A simplified method for determining the number of independent slip systems in crystals

Scripta METALLURGICA et MATERIALIA

Vol.

25, pp. 2395-2398, 1991 Printed in the U.S.A.

Pergamon Press plc All rights reserved

A SIMPLIFIED METHOD FOR DETERMINING THE NUMBER OF INDEPENDENT S L I P SYSTEMS IN CRYSTALS J.D. Cotton and M.J. Kaufman Department of Materials Science and Engineering, University of Florida Gainesville, FL 32611 R.D. Noebe NASA - Lewis Research Center Cleveland, OH 44135 (Received June (Revised August

,1, 1991) 20, 1991)

Introduction Recent efforts to develop smmtural intermetallic compounds have often failed because of inadequate tensile ductility. One of several reasons for low ductility is a lack of five independent slip systems which is necessary to accommodate the arbitrary shape change of a constrained crystal, according to the 'von Mises Criterion' (1). If less than five independent slip systems operate in a polycrystalline material, compatibility at the interface between adjacent grains cannot be maintained and intergranular cracks develop. Therefore, it is important to know the number of independent slip systems in operation for a given alloy as a first measure of the materials potential for ductility. Groves and Kelly (2) examined this problem by applying the yon Mises criterion to a number of differont crystal structures. Their approach to the determination of the number of indeoendent slip systems requires one to choose five different slip systems initially, and then test for independence by evaluating a 5 x 5 determinant of the strain tensor components. If the value of the determinant is zero, the slip systems are independent. This approach can be rather cumbersome when one has a number of physically distinct slip systems from which to choose the five for evaluation. In this paper, an alternative method is presented for determining the number of indepondent slip systems from any number of physically distinct systems. A simple matrix reduction method is demonstrated, which is also capable of evaluating two or more types of slip systems operating simultaneously in a given material. Method First, choose the operative slip system family(ies), {hkl). More than one family may be chosen so that the number of independent slip systems operating can be determined when multiple slip behavior occurs. Then write the strain tensor components for each physically distinct system. "Physically distinct" indicates all unique combinations of bur~,eas vector and slip plane within the operating family. These are simply the swains imposed upon a unit cube by slip upon the given system. From (2): ~xx = hu;

~yy = kv;

exy = 0.5(he + ku) ;

~zz = lw; eyz = 0.5(kw + Iv) ;

ezx = 0.5(lu + hw)

(I)

where the slip plane ffi (hki) and the slip direction = b = [uvw]. The use of Miller indices is merely for convenience when analyzing cubic structures. For non-cubic su'ucuue~ one must make an appmpfia~ wansfca~Jation to orthogonal, equally scaled axes. Set up the matrix, writing one row for each slip system and a column for each of the five calculated swain components. One of the principal tensile strains can be ellralnjted since exx + e ~ + ~zz = 0. Finally, reduce the matrix to row eschelon form, Le. ~ the component values along the diagonal from upper left to lower right equal to one, with only zeros below the diagonal. In the course of reduction, rows may be __~___,~dto one another, as well as multiplied by constants to allow caneellation. The position of rows within the matrix may also be changed. A detailed description of row eschelon reduction is given by (3) or in most linear algebra texts. The number of remaining non-zero rows (the rank of the matrix) equals the number of 2395 0 0 3 6 - 9 7 4 8 / 9 1 $ 5 . 0 0 ÷ .00 C o p y r i g h t ( c ) 1991 P e r g a m o n P r e s s p l c

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independent slip systems. If one keeps track of each row (slip system) during reduction, the independent slip systems are identified by those which arc not eliminated. Examnle As an example, room temperature slip in B2 (CsCl) NiA1 is examined. Slip occurs predominantly on {011 }<100> in this compound (4-6). There are six physically distinct slip systems of the type {011 }<100>, as shown in Table 1, along with the calculated values for the strain tensor components. Table 1. Physically Distinct Slip Systems of the Type {011 }<100>. System (011)[1001 (011)[100] (101)[010] 0701)[010] (110)[001]

exx - ezz 0 0 0 0 0

£yy - Czz 0 0 0 0 0

exy 0.5 -0.5 0.5 -0.5 0

eyz 0 0 0.5 0.5 0.5

OTO)[O01]

0

0

0

-0.5

Ezx 0.5 0.5 0 0 0.5 0.5

Thus, the matrix describing {011 }<100> is: 0 0 0 0 0 0

0 0 0 0 0 0

O.5 -0.5 0.5 -0.5 0 0

0 0 0.5 0.5 0.5 -0.5

O.5 0.5 0 0 0.5 0.5

This matrix is to be reduced to the minimum number of rows required to describe the calculated strains. By multiplying the matrix by two and subtractingappropriaterows rank eschelon form is quickly reached: 0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

