but recently it has gained importance in ceramics (e.g. high temperaturesuperconductors), polymers, and regained interest in the earth sciences. The reason for thelatter is that seismologists have discovered anisotropic wave propagation in large sectors ofthe earth’s interior and a likely cause is preferred orientation of crystals that developed bydeformation during the earth’s long history. This review will highlight some aspects of textureswith focus on new approaches and methods, as well as relevant problems. Metallurgists andceramicists are engaged in texture research to develop materials with favourable properties.In contrast, geologists and geophysicists are using textures to interpret the past. The rationaleis thus reversed. In metallurgy specimens are readily available for analysis, and theoriescan be tested with experiments. Deep-earth materials do not occur on the surface and manyare unstable at ambient conditions. Also, many geological conditions are outside the realmof experiments, particularly the slow strain rates and highly heterogeneous nature of rockformations. Yet, in spite of these differences, methods and approaches are remarkably similar,even though the objects of interest vary greatly in dimension.
This review is intended to provide a brief introduction for physical scientists, not fortexture experts. Some of the important issues are highlighted with examples, and we refernew researchers in the field of texture and anisotropy to important publications. Since theclassic books on metallurgy (e.g. Wassermann and Grewen (1962), Dillamore and Roberts(1965), Hatherly and Hutchinson (1979)) and geology (e.g. Sander (1950), Turner and Weiss(1963)), there have been newer books (e.g. Bunge (1982), Wenk (1985), Kocks et al (2000)),numerous journal articles and particularly research papers in the tri-annual proceedings ofthe International Conferences of Textures of Materials (ICOTOM). These publications needto be consulted for details. Textures in polymers, though important, are only mentionedperipherally (see, e.g. G’Sell et al (1999)). While preparing this review we noticed thatalmost half of the references are in physics journals and fifteen in Nature and Science,illustrating that texture and anisotropy are subjects of core physics as well as of generalinterest.
We try to give a balanced account of recent progress in texture research; however, in thisbroad field it was hard to avoid some emphasis on our own specialities, particularly in theselection of examples which were more readily available. We do not suggest that these are inany way more important.
2. Measurements of textures
2.1. Overview
Interpretation of textures has to rely on a quantitative description of orientation characteristics.Two types of preferred orientations are distinguished: the lattice preferred orientation (LPO)or ‘texture’ (also ‘preferred crystallographic orientation’) and the shape preferred orientation(or ‘preferred morphological orientation’). Both can be correlated, such as in sheet silicateswith a flaky morphology in schists, or fibres in fibre-reinforced ceramics. In many cases they
Texture and anisotropy 1371
are not. In a rolled cubic metal the grain shape depends on the deformation rather than on thecrystallography.
Many methods have been used to determine preferred orientation. Optical methods havebeen extensively applied by geologists, using the petrographic microscope equipped with auniversal stage to measure the orientation of morphological and optical directions in individualgrains (e.g. Phillips (1971)). Metallurgists have used a reflected light microscope to determinethe orientation of cleavages and etch pits (e.g. Nauer-Gerhardt and Bunge (1986)). Withadvances in image analysis, shape preferred orientation can be determined quantitatively andautomatically with stereological techniques. Optical methods of LPO measurements of someminerals have also been automated (Heilbronner and Pauli 1993).
Today diffraction techniques are most widely used to measure crystallographic preferredorientation (e.g. Bunge (1986), Kocks et al (2000)). X-ray diffraction with a pole-figuregoniometer is a routine method. For some applications synchrotron x-rays provide uniqueopportunities. Neutron diffraction offers some distinct advantages, particularly for large bulksamples. Electron diffraction using the transmission (TEM) or scanning electron microscope(SEM) is gaining interest, because it permits one to correlate microstructures, neighbourrelations and texture.
There are two distinct ways to measure orientations. One way is to average over a largevolume of a polycrystalline aggregate. A pole figure collects signals from many crystals andspatial information is lost (e.g. misorientations with neighbours), but also some orientationrelations (such as how x, y, and z-axes of individual crystals correlate). The second methodis to measure orientations of individual crystals. In that case orientations and the orientationdistribution can be determined unambiguously and, if a map of the microstructure is available,the location of a grain can be determined and relationships with neighbours can be evaluated.But compared to the bulk methods, the statistics of such measurements are limited.
2.2. X-ray pole-figure goniometer
X-ray diffraction was first employed by Wever (1924) to investigate preferred orientation inmetals, but only with the introduction of the pole-figure goniometer and use of electronicdetectors did it become a quantitative method (Schulz 1949). Bragg’s Law for monochromaticradiation is applied. The principle is simple: in order to determine the orientation of agiven lattice plane, hkl, of a single crystallite, the detector is first set to the proper Braggangle, 2θ of the diffraction peak of interest, then the sample is rotated in a goniometeruntil the lattice plane hkl is in the reflection condition (i.e. the normal to the lattice planeor diffraction vector is the bisectrix between incident and diffracted beam) (see figure 1). Inthe case of a polycrystalline sample, the intensity recorded at a certain sample orientationis proportional to the volume fraction of crystallites with their lattice planes in reflectiongeometry. Determination of texture can be done on a sample of large thickness and a planesurface on which x-rays are reflected, or on a thin slab which is penetrated by x-rays. Becauseof defocusing effects as the flat sample surface is inclined against the beam, variations in theirradiated volume and absorption intensity corrections are necessary, particularly in reflectiongeometry. In reflection geometry only incomplete pole figures can be measured, usually to apole distance of 80˚ from the sample surface normal.
2.3. Synchrotron x-rays
Conventional x-ray tubes produce a broad beam of relatively low intensity (∼1 mm).A powerful new tool for texture research is synchrotron radiation. In a synchrotron a very
1372 H-R Wenk and P Van Houtte
s
Figure 1. Geometry of a pole-figure goniometer equipped with Soller slits and monochromator.Bragg’s Law applies to lattice planes. The sample is rotated about an axis perpendicular to thesurface.
Figure 2. Synchrotron x-ray diffraction image of a sheet of rolled copper with Debye rings recordedwith a CCD camera at ESRF. Intensity variations immediately display the presence of texture.
fine-focused high-intensity beam of x-rays with monochromatic or continuous wavelengthscan be produced. The unique advantages of high intensity, small beam size (<5 µm) and freechoice of wavelength open a wide range of new possibilities (e.g. Heidelbach et al (1999),Wcislak et al (2002), Wenk and Grigull (2003)). Synchrotron diffraction images recorded byCCD detectors almost instantaneously display the presence of texture expressed in systematicintensity variations along Debye rings, as illustrated for a copper plate (figure 2). While
Texture and anisotropy 1373
(a)
(b)
Figure 3. Geometry of a synchrotron x-ray diffraction experiment in transmission. (a) For a givenreciprocal lattice vector hkl the accessible crystallite orientations lie on a cone which intersectsthe orientation sphere of the sample in a small circle. Diffracted x-rays lie on a cone (shaded)that intersects a planar detector along a circle. The diffraction spot of a single pole is indicated.(b) Diamond anvil cell mounted in radial geometry in the synchrotron beam to display texture inDebye cones. The sample is mounted in a boron gasket with minimal x-ray absorption.
the presence of texture is immediately obvious, elaborate data processing is necessary todetermine texture patterns quantitatively and interpret data in a satisfactory way. Figure 3(a)shows the geometry of a transmission diffraction experiment with incoming x-ray beam, sampleand Debye cone with an opening angle 4θ , on which diffracted x-rays lie. If the sample isstationary only lattice planes hkl that are inclined by an angle (90˚−θ ) to the incoming beamdiffract, and the corresponding reciprocal lattice vectors lie on a cone with an opening angleof 180˚−2θ which intersects the orientation sphere of the sample in a small circle. Coverageof the pole figure from a single image is minimal but even information from such an image canbe sufficient to determine the orientation distribution and then to reconstruct complete polefigures (Wenk and Grigull 2003). The use of high energy is advantageous because of goodpenetration and moderate absorption, as well as small 2θ angles.
Synchrotron analysis is particularly valuable for compounds with weak scattering (e.g.polymers and biological materials) and for investigating local texture variations (Margulieset al 2001). Other applications are in situ observations of texture changes during deformationat high pressure with diamond anvil cells (figure 3(b), e.g. Merkel et al (2002)) and hightemperature (e.g. Puig-Molina et al (2003)) and we will illustrate some examples in a latersection.
1374 H-R Wenk and P Van Houtte
(a) (b)
Figure 4. (a) Schematic of the TOF neutron diffractometer HIPPO at Los Alamos NationalLaboratory. Multiple detector banks are arranged on rings. Each ring (at different 2θ ) recordsreflections of differently oriented lattice planes so that the pole figure is covered simultaneously.A sketch of a (large) person is inserted for scale. (b) Pole figure coverage with thirty 2θ = 40˚, 90˚and 150˚ detectors (Wenk et al 2003).
2.4. Neutron diffraction
Neutron diffraction texture analysis is almost as old as the pole-figure goniometer. It wasfirst applied by Brockhouse (1953) in an attempt to determine magnetic structure in steel.Neutron diffraction texture studies are done either at reactors with a constant flux of thermalneutrons, or with pulsed neutrons at spallation sources. The wavelength distribution of ther-mal neutrons is a broad spectrum with a peak at 1–2 Å, similar to x-rays. A disadvantage ofneutrons is that the interaction of neutrons with matter is low, and long counting times arerequired. Weak interaction is also a great advantage because it provides high penetration andlow absorption, making neutrons suitable for bulk texture investigations of large sample vol-umes. Because of the low absorption, environmental stages (heating, cooling, straining) can beused for in situ observation of texture changes, e.g. during phase transformations. Neutrons arealso sensitive to measuring the orientation of magnetic dipoles, but though this was the originalincentive for Brockhouse, to this day no satisfactory magnetic pole figures have been measured.
A conventional neutron texture experiment at a reactor source uses monochromaticradiation produced with single crystal monochromators. A goniometer rotates the sample tocover the entire orientation range, analogous to an x-ray pole-figure goniometer. To improvecounting efficiency position-sensitive detectors have been applied that record a 2θ spectrumwith many peaks simultaneously. With the advent of pulsed neutron sources it has becomecustomary to use polychromatic neutrons and a detector system that can identify the energy ofneutrons by measuring the time-of-flight (TOF). The new TOF neutron diffractometer HIPPOat Los Alamos (figure 4(a)), dedicated to texture research, has 50 detector panels that recordsimultaneously diffraction spectra from crystals in different orientations (figure 4(b)).
2.5. Transmission electron microscope
The TEM offers excellent opportunities to study textural details in fine-grained aggregates.Like light microscopy, the TEM not only provides information about orientation but also
Texture and anisotropy 1375
about grain shape and, more importantly, about dislocation microstructures indicative of activedeformation mechanisms. There are many applications of Kikuchi patterns for orientationanalysis (e.g. Humphreys (1988)). The indexing procedure has been automated (Schwarzerand Sukkau 1998). A recent study of brass-type shear bands in fcc metals and their influenceon texture evolution has been an excellent example to document microstructural changes withinitial slip, intermediate twinning leading to a final texture (Paul et al 2004).
2.6. Scanning electron microscope
Local orientations can also be measured with the SEM and this technique is becomingvery popular because it does not require much background in texture theory from the user(e.g. Randle and Engler (2000)). Unlike the TEM, the SEM is not restricted to thin areaslocated along the edge of a hole in a small specimen (<2 mm), but enables crystal orientationsto be determined on surfaces of considerable extent. Interaction of the electron beam withthe uppermost surface layer of the sample produces electron back-scatter diffraction patterns(EBSPs or EBSD) that are analogous to Kikuchi patterns in the TEM. EBSPs are captured ona phosphor screen and recorded with a low intensity video camera or a CCD device. A bigadvance came with the automation of pattern indexing and scanning a specimen surface (Wrightand Adams 1992). The sample is translated using a high precision mechanical stage or samplelocations are reached by beam deflection in increments as small as 1 µm. At each position anEBSP is recorded. With a phosphor screen, back-scattered electrons are converted to light,this signal is transferred into a camera. The digital EBSP is then entered into a computer andindexed. Specimen coordinates, crystal orientation, parameters describing the pattern qualityand a parameter evaluating the pattern match are recorded. Then the sample is translated tothe next position and the procedure is repeated. A spatial resolution of less than 0.4 µm canbe reached on a SEM equipped with a field emission gun.
This sounds like an ideal technique. However, it is only applicable to crystals with fairlylow dislocation densities, surface preparation is critical and the automatic indexing procedureis not always reliable. Failure to index and mis-indexing of patterns are both orientation-dependent and can produce texture artefacts. Furthermore there are statistical limitations thatwill be discussed below.
2.7. Comparison of methods
The optimal choice of texture measurements depends on many variables, such as availabilityof equipment, material to be analysed and data requirements.
For routine metallurgical practice and many other applications in materials science andgeology, an x-ray pole-figure goniometer in reflection geometry is generally adequate. It is fast,easily automated and inexpensive both in acquisition and maintenance. Transmission geometryhas been successfully used for texture analysis of sheet silicates in slates and shales (Ho et al1999). Pole figures can only be measured adequately if diffraction peaks are sufficientlyseparated. In geological samples and ceramics, x-ray diffraction is therefore generally limitedto single phase aggregates of orthorhombic or higher crystal symmetry. Synchrotron x-rays areused for in situ experiments at high pressure and temperature, generally of very fine-grainedsamples. Texture changes can be recorded in real time.
With neutron diffraction bulk samples rather than surfaces are measured, coarse-grainedmaterials can be characterized, environmental cells (heating, cooling, straining) are availableand angular resolution is better than for an x-ray pole-figure goniometer because no defocusingoccurs. It is possible to measure complex polyphase composites with many closely spaced
1376 H-R Wenk and P Van Houtte
(a) (b)
Figure 5. (a) Definition of Euler anglesα, β andγ (Roe/Matthies notation) that relate the orthogonalsample coordinate system (a, b, c) and the crystal coordinate system (x, y, z) by three rotations.(b) Corresponding rotations φ1, �, φ2 used in the Bunge convention. Equal area projection.
diffraction peaks. Neutron diffraction data analysis is rather complex and requires considerableexpertise from the user. Also, access to facilities is limited.
Electron diffraction with a TEM is most time-consuming but provides, in addition to crystalorientation, valuable information about microstructures and, at least two-dimensionally, aboutinteraction between neighbours and about heterogeneities within grains. These are importantdata to interpret deformation processes.
Recently EBSPs, measured with the SEM on polished surfaces, have become verypopular. They allow for determination of local orientation correlations. With the possibilityof automation, this technique has become comparable in expense and effort to x-ray diffractionanalysis. There are limitations for samples with many lattice defects. The technique hasalready proved to be essential in the study of recrystallization and grain growth in metals uponannealing, as well of grain fragmentation upon plastic deformation.
3. Data analysis
3.1. Orientation distributions and texture representations
In quantitative texture analysis the coordinate systems of the sample and of the crystal needto be related. This requires three quantities, such as the classical Euler angles that relate twoorthogonal right-handed coordinate systems (sample: a, b, c and crystal: x, y, z) through threerotations (figure 5(a)), or an axis–angle specification that brings the two coordinate systemsto coincidence through a single rotation about a specific axis. (Both these representations oforientations were originally introduced by Euler (1775).) The sample coordinate system inmetals is usually defined by the forming process (e.g. a: rolling direction RD, b: transversedirection TD, c: normal direction ND). For geological samples it is often more arbitrary. Thecrystal coordinate system for symmetries with non-orthogonal axes follows crystallographicconventions (z: [001], y: perpendicular to [001] and [100], x: perpendicular to y and z). Thereare various conventions for the Euler angles. The one in figure 5(a) with α, β and γ is theRoe/Matthies convention. A first rotation (α) around c produces a system a′, b′, c′. The secondrotation (β) around b′ brings c′ to c′′. The final rotation (γ ) around c′′ brings b′′ to b′′′ witha′′′ = x, b′′′ = y and z′′′ = z. The Bunge convention, used in the vast majority of papers onmetals, is related to the Roe/Matthies convention: α = φ1 − 90˚, β = � and γ = φ2 + 90˚.
Texture and anisotropy 1377
Figure 6. The three-dimensional orientation distribution of Euler angles can be viewed as aprobability distribution in cylindrical space of angles α (azimuth in sample coordinates), β (poledistance) and γ (azimuth in crystal space). A pole figure (top) is a two-dimensional projectionof the ODF, a peak intensity in a diffraction of a diffraction pattern (bottom) is proportional to aone-dimensional projection.
Figure 5(b) illustrates the rotations for the Bunge convention. The first rotation (φ1) is aroundc and brings a to a′. The second rotation (φ2) around a′ brings c′ to c′′. And the final rotation(�) around c′′ brings a′′ into a′′′ = x. In the case of a polycrystal, an orientation becomesan orientation probability distribution of the three angular quantities and is described by anorientation distribution function (ODF).
A pole figure is a two-dimensional projection of this three-dimensional distribution andrepresents the probability of finding a pole to a lattice plane (hkl) in a certain sample direction(figure 6). Pole figures are normalized to express this probability in multiples of a randomdistribution (m.r.d.). Depending on the application stereographic or equal area projection ofthe spherical pole density distribution is used. Inverse pole figures are also projections of theODF, but in this case the probability of finding a sample direction relative to crystal directionsis plotted. This is particularly useful for axisymmetric textures (fibre textures) where only onesample direction (the symmetry axis) is of interest.
Conventionally ODFs are represented as two-dimensional sections of a rectangular Eulerspace. Unfortunately it is very difficult to visualize orientations in this representation and thespace is highly distorted. To overcome these deficiencies, cylindrical representations havebeen suggested to represent the crystal orientation distribution relative to sample coordinates(COD) or the sample orientation distribution relative to crystal coordinates (SOD) (Wenk andKocks 1987, Kocks et al 2000). Matthies et al (1990) have introduced special σ -sections withminimal distortion. In this review we will mainly use pole figures and inverse pole figures.
