A misorientation between two cubic crystallites can be described using different orientation
parameters and represented in many symmetrically equivalent ways. It is, however, convenient when
the misorientations between different pairs of crystallites all lie in the same base domain. This paper
presents a computer program which reduces a misorientation given in axis&#x2013;angle or in Euler&#x27;s space
into the base domain of the smallest angle of rotation.

K. Sztwiertnia

J. Pospiech

F. Haessner

Reduction of Misorientations Between Two Cubic Crystals Into the Base Domain of Axis&#x2013;Angle Space

The preferred grain orientation in polycrystals causes a direction dependence of the ultrasonic wave
velocities. In an extensive study, the texture influence on different ultrasonic wave velocities is
measured in series of cold-rolled ferritic steel sheets. Based on that, ultrasonic techniques are
developed to characterize the texture in terms of the three fourth-order expansion coefficients of the
orientation distribution function (ODF). Evaluation equations are given for the application of free
ultrasonic longitudinal and shear waves as well as for the application of ultrasonic SH-waves.Furthermore, linear correlations are found between the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>&#x0394;</mml:mtext><mml:mi>r</mml:mi></mml:mrow></mml:math>- and <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>-values, characterizing the deep drawability of the sheets and the expansion coefficients <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>4</mml:mn><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math> and <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>4</mml:mn><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>, evaluated by ultrasonic techniques. The study demonstrates that ultrasonic techniques hold 
high promise to evaluate the fourth-order expansion coefficients of the ODF and to characterize the deep drawing behavior of
rolled sheets by the use of calibration experiments.

M. Spies

E. Schneider

Nondestructive Analysis of Textures in Rolled Sheets by Ultrasonic Techniques

X-ray diffraction in a lamellar two-phase material is considered. If the lamellae thicknesses are small
compared with the penetration depth of the X-rays then the absorption factor depends on the whole
path of incident and reflected beam in both phases. Particularly, it depends on a rotation <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi></mml:math> about the diffraction vector s additional to the orientation (<mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#x03B1;</mml:mi><mml:mi>&#x03B2;</mml:mi></mml:mrow></mml:math>) of this vector with respect to the sample.
Numerical calculations of the absorption factor <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x03B1;</mml:mi><mml:mi>&#x03B2;</mml:mi><mml:mi>&#x03B3;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> have been carried out for different values of lamellae thicknesses <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>D</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:msup><mml:mi>D</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math>, absorption coefficients <mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>&#x03BC;</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:msup><mml:mi>&#x03BC;</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> and lamellae orientations <mml:math id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#x03C9;</mml:mi><mml:mi>&#x03C1;</mml:mi></mml:mrow></mml:math> with respect to the sample surface.

Y. S. Liu

H. J. Bunge

Numerical Calculation of Anisotropic Absorption Factors for Lamellar Two-Phase Structures

In thin sheet the anisotropy of the internal stress due to constraint by the surrounding parent phase
which will be abbreviated as &#x201C;constraint stress,&#x201D; seems to occur during the martensitic transformation
and the variant selection is expected also in the transformation without external stress. To verify this,
the textures were measured by X-ray pole figure method on the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi></mml:math> and <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B1;</mml:mi></mml:math> phase which were developed
in thin sheet of Fe-30 Ni alloy prepared by severe cold rolling and annealing followed by subzero
quenching to liquid Nitrogen. Experimental data were analyzed with conventional variant selection
models. The results obtained are as follows: (a) Variant selection phenomenon was clearly observed
and the results could be explained with conventional variant selection models. (b) It is expected that
different results occur between the conventional variant selection models when the martensitic
transformation occurs accompanying the strain component of <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>&#x2266;</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math>, where <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math> is the strain along the Bain compression axis and <mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#x03B5;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> is along the perpendicular axis.

Hirofumi Miyaji

Ei-Ichi Furubayashi

Effect of Specimen Size on the Variant Selection in Martensitic Transformation

In an earlier paper (Sakata et al., 1989), it was shown that the 4th and 6th order ODF coefficients
could be successfully derived from Young&#x27;s modulus measurements using the elastic energy method.
However, the values of some of the coefficients fell beyond the expected error ranges. In this study,
more appropriate single crystal elastic constants are selected by means of a fitting procedure. Then
the ODF coefficients are again estimated in the manner described previously. As a result, the values 
of the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>4</mml:mn><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>, <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>6</mml:mn><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>, <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>6</mml:mn><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math> and <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>6</mml:mn><mml:mrow><mml:mn>14</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math> coeffioents, which were somewhat inaccurate in the previous calculation,
are improved considerably. The volume fractions of the principal preferred orientations are then
employed to predict the 8th order coefficients and the fiber components of the l &#x3D; 10 and l &#x3D; 12 (<mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>10</mml:mn><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>,
<mml:math id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>12</mml:mn><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math> and <mml:math id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>C</mml:mi><mml:mn>12</mml:mn><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math>) coefficients. With the aid of the coefficients obtained in this way, both pole and inverse
pole figures are drawn, which are in better agreement with those based on X-rays than when only the
4th order coefficients are employed.

