SOME ASPECTS OF
THE HARDNESS OF METALS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN
DOCTOR IN DE TECHNISCHE WETENSCHAP
AAN DE TECHNISCHE HOGESCHOOL TE
DELFT, OP GEZAG VAN DE RECTOR
MAG-NIFICUS DR O.BOTTEMA, HOOGLERAAR IN DE
AFDELING DER ALGEMENE WETENSCHAPPEN,
VOOR EEN COMMISSIE UIT DE SENAAT TE
VERDEDIGEN OP WOENSDAG 12 DECEMBER
1951, DES NAMIDDAGS TE 4 UUR
DOOR
MAURITIUS ARNOLDUS DU TOIT MEYER
GEBOREN TE BLOEMHOF, TRANSVAAL.
DRUK: EXCELSIORS FOTO-OFFSET, - 'S-GRAVENHAGE
Mrl
no i \
4
'4 ]DIT F-'ROEFSCnRIrT IS G O E D G E K E U R D COOR DE PROMOTOR
^
Aan almal loat daartoe bygedra het om my verblyf in Nederland te veraangenaam, wil ek my erkentlikheid betuig.
Dit is vir my aangenaam om die Snid-Afrikaanse Wetenskaplike en Nywerheidsna-vorsingsraad vir die studiebeurs te bedoaxk.
P R E F A C E
This t h e s i s had i t s o r i g i n in a study of the l i t e r a t u r e on t h e hardness of m e t a l s . I t was r e a l i z e d t h a t t h e r e e x i s t e d a number of gaps on which work was d e s i r a b l e . This led the author
to a study of a few more or l e s s d i s t i n c t aspects of the hard-ness of metals. In preparing t h i s account of the a u t h o r ' s work during the p a s t two y e a r s , he has attempted to review b r i e f l y the most important work r e l a t e d to the subjects studied.
When in 1948, Tabor's work on the i n t e r p r e t a t i o n of hardness was published, a valuable c o n t r i b u t i o n to the understanding of B r i n e l J , Meyer, Vickers and impact hardness was made. T a b o r ' s work i s b r i e f l y summarized in the f i r s t c h a p t e r . L n t i l 1950, l i t t l e t h e o r e t i c a l work on Rockwell hardness was published. The author analyzed the Rockwell method in d e t a i l . His work was published elsewhere. The a u t h o r ' s work on the Rockwell method, the r e l a t i o n between the d i f f e r e n t h a r d n e s s numbers and the effect of p l a s t i c deformation on the Meyer constants i s discussed in the f i r s t chapter.
Solution hardening, work-hardening, the effect of tenperature on the hardness and the p r o p e r t i e s of d i s l o c a t i o n s are so i n t e r -r e l a t e d t h a t i t seemed i n a p p -r o p -r i a t e t o d i s c u s s these e f f e c t s without a b r i e f summary of the p r o p e r t i e s of the l a t t e r . In the second chapter, the d i s l o c a t i o n theory and the r e s u l t s of the f i r s t chapter are applie;d to the e f f e c t of p l a s t i c deformation and temperature on the hardness of pure m e t a l s .
Experimental work on the e f f e c t of a l l o y i n g elements on the hardness of metals (copper, l e a d , aluminium, and s i l v e r ) was done by v a r i o u s a u t h o r s . Nickel o f f e r e d another p o s s i b i l i t y . The effect of eleven alloying elements on the hardness of nickel as well as the effect of temperature and cold work was studied. The experimental r e s u l t s together with a new theory of solution hardening, based on dynamic e f f e c t s of fast moving d i s l o c a t i o n s , are discussed in the t h i r d chapter.
Since the discovery of the unexpected high hardness of c h i l l -c a s t e u t e -c t i -c a l l o y s i n 1908, v a r i o u s e x p l a n a t i o n s for t h i s phaiomenon were put forward. However, many aspects of these
ex-p l a n a t i o n s are oex-pen to serious c r i t i c i s m . For t h i s reason, work on the hardness of c h i l l - c a s t two-phase a l l o y s was s t a r t e d .