1378 H-R Wenk and P Van Houtte
3.2. From pole figures to ODF
A pole figure is a two-dimensional distribution of a crystal direction (e.g. pole to a lattice planehkl) relative to sample coordinates. (A direction is specified by two angles.) As mentionedabove, pole figures can be considered projections of the three-dimensional ODF. There arevarious methods to retrieve the ODF from measured pole figures. One set of methods worksin direct space and uses basically algorithms of tomography. The Williams–Imhof–Matthies–Vinel (WIMV) method, introduced by Matthies and Vinel (1982), is most widely used. Variantsare the vector method (Ruer and Baro 1977), arbitrarily defined cells (ADC) (Pawlik et al1991), and the maximum entropy method (Liang et al 1988, Schaeben 1988). Other methodswork in Fourier space, most notably the harmonic method introduced by Bunge (1965) andRoe (1965). Pole figures are expanded with spherical harmonics. Harmonic coefficients fromthe pole figure expansion can then be used to determine harmonic coefficients of the ODFexpansion. The expansion is carried to a finite order, usually between 22 and 32, providing anangular resolution of about 15–10˚. For sharp textures there remain distinct truncation effectswith subsidiary oscillations, including unrealistic negative ODF values.
All methods yield similar results, at least for ideal test data. However, there issome ambiguity in continuous pole figures as is most transparent for the harmonic method.Pole figures are, by their very nature, centro-symmetric, which means that odd coefficientsin the harmonic expansion vanish. The ODF is not centro-symmetric and therefore requireseven and odd coefficients for a full representation, but the odd coefficients cannot be obtainedfrom pole figures (Matthies 1979). Omissions of odd coefficients can introduce errors inorientation densities in the ODF and add spurious maxima and minima, called ghosts (Kallend2000). This uncertainty is between 10% and 30% for most textures. The uncertainty can bemuch reduced by exploiting the fact that the ODF cannot be negative (see, e.g. Van Houtte(1991)). For sharp textures, there is usually no uncertainty left if an angular resolution ofabout 10˚ is used. At higher resolution, or for weak textures with a high isotropic background(phon), additional assumptions are needed to remove the uncertainty (Mathies and Vinel 1982,Schaeben 1988).
There are many software packages that calculate ODFs from pole figures and performother operations to quantify textures in polycrystals (some examples are BEARTEX, Wenket al (1998), LaboTex, Pawlik et al (1991), MulTex, Helming (1994), POPLA, Kallend et al(1991), TexTools, Resmat Corp.). Details can be obtained from the Internet.
3.3. Use of diffraction spectra
Traditionally texture analysis has relied on pole figure measurements. This is efficient if only afew pole figures are required for the ODF analysis and if diffraction peaks are reasonably strong(relative to background) and well separated. The method becomes increasingly unsatisfactoryfor complex diffraction patterns of polyphase materials and low symmetry compounds withmany closely spaced and partially or completely overlapped peaks. The amount of textureinformation is roughly contained in the product of the number of pole figures (hkl) times thenumber of sample orientations used during the measurement of one pole figure. In conventionalODF analysis one relies on a few pole figures and many sample orientations. Another approachis to use many pole figures and few sample orientations. This is an obvious advantage for TOFneutron diffraction where many diffraction peaks are measured in a continuous spectrum.
Rietveld (1969) proposed a method to use a continuous powder patterns to obtain crystal-lographic information (e.g. Young (1993)) and this method can be expanded to include textureanalysis. In a powder with a random orientation of crystallites, the relative intensities are the
Texture and anisotropy 1379
d [Å]
Figure 7. Spectra of experimentally deformed limestone recorded with TOF neutrons and the 40˚detector banks of HIPPO. Relative intensity variations indicate texture.
same for all sample orientations and are due to the crystal structure. In a textured material, thereare systematic intensity deviations from those observed in a powder as illustrated in figure 7for TOF neutron diffraction data of limestone. Intensities are linked to the crystal structure bymeans of the structure factor, they are also linked to the texture through the ODF (figure 6).
As with the pole figure method described above, texture effects can be implemented in theRietveld method either with Fourier or with direct methods. The finite number of harmonicODF coefficients can be refined in a similar way as crystallographic parameters with a non-linear least squares procedure. With discrete methods ODF values are directly related to peakintensity values in the spectra. The Rietveld method is implemented in software packagesGSAS (Von Dreele 1997) and MAUD (Lutterotti et al 1997) that can be downloaded from theInternet.
3.4. Statistical considerations of single orientation measurements
With single orientation measurements that rely on surface coverage, the number of grains thatcan be measured is limited. This becomes apparent if we consider that a texture function (ODF)with 5˚ resolution has 181 584 cells in the case of triclinic crystal symmetry. Even if we hadthat many grains and a random texture, some ODF cells would have 0 grains, most would have1 grain and there would be cells with 2, 3, 4 or more grains, i.e. the ODF would range between0 and 4 m.r.d. For fewer grains the situation is worse. Either larger cells (and worse angularresolution) have to be used, or data have to be smoothed in a statistically correct way. Forsingle orientations (rather than an averaged diffraction intensity) the statistical fluctuations areexpressed in exaggerated pole densities and a large texture index F2. F2 has been introducedby Bunge (1982) as a bulk measure of texture strength and is equal to the volume-averagedintegral of squared orientation densities over the ODF (equivalent to the ODF-weighted meanof the ODF itself). It is mainly influenced by sharp texture peaks (�1 m.r.d.). Matthies andWagner (1996) have explored the relationship between number of measured grains N and F2,and established an asymptotic 1/N dependence of F2(N) that can be used to determine thesmoothing function for a given sample. It turns out that for any quantitative representation it
1380 H-R Wenk and P Van Houtte
is necessary to measure a large number of grain orientations (not just data points scanned bythe EBSP system).
Thus, EBSP data may provide good qualitative information on texture patterns, butneutron diffraction on bulk samples is often needed to obtain quantitative information ontexture strength. In most published EBSP texture analyses, arbitrary smoothing is applied(e.g. by fitting orientations with harmonic or Gauss functions) and results are, therefore, atbest semi-quantitative.
3.5. From textures to elastic anisotropy
If we know the orientation distribution and single crystal physical properties, then we cancalculate approximate polycrystal physical properties. Of most interest have been elasticproperties, described as a fourth rank tensor. Single crystal elastic constants for many materialsunder a variety of conditions have been compiled (e.g. Simmons and Wang (1971)). Polycrystalelastic properties are obtained by a summation over all contributing single crystals, taking intoaccount their orientation, the relationship with neighbours and microstructure, such as shapeof crystallites, presence of pores and cracks. If there is preferred orientation (non-randomODF), a macroscopic anisotropy will result.
In the case of polycrystal elastic properties, the summation needs to be done to maintaincontinuity across grain boundaries when a stress is applied, and to minimize local stressconcentrations. Recently this has been approached with finite element simulations (e.g.Dawson et al (2001)). In practice the local stress and strain distribution is usually neglectedand the summation is done by simple averages. There are two extreme cases: the Voigtaverage assumes constant strain throughout the material and applies strictly to a microstructurecomposed of laths with a stress applied parallel to the laths. Values of the elastic constantsare maximum. The Reuss average assumes that stress is constant and strictly applies to thecase of a microstructure in which an extensional stress is applied perpendicular to the laths.In this case aggregate elastic constants are at minimum values. There are other averages thatare intermediate between constant strain and constant stress. One example is the Hill average(1952), an arithmetic mean of Voigt and Reuss averages, or the geometric mean (Matthies andHumbert 1993), or a self-consistent average proposed by Kroner (1961).
The relationship between texture, single crystal properties and polycrystal properties isestablished in various ways. One application is to measure bulk elastic anisotropy and estimatefrom it texture. While such a determination is neither accurate nor complete, it can neverthelessbe practical, e.g. to estimate crystal alignment and corresponding flow regimes in the deepearth from seismic anisotropy measurements, or take advantage of destruction-free velocitymeasurements of large engineering components to ascertain specifications and conceivabledamage. A field that is receiving a lot of attention is to determine single crystal elastic propertiesfrom diffraction measurements of elasticity (changes in lattice spacings) on deformed texturedpolycrystals (e.g. Singh et al (1998), Gnaupel-Herold et al (1998), Matthies et al (2001)).Another important application is the estimation of residual stresses in textured materials onthe basis of diffraction data (Van Houtte and De Buyser 1993).
4. Polycrystal plasticity simulations
4.1. General comments
There are several reasons why it is desirable to simulate the development of texture andanisotropy during deformation. For engineering applications an important consideration is
Texture and anisotropy 1381
�2
Figure 8. Hot rolled aluminium (AA5182) and texture simulations with different methods. Shownis the orientation density f (g) along the α-fibre as function of φ2 (Van Houtte and Delannay 1999).
the cost of experiments compared to simulations to obtain information on properties afterspecific treatments. More importantly, if simulations with a physical theory can satisfactorilyexplain experimental results, it is likely that the underlying processes are understood and can begeneralized. Another scientific reason to carry out simulations to predict deformation texturesof metals is the fact, that such simulations usually provide a wealth of additional details, suchas estimations of stored energy, dislocation density, and local heterogeneities of stress andplastic strain, and all this as a function of grain orientation and/or grain neighbourhood. Thesedata are very hard to measure experimentally, but they are desperately needed to understandrecrystallization upon annealing, a phenomenon that is very important in metal processing.In earth science, it is often not possible to reproduce the complex strain paths that occurin nature. Most deformation experiments on rocks are done in compression geometry, andmore recently, also in torsion. Both paths are very special cases that are rarely satisfied inactual geological conditions. A second reason is that conditions in nature often cannot bereproduced in experiments, most significantly strain rates and grain size, e.g. at high pressure.If microscopic mechanisms are known that are active under a given set of conditions and if agood constitutive theory exists, then polycrystal behaviour for any strain path and conditionscan be simulated. Polycrystal plasticity simulations are still far from being perfect as wasrecently demonstrated by Van Houtte and Delannay (1999) for such a simple system as rolledaluminium, where different theories predict different texture patterns and each different fromwhat is actually observed (figure 8).
Different mechanisms can produce or modify texture. Most important is dislocationglide during deformation, which we will discuss in the next section. Also significant isrecrystallization, either dynamic or static, with nucleation of new domains and grain boundarymobility. During phase transformations features of texture patterns are often inherited. (Wewill illustrate examples for metals and rocks in a later section.) Textures may form duringsolidification from a melt, vapour deposition and electro-deposition. If fluids are present,aspects of dissolution and growth in a stress field can have a profound influence on resultingorientation patterns (e.g. Bons and den Brok (2000)). In other systems textures evolve as non-equiaxed rigid particles deform in a viscous matrix (Jeffery 1923, March 1932, Willis 1977).
4.2. Deformation
Deformation of a polycrystal is a very complicated heterogeneous process. When an externalstress is applied to the polycrystal, it is transmitted to individual grains. Dislocations move
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on slip systems, dislocations interact and cause ‘hardening’, grains change their shape andorientation, thereby interacting with neighbours and creating local stresses that need tobe accommodated. To realistically model these processes is a formidable task and onlyrecently have three-dimensional finite element formulations been developed to capture at leastsome aspects (Mika and Dawson 1999). The difficulty is that in real materials local stressequilibrium and local strain continuity are maintained, and this requires local heterogeneityat the microscopic level. Most of the polycrystal plasticity simulations have used highlysimplistic approximations, e.g. that each grain is homogeneous, and yet arrived, at least formoderate strains, at useful results.
There are two extreme assumptions. Taylor (1938) suggested that in modelling plasticdeformations, straining could be partitioned equally among all crystals. This hypothesis hasbeen used extensively for fcc and bcc metals (e.g. Van Houtte (1982), Kocks et al (2000)). Forthis approach even to be viable, the individual crystals must each be able to accommodate anarbitrary deformation, requiring five independent slip systems. While the Taylor assumptionis reasonable for materials comprising crystals with many slip systems of comparable strength,using it in other situations can lead to prediction of excessively high stresses, incorrect texturecomponents, or both. In the Taylor model, high stresses are required to activate slip systems,even in unfavourably oriented grains, and the model is therefore known as an upper boundmodel.
In contrast to the Taylor hypothesis, all crystals in a polycrystal can be required to exhibitidentical stress, given that their behaviour is rate-dependent at the slip system level (a variantof the original Sachs (1928) assumption). The equal stress hypothesis is most effective forpolycrystals comprising crystals with fewer than five slip systems. It has also been used inmodelling the mechanical response of crystals that possess adequate numbers of independentslip systems, but where slip systems display widely disparate strengths. The principal drawbackis that deformation often is concentrated too highly in a small number of crystals, leading toinaccurate texture predictions. With the Sachs approach only the most favourable slip systemsare activated and, therefore, stresses are low. This approach is known as a lower bound model.
Several other approaches have been developed for modelling the heterogeneousdeformation of highly anisotropic polycrystals as is the case for many rocks. For example,Molinari et al (1987) developed the viscoplastic self-consistent (VPSC) formulation for largestrain deformation in which each grain is regarded as an inclusion embedded in a viscoplastichomogeneous equivalent medium whose properties coincide with the average properties ofthe polycrystal. VPSC, mainly in the formulation of Lebensohn and Tome (1993) has beensuccessfully applied to the prediction of plastic anisotropy and texture development of variousmetals (e.g. Tome and Canova (2000)) and geologic materials (e.g. Wenk (1999)). Self-consistent methods have to struggle with the non-linearity of the relation between stress andplastic strain or strain rate, as they implicitly use linearizations of this material model forthe strain field surrounding the ‘inclusion’. The applicability of a self-consistent model toengineering studies can be refined by tuning modelling parameters to better correlate assumedmicroscopic interactions with macroscopic behaviour or with the results of crystal plasticityfinite element simulations as explained below (Gilormini and Michel 1999, Molinari and Toth1994, Molinari et al 2004).
Another modelling approach is to employ finite element methodologies to computedeformations of an aggregate of crystals. In this case local heterogeneity can be taken intoaccount, particularly if grains are discretized with many elements (Kalidindi et al 1992, Bate1999, Mika and Dawson 1999). The boundary value problem resulting from the applicationof homogeneous macroscopic boundary conditions is solved to obtain the deformation ofindividual crystals. In this case the rotation of a grain and its deformation depends both on
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the orientation and the orientations of neighbours. This method is called the crystal plasticityfinite element method (CPFEM).
CPFEM methods are in principle the best models for the simulation of plastic deformationof polycrystalline materials and for the prediction of deformation textures, but they still requirehuge calculation times, even on very powerful computers. This limits their extensive use inengineering applications. For example, in a finite element simulation of the deep drawing ofa car body panel, several tens of thousands of elements need to be used, each representing anentire polycrystal. It is practically impossible to use CPFEM for each of them. The challengethen is to develop polycrystal deformation models which are 103 or 104 times faster thanCPFEM models, but which reach a comparable quantitative accuracy. Much progress towardsthis goal has been achieved by the so-called ‘multi-grain’ models. In such models, theTaylor model is solved not for a single grain, but for an aggregate of several grains. Inthe LAMEL model (Van Houtte et al 2002), two grains are used whereas in the GIA model(Crumbach et al 2001), eight grains are used. This is repeated for many sets taken fromthe microstructure. These models capture the effect of local interactions between particularneighbouring grains on the deformation pattern, which is not possible with conventional self-consistent models, because those always ‘smear out’ the neighbours of a particular grain byaveraging. Surprisingly, the quantitative accuracy of the ODFs predicted by these models foraluminium and steel was found to be comparable to the results of CPFEM models, and farbetter than of any other model (Van Houtte et al 2002, Li and Van Houtte 2002).
It is noteworthy that Raabe and Roters (2004) have tried to solve the calculation timeproblem in a different way: they do not put a true CFEM model in every integration point ofa FE model of a metal forming process (‘FE2 model’), but use the CPFEM code directly forthe forming problem itself. The number of finite elements used for the forming process is notexcessively high. They enter a simplified texture in every integration point, consisting, e.g.of a single grain or a few grains. The orientations of all these grains together do reflect theaverage texture of the material.
All polycrystal plasticity models are comprised of two basic parts: a set of crystal equationsdescribing properties and orientations, and a set of equations that link individual crystalstogether into a polycrystal. The latter set provides the means to combine the single crystalquantities to define the polycrystal response on the basis of physically motivated assumptionsregarding grain interactions.
As a crystal deforms by slip, it undergoes a lattice rotation. These lattice rotations are thecause of development of preferred orientations in aggregates with many component crystals.All polycrystal plasticity models deal with a highly non-linear system. A solution is obtainedby working with a finite number of discrete grains (given as orientations) and deformation isapplied in increments. The deformation is defined by a displacement gradient tensor that maybe constant or may change with deformation. Rotations of all grains are calculated after eachstrain increment and the orientations, as well as their shape and slip system activities, are thenupdated. A comparison of the texture patterns that evolve with those that are experimentallyobserved is a good indicator for the applicability of a model. In addition to texture patternsmicrostructural and mechanical features are also predicted.
4.3. Recrystallization
The relationship between slip and crystal rotations is straightforward. Other processes suchas climb, grain boundary sliding, diffusion in general may also affect orientation distributions.Of particular importance is recrystallization. In deformation studies, recrystallization isthe development of a new grain structure with low dislocation density, either during
1384 H-R Wenk and P Van Houtte
deformation (dynamic recrystallization, Guillope and Poirier (1979)) or after deformation(static recrystallization or annealing, Humphreys and Hatherly (1995)). It is agreed that alsohere energy considerations are responsible for the development of recrystallized domains.Some theories suggest that thermodynamic equilibrium in a non-hydrostatic stress fieldcontrols recrystallization (discussed in various forms by Kamb (1961), Paterson (1973),Green (1980) and Shimizu (1992)), but at least in metals it is clear that strain energiesproduced by accumulations of dislocations far exceed thermodynamic energies imposed bythe stress field (Humphreys and Hatherly 1995). Grains with higher dislocation densitieshave higher stored energies than grains with low dislocation densities (Haessner 1978).Grains with higher stored energies may be consumed through boundary migration bygrains with less stored energy (growth). Alternatively, dislocation-free nuclei may formin grains with high dislocation density and then grow at the expense of others. Thetexture which finally forms is believed to be controlled either by nucleation or by graingrowth. It is possible that grain growth is ‘oriented’, i.e. for some reason grains withcertain crystallographic orientations grow faster than others. In that the case the graingrowth mechanism is likely to control the final texture. Otherwise ‘oriented’ nucleationmay control the final texture. The texture of fresh nuclei may not be random and reflectthe final texture. The last word has not yet been said on this problem (Gottstein 2002). Notonly orientation distributions, but also grain size distributions are important considerations(Shimizu 1999).
