THE PREDICTION OF PLASTIC PROPERTIES OF POLYCRYSTALLINE AGGREGATES OF BCC METALS DEFORMING BY <III> PENCIL GLIDE

" 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 <iii> 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 <iii> slip systems. These relations permit a straightforward computational procedure for determining possible stress states, and insure that external and internal work are equal. the shear stress on the supposedly inactive system is less than the yield stress, that stress state is active.


INTRODUCTION
As Piehler and Backofen have noted, at least three of the four <iii> slip systems must be activated to provide the 5 degrees of freedom necessary to accommodate an arbitrary imposed strain in a BCC crystal deforming by <iii> pencil glide. They obtained closed-form expressions for all stress states which simultaneously operate four <iii> slip systems, and derived conditions for simultaneous operation of three 114 P. R. MORRIS AND S. L. SEMIATIN <iii> slip systems. In the present paper we shall introduce additional relations which must be satisfied for simultaneous operation of three <iii> slip systems. These relations permit a straightforward computational procedure for determining slip planes, shears, rotations, and Bishop and Hill stresses. In addition, the shears associated with the activation of all four slip directions are tabulated. Hence, for these stress states, the internal and external work can be compared, and it can be easily determined whether one of these stress states is activated. In order to be brief, and to avoid duplication of prior work, it will be necessary to make frequent reference to the paper by Piehler and Backofen (PB).

A. Four <IIi> Slip Directions Activated
Piehler and Backofen analyzed cases involving activation of four slip systems by examining slip along arbitrary directions in {iii} planes, the results of which are equivalent to those from analysis of slip along <iii> directions in arbitrary planes containing these directions. The polar form of the yield criterion for pencil glide is 2/ dl =-(cos @a + / sin 8a)S a + (cos 8b-/ sin 8b) Sb +(cos 8c + / sin 8c)Sc (cos @d-/ sin @d) Sd, (3g} / d Sa cos @a + Sb cos @b + Sc cos @c + Sd cos @d (3h) By substituting cos 8a, sin 8a, etc., for Group I (from Table  I Table II. PB Table IV and Tables I and II list the stress states, slip plane orientations, and shears which allow an arbitrary imposed strain to be accommodated by simultaneous slip in four <iii > directions, and satisfy the yield condition for each slip system. We can now compare the external work, 7.
ij deij to the internal work, K 7. Spl Using Bishop i,j p and Hill notation, we may write The equality will apply only for the operative set of slipplanes and shears. 2 Since Eq. (5) involves absolute values of the shears, explicit conditions for operation of Group I, IIIa, IIIb, or IIIc stress state cannot be given. The calculation of the terms in Eq. (5) is, however, straightforward and will be discussed in the section on computational procedure. It is possible that none of the stress states involving activation of four slip directions will satisfy expression (5) with the equality sign. In such situations, stress states which activate three <III> slip directions must be considered.       GROUP IVc 3/ / A -4 K(sin 8 a + sin 8 b) --K(cos @a + cos @b + 2 cos 8 d) 4 K(sin @a + sin @b) + K(cos @a cos @b) C K(cos @b + cos @d) K(sin @b sin @a) + --K(cos @a + cos @b) GROUP IVd 4 K(sin @a + sin @b) + K(cos @a + cos @b + 2 B 3/ 4" K(sin 8 a + sin @b) + K(cos @a -cos 8 b) C --K(cos 8a + cos 8c) For Group IVc, satisfaction of Eq. (9h) requires that 2 cos @a + cos @b + cos @d + /(sin @b sin @d) 0.
For Group IVd, satisfaction of Eq. (9f) requires that cos 8a + 2 cos @b + cos 8c + /(sin 8c For Group IVa, the six equations in Table III together with Eq. (10a) form a system of seven equations in seven unknowns, Sb, s c, s d, e b, e c, 8 d and d2, where X2 Sb cos 8 b, Y a Sb sin 8b, etc. The system is nonlinear, and a solution in closed form has not yet been found. It may, however, be solved to any desired precision by an iterative trial-anderror process, in which an initial value is chosen for d1, e.g., de}a 0, and X i and Y i are determined from Table III.  i0 -When a value of d2 has been found such that the residual of Eq. (10a) is sufficiently small, it is necessary to assure that the stress on the system which has been assumed to be inactive is less than the yield stress. The quadratic forms of the yield conditions for pencil glide are given in PB Table II. A, B, C, F, G and H are calculated for Group IVa, using the expressions in Table IV, and substituted in the left side of the first equation in PB Table II. The result must be less than 9K if the [iii] system is to be inactive. The solutions of Groups IVb, IVc and IVd are accomplished in similar fashion. It may be shown that Eqs. (10) insure that the external work is equal to the internal work. Hence, the stress states and shears defined by Tables III and IV and Eqs. (i0) define those deformations which can accommodate the imposed strain, satisfy the yield condition on three <Iii> slip directions, and give rise to internal work equal to the external work.

C. Computational Procedure
A general procedure to determine the operative stress states, slip plane orientations, and shears which accommodate an arbitrary strain by <iii> pencil glide may now be outlined: The imposed strain-state referred to the macroscopic specimen axes is transformed to obtain the strain tensor relative to the cubic crystal axes, de ij- (2) The stress states and shears corresponding to Groups I, IIIa, IIIb and IIIc may be calculated from the relations in PB Table IV and Table II. If, for any of these groups, the external work is equal to the internal work [Eq. (5) ], the operative stress state and shears have been found. If not, the Group IV stress states must be investigated.
(3) For each Group IV stress state, the X i and Y i are determined from the relations in Table III Table II to assure that this stress state does not give rise to yielding on the supposedly inactive system. If such is the case for some group, it is operative. The stress state so obtained may then be transformed back to the macroscopic specimen axes. If this procedure is repeated for a number of crystallites which represent a given texture, the stresses may be averaged,and one point on the upper-bound yield locus obtained.
An entire upper-bound (isostrain) yield locus can be obtained by following this procedure for a number of imposed strain states.
The use of this procedure for the analysis of the plastic deformation of textured low-carbon sheet steels will be the subject of a subsequent paper. Measurements of r-values and the ratio of plane-strain flow stress to uniaxial flow stress will be compared to isostrain and isostress predictions, and the range of application of these models will be discussed. SUMMARY i. Relations which facilitate analysis of the pencilglide deformation of BCC crystals undergoing an arbitrary shape change have been derived. Stress states which activate all four slip directions or just three slip directions have been investigated.
2. Expressions have been derived for slip-plane orientations and shears for simultaneous slip along four <iii> directions. The expressions for the shears, in conjunction with expressions for the stress states previously calculated by Piehler and Backofen, permit comparison of external and internal work. A given stress state is active if the external and internal work resulting from its activation are equal.
3. In the case of simultaneous slip along three <Iii> directions, expressions have been derived relating X Sa cos 8a, Y Sa sin 8a, etc., to the strain components and one rotation component related to the cubic axes, where s i are the shears along <iii>, and the 8i are the angles between the slip plane normal and <110>. Additional relations among sines and cosines of angles specifying the orientations of active slip planes have been derived. These relations permit a straightforward computational procedure for obtaining possible stress states, and insure that the external work is equal to the internal work. A given stress state involving just three slip directions is active provided the shear stress on the supposedly inactive system is less than the yield stress.