Laboratorium voor Technische Physica, Delft
C O N T E N T S
Chapter I - PHENOMENOLOGICAL THEORIES OP HARDNESS
1. D e f i n i t i o n o f h a r d n e s s 11 2. Meyer and B r i n e l l h a r d n e s s numbers 12
3 . V i c k e r s h a r d n e s s numbers 14 4. P l a s t i c d e f o r m a t i o n and t h e Meyer c o n s t a n t s 14 5. Rockwell h a r d n e s s numbers 20 6. R e l a t i o n between t h e d i f f e r e n t h a r d n e s s numbers 25 7 . R e l a t i o n between h a r d n e s s and s t r u c t u r e s e n s i t i v e p r o -p e r t i e s o f m e t a l s 27 8 . C o n c l u s i o n 28
Chapter I I -THE HARDNESS OF PURE METALS
9. Deformation of metals 29 10. Relation between hardness and s t r u c t u r e i n s e n s i t i v e
p r o p e r t i e s of metals 35 11. Effect of temperature 37
Chapter I I I - THE HARDNESS OF SINGLE PHASE ALLOYS
12. Introduction ' 39 13. The hardness of nickel solid solutions at room
temper-ature 42 14. The effect of temperature on the hardness of nickel
solid solutions 47 15. Effect of cold work on the hardness of nickel solid
solutions -_ 52
16. A theory of solution hardening 57
17. Discussion 63
Chapter IV - THE HARDNESS OP HETEROGENEOUS ALLOYS
18. Introduction 67 19. Experimental procedure 70
20. The hardness of annealed two phase alloys as a
func-tion of the concentrafunc-tion 71 21. Experimental results regarding the hardness of chill»
cast two phase alloys 73
22. Discussion 76 23. Effect of temperature 88 24. Conclusion 88 Appendix 89 Sumnary 92 S a n e n v a t t i n g 94
9
C h a p t e r I
PHENOMENOLOGICAL THEORIES OF HARDNESS 1. D e f i n i t i o n o f h a r d n e s s
The hardness of a metal i s u s u a l l y defined in terms of i t s r e s i s t a n c e to l o c a l i n d e n t a t i o n , A hard i n d e n t e r i s p r e s s e d i n t o the s u r f a c e of t h e metal under a s p e c i f i e d load for a d e f i n i t e time £ind when e q u i l i b r i u m i s a t t a i n e d , a s u i t a b l e measurement i s made of the s i z e or depth of the i n d e n t a t i o n , Of the many e x i s t i n g methods for determining the hardness of metals only the s t a t i c hardness t e s t s will be considered here, Three of the most cornnon methods, the B r i n e l l , the Vickers, and the Rockwell h a r d n e s s t e s t e r s impress e i t h e r a hardened s t e e l b a l l , a diamond pyramid or a diamond cone in the metal being tested^»^•^, The d i f f e r e n t q u a n t i t i e s in \ ^ i c h the hard-ness i s expressed i n these methods ( t h e B r i n e l l , Vickers, and Rockwell hardness numbers) cannot d i r e c t l y be compared with each other, butv^en properly analyzed they are a l l e s s e n t i a l l y a measure of the same property of metals, namely r e s i s t a n c e to plsistic deformation by external forces.
I t will be shown t h a t these d i f f e r e n t hardness numbers can be w r i t t e n in terms of the s t r e s s - s t r a i n curves of m e t a l s . I f t h e s e c h a r a c t e r i s t i c s are known, any of the above h a r d n e s s numbers can be c a l c u l a t e d , and any one of the above mentioned methods can be regarded as s c i e n t i f i c a l l y sound.
In the B r i n e l l t e s t , a hard s t e e l b a l l (diameter 2 r) i s used as the i n d e n t e r and the diameter ( 2 a ) of the r e s u l t i n g indentation i s measured. The hardness i s then expressed as the r a t i o of the load to the curved area of the indentation (Brinell harchess ƒƒ„), or as the r a t i o of the load to the projected area of the indentation (Meyer hardness^ W^).
The r e l a t i o n between the diameter of the indentation and the load P i s given by the following enpirical r e l a t i o n f i r s t d i s -covered by Meyer, *
P= b a^/r"'^ or H^, = (b/n)(a/r)"'^ (1)
1) S . R . W i l l i a m s , Hardness and Hardness Measurements (A. S. M. . 1 9 4 2 ) .
2) H . O ' N e i l l , The Hardness of Metals and i t s Measurements (Lon-don, 1934).
3) V.E.Lysaght, Indentation Hardness T e s t i n g (Reinhold P u b l i s h -i n g Company, 1949).
4) E.Meyer. Zei t . Ver. Deut. Ing. 52, 645, 740, 835, 1908.
where 6 and n are constants for the metal and p r a c t i c a l l y inde-pendent of r . The constant b i n c r e a s e s whereas n decreases with i n c r e a s i n g amounts of cold work. For fully worked metals n i s c l o s e to 2 while for f u l l y annealed pure cubic m e t a l s i t i s approximately 2,5 ( s e e a l s o § 4 ) , The c o n s t a n t s b and n are c a l l e d the Meyer c o n s t a n t s . The material constant b/n i s some-t l S e s c a l l e d some-the Ulsome-timasome-te Meyer hardness {H ),The above r e l a some-t i o n i s v a l i d in the range 0 , 1 < a/r < 0 , 6 ,
In the Vickers t e s t , a diamond square-based pyramid of top angle 136° i s used as i n d e n t e r . The Vickers hardness number i s then defined as the load d i v i d e d by the c o n t a c t area of the i m p r e s s i o n .