-1 I 0 0 0 0

0 -I 1 0 0 0

The resultingexpression is of rank three and therefore only three slipsystems of the original six are independent, one solutionof which is: 0701)[010],(011)[100] and (110)[001]. Itis also possible by thistechnique to calculatethe number of independent systems when more than one family of slip systems is operating,e.g. {011 }<0TI> in addition to {011 }<100>. {011 }<0T1 > contributes six additionalphysically distinctslip systems, for a totalof 12 as shown in Table 2. Subsequent reduction to eschelon form proves that the combination of these two systems provides a totalof five independent slip systems. One combination of these five independent systems which results from the reduction are: 0701)[010], (0T1)[100], (110)[001], (I"01)[101] and

0710)[ii0]. Interestingly,ifsecondary slipoccurs on {100 }<011> insteadof {011 }<0TI >, only three independent systems remain, i.e.{011 }< 100> and {100 }<0 11> arc equivalent systernsand even theirtandem operation does not increasethe number. Thus, the mere addition of a secondary Burger's vector does not insure a greaternumber of independent systems and a technique such as the one outlinedhere isrequired.. In like manner, itmay be shown that either {110} slipor {112} slip will provide five independent slip systems. The operation of secondary slip cannot provide more independent systems in this case, since five is the

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maximum allowed by the number of strain tensor components available. Therefore, the von Mises' criterion may be satisfied in an intermetallie such as NiA1 by either a combination of operative slip families or a single family, provided the proper slip system can be activated. The results of this example are summarized in Table 3 below. Table 2. Physically Distinct Slip Systems of the Type {011 }<100> and {011 }<0TI>. ]Slip System (011)[100] (011)[100]

exx - ezz 0 0

~yy - Ezz 0 0

~xy 0.5 -0.5

Eyz 0 0

¢-zx 0.5 0.5

(1001010]

0

001)[010] (110)[001] (IT0)[001] (011)[011] (01"1)[011] (101)[101] (T01)[101] (llO)[TIO] 010)[110]

0 0 0 -I -1 -2 -2 -1 -1

0

0.5

0.5

0

0 0 0 -2 -2 -1 -1 1 1

-0.5 0 0 0 0 0 0 0 0

0.5 0.5 -0.5 0 0 0 0 0 0

0 0.5 0.5 0 0 0 0 0 0

Table 3. Slip Families and Resulting Number of Independent Slip Systems Slip Famil~,0es) {011 }<100> {011 } <0"1"1> {011}<100> + {011}<0TI> {100)<011> {011)<100> + (100}<011> {011 }<1 IT> {112}<11"1">

# Ph~,sicall~ Distinct 6

# Independent 3

6

2

12 6 12

5 3 3

12

5

12

5

One may also consider twinning as an additional deformation mode, with [uvw] -- twin direction and (hkl) = twin plane. However, the non-equivalence of twinning ease in the positive and negative directions only allows one to determine whether twinning cannot provide extra deformation modes. That is, if this analysis shows an increase in the number of independent systems due to twinning, only the possibility of additional deformation modes has been proven. A different approach, such as that by Goo and Park (7) must be utilized in this case. Summarv This paper describes a method of determining the number of independent slip systems for any family or combination of families of slip systems which is more direct than previous approaches. Specifically, if the operative slip system(s) are known for a material, one may determine in a straightforward manner if the material is slip system-deficient by this technique. In addition, one can determine if twinning may contribute additional deformation modes. Acknowledecmcnts Particular thanks are due Dr. S.M. Arnold of the N A S A - Lewis Research Center and Prof. F. Ebrahimi of the University of Florida Department of Materials Science and Engineering for their helpful comments and also to Prof. Y.K. Ran of the University of Washington Department of Materials Science and Engineering for instruction in the technique. This work was performed in the course of work funded by the NASA - Lewis Research Center under grant NAG 3-1079.

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References 1. 2. 3. 4. 5. 6. 7.

R. Von Mises, Z. Aneew. Math. Mech.. 8 (3), 1928, 161. G.W. Groves and A. Kelly, ~ , 81, 1963, 877. Y.IC Rao, Thermodynamics of Metallurgical Processes, Cambridge University Press, 1985, 903. R.T. Pascoe and C.W.A. Newey, Phys. Star. Sol.. 29, 1968, 357. RJ. Wasilewski, S.R. Butler and J.E. I-lanlon, Trans. Metall. Soc. AIME, 245, 1967, 1357. A. Ball and R.E. Smallman,~ 14, 1966, 1517. E. Goo and ICT. Park, fU~LMfJa~, 23, 1989, 1053.

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