The classical three-dimensional analysis (O.D.F.) has been extended in such a manner as to obtain the representation of the texture, and, more precisely, the distribution of the slip systems &#x3C;111&#x3E; <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mi>k</mml:mi><mml:mi>&#x2113;</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math> in the planes where shear stresses induced in equibiaxial expansion are maximum.Quantification of the slip systems configuration allows the development of a deformation model which takes into account the factors influencing the degree of stretchability of the steels. From this model it is possible to define true hardening rates and equibiaxiality coefficients characterizing the ability of the steel to distribute the deformation in the macroscopic shear planes.In association with the resistance to thinning defined by the configuration of the slip systems, the theoretical treatment makes it possible to grade different steels according to their experimental behavior.

C. Gaspard

P. Messien

J. Mignon

T. Greday

Plastic Deformation and Crystalline Texture in Equibiaxial Expansion

A generalization of the O.D.F. representation has been proposed in such a way that its calculation is possible in any axes system of a material provided that is known in one particular system. Using an appropriate rotation matrix T&#x28;a,b,c&#x29; it is possible to compute crystallographic orientations <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>hk</mml:mtext><mml:mi>&#x2113;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
&#x5B;uvw&#x5D; in any macroscopic plane of the material. This calculation method leads also to a direction distribution function <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mprescripts /><mml:mi>&#x03C0;</mml:mi><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mmultiscripts><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x03C8;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03B1;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> which gives the angular distribution of the crystallographic direction <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>u</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math> in a macroscopic plane of the material.This analysis has been applied to the representation of texture and of slip directions in planes at 45 degrees from the directions RD, ND and TD of a mild steel deformed in equi-biaxial expansion.

Generalization of the Orientation Distribution Function: Determination of Crystalline Orientations in any Section of the Specimen

The Bishop and Hill-type isostrain analysis for the deformation of BCC crystals by pencil glide has been re-examined. Expressions have been derived for slip-plane orientations and shears for simultaneous slip along four &#x3C;111&#x3E; directions. The expressions for shears, in conjunction with expressions for the stress states previously calculated by Piehler and Backofen, permit comparison of external and internal work, which must be equal for the active stress state. Additional relations are introduced which must be satisfied for simultaneous operation of three &#x3C;111&#x3E; slip systems. These relations permit a straightforward computational procedure for determining possible stress states, and insure that external and internal work are equal. If for a possible stress state involving activation of three slip directions, the shear stress on the supposedly inactive system is less than the yield stress, that stress state is active.

Peter R. Morris

S. L. Semiatin

The Prediction of Plastic Properties of Polycrystalline Aggregates of BCC Metals Deforming by &#x3C;111&#x3E; Pencil Glide

The recrystallization textures of copper and different Cu-Zn alloys as well as the rolling and recrystallization textures of silver of varying purity were investigated as a function of the rolling temperature. In all cases in which the pure copper type rolling texture was present the cube texture was found as recrystallization texture, whereas in the case of the pure brass type rolling texture the brass type recrystallization texture (326) <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>83</mml:mn><mml:mover accent="true"><mml:mn>5</mml:mn><mml:mo stretchy="true">&#x00AF;</mml:mo></mml:mover></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math> developed. In the transition range a large number of well defined and reproducible recrystallization orientations occurred. The high accuracy of the present pole figure measurements allowed a detailed discussion of the results with regard to the mechanisms of formation of the recrystallization textures.

Ursula Schmidt

Kurt L&#xFC;cke

Recrystallization Textures of Silver, Copper and &#x3B1;-Brasses With Different Zinc-Contents as a Function of the Rolling Temperature

In order to evaluate pole-figure measurements quantitatively, one
needs the normalization factor which reduces measured intensity
values to multiples of the random density. This factor may be
determined experimentally by measuring the intensities of a random
sample or it may be calculated by integrating over the whole
pole-figure or its asymmetric unit. If pole-figure values are not
available in the whole angular range <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>&#x3C6;</mml:mi><mml:mo>&#x2264;</mml:mo><mml:msup><mml:mrow><mml:mn>90</mml:mn></mml:mrow><mml:mo>&#xB0;</mml:mo></mml:msup></mml:mrow></mml:math> (incomplete pole-figures), then the calculation is in
general much more difficult and it usually presumes the knowledge
of several pole-figures.In the case of fiber textures (axial symmetry), consisting of only
a few strongly preferred orientations with the crystal directions
<mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>&#x2329;</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:mi>w</mml:mi></mml:mrow><mml:mo>&#x0232A;</mml:mo></mml:mrow></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math> parallel to the axis of symmetry, the normalization factor and hence the volume fractions of the components i may be calculated in a rather simple way requiring only one, possibly incomplete, pole figure.