The deformation state of a grain depends on its orientation and its history and can thusbe predicted with polycrystal plasticity theory. The changes in texture and grain size thatoccur during annealing, and their dependence on microstructural mechanisms provides alogical link to develop detailed recrystallization models, which couple deformation modelswith probabilistic laws to simulate recovery and recrystallization. Among them, a modeldeveloped by Radhakrishnan et al (1998) couples the finite element method with the MonteCarlo technique so as to account for local effects in aggregates.
Deformation simulations with the self-consistent model generate a population of grainswith a variation in deformation and a variation in dislocation density. Whether a grain which is‘hard’ due to a high Taylor factor and as a results features a smaller strain, will be found to havea smaller or higher than average dislocation density depends on the value of tangent or secantmodulus used in the self-consistent model, and whether an anisotropic grain-by-grain hardeningmodel has been implemented in it. Note that according to physics, the dislocation density is anon-monotonically increasing function of the product of the Taylor factor with the local strain.Results of such models have been used in an empirical model to simulate texture changesduring recrystallization (Wenk et al 1997). The microstructural hardening of slip systemsduring deformation provides an incremental strain energy to grains after each deformation step.Grains with a high stored energy are likely to be invaded by their neighbours with a lower storedenergy. In the model the stored energy of each grain is compared with the average stored energyof the polycrystal. If the stored energy of a grain is lower than the average, it grows; if it is higher,it shrinks. If nucleation in dislocation-free domains accompanies boundary migration, a highlydeformed parent grain divides upon reaching a threshold strain rate and produces a dislocationfree nucleus. The nucleus (which may be a subgrain or a bulge in a grain boundary) takes onthe current orientation of the parent at the time of its formation, but its dislocation density isreset to zero. This has an effect on the subsequent evolution, because these domains with lowdislocation density can grow much faster. Nucleation takes place if the strain increment exceedsa threshold value. If nucleation dominates over growth, grains with high dislocation densitywill preferentially determine the final texture. With these parameters growth and nucleation canbe balanced.
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The model was applied to simulate static and dynamic recrystallization textures inhexagonal metals (Solas et al 2001, Puig-Molina et al 2003) and geologic materials such asquartz (Takeshita et al 1999), calcite (Lebensohn et al 1998), olivine (Wenk and Tome 1999)and ice (Thorsteinsson 2002). Some examples will be given below. Admittedly, currentlyavailable recrystallization models are highly simplistic approaches to the complex and stillpoorly understood process of recrystallization, but they provide methods to estimate changesin bulk anisotropy (Gottstein 2002).
5. Important texture types in metals
By far most effort has been made in the characterization of metal textures and amongthose aluminium and steel received much attention. A majority of texture researchersinvestigate conditions leading to favourable textures for particular applications. Manyindustrial processing facilities characterize texture patterns to achieve required standards. Oneof the best known examples is steel sheet or aluminium alloy sheet for car body panels,where severe requirements concerning desired anisotropy have been set by the car industryfor decades. In the case of thin-walled materials such as aeroplane components and beveragecans texture optimization is crucial to ensure satisfactory mechanical properties at a minimalcost as well as minimal material loss in manufacturing. While many details are known aboutmetal textures, there is a relatively small variety of fundamental types, compared to ceramicsand geological materials. Crystal structures are basically fcc, bcc and hcp, not counting somerare ordered alloys. Deformation is principally by rolling, extrusion (of wires) and rarelycompression. Deformation may occur at cold or hot conditions and is often followed byannealing. In this section we will introduce the main texture types that are observed andhighlight a few issues. There are several reviews on the diversity of metal texture types thatshould be consulted, among them Dillamore and Roberts (1965), Hatherly and Hutchinson(1979), Mecking (1985), Rollett and Wright (2000) and Wassermann and Grewen (1962).
5.1. Fcc metals
The weakest slip systems in fcc metals are {111}〈110〉, consisting of 12 symmetricallyequivalent variants. Activity of these systems produce a characteristic texture pattern duringrolling, which is illustrated as pole figures and orientation distribution sections for copper(figures 9(a) and 10). The large number of slip systems makes it easy to achieve compatibilityand the rolling texture can be well explained with the Taylor theory, especially if individualslip systems are allowed to harden, according to their activity (Kocks and Mecking 2003)and if allowance is made for some heterogeneity across small grain boundaries (Honneff andMecking 1981). As is obvious, the texture pattern is complex, even for such a simple case whereonly a single family of slip systems is active. The reasons for this complexity are obvious ifone considers polycrystal plasticity with characteristic rotations of individual orientations thatvary in direction and amount. Crystal orientations rotate through orientation space and thoseregions in orientation space where rotations are smallest and collect dynamically large grainnumbers generally correspond to high orientation densities; regions where rotation incrementsare large correspond to depleted orientation densities. A maximum in the ODF is rarely astable orientation where all rotations converge. More often it is a transient where orientationstemporarily accumulate.
In order to simplify the description and facilitate quantitative comparisons, the three-dimensional orientation distribution of the rolling texture has been divided into idealizedcomponents, defined as orientations with a lattice plane normal (hkl) in the normal direction
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Figure 9. Pole figures of rolled fcc metals. (a) Rolled copper determined from a singlesynchrotron image measured at ESRF. (b) Brass measured with an x-ray pole-figure goniometer.(c) Recrystallized copper measured with a pole-figure goniometer and displaying mostly the cubetexture with a minor component due to twinning. Equal area projection, logarithmic pole densityscale. Ideal components are indicated in (a): � cube, ♦ Goss, × S, ◦ copper, • brass.
and a lattice direction [uvw] in the rolling direction. This description dates back to the dawn oftexture analysis (e.g. Polanyi and Weissenberg (1923)), long before the ODF was introduced(Bunge 1965) and was a first step in quantifying the three-dimensional aspect of textures.Table 1 summarizes the main fcc texture components. They are also shown with symbols infigures 9(a) and 10.
Some words of caution are appropriate about the ideal component description. First,components are distributions centred on the ideal orientation. Polycrystal plasticitydemonstrates that the real distributions are much more complex and generally asymmetrical.Second, particularly geologists have been tempted to associate ideal components withdeformation mechanisms, assuming that the slip plane normal is parallel to the normaldirection and the slip direction parallel to the rolling direction. In fcc metals there is no{111}〈110〉 component, the main slip system, and in most crystals several slip systems areactive simultaneously! There are a few cases of low symmetry crystals and deformationconditions where such a simplistic interpretation applies but in most cases it does not (see, e.g.Wenk and Christie (1991)).
Two main fcc rolling textures are distinguished, the copper type with Cu, S and brasscomponents (figure 9(a)), and brass with brass and Goss (figure 9(b)). In low stacking faultmetals such as brass mechanical twinning on {111}〈112〉 accompanies slip and this producescharacteristic texture differences. There is a range of intermediate types that depend on thecomposition and temperature, both affecting the stacking fault energy (e.g. Alam et al (1967)).
More refined idealized descriptions have been introduced, such as ‘texture fibres’corresponding to scattering about a line in orientation space. Such ‘skeleton lines’ may be
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Figure 10. ODF of rolled copper (same sample as shown in figure 9(a), shown as γ -sections(Roe/Matthies convention) (COD: crystal orientation distribution relative to sample coordinates).Orthorhombic sample symmetry is assumed. Ideal components are indicated in (a): � cube,♦ Goss, × S, ◦ copper, • brass. (a) Contoured rectangular sections. (b) Polar coordinates witha less distorted orientation space.
Table 1. Ideal texture components for rolled fcc metals.
a Crystallographic equivalent angles inside the unit zone for orthorhombic sample symmetry (as inrolling).
defined on the basis of crystal-sample geometry (and have also been used way back, e.g.Glocker (1924)) or extend irregularly through the ODF along regions of high densities. Theα-fibre, with 〈110〉 parallel to the rolling direction (0˚, 0–90˚, 45˚, Bunge convention), connectsbrass and Goss components, the β-fibre connects copper and brass components (Hirsch andLucke 1988), the γ fibre has {111} parallel to the normal direction (0–90˚, 55˚, 45˚, Bungeconvention).
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Figure 11. Plot of volume fractions of ideal components for copper with increasing recrystallization(Rollett and Wright 2000).
Figure 12. Inverse pole figures illustrating axial deformation of fcc metals. (a) Copper deformedin compression to a strain of 0.68 (courtesy C Necker). (b) Gold wire (Wenk and Grigull 2003).Equal area projection, logarithmic contours.
The orientation density that is associated with a fibre or spherical component can then beused to follow changes, e.g. during recrystallization, where the cube component dominatesover others (figure 9(c)). This fcc recrystallization texture has been explained as preferentialnucleation on shear bands with high misorientations (Beaudoin et al 1996). Often a minor{111}〈112〉 twin component is observed. With the component method the texture changesduring annealing can be quantified (figure 11). Puig-Molina et al (2003) observed withsynchrotron x-rays in a furnace the in situ evolution of the cube component, which occurredwithin a few minutes around 300˚C.
Relatively little work has been done on fcc torsion textures (simple shear). Canova et al(1984) proposed ideal orientations based on Taylor simulations. Van Houtte (1981), Stout et al(1988) and Hughes et al (2000) documented considerable variation with material, particularlystacking fault energy. At a first glance simple shear textures resemble rolling textures butrotated against the sense of shear.
In axisymmetric compression fcc materials such as copper display a strong 110 maximumparallel to the compression direction (figure 12(a)). In extension, e.g. wire drawing, a111 maximum in the extension direction develops as illustrated for gold (figure 12(b)).Axisymmetric textures are best illustrated in inverse pole figures.
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Figure 13. Texture variations for electro-deposited nickel produced by deep etch lithography (theLIGA process) as function of pH and deposition rate (Dini 1993).
There has been much interest in textures formed by electro-deposition, e.g. in theproduction of microactuators where orientation patterns are of considerable importance.Artificially patterned structures have been produced by deep etch lithography (the LIGAprocess) and applied extensively to Ni and Ni–Fe alloys. Strong texture development hasbeen observed and variations depend on deposition conditions (figure 13, see, e.g. Van Ackeret al (1994), Dini (1993) and Buchheit et al (2002)).
5.2. Bcc metals
The most common deformation mode in bcc metals is {110}〈111〉 slip, which is a transpositionof slip plane and slip direction with respect to fcc metals. This correspondence produces someanalogies: compression textures for bcc are similar to extension textures for fcc. Rollingpole figures for bcc can be obtained from those of fcc by reversing extension direction(rolling direction) and compression direction (normal direction). But in detail the situation ismore complicated, because bcc metals also slip on other planes than {110} in the 〈111〉direction(Christian 1970) and this can be described by the so-called ‘pencil glide model’ (Rosenbergand Piehler 1971, Van Houtte 1984). Table 2 lists the main bcc texture components due torolling and annealing. They have the convenient property that they can all be represented ina ϕ2 = 45˚ section of the ODF (Bunge notation), as shown in figure 14(a). Such sections arewidely used in the steel literature to represent textures and figure 14(b) shows an example ofcold-rolled steel.
The bcc texture development has been most extensively studied for steel and there isconsiderable variation with deformation conditions and composition (Ray et al 1993). Additionof 3% Si profoundly enhances the texture as illustrated by orientation density variations alongthe α fibre axis, both in cold and hot rolled steel (figure 15).
At least qualitatively deformation textures of rolled fcc and bcc metals can be wellsimulated with the Taylor model, since many slip systems are available to satisfy compatibilitywithout undue stress concentrations. However, this is not the case for axial deformation of fccmetals in compression and bcc metals in tension with a lack of slip systems to deform crystalsto an axial shape. In this case the local deformation is in plane strain, resulting in a ‘curling’microstructure (Hosford 1964).
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Table 2. Ideal texture components for rolled and annealed bcc metals.
a Crystallographic equivalent angles inside the unit zone for orthorhombic sample symmetry (as inrolling).b See figure 14(a).
Figure 14. (a) Texture components in rolled bcc metals displayed in a φ1 − �, φ2 = 45˚ sectionof the ODF. The relative orientation of a cube is illustrated (see table 2). (b) φ2 = 45˚ section ofthe ODF for cold-rolled steel.
5.3. Hcp metals
Texture development in hexagonal metals has received a lot of attention because of theapplications of zirconium alloys in the reactor industry and titanium alloys as structuralmaterials in aerospace. In contrast with cubic metals twinning is always significant andcorrespondingly textures are more complex. Because of the low crystal symmetry severalfamilies of slip systems need to be activated. The critical shear stresses depend on composition(particularly the C/A ratio), temperature and degree of deformation (including hardeningcharacteristics, e.g. Chin and Mammel (1969), Thornburg and Piehler (1975), Tenckhoff (1988)and Philippe (1994)). Major deformation systems are listed in table 3. A typical rolling textureof titanium is shown in figure 16(a). It is characterized by a spread-out c-axis maximum aroundthe normal direction and (1010) poles concentrated in the rolling direction. Lebensohn andTome (1993) have modelled texture evolution of hcp metals with the self-consistent theoryand could explain experimental patterns as a result of mainly prismatic and some pyramidal
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Figure 15. Texture variations in bcc iron as function of Si content. Shown are orientation densityvariations along the α-fibre (Sudo et al 1981).
Table 3. Main deformation systems in hcp metals. The critical resolved shear stress ratios (crss)are typical for zirconium deformed at low temperature.
slip, combined with compressive and tensile twinning. Twinned orientations have a distinctpattern and c-axes concentrate near the normal direction (figure 17), whereas grains with c-axesat large angles to the normal direction (near the transverse direction) do not twin.
During recrystallization subtle changes are less dramatic than in fcc metals (figure 16(b))and those changes are best revealed in difference pole figures (figure 16(c)). They indicate thatgrains with c-axes slightly tilted to the normal direction and a = (1120) parallel to the rollingdirection become dominant in the recrystallization texture (see also Wagner et al 2002). Theseare orientations that are most heavily twinned (figure 17) and presumably twinned regionswith high surface energy and heterogeneity are favoured nucleation sites and those nucleieventually grow.
5.4. Phase transformations
Metallic compounds undergo phase transformations when subjected to temperature–pressurechanges. If a polycrystalline material is textured then the new phase may inherit textureinformation from the parent phase. Orientation relations have been suggested based onanalogies in crystal structures (e.g. Kurdjomov and Sachs (1930) for bcc→fcc and Burgers(1934) for hcp→bcc) (table 4). The orientation relations suggest that in correspondingstructures close-packed or nearly close-packed lattice planes are parallel, and that close-packed
1392 H-R Wenk and P Van Houtte
(a)
(b)
(c)
Figure 16. Textures of rolled titanium. In situ synchrotron measurements with a vacuum furnace.(a) 200˚C, before the onset of recrystallization, (b) at 700˚C after recrystallization, (c) differencepole figure (starting material-recrystallized material). Equal area projection (Puig-Molina et al2003). Logarithmic contours for (a) and (b), linear contours for (c).
Figure 17. Texture simulations for hcp deformation using the VPSC theory during rolling to 50%.+ symbols indicate grains that have not twinned, × are grains that have twinned at least once;symbol size is proportional to the deformation of individual grains. (001), (100) and (110) polefigures are shown in equal area projection (Wenk et al 2004).
Table 4. Orientation relations during phase transformations and number of variants.
directions are parallel. In each case there are several symmetrically equivalent variants. If allvariants were applied the texture would more or less randomize. There appears to be a strongvariant selection in most of these transformations, regardless of whether the transformationsoccur by a martensitic (shear) mechanism or by nucleation. There is much interest, because
Texture and anisotropy 1393
(a)
(b)
(c)
Figure 18. In situ observation of texture changes during phase transformations, measured withthe HIPPO TOF neutron diffractometer at Los Alamos. (a)–(c) ultra low carbon iron (Wenk,unpublished), (d)–(f) zirconium (Wenk et al 2004). (a) 800˚C, bcc; (b) 950˚C, fcc; (c) 400˚C, bccafter cooling; (d) 650˚C, hcp; (e) 950˚C, bcc; (f) 650˚C, hcp after cooling. Pole figures in equalarea projection. Rolling (RD), normal (ND) and transverse directions (TD) are indicated.
the ‘texture memory’ is relevant for novel shape memory alloy applications (such as nitinol,e.g. Vaidyanathan et al (2001)). The selection principles are poorly understood, even for thetechnologically important bcc→fcc transformation in steel which has been studied in mostdetail (Ray and Jonas 1990). It is established that microstructure, composition and stress allhave an influence. The reason for the lack of knowledge is partially the difficulty of measuringtextures at high and low temperatures in situ. This is changing with the availability of newneutron and synchrotron facilities that can accommodate vacuum furnaces.
We will have a closer look at the bcc→fcc transformation in ultra low carbon iron, whereit has recently become possible to measure the high temperature fcc texture (950˚C) in situby neutron diffraction. At 800˚C we find a typical rolling texture (figure 18(a)). After thetransformation (800˚C) we observe a Kurdjomov–Sachs relationship with the fcc (110) polefigure roughly corresponding to the bcc (111) pole figure (figure 18(b)). Upon cooling to 400˚Cthe texture returns almost exactly to the initial bcc texture (figure 18(c)), documenting a texturememory and variant selection. Similar relationships have been documented for the hcp→bcctransformation. At 650˚C the texture of zirconium is similar to that of titanium (figures16(a) and 18(d)). The high-temperature (bcc) texture, measured at 950˚C (figure 18(e)),is related to the low-temperature (hcp) texture by the Burgers relation, but with a strongvariant selection. Interestingly this experimental texture is different from that predictedon the basis of models (e.g. Ciurchea et al 1996, Gey and Humbert 2002). The cubictransformation texture is best explained if preferential nucleation of the bcc phase takesplace in the most highly twinned hcp grains (figure 17), similar to the recrystallization casedescribed above. After cooling the new hcp texture closely resembles the original texture(figure 18(f )), also here with a strong memory, probably imposed by stresses from neighbouringgrains.
1394 H-R Wenk and P Van Houtte
6. Ceramic textures
Polycrystalline ceramic materials are either produced in bulk or as thin coatings on substrates.Many of the bulk processing methods, such as tape casting, extrusion and injection mouldinginvolve deformation that may produce textures. Whereas in metals most textures develop bydeformation, recrystallization and casting (growth from melt), more mechanisms are active inceramics. Processes such as cold pressing (‘green body pressing’), hot pressing (densificationand phase transformation), surface grinding, epitaxial and topotaxial growth can all lead totextures in ceramics. Ceramic coatings display often clear epitaxial or topotaxial relationshipsbetween substrate and film. While textures are often weaker than in metals, ceramics havestrongly anisotropic properties and optimization of textures has become increasingly important.
