Relation between axial stresses and texture development during torsion testing: A simplified theory
Abstract
A method is demonstrated for describing the anisotropy of textured cubic polycrystals in terms of the ellipsoidal yield surfaces originally proposed by Hill. The axes of these ellipsoids are first selected to coincide with those of the ideal orientations; the coefficients describing the ellipsoid are then found by a minimization procedure involving the single crystal (Bishop and Hill) yield surface. The various components of the stress tensor are calculated for the fixed end torsion test. It is shown that:
- 1.
(i) two texture components ( and ) do not lead to an axial stress, whatever the angle of rotation φ with respect to the specimen axis;
- 2.
(ii) the ideal orientations , C and produce small axial stresses, the magnitude and sign of which depend on φ;
- 3.
(iii) the orientations , , D1 and D2 produce large axial stresses, which are relatively independent of φ.
Résumé
Une description du comportement anisotrope des agrégats texturés de cristaux cubiques est developpée à partir des frontières d'écoulement ellipsoïdales initialement proposées par Hill. Les axes de ces ellipsoïdes sont choisis en coïncidence avec ceux des orientations idéales; les paramètres déterminant la forme de l'ellipsoïde sont alors déterminés par une procédure d'optimisation par rapport à la frontière d'écoulement (Bishop et Hill) du monocristal. Les différentes composantes du tenseur des contraintes sont calculées dans le cas de l'essai de torsion à longueur fixée. On montre que:
- 1.
(i) deux composantes de la texture ( et ) ne produisent pas de contrainte axiale quel que soit leur angle de rotation φ par rapport aux axes de l'éprouvette;
- 2.
(ii) les orientations idéales , C et induisent des contraintes axiales faibles, dont l'amplitude et le signe dépendent de φ;
- 3.
(iii) les orientations , , D1, et D2 produisent des contraintes axiales importantes, qui varient peu avec φ.
Zusammenfassung
Es wird eine Methode vorgeschlagen, mit der die Anisotropie texturbehafteter Polykristalle mit der von Hill ursprünglich vorgeschlagenen Ellipsoid-Oberfläche beschrieben werden kann. Die Achsen dieser Ellipsoïde werden zuerst so gewählt, daβ sie mit denen der idealen Orientierungen zusammenfallen; die das Ellipsoid beschreibenden Koeffizienten werden dann mit einer Minimalisierungsprozedur, welche die Flieβoberfläche der Einkristalle (Bishop und Hill) enthält, aufgefunden. Die verschiedenen Komponenten des Spannungstensors werden für den Torsionsversuch mit festem Ende berechnet. Es wird gezeigt, daβ
- 1.
(i) zwei Texturkomponenten ( und ) zu keiner achsialen Spannung führen, wei immer auch der Rotationswinkel φ bezüglich der Probenachse ist;
- 2.
(ii) die idealen Orientierungen , C und kleine achsiale Spannungen erzeugen, deren Gröβe und Vorzeichen von φ abhängen;
- 3.
(iii) die Orientierungen , , D1, und D2 groβe achsiale Spannungen erzeugen, welche vergleichsweise wenig von φ abhängen.
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