\Wien conical or pyramidal i n d e n t e r s are used as in the Ludwik and Vickers hardness t e s t s r e s p e c t i v e l y , a s i n p l e r r e l a t i o n be-tween load and s i z e of the indentation i s observed. Over a wide range of e^qserimental condition i t i s found t h a t
P M k a ^
for an i n d e n t e r of fixed angle. The power of a i s fixed but k depends on the angle of the cone or pyramid.
Rockwell hardness numbers are based on the a d d i t i o n a l depth t o which an i n d e n t e r i s driven i n t o a metal by a heavy load, beyond the depth to which the same indenter has been driven by a l i g h t load under a r b i t r a r y but d e f i n i t e conditions (for f u l l e r discussion see § 5 ) .
2, Meyer and B r i n e l l h a r d n e s s numbers
According to T a b o r ' , the following mechanisms involved in the Brinell t e s t , may be put forward, Wien the indenter presses on the s u t f a c e , the metal i s f i r s t deformed e l a s t i c a l l y . The s t r e s s e s soon reach the e l a s t i c l i m i t of the metal and p l a s t i c flow o c c u r s . As p l a s t i c flow c o n t i n u e s , work-hardening occurs and the e l a s t i c l i m i t i n c r e a s e s . This process continues u n t i l the s t r e s s e s are d i s t r i b u t e d over an iirpression of such dimen-s i o n dimen-s t h a t the dimen-s t r e dimen-s dimen-s e dimen-s are w i t h i n the e l a dimen-s t i c l i m i t of the deformed m e t a l . At t h e end of the i n d e n t a t i o n p r o c e s s , when p l a s t i c flow has ceased, thewiiole of the load i s bom by e l a s -t i c s -t r e s s e s in -the m a -t e r i a l . I f -the load i s removed -t h e r e i s an e l a s t i c recovery of the b a l l and an e l a s t i c " s h a l l o w i n g " of the i n d e n t a t i o n .
5) D.Tabor, P r o c . R o y . S o c . 1 9 2 , 247, 1948.
*) Secondary p l a s t i c e f f e c t s and c r e e p a r e n o t c o n s i d e r e d h e r e . 12
I f the same load i s reapplied, the surfaces deform e l a s t i c a l l y u n t i l they j u s t f i t over the diameter of the impression. The e l a s t i c s t r e s s e s now reach the l i m i t s vihich the deformed metal around the impression can stand. I f the load i s increased, the s t r e s s e s exceed the e l a s t i c l i m i t and further flow occurs. There i s f u r t h e r work-hardening and the p r o c e s s continues u n t i l the s t r e s s e s are d i s t r i b u t e d over a l a r g e r impression and so f a l l again within the e l a s t i c l i m i t ,
Tabor* has shown e x p e r i m e n t a l l y , t h a t i f the t r u e s t r e s s -s t r a i n curve for annealed metal-s can be repre-sented by
cr= f ( e ) , (2)
«Cereals the true tensile stress and e the plastic elongation, then the Meyer hardness number of the metal is given by
Hi^= 2.8 f (p a/r) (3)
where p i s a c o n s t a n t , approximately equal to 0,20, being the same for d i f f e r e n t metals and independent of the s t r u c t u r e of the metal. The quantity p a/r may t h e r e f o r e be c a l l e d the " r e -p r e s e n t a t i v e -p l a s t i c deformation" of a s-pherical indentation,
If, however, the metal has previously been cold worked by an amount fig, i t may be considered as an unworked metal t h a t has undergone a t o t a l p l a s t i c deformation equal to the sum* of the i n i t i a l deformation e^ and any subsequent deformation e. In view of t h i s , (2) can be w r i t t e n as
^ = f(e„-^e). (4)
S i m i l a r l y , Tabor has extended (3) t o hold for metals t h a t have been previously cold-worked. Thus, according to Tabor,
Hf,= 2.8 f (e^ + p a/r). (5)
Since ƒ (^e) i n c r e a s e s as e i n c r e a s e s , Meyer hardness numbers will i n c r e a s e with i n c r e a s i n g a / r . Geometrical s i m i l a r impres-sicms ( i , e . impressions for which a/r = constant) w i l l y i e l d the same values for Meyer hardness numbers, independent of the r a d i u s of the b a l l . This equation can be used to c o - o r d i n a t f hardness measuranents with various loads and b a l l diameters oit
a given specimen*.
•) This i s of course only true when no recovery effects are ef-f e c t i v e .