Compared to metals and to geological materials the literature on texture developmentin ceramics is surprisingly sparse. Some of the pioneering work was that of Pentecostand Wright (1964) who demonstrated, with pole figures, an alignment of crystallites in pressedpowders of plate-shaped Al2O3 and needle-shaped BeO. Textures in ceramics were reviewedby Bunge (1991), documenting that research in this field is growing as new materials are beingmanufactured with critical properties, brittle, ductile and electrical. The following discussionhighlights some systems, dividing ceramics into two sections: bulk ceramics and films.
6.1. Bulk ceramics
6.1.1. α-alumina (Al2O3). Trigonal aluminium oxide (point group 32/m, mineral namecorundum) is the most widely used structural ceramic, such as for lamp envelopes, sparkplugs and substrates for integrated circuits. Hot forging and deep drawing experiments havedocumented that large scale deformation is a viable means of forming alumina ceramics. Inhot forging intracrystalline slip is active and contributes to texture development (Heuer et al1980). The most important production method is green processing of powders (die pressing,slip casting, tape casting, extrusion) and since alumina particles are generally plate-shaped,textures are produced (Dimarciello et al 1972, Bocker et al 1991, Raj and Cannon 1999).The powder processing is followed by sintering to produce cohesion by grain growth, whichoften enhances the texture. Since thermal expansion is anisotropic, small residual stresses areintroduced during thermal cycling in highly textured alumina which can lead to microfracturesand eventual brittle failure (Lee et al 1993).
6.1.2. Silicon nitride (Si3N4). Silicon nitride is a ceramic material with numerous newapplications in automotive components and machining tools due to its high stiffness, strengthand hardness, coupled with good fracture toughness and thermal shock and corrosion resistance.It occurs in two hexagonal forms, α and β. The high-temperature β-phase forms rod-shapedneedles, elongated along the c-axis, which can be oriented by hot-pressing or forging (Walkeret al 1995). The textured materials have very anisotropic fracture properties (Willkens et al1988). In accordance with the grain shape (needles and platelets, respectively), the preferredorientation is ‘inverse’ to that of alumina, i.e. (0001) poles are at high angles to the direction ofprincipal compression, in axial and plane-strain compression. The fracture toughness generallyincreases and becomes more anisotropic with texture development.
6.1.3. Zirconia (ZrO2). Zirconia occurs in a cubic (when doped with Y), a tetragonal (high-temperature) and a monoclinic (lower temperature) structure (mineral name baddeleyite). Thetetragonal to monoclinic phase transformation can be stress-induced (martensitic) and is greatly
Texture and anisotropy 1395
influenced by crystallite orientation (Muddle and Hannink 1986). While the texture of themonoclinic phase is most relevant to generating strains and plastic work, the texture of thetetragonal phase can be advantageously tailored to enhance transformability and toughness(Bowman and Chen 1993). The tetragonal parent phase transforms into the monoclinic phaseby simple shear on the plane (100) in the [001] direction. In zirconia texture is produced duringthe phase transformation directly by the selective stress-induced transformation and does notrequire the parent material to be textured.
6.1.4. Ceramic matrix composites. During many investigations of composite ceramics,it became apparent that reinforcement textures are extremely important for mechanicalproperties. Yet a quantitative characterization of textures is generally lacking. Experimentalwork on alumina–SiC whisker composites has shown that little texture development is observedin the alumina matrix under conditions that have been shown to produce strong preferredorientation in pure alumina (Sandlin and Bowman 1992). By contrast, whiskers attainstrong preferred orientation due to their elongated grain shape. Zirconia ceramics have beenreinforced with alumina platelets to increase fracture toughness by crack deflection and cracksarresting at platelets (Li and Sorensen 1995). These mechanisms depend on textures. Ceramiccomposites consisting of a SiC matrix, reinforced by SiC fibres have recently been developedfor thermostructural applications. (SiC exists both in a cubic form, β, and a hexagonalform, α.) The advantage of such composites lies in the low density, the high mechanicalstrength and rigidity and their chemical inertness. Diot and Arnault (1991) documented thatthe SiC matrix has 〈111〉 directions preferentially aligned parallel to the fibre axis of thecomposite. Such a texture has the lowest surface energy between matrix and fibres, and ispreferred.
6.1.5. Bulk high-temperature superconductors. The conductivities of the high-temperaturecuprate superconductors are highly anisotropic and largely confined to the (001) Cu–O plane(e.g. Tuominen et al (1990)). Techniques for developing strong preferred orientations inmacroscopic samples have been successful in improving critical current densities in many ofthese materials (e.g. Kumakura (1991)).
Among the cuprate oxides, texture development is best documented for the Y-123compounds (YBa2Cu3O7−x). Significant degrees of texture can be produced by high-temperature plastic deformation of polycrystalline pellets (Wenk et al 1989, Chen et al 1993).[001] axes of Y-123 align themselves with the compression direction, due to intracrystallineslip (figure 19(a)). Other techniques to produce texturing in Y-123 are melt textured growth(e.g. Selvamanickam and Salama (1990)) and alignment in a magnetic field (e.g. deRango et al(1991)).
Other compounds of interest are Bi-2223 (Bi2Sr2Ca2Cu3Ox) and Bi-2212(Bi2Sr2CaCu2Ox). The plate-like crystal morphology makes these materials suitable for green-body processing (Steinlage et al 1994). Plate-like crystals are preferentially oriented with [001]axes (normal to the plane of the plate) parallel to the compression direction (figure 19(b)) andpreferred orientation develops largely by rigid body rotations.
Unfortunately techniques using axial stress produce a relatively low degree of crystalliteorientation. More successful has been a method to sheath Bi-superconductors in silver tubesand fabricate a tape or wire and then thermally treating it (Sandhage et al 1991, Wenk et al1996). Silver is used as the sheath material due to its chemical compatibility with Bi-2212and 2223. It also supplies ductility and protection for the tape or wire for use in potentialapplications such as long power transmission lines and superconducting coils.
1396 H-R Wenk and P Van Houtte
(a) (b)
Figure 19. Texture development in high-temperature superconductors (inverse pole figures,assuming tetragonal crystal symmetry). (a) Hot compression of Y-123 (Wenk et al 1989),(b) Bi-2212 multifilamentary tape encased in silver (Wenk et al 1996).
6.2. Thin films and coatings
In thin films the importance of texture is obvious (e.g. Szpunar (1996)). Texture influenceselastic properties and thermal expansion that are essential for the mechanical stability of films.Also, in the case of superconducting films, electrical properties are intimately linked to crystalorientation. Qualitative texture information on thin films has been extracted from powderdiffractometry. Only recently have pole figure measurements and orientation distributionanalysis been used to quantify crystallite orientation. There are two types of film textures.The first type consists of films and coatings on polycrystalline or amorphous substrates inwhich the influence of the substrate is minor and the texture is formed mainly by anisotropicgrowth. These are generally axially symmetric fibre textures. Of a second type are epitaxialfilms deposited on a structurally related single crystal. In those the texture is usually extremelystrong, approaching a single crystal and is controlled by the match between the two crystalstructures.
6.2.1. Silicon and diamond. The broad range of applications of polycrystalline silicon filmsin microelectronics and in the fabrication of micromechanical structures for use in actuatorsand sensors demands that the material be well characterized. From the mechanical point ofview, the elastic properties and internal stresses of films during thermal cycling are texturedependent and need to be optimized for satisfactory performance.
Silicon films deposited from vapour on (111) single crystal substrates on an amorphoussilicon dioxide layer show a large variation in texture which is mainly controlled by temperatureand silane pressure (e.g. Joubert et al (1987), Wenk et al (1990)). At high silane pressure andlow temperature, Si-films are amorphous (figure 20(a)). With increasing temperature andlower pressure textures change from a strong {113/112} fibre to {110} (figure 20(b)) and thento {100} (figure 20(c)). At lowest pressure and highest temperature crystallites are alignedrandomly. Low-temperature films show tensile stresses (up to 700 MPa), high-temperaturefilms display compressive stresses (up to 600 MPa). Combinations of the texture types can beused to attain overall equilibrium. Texture types in polycrystalline Si-films can be correlatedwith microstructures. The {110} texture in the 620–650˚C range is due to the columnargrains, which share 〈110〉 as the growth direction (Krulevitch et al 1991). 〈110〉 and 〈112〉are fast growth directions for silicon that crystallizes out of an amorphous state. At low
Texture and anisotropy 1397
Figure 20. Si films deposited on a single crystal Si wafer, covered with amorphous silicon dioxide.(a) Variation in texture as function of temperature/silane-pressure. (b) and (c) Typical texturesrepresented in inverse pole figures. Stereographic projection, contour interval for (a) and (c)0.25 m.r.d., for (b) 0.5 m.r.d., shaded above 1 m.r.d. (Wenk et al 1990).
temperature twinned grains oriented with the 〈110〉 direction close to the film normal surviveand become columnar in shape. At higher temperatures, when twinning does not accompanycrystallization, 〈100〉 is the direction of fastest crystal growth. For low stress 700˚C filmsdeposited at 300 mTorr, there is a lower driving force for twinning, resulting in columnargrains elongated along 〈100〉 and consequently a predominantly {100} texture.
Recently diamond coatings have received attention. Also here the texture varies withfabrication conditions and can be very strong (e.g. Brunet et al (1996), Helming et al (1995)).
6.2.2. Nitride, carbide and oxide coatings. Coatings are often applied to metal and ceramicsubstrates to improve properties for various applications. They may improve heat and wearresistance, lower frictional forces, prevent chemical corrosion resistance or simply add to thedecorative appearance. Carbides and nitrides of transition metal coatings have led to significantimprovements of cutting tools for turning and milling operations. Coatings are often preparedby arc evaporation and deposited from a gaseous phase on to a substrate, e.g. tungsten carbide(WC). Leonhardt et al (1982) describe strong {111} fibre textures for TiC, typical of high-temperature deposition. At lower deposition temperature, a {100} fibre is also observed.Similar textures are observed in TiN and HfN coatings on WC (figures 21(a) and (b)). Sue andTroue (1987) have documented that (100) textures reduce the erosion rate compared with (111)(figure 21(c)).
Textures of oxide coatings on polycrystalline substrates appear to be mainly controlledby growth kinetics rather than by the texture of the substrate material. Zirconium oxide hasa thermal expansion coefficient similar to that of stainless steel and thus only a small thermalmismatch when deposited on metallic substrates. Because of its high hardness and toughness,cubic yttrium stabilized zirconia (YSZ) is one of the most widely used protective coatings inthe automotive, aeronautical and cutting tool industry (Rhys-Jones 1990).
A survey of ferroelectric Pb2ScTaO6(PST) films on various substrates revealed a strong111 fibre texture when deposited on a (100) silicon crystal coated with platinum (figure 22(a))and a 001 fibre on (110) Al2O3 (figure 22(b)) (Chateigner et al 1997). The ferroelectric andfatigue behaviours of such films depends on crystallite orientation (Kim et al 1994).
1398 H-R Wenk and P Van Houtte
Figure 21. Textures in HfN (a) and TiN (b) tool coatings, represented as inverse pole figures,stereographic projection. (c) Dependence of the alumina particle jet erosion rate on a TiN coating asa function of texture, evaluated as (111)/(200) diffraction peak intensity ratio (Sue and Troue 1987).
Figure 22. Inverse pole figures of PST films on substrates. (a) on platinum coated (100) Si, (b) on(110) Al2O3. Equal area projection (Chateigner et al 1997).
6.2.3. Epitaxial films. Much research has been invested in producing epitaxialsuperconducting films on various single crystal substrates, in an effort to obtain electronicdevices with favourable electrical properties. TEM studies have documented that the filmsconsist of a polycrystal with small crystallites but with a very high degree of preferredorientation. The orientation of the crystallites is controlled by epitaxy and growth velocity.The balance between the two factors is primarily dependent on the distance from the substratesurface (and thus the foil thickness) and the temperature and partial pressures at which thecrystals grow. Because superconductivity in HTS ceramics is restricted to the Cu-plane, it isimportant to have the corresponding lattice plane (001) parallel to the film. The goal is to findconditions at which relatively thick films have a favourable crystal alignment with the substrate.
Most work so far has been done on Y-123 films on various substrates. In this system ithas been useful to measure pole figures of the combined 102 and 012 reflections of Y-123because this reflection does not coincide with the substrate. Two orientations are of primaryinterest: the C-orientation has c-axes of Y-123 parallel to the foil normal and the A-orientationhas c-axes in the foil plane and a-axes of Y-123 in the foil normal (Heidelbach et al 1992).The two orientations are expressed by 102 peaks at pole distances of 56˚ (C-orientation)and 34˚ (A-orientation), respectively (figure 23(a)). The a-axes of Y-123 crystallites in theC-orientation are aligned either parallel or at 45˚ to those of the substrate depending on thebest match in the oxygen sublattice of film and substrate in the perovskite, periclase andfluorite structures (Tietz et al 1989). In these very strong epitaxial textures, where widths oftexture peaks are less than 5˚, pole figures measured with 5˚ × 5˚ angular increments give onlyqualitative information about orientation relationships. To obtain quantitative values, texture
Texture and anisotropy 1399
Figure 23. (a) 102 pole figure of a thin film of Y-123, laser ablated on (001) LaAlO3 at 710˚C.Peaks from A orientation (with [100] normal to the substrate) and C-orientation (with [001] normalto the substrate) are indicated. (b) Change of integrated C/A peak intensity ratio as function ofdeposition temperature (Wenk et al 1996).
peaks need to be scanned on a much finer grid and effective peak intensities can be integrated.The ratio of the volume of crystals with c-axes in the foil plane to that of crystals with a-axesparallel to the foil plane (C/A ratio) on (001) LaAlO3 shows a steady increase with depositiontemperature (figure 23(b)). A high C/A ratio is obviously preferred.
Deposition on single crystal substrates is adequate for small electronic devices. Manyother applications require larger structures and flexibility. Here laser deposition of thicksuperconducting films on textured metallic substrates has been employed, particularly biaxiallytextured Ni-tapes (Wu et al 1995, Yang 1998).
7. Textures in minerals and rocks
Geological polycrystals are referred to as rocks. Some rocks are monomineralic such as marble,composed of calcite, and quartzite, composed of quartz. Many rocks are polymineralic withexamples such as granite, composed of feldspar, quartz and mica, and peridotite, composed ofolivine and pyroxene. We will introduce a few minerals where texture development has beeninvestigated in detail and patterns of preferred orientation have added to a better understandingof deformation in the earth. Calcite is one of the best studied minerals with strong texturesdeveloping by slip, mechanical twinning and recrystallization. Also quartz has been studiedextensively, but many aspects of texture development remain enigmatic. The two trigonalminerals are also of relevance to metallurgists since they deform on similar slip systems ashexagonal metals. Orthorhombic olivine is important because of its significance for convectiveflow in the earth’s mantle and the seismic anisotropy that is observed. Sheet silicates are agroup of minerals with a distinct platy morphology and their alignment during deformationor compaction has been modelled as rigid particle rotations in a viscous medium. Texturesin ice have been investigated to better understand glacier flow and deformation of the largeice shields, as well as the rheology of the large outer planets. An interesting model systemis halite (NaCl) and isostructural periclase (MgO) with high plastic anisotropy. Following abrief survey of these minerals, we will illustrate two geological applications: the use of texturein unravelling strain history and the importance for understanding seismic anisotropy.
Earth materials differ from metals in several respects. Whereas ODFs of cubic metalscan be represented in a small orientation volume and share many common features that can
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Figure 24. Graph of normalized shear strain rate plotted against shear stress, illustrating the effectof the stress exponent n on the deformation behaviour. High strain rate sensitivity, approaching aviscous behaviour, is present in minerals (n = 3–9), compared to metals (n ∼ 99).
be catalogued (e.g. as component volume fractions), ODFs of low symmetry materials requirelarge volumes, are specific in each case, with few common features. The sample coordinatesof geological specimens are generally not a priori defined and it is impossible to visualize anODF in a different orientation. For these reasons ODFs, though essential for interpretations,calculations of properties and texture simulations, are often difficult to interpret and pole figuresremain an important representation.
Minerals have a high strain rate sensitivity (low stress exponent; 3–9 versus >99 in metals)and differently oriented crystals may deform at very different rates and long before the criticalshear stress is reached (figure 24). Minerals were the incentive to introduce strain sensitivity inpolycrystal plasticity. Minerals have lower symmetry and thus fewer equivalent slip systems.This causes a high plastic anisotropy. Both features suggest that deformation of mineralaggregates is heterogeneous and this has to be taken into account in models.
It must be emphasized that texture is the result of the accumulated strain history, whichis particularly significant for geological situations where the strain path often changes inthe course of a long history. In simulations such changes can be easily included and textureevolution, for example, in a highly heterogeneous system such as a convection cell in theearth’s mantle, can be simulated.
7.1. Calcite (CaCO3)
Calcite rocks (limestone and marble) have long been a model system for experimental rockdeformation. The classic Yule marble studies (e.g. Turner et al (1956)) were the basis for aquantitative interpretation of texture development in minerals by slip and twinning. Largelybased on single crystal studies, the main deformation systems were established: slip onthe rhombs r = {1014}〈2021〉 and f = {1012}〈0221〉〈2201〉 and mechanical twinning one = {0118}〈4041〉. (Note that there is a preferred sense to slip on the rhombohedral systems.)In axial compression experiments, regular texture variations were observed with temperature,pressure, strain rate and grain size (Wenk et al 1973, Schmid et al 1977). At low temperature
Texture and anisotropy 1401
Figure 25. Calcite deformed in axial compression and textures represented in inverse pole figuresof the compression direction. (a) Experimental texture observed for fine-grained limestone at lowtemperature; (b) Taylor simulations for fully imposed compatibility fail to predict the experimentaltexture; (c) relaxation of constraints for curling are in much better agreement (Wenk et al 1986).(d) Experimental high temperature texture with significant grain growth (Wenk et al 1973).(e) Results from a recrystallization simulation that show with symbol size the degree of deformationof differently oriented grains, evaluated with the self-consistent model (Lebensohn et al 1998).
a texture with a maximum of compression axes near c = (0001) and a shoulder towards thenegative rhomb e = (0118) develops (figure 25(a)). Taylor simulations failed to reproducethe low temperature texture (figure 25(b)), until it was recognized that local plane straindeformation was energetically favourable over axial deformation, analogous to curling in cubicmetals (Hosford 1964). When Taylor conditions are relaxed the correct texture is predicted(figure 25(c)). At high temperature, where grain growth indicates recrystallization, the texturehas a maximum at high-angle positive rhombs (figure 25(d)). The high temperature texturecan be explained as due to grain growth of the least deformed grains (figure 25(e)). As we willsee later, this is rather exceptional in minerals. In most cases dynamic recrystallization favoursthe most highly deformed grains.
