Calculation of the taylor factor and lattice rotations for bcc metals deforming by pencil glide
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H. R . P I E H L E R
Values of the Taylor factor M and the corresponding lattice rotations for tension or compression have been computed for grains of various orientations postulated to slip on arbitrary planes in (Iii) directions. The stresses in each grain were first obtained by numerically maximizing the work expression subject to the constraints of the pencil-glide yield expressions. The derived value ofMave = 2.733 is slightly lower than in the case of mixed {I I0}, {i12}, a n d {123} , and {123}(Iii) slip. Furthermore, the average value ofM ac3182
Axis Of
[111]
cording to the present calculations, 2.733, is not much less than values computed by restricted-glide models with large numbers of slip systems. Chin and Mammel obtaineda value of 2.754 for their mixed slip model; Hutchinson computeda value of 2.748 for the case of twenty equally spaced planes for each (111> slip direction.II The magnitude of the difference between pencil glide (the limiting case of an infinite number of possible slip planes) and Hutchinson's model is less than the 1 pct he estimated. These results are summarized in Table I. T e x t u r e Prediction P r e d i c t e d lattice r o t a t i o n s for c o m p r e s s i o n of o r i e n t a t i o n s a g a i n r e p r e s e n t e d in the 8-(p c o o r d i n a t e s y s t e m a r e s h o w n in Fig. 2. T h e p e n c i l - g l i d e h y p o t h e s i s u n i q u e l y p r e d i c t s lattice r o t a t i o n s w i t h o u t r e c o u r s e to a n y a d d i t i o n a l a s s u m p t i o n s e x c e p t the p r i n c i p l e of m a x i m u m w o r k . In this r e s p e c t , p e n c i l g l i d e d i f f e r s c o n s i d e r a b l y f r o m r e s t r i c t e d g l i d e , for e v e n the ass u m p t i o n ofm i x e d { 1 1 0 } ( 1 1 1 > {112} (111), and {123} (Ii I> slip fails to predict unique slip combinations over half the stereographic triangle. 2'3The reasons for this have been pointed out by Piehler and Backofen. 8 As expected, the results show that, in compression, (110> is a point of unstable equilibrium and all grains rotate to (Iii> or (100>. Since the slip process is assumed to be reversible, the results for tension are exactly the reverse of those for compression and all grains tend to rotate toward (110>. Where unique results are predicted for mixed {110, 112, and 123} (Iii> slip, 3 the lattice rotations are almost exactly the same as those calculated here. The data of Ref. 3 predict a
TableI. Average ValuesofMforVariousSlip Modes No. of Slip Planes per D/rection Maye
Slip Mode {110} {112} {123} mixed {110, 112, and 123} approximate pencil glide pencilglide
3 3 6 12 20 ~
3.067 2.954 2.803 2.754 2.748 2.733
Ref. 6,2 2 2 2 11 present work
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Fig. 1--Contours of constantM for pencil glide. T h e h e a v y s o l i d lines s e p a r a t e regions w i t h i n which s t r e s s
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