Ivan Tomov

H. J. Bunge

An Analytical Method for the Quantitative Determination of the Volume Fractions in Fiber Textures

The important class of metallurgical treatments known as thermomechanical treatment &#x28;TMT&#x29; is especially appropriate for textural studies because the crystallographic disposition of the matrix largely determines the final properties of the material. The current work utilized regular and inverse pole figure measurements of the ferrite matrix in TMT eutectoid steel to monitor the effects of various processing steps such as degree of cold rolling and the temperature and time of rapid annealing. Textural results provided the key to understanding fatigue life differences between longitudinally and transversely oriented sheet samples. The most favored ferrite slip system was not well oriented to act in the transverse samples. This in turn retarded the slip which nucleates fatigue cracking and hence extended the total fatigue life of transverse samples.

A. Querales

J. G. Byrne

Textural Studies of Thermomechanically Treated Eutectoid Steel

The choice of the particular angular parameters <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x03A8;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03B6;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> used in the vector method of texture analysis is justified and the relstionships for expressing any ideal orientation <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#x03A8;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03C9;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#x03B6;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> in terms of &#x28;hkl&#x29; &#x5B;uvw&#x5D; and vice versa are given. Examples of orientation distribution are presented with constant sections of each of the three angular parameters.

D. Ruer

A. Vadon

R. Baro

Analysis of Orientation Distribution Plots Obtained With the Vector Method for Cubic Polycrystals

Samples of molybdenum TZM alloy in sheet form have been investigated in the warm-worked condition and after recrystallisation heat treatments to produce a range of different grain sizes. Orientation distributions were calculated from measured pole figure data and specimens were cut at various angles around the sheet to determine the planar variations in Young&#x27;s modulus and microyield stress. Elastic moduli have been calculated for the materials on the basis of several different models using the preferred orientation data. Excellent agreement is found between the measured moduli and those calculated according to the Eshelby-Kr&#xF6;ner-Kneer model. Variations in stress level from grain to grain were also calculated according to this model and used to predict the initiation of plastic flow, i.e. the microyield point. These predictions also agree closely with measured values. Variation of grain size did not have a major influence on elasticity or microyielding.

A. Kumar

W. B. Hutchinson

An Analysis of Elastic Anisotropy and Microyielding in Polycrystalline Molybdenum TZM

The effects of carbon content on the crystallographic texture, plastic anisotropy, and mechanical properties of cold-rolled and annealed sheets of 0.07 percent phosphorus steels have been studied. Both vacuumand air-melted laboratory heats were used. Results show that although the strength of the sheet increased with increasing carbon content, the <mml:math id="E1" 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> value decreased. The detrimental effect of carbon on <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> value was more pronounced after annealing at an intercritical temperature of 780&#xB0;C &#x28;1435&#x2109;&#x29; than at a subcritical temperature of 710&#xB0;C &#x28;1310&#x2109;&#x29;. These and other observations, including variations in texture, anisotropy, strength, work hardening, and grain size, are discussed.

Hsun Hu

Effect of Carbon Content on Annealing Texture, Plastic Anisotropy, and Mechanical Properties of 0.07&#x25; Phosphorus Sheet Steels

The plastic deformation of low carbon sheet steels has been analyzed using microplasticity models and Hill&#x27;s theory of anisotropic continuum plasticity. The analytical methods used to determine the crystallite stress states which must be generated for the isostress &#x28;lower-bound&#x29; and isostrain &#x28;upper-bound&#x29; cases are briefly discussed and then applied to predict yield loci for three cold-rolled, annealed low-carbon steels &#x28;enameling iron, aluminum-killed steel, and interstitial-free steel&#x29; whose textures were characterized by crystallite orientation distribution functions. While measured r-values correlate best with upper-bound yield loci predictions for enameling iron, those for the aluminum-killed steels show better agreement with lower-bound yield loci predictions. In the case of the interstitial-free steel, measured r-values could not be accounted for by either upper- or lower-bound predictions. The yield loci measurements and predictions are reviewed in the context of Hill&#x27;s continuum theory.

P. R. Morris

H. R. Piehler

Microplasticity Predictions of r-Values and Yield Loci of Low Carbon Sheet Steels Deforming by &#x3C;111&#x3E; Pencil Glide

It is shown, theoretically and with numerical examples, that the orientation distribution function may vary between zero and two or even more times random while a corresponding pole figure is completely random.

C. Esling

Non-Random Orientation Distribution Functions With Random Pole Figures

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Texture, Stress, and Microstructure