Since calcite rocks such as limestone and marble are mechanically ductile compared toother minerals, deformation experiments other than axial compression can be performed moreeasily. Noteworthy are plane-strain compression experiments (referred to as ‘pure shear’ in thegeological literature) with three mirror planes in the pole figure (figure 26(b)) and simple shearwith only one mirror plane and a 2-fold axis in the pole figure (figure 26(c)) as compared to axialcompression experiments which yield a fibre texture (figure 26(a)). These experimental polefigures are good examples to illustrate that the symmetry of pole figures reflects the symmetry of
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Figure 26. Calcite (0001) pole figures of experimentally deformed calcite limestone illustratinghow the symmetry of the pole figure relates to the symmetry of the strain path: (a) axisymmetriccompression (Wenk et al 1986), (b) plane-strain compression (Wagner et al 1982), (c) simple shear(Kern and Wenk 1983). Some contour levels (in m.r.d.) are marked. Compression, extension andshear directions are indicated by arrows.
the deformation history (strain path) which has been a main criterion for interpreting textures innaturally deformed rocks (Paterson and Weiss 1961). Dynamic recrystallization was producedin torsion experiments to large strains (Pieri et al 2000).
Naturally deformed calcite rocks, limestones and marbles, often display strong preferredorientation with a c-axis maximum more or less perpendicular to the schistosity plane. Wewill return to the interpretation of natural calcite textures later in this review.
7.2. Quartz (SiO2)
Low quartz (α) is trigonal, point group 32. At high temperature α-quartz transforms tohexagonal β-quartz (point group 622). Both low and high quartz lack a centre of symmetryand exist in a right- and a left-handed form. The absence of a centre of symmetry is the reasonfor properties such as piezoelectricity in polycrystalline quartz (e.g. Parkhomenko (1971)).In investigations of preferred orientation by diffraction methods, enantiomorphism cannot beidentified and a higher crystal symmetry is used by adding a centre of symmetry to pointgroup 32. This produces point group 32/m, which is that of calcite and is generally used inrepresentations.
Slip systems have been established by laboratory deformation of oriented single crystals(e.g. Blacic (1975)). Basal (0001) 〈1120〉 slip dominates at low temperature. At highertemperature prismatic slip {1010} [0001] becomes active. These well established slip systemshave a hexagonal symmetry, and if they alone were active, the orientation distribution inpolycrystals should also display hexagonal symmetry and this is generally not the case. Oneexplanation is that trigonality is introduced by mechanical Dauphine twinning (Zinserling andShubnikov 1933). Dauphine twinning is geometrically a two-fold rotation about the c-axis butcan be produced by a slight distortion of the lattice, either during the α–β phase transformationor by shear. It does not affect the orientation of c and a-axes. Dauphine twinning is a mechanismto produce ‘trigonal textures’.
An example of a trigonal quartz texture in a naturally deformed mylonite is shownin figure 27(a)). The trigonality may have been introduced during crystallization understress (Tullis and Tullis 1972). Twins in naturally deformed quartzite have actually beenobserved with EBSP orientation imaging analysis (Heidelbach et al 2000). Upon heatingabove 573˚C at ambient pressure, quartz transforms to hexagonal symmetry and pole figuresfor positive r = {1011} and negative rhombs z = {0111} that were different for trigonal
Texture and anisotropy 1403
Figure 27. Texture changes during phase transformations in quartzite. In situ measurements withTOF neutrons. (a) Initial trigonal texture of a single crystal type with broad scattering (500˚C).(b) Hexagonal texture at 650˚C where the two rhombs 1011 and 0111 become equivalent. (c) Uponcooling to 500˚C the texture returns to the original orientation variant. Equal area projection.
quartz are now equal (figure 27(b)). Interestingly, upon cooling the material returns almostexactly to the starting texture (figure 27(c)), similar to the case of zirconium described earlier(figure 18(d)–(f)). It is speculated that the memory is imposed by stresses from neighbouringgrains since α-quartz is elastically strongly anisotropic and these stresses produce a selectionof the energetically favourable twin orientation.
Texture development and deformation in naturally and experimentally deformed quartziteshave been reviewed (Wenk 1994) and we only summarize some salient aspects. Polycrystallinequartz has been the subject of numerous experimental studies, mainly in axial compression.Figure 28 shows typical texture types developing during plastic deformation of fine grainedquartz. At lower temperatures and high strain rates, c-axes concentrate near the compressionaxis (figure 28(a)). At higher temperature and slower strain rates a distinct maximum developsnear r = {1011} (figure 28(b)). The distinct difference between positive rhombs (h0hl) andnegative rhombs (0hhl) (e.g. in figure 28(b)), can be at least partially attributed to mechanicalDauphine twinning.
Experimental investigations also addressed the behaviour of quartz during dynamicrecrystallization. Mainly based on microstructures, Hirth and Tullis (1992) identified threedifferent mechanisms with increasing temperature, decreasing strain rate and thus decreasingflow stress: (1) strain induced grain boundary migration; (2) progressive subgrain rotation and(3) progressive subgrain rotation with rapid boundary migration at the highest temperatures.Changes in textures are characteristic of these mechanisms (Gleason et al 1993).
1404 H-R Wenk and P Van Houtte
(a) (b)
Figure 28. Typical texture patterns in axially compressed flint (a fine-grained aggregate of quartz)represented as inverse pole figures of the compression direction. At low temperature and high strainrates there is a maximum near c = 0001. At high temperature the maximum is shifted towardsr = 1011 (Green et al 1975).
Many naturally deformed quartzite specimens have been examined. Most of theseinvestigations relied on measurements of [0001] axes with a petrographic microscope, equippedwith a universal rotation stage. The quartz texture types have been classified by Sander(1950) into c-axis maximum fabrics at low temperature with the maximum perpendicular tothe foliation plane (figure 29(a)) and at higher temperature with pervasive recrystallizationwith the maximum in the foliation plane and perpendicular to the lineation (stretchingdirection) (figure 29(b)), small-circle girdles (figure 29(c)) and asymmetric crossed girdles(figure 29(d)). In all the representations the schistosity plane is horizontal and the lineationdirection pointing to the right). The texture types are characteristic of formation conditions,particularly temperature (metamorphic grade). At low temperature and dominant basal slip thec-maximum perpendicular to the foliation plane develops (figure 29(a)). The crossed girdletype is characteristic of simple shear deformation with basal and prismatic slip active (Takeshitaet al (1999), figure 29(d)). The type with the strong c-axis maximum in the intermediatefabric direction is characteristic of mylonites, intensely deformed rocks (figure 29(b), also27(a)). The microstructure indicates recrystallization and plasticity simulations can explainthis common texture type (Wenk et al 1997). In plane strain compression c-axes rotatetowards the compression direction (figure 30(a)). However the most deformed grains are inthe intermediate strain direction and if nucleation is allowed in these subordinate orientations,and the nuclei grow, they dominate the recrystallization texture (figure 30(b)). The cause ofthe small circle fabric is still unclear (figure 29(c)). It occurs at highest metamorphic grade(granulite facies) and may be due to activation of rhombohedral slip systems.
7.3. Olivine (Mg2SiO4)
The orthorhombic mineral olivine is the major constituent of the upper mantle of the earthand has therefore been of long standing importance for understanding geodynamic processes.Natural olivine textures have been compiled by Ben Ismaıl and Mainprice (1998). By far themost common texture type has (010) poles nearly perpendicular to the foliation plane and [100]axes subparallel to the lineation direction (figure 31(a)). Texture development is attributed todeformation in the upper mantle at depth and the mantle rocks have later been juxtaposed inthe crust where they can be sampled.
Slip systems in olivine have been established in single crystal experiments (e.g. Raleigh(1968), Nicolas et al (1973), Bai et al (1991)). At high temperature (010)[100] is the
Texture and anisotropy 1405
Figure 29. Typical quartz textures of naturally deformed quartzites represented as (0001) polefigures; Z is the pole to the foliation, X the lineation, and Y (centre) is the intermediate fabricdirection. (a) Quartz layer in limestone (Thurnpass, Tirol), (b) Typical Y-maximum texture,common in mylonites (Melibokus, Odenwald Germany); (c) small-circle girdle observed ingranulites (Burgstadt, Saxony); (d) crossed girdle in quartz lens in marble (Hintertux, Tirol)(Sander 1950).
Figure 30. VPSC simulations of texture changes during deformation and recrystallization ofquartz in pure shear plane strain. (0001) = c pole figures, shortening direction is vertical, extensiondirection horizontal. (a) Texture development during deformation. (b) and (c) Texture developmentif deformation is accompanied by recrystallization with significant nucleation. Initially new grainsnucleate (×-symbols) and increasingly dominate the texture (c). Symbol size is proportional tograin size (Wenk et al 1997).
preferred slip system and simulations reproduce the high temperature texture observed inmantle peridotites (figure 31(b)). Zhang and Karato (1995) produced simple shear texturesand observed at moderate strain a [100] maximum displaced from the shear direction againstthe sense of shear and, at higher strain with pervasive recrystallization, a more symmetrical
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Figure 31. (a) Naturally deformed olivine from peridotite occurring as xenoliths in basalt fromSouth Africa and originating from the upper mantle. (b) Based on known slip systems and assumingpure shear deformation this texture can be simulated with the VPSC theory.
texture with [010] perpendicular to the shear plane and [100] in the shear direction. Thistexture transition can be explained with polycrystal plasticity models, again, as in the case ofquartz, assuming that the most highly deformed orientations are favoured (Wenk and Tome1999). We will return to olivine in the context of mantle convection.
7.4. Sheet silicates
Sheet silicates such as micas (muscovite, biotite, chlorite, etc) and clay minerals (illite, smectite,kaolinite, etc) occur commonly in many deformed sedimentary (mudstone, shale, e.g. Ho et al(1999)) and metamorphic rocks (slate, schists, e.g. O’Brien et al (1987)). Sheet silicatesare characterized by a plate-like morphology, parallel to the basal (001) lattice plane. Thismorphological anisotropy largely controls the orientation changes of sheet silicates duringcompaction and subsequent deformation. They are important because single crystals displayextreme anisotropy of physical properties such as seismic wave propagation and rocks in whichsheet silicates are oriented are also highly anisotropic (e.g. Kern and Wenk (1990)).
The orientation behaviour of non-equiaxed, rigid particles in a viscous medium duringdeformation has been modelled by Jeffery (1923). In the extension of March (1932) and Willis(1977), in which particles are considered infinitely oblate (or infinite needles), the texture hasbeen directly correlated with the finite strain. In spite of the severe constraints of the modelwhich rarely apply to real systems, the predicted strain/texture relationship is often in goodagreement with independent strain markers (e.g. Oertel (1983)) and experiments (Tullis 1976).
The interpretation of (001) sheet silicate pole figures is based on their symmetry, theirintensity and their geometrical relationship with regard to the mesoscopic structural elementssuch as bedding, cleavage and lineation (Weber 1981). In pure compaction, pole figures areaxially symmetric with a maximum parallel to the bedding pole (figure 32(a)). In deformationtextures axially symmetric pole figures, centred around the cleavage pole, indicate flattening
Texture and anisotropy 1407
Figure 32. Examples of (001) sheet silicate pole figures. (a)–(c) are illite/muscovite pole figuresfrom shales and slates in the Variscan fold-and-thrust belt in Belgium (Sintubin 1994); (d) is biotitefrom mylonites in Southern California which were partially deformed in simple shear (O’Brienet al 1987). Equal area projection. (a) Axisymmetric compaction texture (maximum is 6.4 m.r.d.),(b) axisymmetric deformation textures (maximum is 20 m.r.d.), (c) orthorhombic deformationtexture (maximum is 10 m.r.d.), (d) asymmetric deformation texture produced by simple shear(maximum is 14.5 m.r.d.).
(figure 32(b)), whereas orthorhombic textures imply a plane-strain component (figure 32(c)).In some heavily deformed metamorphic rocks (mylonites), pole figures are often asymmetric(figure 32(d)) and this has been attributed to a simple shear component in the deformationhistory (O’Brien et al 1987).
7.5. Ice (H2O)
Deformation of hexagonal ice–I has been the subject of numerous investigations and it has longbeen known that preferred orientation develops during the flow of glaciers and deformation ofthe large polar ice sheets, with c-axes oriented perpendicular to the surface (e.g. Kamb (1959),Duval (1979), Lipenkov et al (1989), Tison et al (1994), Alley et al (1995)). These textures arelargely attributed to flattening in a glide and climb regime. Locally components of simple shearhave been documented and in polar ice sheets there are layers with pervasive recrystallization.In experimental studies it was observed that during deformation and dynamic recrystallizationc-axes align parallel to the compression direction (Duval 1981). Fabric development of ice hasbeen modelled for regimes of deformation (Castelnau et al 1996) as well as recrystallization
1408 H-R Wenk and P Van Houtte
(Thorsteinsson 2002). Texture development greatly influences the anisotropic flow behaviourand has received a lot of interest (Duval et al 1983, Lliboutry and Duval 1985, Paterson 1991).Most of these preferred orientation studies were performed close to the melting point in thepolymorph ice–I. Only recently has ice been deformed at low temperature and high pressurein the stability field of other polymorphs, relevant for the dynamics of outer planets and theirsatellites. Bennett et al (1997) have analysed ice textures by neutron diffraction at 77 K andobserved that ice–I deformed at low temperatures also has c-axes parallel to the compressiondirection, whereas the high pressure polymorph ice–II has a-axes parallel to the compressiondirection.
7.6. Halite (NaCl) and periclase (MgO)
The deformation behaviour of halite (‘rock salt’, NaCl) and related minerals has beenextensively investigated, mainly because salt mines are being considered as suitable nuclearwaste repositories and their long term stability needed to be evaluated. Most of these studiesemphasize mechanical properties but texture plays obviously an important role. The review ofKern and Richter (1985) still gives a fairly up-to-date overview to which interested readers arereferred. In compression experiments {110} fibre textures are generally produced (Kern andBraun 1973) and in extrusion experiments at low temperature a {100} fibre dominates (Skrotzkiand Welch 1983). Salt textures in natural settings described by Goeman and Schumann (1977),Kampf et al (1986) and Scheffzuk (1996) document a {100} maximum perpendicular to thefoliation plane.
Ductility of single crystals and deformation mechanisms were investigated by Carterand Heard (1970). At low temperature {110}〈110〉 slip is dominant. At higher temperature{100}〈011〉 and {111}〈110〉 become equally active. Salt deformation is of interest forpolycrystal plasticity because, though this mineral is cubic and has many slip systems, itis plastically highly anisotropic. This is because the favoured slip system {110}〈110〉 hasonly two independent variants and harder slip systems are required to deform an aggregatehomogeneously. In lower bound models all deformation is concentrated on soft systems,whereas upper bound models equally activate hard systems to achieve compatibility. Thusdifferent plasticity models give entirely different texture results (Wenk et al 1989). FEM hasrecently been applied to halite (Lebensohn et al 2003) and predictions from different modelshave been compared with new experiments performed in axial extension. Differences betweenthe models are best visible in the grain shape distribution as illustrated for d23 strain ratecomponents in figure 33. For homogeneous deformation (Taylor) all grains have the samestrain rate (circle in centre). For VPSC there is a very large spread, some grains deformingover 10 times more than others. For FEM the spread is considerably smaller and similar tothat which is actually observed.
The halite crystal structure is common to many ionic compounds. Of particular interest togeophysics is periclase (MgO) because it is a likely component of the earth’s lower mantle andsignificant for its rheology. With radial diamond anvil experiments, texture development inaxial deformation can be studied in situ at high pressure. Figure 32(a) is an inverse pole figurefor MgO deformed at 47 GPa. The axial stress component is estimated to be 8.5 GPa. Thetexture displays a strong (001) fibre component (figure 34(a)) which can only be explained ifexclusively {110}〈110〉 slip is active as simulated with the self-consistent model (figure 34(b)).If the Taylor model is used, compatibility requires that harder slip systems become activatedand the texture changes profoundly (figure 34(c)). It is to be expected that at lower mantleconditions, in analogy to halite at higher temperature, several slip systems become active.Indeed, texture patterns produced in torsion experiments on magnesiowuestite (Mg, Fe)O by
Texture and anisotropy 1409
Figure 33. Polycrystal plasticity simulations for halite. Shown are the off-diagonal strain ratecomponents d23 predicted at the initial step of each simulation. For Taylor assumption all grainsdeform at the same rate (dot in centre), for VPSC there is a large spread and for FEM (HEP forhybrid element polycrystal) there is a moderate spread (Lebensohn et al 2003).
Figure 34. Inverse pole figures for MgO, deformed in axial compression. (a) In situ diamondanvil experiment at 20 GPa. To the right are simulations with {110}[110] slip highly favoured with(b) the self-consistent theory and (c) the Taylor theory (Merkel et al 2002). Logarithmic contours.
Stretton et al (2001) and shear experiments by Yamazaki and Karato (2002) suggest activityof {111}, {110} and {100} slip systems, all with the [110] slip direction.
7.7. Polymineralic rocks
Most texture research in geology was done on monomineralic rocks such as quartzites andmarbles. By contrast most of the naturally occurring deformed rocks are polymineralic and thisadds great complexities to characterization and interpretation. For example, granite, consistingof feldspars, quartz and mica, shows initially an almost random orientation distribution
1410 H-R Wenk and P Van Houtte
Figure 35. Deformation of granite in the Santa Rosa mylonite zone in Southern California.Pole figures of (a) (001) biotite (a mica mineral) and (b) (11−20) quartz in granite whichwas progressively deformed to mylonite and phyllonite. Determination by neutron diffraction.Equal area projection. Contours for biotite are 0.5, 1, 1.5, 2, 2.5, 3, 4, . . . 14 m.r.d., for quartz0.5, 0.75, 1, 1.25, . . . m.r.d., dot pattern below 1 m.r.d. (Wenk and Pannetier 1990).
