Horst Quade

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Because of the superposition of pole figures corresponding to symmetrically
equivalent crystal directions, only the reduced orientation distribution function <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover accent="true"><mml:mi>f</mml:mi><mml:mo>&#x223C;</mml:mo></mml:mover><mml:mrow><mml:mo>(</mml:mo><mml:mi>g</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
can be obtained directly by pole figure inversion. The additional information
contained in the positivity condition of the ODF allows, however, the determination
of an approximation to the &#x201C;indeterminable&#x201D; part and hence of the complete ODF
<mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>g</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, if the texture has sufficiently large zero-ranges. The application of the method
and the accuracy of the results was tested using two theoretical and one experimental
textures. The accuracy of the complete ODF depends on the size of the zero-range,
the errors in its determination, and on the errors, experimental and truncational, of
the reduced ODF. The &#x201C;physical zero&#x201D; used in order to determine the zero-range is
defined according to the statistical error of the pole figure measurement.

H. P. Lee

H. J. Bunge

C. Esling

Determination of the Complete Orientation Distribution Function by the Zero-Range Method

The crystallographic orientation distribution and the geometrical lamellae orientation
distribution in lamellar eutectics are, in general, not independent of each other. The
combined orientation-lamellae distribution function depends on five angular parameters.
X-ray diffraction in such eutectics may exhibit an anisotropic macroscopic
absorption factor if the penetration depth of the X-rays is large compared with their
planar size. As a consequence, the reflected X-ray intensity may depend on a third
angle <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi></mml:math>, i.e. a rotation of the sample about the diffraction vector s additionally to the
usual pole figure angles <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B1;</mml:mi></mml:math>, &#x3B2; which describe the orientation of the diffraction vector s
with respect to the sample coordinate system. It is thus necessary to measure
three-dimensional generalized pole figures instead of conventional two-dimensional
ones.

Texture and Lamellae Distribution Functions in Lamellar Eutectics

We describe a computer procedure which allows us to introduce experimental initial
textures as starting conditions for texture simulation (instead of a theoretical random
texture). We apply the procedure on two batches of copper with weak initial textures
and on fine-grained and coarse-grained aluminium with moderately strong initial
textures. In copper the initial texture turns out to be too weak to have any significant
effect. In aluminium the initial texture has a very significant effect on the simulated
textures&#x2014;similar to the effect it has on the experimental textures. However, there
are differences between the simulated and the experimental aluminium textures that
can only be explained as a grain-size effect. Possible future applications of the
procedure are discussed.

T. Leffers

D. Juul Jensen

Evaluation of the Effect of Initial Texture on the Development of Deformation Texture

Four kinds of cubic invariant spherical surface harmonics are introduced. It has been
shown previously that these harmonics occur in the equations relating measured diffraction
(line-shift) elastic strain and macro-stresses generating these strains for the case of
textured cubic materials. As a consequence, these harmonics are important for the
determination of unknown macro-stress tensor components if the o.d.f. expansion
coefficients are known. On the other hand, they play a role in the determination of
unknown o.d.f. expansion coefficients if the macro-stresses are known using for instance,
a tensile test device on the diffractometer. Then, even the odd-order o.d.f. expansion
coefficients can be obtained. In this paper, special attention is given to the mathematical
construction of the cubic harmonics, the physical symmetry requirements they are subject
to and some examples are given exhibiting both even and odd order harmonics.

C. M. Brakman

Cubic Invariant Spherical Surface Harmonics in Conjunction With Diffraction Strain Pole-Figures

The analysis of misorientations has up to now usually be carried out by comparing the
values obtained experimentally using two dimensional distributions of rotation axes or
rotation angles with the distribution calculated by Mackenzie for the statistically random
case. In this paper the presentation of the distribution of the misorientations is based
on the three dimensional orientation distribution function (ODF) (as described by Bunge).
The new function is termed the misorientation distribution function (MDF) to differentiate
it from the ODF. The advantages in using this function are presented and illustrated
by three MDF&#x27;s derived from the work of Haessner, Pospiech and Sztwiertnia.

J. Pospiech

K. Sztwiertnia

F. Haessner

The Misorientation Distribution Function

The mean orientation change &#x3B4;g of crystals of the orientation g in a polycrystalline
material due to plastic deformation can be described (under certain assumptions) by an
orientation flow field &#x3B4;g(g) &#x3D; d&#x3B7;&#xB7;<mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>&#x03BD;</mml:mtext></mml:math>(g) where
d&#x3B7; is the absolute value of a small deformation step.
The flow field <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>&#x03BD;</mml:mtext></mml:math>(g) is a vector field in the orientation space g &#x3D; &#x7B;&#x3C6;1, &#x3A6;, &#x3C6;2&#x7D;
which must obey a continuity equation. The flow field describes the texture changes due to
plastic deformation. The components of the flow vector, as a function of the orientation g, 
can be represented in terms of a series expansion which must obey certain symmetry
conditions. As an example, the flow field calculated according to the Taylor theory for
&#x7B;111&#x7D;<mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>&#x2329;</mml:mo><mml:mrow><mml:mn>110</mml:mn></mml:mrow><mml:mo>&#x0232A;</mml:mo></mml:mrow></mml:mrow></mml:math> glide was calculated in 5-degree steps in the orientation space.

E. Dahlem

H. Klein

The Development of Deformation Textures Described by an Orientation Flow Field

The calculation of orientation distribution functions from incomplete pole figures can
be carried out by a least squares approximation of the texture coefficients <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mtext>C</mml:mtext><mml:mi>l</mml:mi><mml:mrow><mml:mi>&#x03BC;</mml:mi><mml:mtext>&#x03BD;</mml:mtext></mml:mrow></mml:msubsup></mml:math> and the normalization factors <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mtext>N</mml:mtext><mml:mrow><mml:mi>h</mml:mi><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> to the available experimental data. This procedure is less
susceptable to instabilities due to experimental errors if the normalization factors can
be calculated independently of the coefficients <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mtext>C</mml:mtext><mml:mi>l</mml:mi><mml:mrow><mml:mi>&#x03BC;</mml:mi><mml:mtext>&#x03BD;</mml:mtext></mml:mrow></mml:msubsup></mml:math>. In the case of cubic materials, the relationship <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mtext>F</mml:mtext><mml:mn>2</mml:mn><mml:mn>0</mml:mn></mml:msubsup></mml:mrow></mml:math> &#x3D; 0 to be fulfilled by pole figure values provides an independent condition
for the calculation of the normalization factor. This condition can still be improved by
taking the slopes of the pole density curves at <mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B1;</mml:mi></mml:math> &#x3D; <mml:math id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x03B1;</mml:mi><mml:mrow><mml:mi>max</mml:mi><mml:mo>&#x2061;</mml:mo></mml:mrow></mml:msub></mml:math> and <mml:math id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B1;</mml:mi></mml:math> &#x3D; 90&#xB0; into account. An
economic way to consider the slope in the pole figures is to use a cubic spline interpolation.

M. Dahms

H.-J. Bunge

Calculation of the Normalization Factor of Incomplete Pole Figures by Cubic Extrapolation

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Texture, Stress, and Microstructure