Kei Sakata

Dominique Daniel

John J. Jonas

Estimation of 8th, 10th and 12th
Order ODF Coefficients From Elastic Properties in Cold Rolled Steel Sheets by Adjustment of Single Crystal Elastic Constants

True cross-rolling and pseudo cross-rolling (with only one change of the rolling direction after half of
the total deformation degree) was investigated in an alloy AlMn1 up to 93&#x25; deformation. The texture
formation was studied in terms of ODF. After true cross-rolling (multi-stage rolling) a strong
two-component ideal orientation near (011)&#x5B;322&#x5D; was found with maximum densities up to 60 times
random. Pseudo cross-rolling (two-stage rolling) resulted in weaker, but still strong deformation
textures with maximum densities up to twenty times random which were intermediate between
unidirectional and true cross-rolling textures. In both cases, the originally present cube texture
decreased continuously with increasing deformation degree.

A. B&#xF6;cker

H. Klein

Development of Cross-rolling Textures in AlMn1

Three fcc Ni&#x2013;Co alloys with different stacking fault energies (SFE&#x27;s) were cold rolled 95&#x25; and their textures
were characterized by the orientation distribution function (ODF) method. BCC transformation
textures were calculated from these experimental textures using three different orientation
relationships for the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#x3B3;</mml:mi><mml:mo>&#x2192;</mml:mo><mml:mi>&#x3B1;</mml:mi></mml:mrow></mml:math> transformation. The transformed ODF&#x27;s derived from the Bain
relationship were much sharper than the ones deduced from the Kurdjumov&#x2013;Sachs (K&#x2013;S) or the
Nishiyama&#x2013;Wassermann (N&#x2013;W) relations. The ferrite texture determined on a controlled rolled steel,
heavily deformed in the unrecrystallized <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3B3;</mml:mi></mml:math> region, agrees reasonably well with the bcc texture
calculated using the K&#x2013;S relation from the rolled Ni&#x2013;Co alloy with similar SFE. The major texture
components of the ferrite, namely &#x7B;332&#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>113</mml:mn></mml:mrow><mml:mo>&#x0232A;</mml:mo></mml:mrow></mml:mrow></mml:math> and &#x7B;311&#x7D;<mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>&#x2329;</mml:mo><mml:mrow><mml:mn>011</mml:mn></mml:mrow><mml:mo>&#x0232A;</mml:mo></mml:mrow></mml:mrow></mml:math>, are found to originate from the two
major rolling texture components of the austenite, i.e. the &#x7B;110&#x7D;<mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>&#x2329;</mml:mo><mml:mrow><mml:mn>112</mml:mn></mml:mrow><mml:mo>&#x0232A;</mml:mo></mml:mrow></mml:mrow></mml:math>(Bs) and &#x7B;112&#x7D;<mml:math id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>&#x2329;</mml:mo><mml:mrow><mml:mn>111</mml:mn></mml:mrow><mml:mo>&#x0232A;</mml:mo></mml:mrow></mml:mrow></mml:math>(Cu),
respectively. Such ferrite transformation from heavily deformed austenite seems to follow the K&#x2013;S
relationship without any variant selection. By contrast, the texture of the martensite produced from
deformed austenite appears to involve significant amounts of variant selection.

R. K. Ray

Ph. Chapellier

J. J. Jonas

Correlations Between the Rolling Textures in FCC Ni&#x2013;Co Alloys and the BCC Transformation Textures in Controlled Rolled Steels

The elastic properties of polycrystals depend on the single-crystal elastic constants of the crystallites
which build up the polycrystal and on the manner in which the crystallites are connected. Because of
the technical importance of polycrystalline materials a lot of papers deal with the problem to calculate
effective elastic constants of polycrystals from single-crystal and structure properties. This paper gives
a review concerning the most important theories and methods respecting this matter.

S. Hirsekorn

Elastic Properties of Polycrystals: A Review

From the first forged turbine blades made of iron base alloys to the present nickel base single-grain
turbine blades and vanes manufactured by directional solidification, an enormous amount of research
has been directed to attaining the hottest possible combustion chamber temperatures in jet engines.
Temperature has been increased by about 15 K each year for the last two decades, improving the
thermodynamic efficiency of the engines. The more recent developments concern the manufacturing
of single-grain parts made of nickel base superalloys with large amount of the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi></mml:math>&#x2032; hardening phase.This paper first presents the directional solidification process used to produce single-grain parts, the
formation of as-cast microstructures and the defects that can arise during solidification. In the second
part the thermal treatments that are applied to the nickel base superalloys in order to enhance their
mechanical properties are detailed. The effect of crystallographic orientation and of the <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi></mml:math>/<mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B3;</mml:mi></mml:math>&#x2032;
microstructure on the mechanical properties is briefly presented, as well as the. microstructural
changes that can possibly arise during service.

Jacques Lacaze

Alain Hazotte

Directionally Solidified Materials:
Nickel-base Superalloys for
Gas Turbines

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Texture, Stress, and Microstructure