(figure 35(a)). With increasing deformation at metamorphic conditions mica and quartz attaina strong texture in mylonite (figure 35(b)). During progressive deformation to phyllonite,which includes grain size reduction, the mica texture further increases, whereas the quartztexture attenuates (figure 35(c)). The texture change in quartz was attributed to a change indeformation mechanisms: dislocation glide accompanied by recrystallization during the firststage, and superplastic flow during the second stage. In polymineralic eclogite from highpressure rocks of Dabie Shan, China, omphacite (pyroxene) develops a strong texture whereasthe garnet component has a random orientation distribution (Wenk et al 2001). A challenge fortexture research has been to determine preferred orientation of triclinic plagioclase, a commonconstituent in many rocks (Dornbusch et al 1994, Siegesmund et al 1994, Xie et al 2003). Allof these studies relied on neutron diffraction and some on the Rietveld method to deconvolutethe very complex diffraction patterns.
There are few experiments on texture development in polymineralic rocks. Tullis andWenk (1994) found that addition of mica to quartz reduces the strength of preferred orientationof quartz, presumably due to preferential sliding on sheet silicates and a similar behaviour wasobserved in limestone–halite aggregates (Jordan 1987).
The understanding of plasticity in polyphase materials is still very rudimentary. Differentphases may have very different mechanical properties and corresponding aggregates willdeform very heterogeneously. Microstructural investigations of granitic rocks (Tullis 2002)distinguish two regimes, one with interconnected strong or weak phases and another one withinterconnected weak layers. It appears that with increasing deformation granites switch from
Texture and anisotropy 1411
Figure 36. Texture of calcium hydroxide on cement paste-aggregate interface. The plot of 0001pole density against angle to surface interface illustrates the effect of adding 5% silica fume to thepaste (none) and of ageing (increase in texture strength) (Detwiler et al 1988).
a frame structure to a layer structure in which strain is localized in bands (e.g. Herwegh andHandy (1996)). Similar processes have been described in engineering materials (Raj andGhosh 1981).
Texture development in polymineralic metamorphic rocks cannot be explained purely withpolycrystal plasticity. Models must include processes of dissolution and growth (Spiers andTakeshita 1995), interaction with aqueous solutions (Karato et al 1986), chemical reactionsand phase transformations. Apart from orientation state and dislocation microstructure, thechemical composition, grain size and shape are important parameters (Wang 1994, Heilbronnerand Bruhn 1998, Shimizu 1999). Because of these complexities there are still many unresolvedquestions about texture development in such common rocks as gneiss and amphibolite. Aremica and hornblende crystals aligned due to rigid body rotations of particles with anisotropicgrain shape, is the alignment the result of crystal plasticity or is it due to growth, perhaps understress?
7.8. Cement minerals
Concrete consists of particles of gravel and sand (‘aggregate’) that are connected by a varietyof hydrated minerals occurring in the cement paste. Of particular importance for the strengthof concrete is a thin layer of calcium hydroxide (portlandite) directly adjacent to aggregateparticles. Detwiler et al (1988) have documented that this layer shows a strong growth texturewith c-axes aligned perpendicular to the aggregate surface. The strength of the texture increaseswith ageing (figure 36) which appears to be preferable. Other cement minerals, such asettringite may also show local preferred orientation that influences not only the strength butalso the permeability by solutions and thus affects the durability.
Rather unexpectedly it was observed that the durability of concrete correlates amazinglywith texture characteristics of aggregate rocks (Monteiro et al 2001). Deformation of granite,such as that illustrated in figure 35 greatly enhances the alkali silica reaction that causes concreteto prematurely deteriorate. Texture is a bulk measure of the deformation state and it appearsthat highly deformed quartz is very unstable.
1412 H-R Wenk and P Van Houtte
Figure 37. Structure of Earth. Depth, pressure and temperature are indicated as well as mainphases in the different units.
7.9. Earth structure
The Earth is composed of compositionally distinct units with characteristic properties. Theyare summarized in figure 37, which also gives information on temperature and pressure. Inthe hydrosphere ice is significant and texture forms during flow of glaciers and the large polarice caps. The crust shows the greatest diversity of minerals (over 3000) that are accessibleto direct observation in igneous, metamorphic and sedimentary rocks. The minerals of mostinterest are calcite, quartz and mica. They have been discussed earlier. Structural geologistsuse textures to unravel the deformation history during mountain building and we will illustratethis with an example below. The deeper earth (mantle) is dominantly composed of Mg, Si,and O: olivine (Mg2SiO4) in the upper mantle, spinel-like structures ringwoodite (Mg2SiO4)
and wadsleyite (Mg2SiO4) in the transition zone, perovskite (MgSiO3) and periclase (MgO)in the lower mantle. At higher pressure phases tend to have simpler crystal structuresthan minerals in the crust, as established by high pressure experiments and theory (e.g.Fiquet (2001)). Textures have become of special significance with the discovery of seismicanisotropy.
Seismic anisotropy was first recorded in the shallow upper mantle beneath Hawaii (Hess1964), observing that surface waves travel about 10% faster in the E–W direction than N–S.Since then maps of azimuthal anisotropy have been constructed for the uppermost mantlebeneath oceanic lithosphere (figure 38). Fast wave propagation directions overall correspondto flow directions as implied from plate motions. The anisotropy pattern has been refined andit became apparent that anisotropy in the upper mantle varies greatly with depth (e.g. Silver(1996), Montagner and Guillot (2002)).
Texture and anisotropy 1413
Figure 38. Distribution of velocities for Rayleigh waves. The lines correspond to the maximumvelocities at a depth of about 100 km. Spreading ridges are indicated (Montagner and Tanimoto1990).
Anisotropy was documented in layers of the transition zone (400–700 km) where olivinebreaks down to high pressure phases. Phase transformations in this zone are associated withlarge volume changes and this may be the cause of deep-focus earthquakes (Green and Houston1995). Just above the 660 km discontinuity anisotropy is pronounced and may be related tointense deformation associated with those phase transitions (Trampert and Heijst 2002). Butcontrary to the regular pattern of anisotropy in the uppermost mantle, anisotropic regions inthe transition zone are confined, vary in extent and depth and are to some degree associatedwith subduction (Fouch and Fisher 1996).
There is little evidence for anisotropy between 700 and 2700 km, but this may be partiallydue to limited crossing ray coverage. However, the deep mantle, in the vicinity of the boundarywith the core (D′′ layer), reveals itself as a fascinating and heterogeneous area of the earth.Geodynamic modelling suggests very strongly deformed subducting slabs (McNamara et al2001) and seismologists have observed anisotropy that could be due to texturing (e.g. Fouchet al (2001), Vinnik et al (1998)).
The outer core is liquid and thus isotropic, but it is firmly established, both from body wavesand free oscillation observations, that the solid inner core is again anisotropic. CompressionalP-waves travel 3–4% faster along the (vertical) axis of the earth, than in the equatorial plane(e.g. Song (1997)).
In the next sections we will first show an example of how texture can be used by structuralgeologists to infer the tectonic deformation history in the crust. We will then explore howtexture analysis and polycrystal plasticity contribute to a better understanding of deformationand anisotropy in the deep earth.
7.10. Textures as indicators of strain history
For structural geologists the aim is often to unravel the detailed strain history and textures aresignificant because they are generally sensitive to the path. For example, has a tectonic zone
1414 H-R Wenk and P Van Houtte
Figure 39. Cartoon comparing crustal deformation by coaxial thinning (top) and non-coaxialshearing (bottom). The finite strain (shape of ellipse) may be the same, but in simple shear theellipse is inclined by an angle θ to the shear plane.
Figure 40. (0001) pole figures of calcite obtained with the Taylor model for 100% equivalent strainand using resolved shear stress ratios corresponding to low temperature deformation. Pure shearon left, mixed modes in the centre, and simple shear on right. Note the increasing asymmetry ofthe maximum (inclined by ω to the shear plane normal). Sense of shear is indicated (from Wenket al 1987).
been subject to non-coaxial shearing in a shear zone (figure 39, bottom) or coaxial crustalthinning (figure 39, top)? Both paths can lead to an identical finite strain. Textures provide ameans to quantify the amount of non-coaxial deformation. We have already above seen howthe deformation geometry affects the symmetry of the texture pattern of calcite (figure 26).For coaxial deformation one expects orthorhombic pole figures, whereas a non-coaxial path islikely to produce monoclinic pole figures.
Texture patterns of calcite in deformed marbles can be used to determine the partitioningof deformation into a coaxial deformation component and a non-coaxial (simple shear)component. The philosophy is to first develop a deformation model and test it by comparingresulting texture patterns with experiments. Once mechanisms are established and the modeladequately predicts the experiment, one can then apply the model to any arbitrary strain path,including those that cannot be approached experimentally. For calcite, calculated (0001)pole figures for plane strain document a symmetrical (orthorhombic) pure shear pole figureand an asymmetrical (monoclinic) simple shear pole figure (figure 40) and this agrees withexperiments (Kern and Wenk 1983, Pieri et al 2000). Simulations can provide intermediate
Texture and anisotropy 1415
Figure 41. (a) Determinative diagram with angle of asymmetry ω against strain partitioningfactor as obtained from Taylor simulations (figure 40). Results for (b) marble mylonites from corecomplexes of the American Cordillera (Erskine et al 1993), and (c) various limestones from Alpinespreading nappes (Ratschbacher et al 1991).
states and the relative amount of simple shear can be quantified by measuring the angle ofasymmetry ω between the (0001) maximum and the shear-plane normal. One can constructan empirical determinative diagram to assess the amount of simple shear from the asymmetryof the (0001) texture maximum (figure 41(a)).
In practice, geologists collect oriented rock samples in the field, then measure pole figuresin the laboratory relative to geological coordinates, such as schistosity plane and lineationdirection which define the macroscopic shear plane and shear direction, respectively. From theasymmetry of the 0001 pole figure maximum relative to the shear plane the sense of shear can beinferred. From the angle of asymmetry ω and using the determinative diagram in figure 41(a),the strain partitioning can be estimated. Whereas many marbles in core complexes of theWestern United States show almost symmetrical patterns of (0001) axes (Erskine et al (1993),figure 41(b)) and presumably formed largely by coaxial crustal extension, limestones from thespreading nappes in the Alps have generally highly asymmetric texture patterns attributed toshearing on thrust planes (Dietrich and Song (1984), Ratschbacher et al (1991), figure 41(c)).
7.11. Anisotropy in the deep earth
The mantle is not exposed on the surface of the earth. However, during solid state convectionof the mantle and tectonic activity, smaller and larger parts of upper mantle rocks have beenlocally juxtaposed within the crust and can be sampled. The rocks are mainly peridotite,composed largely of olivine and subordinate pyroxene. Locations where mantle peridotitescan be sampled are in Oman (studied extensively by Boudier and Nicolas (1995)), as inclusionsin volcanic rocks from Africa, and several other places. As has been described earlier by farthe most common olivine texture type has (010) poles nearly perpendicular to the foliationplane and [100] axes subparallel to the lineation direction (figure 31(a)).
Deformation mechanisms in olivine have been studied in the laboratory and, knowingslip systems and their activity, it is possible to predict texture development for a given strainpath. In figure 31(b) we illustrated this for plane strain compression. We can now apply thesame method to the larger system of upper mantle deformation. In the mantle large cells ofconvection are induced by instabilities and driven by temperature gradients (e.g. Bunge et al
1416 H-R Wenk and P Van Houtte
Figure 42. Simulated texture development during convective flow in the upper mantle. Streamlinewith (100) olivine pole figures (1000 grains) at five different locations. During upwelling (left)a strong texture develops and is modified during spreading and subduction (Dawson and Wenk2000).
(1998)). The strain distribution in a convection cell is highly heterogeneous. Dawson andWenk (2000) have used the finite element method that incorporates as a constitutive equationpolycrystal plasticity to investigate the development of anisotropy during mantle convection.Figure 42 follows texture development of olivine in the upper mantle along a streamline in[100] pole figures. A strong texture develops rapidly during upwelling (B). The preferredorientation stabilizes during spreading (C, D) and attenuates during subduction (E). The polefigures are distinctly asymmetric due to the component of simple shear. While the finite strainalong a streamline increases monotonically, the texture does not. Knowing textures patternsover the upper mantle, one can then average anisotropic elastic properties and from thoseevaluate seismic wave velocities. The model suggests large variations, up to 15%, comparableto those that have actually been observed by seismologists.
Even though in detail texture development of olivine is complex and not simply analignment of slip directions with flow lines, the overall seismic anisotropy of the uppermostmantle can be reasonably explained as a result of texturing during upwelling along ridges(Blackman et al 2002).
Much less is known about the deeper Earth. Deformation experiments are more difficultbecause pressures are beyond conditions reached by ordinary mechanical devices. Oneapproach has been to use analogue systems with phases of similar structures and bondingbut different composition that are stable at lower pressure and deform at lower temperature.For example, for MgSiO3 perovskite CaTiO3 has been used (Karato et al 1995). An analoguefor periclase (MgO) and magnesiowuestite (FeMgO2) is halite (NaCl). But analogues areof questionable value when it comes to deformation mechanisms, since slip depends onthe detailed electronic structure and bonding characteristics.
For non-quenchable phases in situ observations are required and here texture comes toplay a crucial role because, contrary to microstructure, texture can be measured in situ athigh pressure, for example, with diamond anvil experiments (Merkel et al 2002). If we know
Texture and anisotropy 1417
Figure 43. Temperature distribution (grey shades) during simulated subduction of a slab into thelower mantle (dark: cold, white: medium, dark: hot, as indicated) (McNamara et al 2001).
Figure 44. Simulation of texture development of periclase (MgO) during slab subduction into thelower mantle along a streamline of the McNamara et al (2001) model (figure 43) at 4 differentdepths. {100} pole figures. It is assumed that {110} and {111) slip are equally active.
texture patterns, we can infer deformation mechanisms. During subduction of upper mantleslabs into the lower mantle, geodynamic modelling suggests heterogeneous deformation withcomplicated streamlines (figure 43). Using slip systems that are assumed to be active at hightemperature and pressure, we can then again predict texture evolution along such a streamline inthe subducting slab with increasing depth (figure 44). Simulated textures for MgO (representedas {100} pole figures) are strong at depths of 2000 km, but to produce seismic anisotropy,substantial single crystal elastic anisotropy is also required. At high pressure and temperatureMgO is in fact strongly anisotropic (e.g. Karki et al (1999)). Curiously, at intermediatepressures single crystal anisotropy of MgO is minimal, which may explain the absence ofsignificant anisotropy in the intermediate lower mantle.
The main component of the inner core is an iron-rich alloy, probably with a hexagonalclose packed (ε-iron) structure (e.g. experiments by Shen et al (1998), and first principlescalculations by Wasserman et al (1996)). As was mentioned, seismic waves travel about3–5% faster along the N–S axis of the inner core than in the equatorial plane. There is generalagreement that the reason for seismic anisotropy is an alignment of crystals but there are
1418 H-R Wenk and P Van Houtte
Figure 45. Inverse pole figures for ε-iron (hcp) deformed in compression. (a) In situ diamondanvil texture determination at 220 GPa. (b) Texture simulation with the self-consistent theory forconditions that favour basal slip, 50% strain. Equal area projection, logarithmic contours (Wenket al 2000).
many ideas about the processes that lead to such an alignment. Wenk et al (2000) presenteda model for core texturing during convection. Another possibility is solidification texturingat the boundary with the liquid outer core (Bergman 1997). Finally the earth’s magnetic fieldmay produce stresses that deform the material and thus produce crystal rotations (Karato 1999,Buffett and Wenk 2001). Since the core is so remote, and many conditions are poorly known,convincing cases can be made for each mechanism.
Here we will only illustrate the example of the magnetic field. The largest electromagnetic(Maxwell) shear stresses in the Earth’s geodynamo arise from the combined influence ofthe radial and azimuthal components of the magnetic field and are on the order of severalpascals. Strain gradually accumulates to about 50% over 1 million years as the inner coregrows by solidification. The azimuthal component of the Maxwell stress, which is aboutan order of magnitude larger than the radial component, imposes a strong simple sheardeformation.
A prerequisite for modelling this deformation is the knowledge about slip systems that areactive in ε-iron. This phase is not stable at ambient conditions and deformation experimentsneed to be performed at high pressure. This can be done again with diamond anvil cells.A comparison of texture patterns that were observed at pressures close to those in the innercore (220 GPa) (figure 45(a)) with polycrystal plasticity simulations (figure 45(b)) can beused to infer slip systems. A strong c-axis maximum near the compression direction is onlycompatible with significant basal slip, consistent with ab initio predictions (Poirier and Price1999).
With such information about intracrystalline mechanisms one can then apply the Maxwellstresses to the solid core and predict orientation patterns for different locations. Indeed a strongtexture develops, particularly in the outer parts of the inner core (Buffett and Wenk (2001),figure 46(a)). A next step is to average single crystal elastic properties over the simulatedorientation distributions. Ab initio calculations of Steinle-Neumann et al (2001) for coretemperature (6000 K) and pressure reveal strong anisotropy for the single crystal, with fastestP-wave velocities perpendicular to the c-axis and more than 15% slower than that parallel tothe c-axis (figure 46(b)). These single crystal elastic constants can then be averaged with thetexture predicted from Maxwell stresses (figure 46(c)) and the pattern suggest indeed a smallanisotropy with faster velocities parallel to the N–S axis. The example illustrates that evenfor such remote places as the centre of the earth a combination of experimental techniques
Texture and anisotropy 1419
Figure 46. Anisotropy development in the inner core of the earth assuming that the material isε-iron (hcp) and was deformed by Maxwell stresses. (a) (0001) pole figure for one location in thecore. (b) and (c) Anisotropy of velocities of compressional elastic waves (P). (b) Single crystalP-velocities for 330 GPa and 6000 K based on first principles calculations (Steinle-Neumann et al2001); (c) P-velocities for the aggregate shown in (a) obtained by averaging of the single crystaltensor. The range of grey shades is 5%, equal area projection, linear contours (Buffett and Wenk2001).
and theories that are commonly applied in materials science can be used to better understandstructural features.
8. Textures in mineralized biological materials
Texture research has focused on engineering and earth materials such as metals, ceramics,polymers and rocks. However, there is increasing awareness that texture is significant inorganisms, both in the organization of protein crystals as well as in mineralized skeletons.In such skeletons texturing has a direct mechanical support function and organisms optimizetexture patterns for particular environments. We will briefly illustrate a few examples ofmollusc shells and bones.
8.1. Nacre of mollusc shells (aragonite)
Mollusc shells are composed either of calcite or aragonite, trigonal and orthorhombicpolymorphs of CaCO3, respectively, and show a wide variety of textures. In many molluscsan outer shell is composed of calcite and the inner shell of aragonite. This aragonite nacre(‘mother of pearl’) displays a brick-like microstructure. Texture analysis indicates that in nacreof most molluscs, c-axes are oriented more or less perpendicular to the surface of the shellbut a-axes display characteristic patterns, either resembling a single crystal as in the bivalvePinctada (figure 47(a)) and shells of most land snails, or they display a texture pattern with{110} twinning as in Nautilus (figure 47(b)), or they spin randomly about the c-axis with a[001] fibre texture as in Haliotis (Abalone, figure 47(c)). Chateigner et al (2000) surveyed alarge number of mollusc species and concluded that the diverse texture patterns are related tophylogeny and presumably the protein type. Since [100] is the stiffest crystal direction, sucha texture produces an optimal strength of the shell. Mollusc shells have received considerableattention because of their bio-mimetic properties.
8.2. Bones (apatite)
Texture patterns in biological materials, including carbonates, silica minerals and phosphates,composing shells, skeletons, bones and teeth are a new field of endeavour, with very fewquantitative investigations (Lowenstam and Weiner 1989). The structure of bone and teeth
1420 H-R Wenk and P Van Houtte
(a) (b) (c)
Figure 47. Growth textures of aragonite in nacre from mollusc shells. (010), and (001) pole figures,projected on the shell surface. Equal area projection, logarithmic contours 0.5, 0.7, 1, 1.4 m.r.d.,etc, dot pattern below 1 m.r.d. (a) Nacre of the bivalve Pinctada maxima (Oyster) with more or less asingle component texture, (b) nacre of Nautilus macromphalus with a pseudohexagonal pattern dueto twinning on 110, (c) nacre of Haliotis cracherodis (Abalone) with an axial texture (Hedegaardand Wenk 1997).
Figure 48. Pole figures of hexagonal hydroxyapatite in tissue of a dinosaur tendon. The texturewas measured by synchrotron radiation and analysed with the Rietveld method (Lonardelli et al2004). The tendon axis is in the centre of the pole figure. Contour levels logarithmic; some areindicated (maximum 3.2 m.r.d.). Equal area projection.
is locally very heterogeneous and crystallite size is very small. Only with the advent ofmicrofocus x-ray beams at synchrotron sources have these materials become accessible toquantitative analysis. While in turkey and dinosaur tendon c-axes of pseudohexagonal needle-shaped hydroxyapatite crystals are aligned parallel to the tendon axis, a bovine ankle bonedisplays a strong alignment of c-axes parallel to the surface of the bone (Wenk and Heidelbach1998). Figure 48 shows a c-axis pole figure of apatite in a well-preserved dinosaur tendon, with
Texture and anisotropy 1421
a very strong crystallite alignment. Texturing is important in bone implants as was documentedby Benmarouane et al (2004) in a neutron diffraction study. Single-cell organisms, calledForaminifera, exploit in their calcite skeleton all possible preferred orientation patterns, andalso a random arrangement of calcite laths (Haynes 1981).
9. Conclusions
While we were preparing this review, we surveyed a large amount of literature. It hasbeen impressive to see how much information the classic books of Wassermann on metals(1939, 1962) and Sander on rocks (1950) already contain. Different ideas on texture-formingprocesses and mechanisms had been proposed over many years (such as Sachs (1928),Kurdjomov and Sachs (1930), Burgers (1934), Taylor (1938), Calnan and Clews (1950),Hill (1952), Bishop (1954), Eshelby (1957) to name just a few). The next twenty years,between 1960 and 1980, brought enormous advances, mainly because of the new possibilitiesof quantifying texture data. This was only possible through introduction of high-speedcomputers for data processing and digital electronics for diffraction measurements. Thesesame developments produced a big leap in structural crystallography. The years will beremembered for the introduction of the ODF (Bunge 1965, Roe 1965), the first polycrystalplasticity simulations (Hutchinson 1970, Siemes 1974, Van Houtte and Aernoudt 1975, Listeret al 1978) and TEM investigations to establish deformation mechanisms (e.g. Groves andKelly (1963), Weertman (1968), Kocks (1970), Ashby (1972)). Between 1980 and 2000 camea period of refinements. It is also the time when the quantitative methods, developed in theprevious years, became available for a large group of texture researchers and became generallyaccepted. Calculations of the ODF were refined (e.g. Matthies and Vinel (1982), Van Houtte(1983, 1991), Schaeben (1988), Dahms (1992)), polycrystal plasticity models allowed forheterogeneity (e.g. Molinari et al (1987), Canova et al (1992), Asaro and Needleman (1985),Mathur and Dawson (1989), Van Houtte et al (2002)). On the experimental side, a new toolwas introduced, EBSPs (Dingley 1981) with the SEM and its automation (Wright and Adams1992). With this method not only bulk texture but local textures and misorientation analysisbecame available for characterization.
What will the future bring? Of course predicting directions of research is purelyspeculative. Nevertheless there are numerous new opportunities to extend texture researchinto new directions. With computers available in every lab that are many orders of magnitudefaster that the mainframes of the 1960s, polycrystal plasticity modelling can now be muchmore sophisticated, and with finite element models local intragranular heterogeneity canbe accounted for (e.g. Mika and Dawson (1999)). Such models will probably replace theconventional one-grain one-orientation models and can be applied to polyphase materials forwhich still no satisfactory polycrystal plasticity simulations exist. A weak link of simulationsis the uncertainty about microstructural hardening (Kocks and Mecking 2003). Quantificationon the local scale of slip systems is generally beyond experiments and here first principlessimulations of dislocation dynamics and dislocation interaction will contribute greatly to arealistic assessment of slip activity (e.g. Tang et al (1998)). So far the texture communityand the internal stress community have been separated, yet both effects are highly correlatedand satisfactory interpretation relies on a combined approach. Particularly with new neutrondiffraction facilities simultaneous texture and strain measurements are now within reach (e.g.Walther et al (2000)).
With new instrumentation at neutron diffraction and synchrotron facilities in situ texturemeasurements at high and low temperatures and high pressure can now be performed routinely.This is significant to understand phase transformations, memory effects and variant selection.
1422 H-R Wenk and P Van Houtte
For geophysics the unprecedented possibilities to perform deformation experiments at anyconditions in the earth and study in situ texture changes is revolutionizing a field andmineralogists that were previously mainly concerned with phase relations and crystal structuresare becoming aware of preferred orientation.
We predict that a whole new community of synchrotron users will become texture clients.Many images on inorganic and organic materials reveal texture and this needs to be takeninto account by crystallographers that rely on normalized diffraction intensities. With modernRietveld codes that are capable of processing such images and extract texture information,crystal structure analysis can be quantified. Also with synchrotrons and a fine-focused beamof 1–10 µm local texture variations can be investigated such as in biomaterials. Data acquisitionis extremely fast and the method is available for so many in situ applications, including stress,that there will be breakthroughs, similar to EBSD in the 1980s.
The discussion has illustrated that texture research is indeed a wide multidisciplinaryfield dealing with an intrinsic material property. It connects many domains of sciencewith metallurgy, ceramics, polymer science, geology, geophysics and biology as just a fewhighlights. The field is old but has advanced tremendously during the last decade, thanksto new experimental techniques and to sophisticated modelling that became possible withmodern computing resources. Yet it remains a field with many unsolved problems that willkeep scientists occupied for years to come.
References
Alam R, Mengelberg H D and Luecke K 1967 Z. Metallkunde 58 867Alley R B, Gow A J and Meese D A 1995 J. Glaciol. 41 197Asaro R J and Needleman A 1985 Acta Metall. 33 923Ashby M F 1972 Acta Metall. 20 887Bai Q, Mackwell S J and Kohlstedt D L 1991 J. Geophys. Res. 96 2441Bate P 1999 Phil. Trans. R. Soc. A 357 1589Beaudoin A J, Mecking H and Kocks U F 1996 Phil. Mag. A 73 1503Ben Ismaıl W and Mainprice D 1998 Tectonophysics 296 145Benmarouane A, Hansen T and Lodini A 2004 Physica B at pressBennett K, Wenk H-R, Durham W B and Stern L A 1997 Phil. Mag. A 76 413Bergman M I 1997 Nature 389 60Bishop J F W 1954 J. Mech. Phys. Solids 3 130Blacic J D 1975 Tectonophysics 27 271Blackman D K, Wenk H-R and Kendall J-M 2002 Geochem. Geophys. Geosyst. 3 10.1029Bocker A, Brokmeier H G and Bunge H J 1991 J. Eur. Ceram. Soc. 7 187Bons P D and den Brok B 2000 J. Struct. Geol. 22 1713Boudier F and Nicolas A 1995 J. Petrol. 36 777Bowman K J and Chen I-W 1993 J. Am. Ceram. Soc. 76 113Brockhouse B N 1953 Can. J. Phys. 31 339–55Brunet F, Germi P, Pernet M and Wenk H-R 1996 Textures of Materials ICOTOM-11 ed Z Liang et al (Beijing:
International Academic) p 1105Buchheit T E, Lavan D A, Micjael J R, Christenson T R and Leith S D 2002 Metall. Materials Trans. A 33 539Buffett B and Wenk H-R 2001 Nature 413 60Bunge H-J 1965 Z. Metallkunde 56 872Bunge H-J 1982 Texture Analysis in Materials Science—Mathematical Methods (London: Butterworths)Bunge H-J (ed) 1986 Experimental Techniques of Texture Analysis (Oberursel, Germany: DMG Informationsgemein-
schaft) p 442Bunge H-J 1991 Textures Microstruct. 14–18 283Bunge H-P, Richards M A, Lithgow-Bertelloni C, Baumgardner J R, Grand S P and Romanowicz B 1998 Science
280 91Burgers W G 1934 Physica 1 561Calnan E A and Clews C J B 1950 Phil. Mag. 41 1085
Texture and anisotropy 1423
Canova G R, Kocks U F and Jonas J J 1984 Acta Metall. 32 211Canova G R, Wenk H-R and Molinari A 1992 Acta Metall. Mater. 40 1519Carter N L and Heard H C 1970 Am. J. Sci. 269 193Castelnau O, Duval P, Lebensohn R A and Canova G R 1996 J. Geophys. Res. 101 13851Chateigner D, Hedegaard C and Wenk H-R 2000 J. Struct. Geol. 22 1723Chateigner D, Wenk H-R, Barber D J and Patel A 1998 Ferroelectrics 19 121Chin G Y and Mammel W L 1969 Trans. Metall. Soc. AIME 245 1211Christian J W 1970 Proc. ICSMA 2: 2nd Int. Conf. on Strength of Metals and Alloys vol 1 (American Society for
Metals) p 29Ciurchea D, Pop A V, Gheorghiu C, Furtuna I, Todica M, Dinu A and Roth M 1996 J. Nucl. Mater. 231 83Crumbach M, Pomana G, Wagner P and Gottstein G 2001 Proc. 1st Joint Conf. on Recrystallisation and Grain Growth
ed G Gottstein and D A Molodov (Berlin: Springer) p 1053Dahms M 1992 Textures Microstruct. 19 169Dawson P, Boyce D, MacEwen S and Rogge R 2001 Mater. Sci. Eng. A 313 123Dawson P R and Wenk H-R 2000 Phil. Mag. A 80 573deRango P, Lees M, Lejay P, Sulpice A, Tournier R, Ingold M, Germi P and Pernet M 1991 Nature 349 770Detwiler R J, Monteiro P M, Wenk H-R and Zhong Z 1988 Cement Concrete Res. 18 823d’Halloy O J J 1833 Introduction a la Geologie (Paris: Levrault)Dietrich D and Song H 1984 J. Struct. Geol. 6 19Dillamore I L and Roberts W T 1965 Metall. Rev. 10 271Dimarciello F I, Key P L and Williams J C 1972 J. Am. Ceram. Soc. 55 509Dingley D J 1981 Scanning Electron Microsc. 4 273Dini J W 1993 Electrodeposition. The Materials Science of Coatings and Substrates (Park Ridge, NJ: Noyes)
p 367Diot C and Arnault V 1991 Textures Microstruct. 14–18 389Dornbusch H J, Skrotzki W and Weber K 1994 Textures of Geological Materials ed H J Bunge et al (Oberursel,
Germany: Deutsch. Gesell. Metallkunde) p 187Duval P 1979 Bull. Mineral. 102 80Duval P 1981 J. Glaciol. 27 129Duval P, Ashby M F and Anderman I 1983 J. Phys. Chem. 87 4066Erskine B G, Heidelbach F and Wenk H-R 1993 J. Struct. Geol. 15 1189Eshelby J 1957 Proc. R. Soc. Lond. A 241 376Euler L 1775 Nov. Comm. Acad. Sci. Imp. Petrop. 20 189Fiquet G 2001 Z. Kristallogr. 216 248Fouch M J and Fisher K M 1996 J. Geophys. Res. 101 15987Fouch M J, Fisher K M and Wysession M E 2001 Earth Planet. Sci. Lett. 190 167Gey N and Humbert M 2002 Acta Mater. 50 277Gilormini P and Michel J C 1999 Eur. J. Mech. A Solids 17 725Gleason G C, Tullis J and Heidelbach F 1993 J. Struct. Geol. 15 1145Glocker R 1924 Z. Metallkunde 16 180Gnaupel-Herold T, Brand P C and Prask H J 1998 J. Appl. Cryst. 31 929Goeman U and Schumann H 1977 Z. Kali- und Steinsalz 7 171Gottstein G 2002 Proc. ICOTOM 13, Mater. Sci. Forum 408–412 1Green H W 1980 Phil. Mag. A 41 637Green H W, Griggs D T and Christie J M 1970 Experimental and Natural Rock Deformation ed P Paulitsch (Berlin:
Springer) p 272Green H W and Houston H 1995 Annu. Rev. Earth Planet. Sci. 23 169Groves G W and Kelly A 1963 Phil. Mag. 8 877G’Sell C, Dahoun A, Royer F X and Philippe M J 1999 Modelling Simul. Mater. Sci. Eng. 7 817Guillope M and Poirier J-P 1979 J. Geophys. Res. 84 5557Haessner F 1978 Recrystallization of Metallic Materials (Stuttgart: Riederer)Hatherly M and Hutchinson W B 1979 An Introduction to Textures in Metals, Monograph No 5 (London: The Institution
of Metallurgists) p 76Hedegaard C and Wenk H-R 1997 J. Mollusc Studies 64 133Heidelbach F, Kunze K and Wenk H-R 2000 J. Struct. Geol. 22 91Heidelbach F, Riekel C and Wenk H-R 1999 J. Appl. Cryst. 32 841Heidelbach F, Wenk H-R, Muenchausen R E, Foltyn S, Nogar N and Rollett A D 1992 J. Mater. Res. 7 549Heilbronner R P and Bruhn D 1998 J. Struct. Geol. 20 695
1424 H-R Wenk and P Van Houtte
Heilbronner R P and Pauli C 1993 J. Struct. Geol. 15 369Helming K 1994 Proc. 10th Int. Conf. on Texture of Materials (Clausthal) Mater. Sci. Forum 157–162 363Helming K, Geier S, Schreck M, Hessmer R, Stritzker B and Rauschenbach B 1995 J. Appl. Phys. 77 4765Herwegh M and Handy M R 1996 J. Struct. Geol. 18 689Hess H H 1964 Nature 203 629Heuer A H, Tighe N J and Cannon R M 1980 J. Am. Ceram. Soc. 63 53Hill R 1952 Proc. Phys. Soc. A 65 349Hirsch J and Lucke K 1988 Acta Metall. 36 2863Hirth G and Tullis J 1992 J. Struct. Geol. 14 145Ho N C, Peacor D R and Van der Pluijm B A 1999 Clays Clay Miner. 47 495Honneff H and Mecking H 1981 6th Int. Conf. on Textures of Materials (Tokyo: The Iron and Steel Institute of Japan)
p 347Hosford W F 1964 Trans. Metall. Soc. AIME 230 12Hughes D A, Lebensohn R, Wenk H-R and Kumar A 2000 Proc. R. Soc. Lond. A 456 1Humphreys F J 1988 8th Int. Conf. on Textures of Materials ed J S Kallend and G Gottstein (Warrendale, PA:
The Metallurgical Society) p 171Humphreys F J and Hatherly M 1995 Recrystallization and Related Annealing Phenomena (Oxford: Oxford University
Press)Hutchinson J W 1970 Proc. R. Soc. Lond. A 319 247Jeffery G B 1923 Proc. R. Soc. Lond. A 102 161Jordan P G 1987 Tectonophysics 135 185Joubert P, Loisel B, Chouan Y and Haji L 1987 J. Electrochem. Soc. 134 2541Kalidindi S R, Bronkhorst C A and Anand L 1992 J. Mech. Phys. Solids 40 537Kallend J S 2000 Texture and Anisotropy. Preferred Orientations in Polycrystals and Their Effect on Materials
Properties 2nd paperback edn, ed U F Kocks et al (Cambridge: Cambridge University Press) chapter 3, p 102Kallend J S, Kocks U F, Rollett A D and Wenk H-R 1991 Mater. Sci. Eng. A 132 1Kamb W B 1959 J. Geophys. Res. 64 1891Kamb W B 1961 J. Geophys. Res. 66 259Kampf H, Ertel A, Bankwitz P, Betzl M and Zanker G 1986 Z. Angew. Geol. 33 104Karato S-I 1999 Nature 402 871Karato S-I, Zhang S and Wenk H-R 1995 Science 270 458Karki B B, Wentzcovitch R M, de Gironcoli S and Baroni S 1999 Science 286 1705Kern H and Richter A 1985 Preferred Orientation in Deformed Metals and Rocks: An Introduction to Modern Texture
Analysis ed H-R Wenk (Orlando, FL: Academic) p 317Kern H and Wenk H-R 1983 Contrib. Mineral. Petrol. 83 231Kern H and Wenk H-R 1990 J. Geophys. Res. 95 11213Kim C J, Yoon D S, Lee J S, Choi C G, Lee W J and No K 1994 J. Appl. Phys. 76 7478Kocks U F 1970 Metall. Trans. 1 1121Kocks U F and Mecking H 2003 Prog. Mater. Sci. 48 171Kocks U F, Tome C N and Wenk H-R 2000 Texture and Anisotropy. Preferred Orientations in Polycrystals and Their
Effect on Materials Properties 2nd paperback edn (Cambridge: Cambridge University Press)Kroner E 1961 Acta Metall. 9 155Krulevitch P, Nguyen Tai D, Johnson G C, Wenk H-R and Gronsky R 1991 Evolution of Thin-Film and Surface
Microstructure vol 202, ed C V Thompson et al (Pittsburgh, PA: Mater. Res. Soc.) p 167Kumakura H, Togano K, Maeda H, Kase J and Morimoto T 1991 Appl. Phys. Lett. 58 2830Kurdjomov G and Sachs G 1930 Z. Physik 64 325Lebensohn R A, Dawson P R, Wenk H-R and Kern H 2003 Tectonophysics 370 287Lebensohn R A and Tome C N 1993 Acta Metall. Mater. 41 2611Lebensohn R A, Wenk H-R and Tome C N 1998 Acta Mater. 46 2683Lee C S, Duggan B J and Smallman R E 1993 Phil. Mag. Lett. 68 185Leonhardt A, Schlafer D, Seidler M, Selbmann D and Schonherr M 1982 J. Less-Common Met. 87 63Li S and Van Houtte P 2002 Aluminium 78 918Li W-Y and Sorensen O T 1995 Textures Microstruct. 24 53Liang Z, Wang F and Xu J 1988 8th Int. Conf. on Textures of Materials ed J S Kallend and G Gottstein (Warrendale,
PA: The Metallurgical Society) p 111Lipenkov V Y, Barkov N I, Duval P and Pimienta P 1989 J. Glaciol. 35 392Lister G S, Paterson M S and Hobbs B E 1978 Tectonophysics 45 107Lliboutry L and Duval P 1985 Annu. Geophys. 3 207
Texture and anisotropy 1425
Lonardelli I, Wenk H-R, Luterrotti L and Goodwin M 2004 J. Synchr. Res. submittedLowenstam H A and Weiner S 1989 On Biomineralization (New York: Oxford University Press)Lutterotti L, Matthies S, Wenk H-R, Schultz A J and Richardson J W 1997 J. Appl. Phys. 81 594March A 1932 Z. Kristallogr. 81 285Margulies L, Winther G and Poulsen H F 2001 Science 291 2392Mathur K K and Dawson P R 1989 Int. J. Plast. 5 67Matthies S 1979 Phys. Status Solidi b 92 K135Matthies S, Helming K and Kunze K 1990 Phys Status Solidi b 157 71Matthies S, Helming K and Kunze K 1990 Phys Status Solidi b 157 489Matthies S and Humbert M 1993 Phys. Status Solidi b 177 K47Matthies S, Merkel S, Wenk H-R, Hemley R J and Mao H-K 2001 Earth Planet. Sci. Lett. 194 201Matthies S and Vinel G W 1982 Phys. Status Solidi b 112 K111Matthies S and Wagner F 1996 Phys. Status Solidi b 196 K11McNamara A K, Karato S-I and van Keken P E 2001 Earth Planet. Sci. Lett. 191 85Mecking H 1985 Preferred Orientation in Deformed Metals and Rocks: An Introduction to Modern Texture Analysis
ed H-R Wenk (Orlando, FL: Academic) p 267Merkel S, Wenk H-R, Shu J, Shen G, Gillet P, Mao H-K and Hemley R J 2002 J. Geophys. Res. 107 2271Mika D P and Dawson P R 1999 Acta Mater. 47 1355Molinari A, El Houdaigui F and Toth L S 2004 Int. J. Plast. 20 291Molinari A, Canova G R and Ahzi S 1987 Acta Metall. 35 2983Molinari A and Toth L 1994 Acta Metall. Mater. 42 2453Montagner J P and Guillot L 2002 Plasticity of Minerals and Rocks, Reviews in Mineralogy and Geochemistry
ed S-Y Karato and H-R Wenk (Mineral. Soc. America) p 353Montagner J P and Tanimoto T 1990 J. Geophys. Res. 95 4797Muddle B C and Hannink R H J 1986 J. Am. Ceram. Soc. 69 547Nauer-Gerhardt C U and Bunge H J 1986 Experimental Techniques of Texture Analysis ed H J Bunge (Oberursel,
Germany: Deutsch. Gesell. Metallkunde) p 125Nicolas A, Boudier F and Boullier A-M 1973 Am. J. Sci. 273 853O’Brien D K, Wenk H-R, Ratschbacher L and You Z 1987 J. Struct. Geol. 9 719Oertel G 1983 Tectonophysics 100 413Parkhomenko E I 1971 Electrification Phenomena in Rocks (Transl. G V Keller (New York: Plenum)) p 285Paterson M S 1973 Rev. Geophys. Space Phys. 11 355Paterson M S 1989 Rheology of Solids and of the Earth ed S I Karato and M Toriumi (Oxford: Oxford University
Press) p 107Paterson M S and Weiss L E 1961 Geol. Soc. Am. Bull. 72 841Paterson W S B 1991 Cold Reg. Sci. Tech. 20 75Paul H, Morawiec A, Bouzy E, Fundenberger J J and Piatkowsky A 2004 Metall. Mater. Trans. A at pressPawlik K, Pospiech J and Lucke K 1991 Textures Microstruct. 14–18 25Pentecost J L and Wright C H 1964 Advances in X-Ray Analysis vol 7, ed M Mueller et al (New York: Plenum)
p 174Philippe M J 1994 Textures of Materials ICOTOM-10 ed H J Bunge (Switzerland: Trans. Tech.) p 1337Phillips W R 1971 Mineral Optics, Principles and Techniques (San Francisco: Freeman) p 249Pieri M, Stretton I, Kunze K, Burlini L, Olgaard D L, Burg J-P and Wenk H-R 2000 Tectonophysics 330 119Polanyi M and Weissenberg K 1923 Z. Techn. Phys. 4 199Puig-Molina A, Wenk H-R, Berberich F and Graafsma H 2003 Z. Metallkunde 94 1199Raabe D and Roters F 2004 Int. J. Plast. 20 339Radhakrishnan B, Sarma G B and Zacharia T 1998 Acta Mater. 46 4415Raj P M and Cannon W R 1999 J. Am. Ceram. Soc.Raj R and Ghosh A K 1981 Acta Metall. 29 283Raleigh C B 1968 J. Geophys. Res. 73 5391Randle V and Engler O 2000 Introduction to Texture Analysis: Macrotexture, Microtexture and Orientation Mapping
(London: Gordon and Breach Science)Ratschbacher L, Wenk H-R and Sintubin M 1991 J. Struct. Geol. 13 369Ray R K, Jonas J J and Hook R E 1993 Int. Mater. Rev. 39 129Ray R and Jonas J J 1990 Int. Mater. Rev. 35 1Rhys-Jones T N 1990 Surf. Coat. Technol. 43/44 402Rietveld H M 1969 J. Appl. Cryst. 2 65Roe R-J 1965 J. Appl. Phys. 36 2024
1426 H-R Wenk and P Van Houtte
Rollett A D and Wright S I 2000 Texture and Anisotropy. Preferred Orientations in Polycrystals and Their Effect onMaterials Properties 2nd paperback edn, ed U F Kocks et al (Cambridge: Cambridge University Press) chapter 5,p 178
Rosenberg J M and Piehler H R 1971 Met. Trans. 2 257Ruer D and Baro R 1977 Adv. X-Ray Anal. 20 187Sachs G 1928 Z. Ver. Deutsch. Ing. 72 734Sander B 1950 Einfuhrung in die Gefugekunde der Geologischen Korper vol 2 (Vienna: Springer)Sandhage K H, Riley G N Jr and Carter W L 1991 JOM 43 21 (The Min. Met. & Mater. Soc.)Sandlin M S and Bowman K J 1992 Ceram. Eng. Sci. Proc. 13 661Schaeben H 1988 Phys. Status Solidi b 148 63Scheffzuk C 1996 Z. Geol. Wiss. 24 403Schmid S M, Boland J N and Paterson M S 1977 Tectonophysics 43 257Schulz L G 1949 J. Appl. Phys. 20 1033Schwarzer R A and Sukkau J 1998 Mater. Sci. Forum 273–275 215Selvamanickam V and Salama K 1990 Appl. Phys. Lett. 57 1575Shen G, Mao H-K, Hemley R J, Duffy T S and Rivers M L 1998 Geophys. Res. Lett. 25 373Shimizu I 1992 J. Geophys. Res. 97 4587Shimizu I 1999 Phil. Mag. A 79 1217Siegesmund S, Helming K and Kruse R 1994 J. Struct. Geol. 16 131Siemes H 1974 Contrib. Mineral. Petrol. 43 149Silver P G 1996 Annu. Rev. Earth Planet. Sci. 24 385Simmons G and Wang H 1971 Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook
2nd edn (Cambridge, MA: MIT Press) p 370Singh A K, Mao H-K, Shu J and Hemley R J 1998 Phys. Rev. Lett. 80 2157Sintubin M 1994 Textures of Geological Materials ed H-J Bunge et al (Oberursel, Germany: DMG
Informationsgesellschaft) p 221Solas D E, Tome C N, Engler O and Wenk H-R 2001 Acta Mater. 49 3791Song X 1997 Rev. Geophys. 35 297Spiers C J and Takeshita T (ed) 1995 Influence of fluids on pressure solution in rocks Tectonophysics 245 (special
issue)Steinlage G, Roeder R, Trumble K, Bowman K, Li S and McElresh M 1994 J. Mater. Res. 9 833Steinle-Neumann G, Stixrude L, Cohen R E and Gulseren O 2001 Nature 413 57Stout M G, Kallend J S, Kocks U F, Przystupa M A and Rollett A D 1988 8th Int. Conf. on Textures of Materials
ed J S Kallend and G Gottstein (Warrendale, PA: The Metallurgical Society) p 479Stretton I, Heidelbach F, Mackwell S and Langenhorst F 2001 Earth Planet. Sci. Lett. 194 229Sudo M, Hashimoto S and Tsukatani I 1981 6th Int. Conf. on Textures of Materials (Tokyo: The Iron and Steel Institute
of Japan) p 1076Sue J A and Troue H H 1987 Surf. Coat. Technol. 33 169Szpunar J A 1996 Texture of Materials ICOTOM-11 ed Z Liang et al (Beijing: International Academic) p 1095Takeshita T, Tome C N, Wenk H-R and Kocks U F 1987 J. Geophys. Res. B 92 12917Takeshita T, Wenk H-R and Lebensohn R 1999 Tectonophysics 312 133Tang M, Kubin L P and Canova G R 1998 Acta Mater. 46 3221Taylor G I 1938 J. Inst. Metals 62 307Tenckhoff E 1988 Deformation Mechanisms, Texture, and Anisotropy in Zirconium and Zircaloy (Philadelphia, PA:
ASTM)Thornburg D R and Piehler H R 1975 Metall. Trans. A 6 1511Thorsteinsson T 2002 J. Geophys. Res. 107 148Tietz L A, Carter C B, Lathrop D K, Russek S E, Buhrman R A and Michael J R J 1989 J. Mater. Res. 4 1072Tison J-L, Thorsteinsson T, Lorrain R D and Kipfstuhl J 1994 Earth Planet. Sci. Lett. 125 421Tome C and Canova G 2000 Texture and Anisotropy. Preferred Orientations in Polycrystals and Their Effect on
Materials Properties 2nd paperback edn, ed U F Kocks et al (Cambridge: Cambridge University Press) chapter 11,p 466
Tullis T E 1976 Geol. Soc. Am. Bull. 87 745Tullis J 2002 Plasticity of Minerals and Rocks, Reviews in Mineralogy and Geochemistry ed S-Y Karato and H-R Wenk
(Washington, DC: Mineral. Soc. America) p 51Tullis J, Christie J M and Griggs D T 1973 Geol. Soc. Am. Bull. 84 297Trampert J and Heijst H J 2002 Science 296 1297Tullis J and Tullis T 1972 Am. Geophys. U. Geophys. Monogr. 16 67
Texture and anisotropy 1427
Tullis J and Wenk H-R 1994 Mater. Sci. Eng. A 175 209Tuominen M, Goldman A M, Chang Y Z and Jiang P Z 1990 Phys. Rev. B 42 412Turner F J, Griggs D T, Clark R H and Dixon R H 1956 Geol. Soc. Am. Bull. 67 1259Turner F J and Weiss L E 1963 Structural Analysis of Metamorphic Tectonites (New York: McGraw-Hill)Vaidyanathan R, Bourke M A M and Dunand D C 2001 Metall. Mater. Trans. A 32 777Van Acker K, De Buyser L, Celis J P and Van Houtte P 1994 J. Appl. Cryst. 27 56Van Houtte P 1981 6th Int. Conf. on Textures of Materials (Tokyo: The Iron and Steel Institute of Japan) p 428Van Houtte P 1982 Mater. Sci. Eng. 55 69Van Houtte P 1983 Textures Microstruct. 6 1Van Houtte P 1984 7th Int. Conf. on Textures of Materials ed C M Brakman et al (Netherlands: Soc. for Materials
Science) p 7Van Houtte P 1991 Textures Microstruct. 13 199Van Houtte P and Aernoudt E 1975 Z. Metallkunde 66 202Van Houtte P and Aernoudt E 1975 Z. Metallkunde 66 303Van Houtte P and De Buyser L 1993 Acta Metall. Mater. 41 323Van Houtte P and Delannay L 1999 Proc. 12th Int. Conf. on Textures of Materials ed J A Szpunar (Ottowa: NRC
Research Press) p 623Van Houtte P, Delannay L and Kalidindi S R 2002 Int. J. Plast. 18 359Vinnik L, Breger L and Romanowicz B 1998 Nature 393 564Von Dreele R B 1997 J. Appl. Cryst. 30 517Wagner F, Bozzolo N, Van Landuyt O and Grosdidier T 2002 Acta Mater. 50 1245Wagner F, Wenk H-R, Kern H, Van Houtte P and Esling C 1982 Contrib. Mineral. Petrol. 80 132Walker T, Mattern N and Herrmann M 1995 Textures Microstruct. 24 75Walther K, Scheffzuk Ch and Frischbutter A 2000 Physica B 276–278 130Wang J N 1994 J. Struct. Geol. 16 961Wassermann G and Grewen J 1962 Texturen Metallischer Werkstoffe 2nd edn (Berlin: Springer)Wasserman E, Stixrude L and Cohen R E 1996 Phys. Rev. B 53 8296Wcislak L, Klein H, Bunge H-J, Garbe U, Tschentscher T and Schneider J R 2002 J. Appl. Cryst. 35 82Weber K 1981 Tectonophysics 78 291Weertman J 1968 Trans. Am. Soc. Met. Eng. 61 681Wenk H-R 1994 Reviews in Mineralogy, Silica Physical Behavior, Geochemistry and Materials Applications vol 29,
ed P J, C T Prewitt and G V Gibbs (Washington, DC: Mineral. Soc. America) p 177Wenk H-R 1999 Modelling Mater. Sci. Eng. 7 699Wenk H-R (ed) 1985 Preferred Orientation in Deformed Metals and Rocks: An Introduction to Modern Texture
Analysis (Orlando, FL: Academic) p 610Wenk H-R and Christie J M 1991 J. Struct. Geol. 13 1091Wenk H-R and Grigull S 2003 J. Appl. Cryst. 36 1040Wenk H-R and Heidelbach F 1998 Bone 24 361Wenk H-R and Kocks U F 1987 Metall. Trans. A 18 1083Wenk H-R and Pannetier J 1990 J. Struct. Geol. 12 177Wenk H-R and Phillips D 1992 Physica C 200 105Wenk H-R and Tome C N 1999 J. Geophys. Res. 104 25,513Wenk H-R, Chakmouradyan A and Foltyn S 1996 Mater. Sci. Eng. A 205 9Wenk H-R, Lutterotti L and Vogel S 2003 Nucl. Instrum. Methods A 515 575Wenk H-R, Lonardelli I and Williams D 2004 Acta Mater. 52 1899Wenk H-R, Baumgardner J, Lebensohn R and Tome C 2000 J. Geophys. Res. 105 5663Wenk H-R, Canova G, Brechet Y and Flandin L 1997 Acta Mater. 45 3283Wenk H-R, Canova G, Molinari A and Mecking H 1989 Acta Metall. 37 2017Wenk H-R, Matthies S, Hemley R J, Mao H-K and Shu J 2000 Nature 405 1044Wenk H-R, Kern H, Van Houtte P and Wagner F 1986 Mineral and Rock Deformation: Laboratory Studies vol 36,
ed B E Hobbs and H C Heard (Am. Geoph. U. Geophys. Monogr.) p 287Wenk H-R, Matthies S, Donovan J and Chateigner D 1998 J. Appl. Cryst. 31 262Wenk H-R, Pannetier J, Bussod G and Pechenik A 1989 J. Appl. Phys. 65 4070Wenk H-R, Venkitasubramanyan C S, Baker D W and Turner F J 1973 Contrib. Mineral. Petrol. 38 81Wenk H-R, Sintubin M, Huang J, Johnson G C and Howe R T 1990 J. Appl. Phys. 67 572Wenk H-R, Takeshita T, Bechler E, Erskine B G and Matthies S 1987 J. Struct. Geol. 9 731Wenk H-R, Chateigner D, Pernet M, Bingert J, Hellstrom E and Ouladdiaf B 1996 Physica C 272 1Wenk H-R, Cont L, Xie Y, Lutterotti L, Ratschbacher L and Richardson J 2001 J. Appl. Cryst. 34 442
1428 H-R Wenk and P Van Houtte
Wever F 1924 Z. Phys. 28 69Willis D G 1977 Geol. Soc. Am. Bull. 88 883Willkens C A, Corbin N D, Pujari V K, Yeckley R L and Mangaudis M J 1988 Ceram. Eng. Sci. Proc. 9 1367Wright S I and Adams B L 1992 Metall. Trans. A 23 759Wu X D, Foltyn S R, Arendt P N, Blumenthal W R, Campbell I H and Cotton J D 1995 Appl. Phys Lett. 67 2397Xie Y, Wenk H-R and Matthies S 2003 Tectonophysics 370 269Yamazaki D and Karato S 2002 Phys. Earth Planet. Int. 131 251Yang C Y 1998 Physica C 307 87Young R A 1993 The Rietveld Method (Oxford: Oxford University Press) p 298Zhang S and Karato S-Y 1995 Nature 375 774Zinserling K and Shubnikov A 1933 Z. Kristallogr. 85 454