Since, to a very good degree of approximation
the same considerations apply for Brinell hardness numbers,
3 . V i c k e r s h a r d n e s s numbers
For pyramidal and conical i n d e n t e r s , the i n d e n t a t i o n s are g e o m e t r i c a l l y s i m i l a r whatever the s i z e of the i n d e n t a t i o n . As a r e s u l t , the " r e p r e s e n t a t i v e p l a s t i c deformation" of these i n d e n t a t i o n s will be independent of the load. Tabor has shown t h a t the " r e p r e s e n t a t i v e p l a s t i c deformation" of Vickers pyra-midal indentations i s equal t o 0 , 0 8 , Therefore, Vickers hardness numbers are given by*
Hy= 3.0 ƒ (e^ + 0.08). (6)
I t i s therefore seen t h a t Vickers hardness numbers are indepen-dent of the load.
4. P l a s t i c d e f o r a a t i o n and t h e Meyer c o n s t a n t s * *
The true s t r e s s - s t r a i n curve for metals i s , to a good degree of approximation and over an appreciable range of deformation, given by*
a = g e« (7) where g emd x are consteints. Other e m p i r i c a l r e l a t i o n s were
also suggested. However, (7) will be employed in t h i s discussion as i t i s p a r t i c u l a r l y simple and s u i t e d to the present purpose. For annealed, p o l y c r y s t a l l i n e cubic m e t a l s x has a v a l u e of approximately 0 . 5 , whereas for annealed p o l y c r y s t a l l i n e hexa-gonal metals x has a value q)proximately *** equal to 0 . 3 . Higji values of x (of about 0,5) mean t h a t the metal work-hardens r a p i d l y while a lower value of x means t h a t the metal work-hardens a t a lower r a t e . This value of x for cubic m e t a l s i s a l s o p r e d i c t e d by T a y l o r ' s simple theory of w o r k - h a r d e n i n g ' .
• For a d i s c u s s i o n of t h e c o n s t a n t s 2.8 and 3.0 i n (5) and (6) see T a b o r ' s p u b l i c a t i o n .
**A s h o r t account of t h i s p a r a g r a p h w i l l be p u b l i s h e d i n N a t u r e . •**Own measurements on Zn and Cd for v a l u e s of e up to 40%.
6) A . N a d a i , P l a s t i c i t y (Mc Gr aw H i l l , 1 9 3 1 ) ; R . H i l l , The Ma-t h e m a Ma-t i c a l Theory of P l a s Ma-t i c i Ma-t y ( O x f o r d , 1 9 5 0 ) ; C h . C r u s s a r d and B . J a o u l , Rev.Mét. 47, 489, 1950.
The lower value of x for hexagonal metals i s not s u r p r i s i n g , since i t i s known t h a t hexagonal metals work-harden l e s s rapidly than cubic metals*.
By combining (3) and (7) and taking the appropriate value of X for annealed cubic metals, one obtains
H„=2.8 g p^ (a/rf^.
This i s of the same form as (1) for annealed cubic metals ( i . e . when n = 2, 5),
As b e f o r e , i f the metal has been cold worked by an amount e , (7) may be w r i t t e n as
<^= S (e„ + e)*- (8)
I f the s t r e s s - s t r a i n c h a r a c t e r i s t i c s of a metal are t r e a t e d in t h i s manner, taking i n t o account the i n i t i a l p l a s t i c deformation e , then the c o n s t a n t s g and x are independent of the p l a s t i c deformation. I f t h i s i s not taken i n t o account, (8) may be ap-proximated by an equation of the form ( 7 ) . In t h i s case, as i s e a s i l y seen from the form of the s t r e s s - s t r a i n curve, g will be an increasing function and x a decreasing function of the p i a s -t i c deforma-tion. In -the l i m i -t i n g case when -the me-tal has been cold worked to such an extent that i t does not appreciably work-harden einy further, x ~ 0 and a w i l l simply be equal to a con-s t c n t . In t h i con-s cacon-se then
Hm = 2.8 X constant,
This i s of the same form as (1) for f u l l y worked metals ( i . e . irfien r» = 2 ) . Further as follows from ( 4 ) , (5) and (8)
HM = 2.8 g(e^ + p a/r)^ (9)
which, according to the above d i s c u s s i o n , can be ^ p r o x i m a t e d by
"M = 2 . S g (ej (P a/r)' ('o> (10) wiiere g (e ) i s an i n c r e a s i n g and x (e ) a decreasing function
of the i n i t i a l p l a s t i c deformation. This i s exactly the r e l a t i o n ( 1 ) . I t follows from (10) and (1) t h a t
8. W.Boas, llie P h y s i c s of Metals and Alloys (Melbourne U n i v e r s i t y P r e s s , 1946).
6 = 2.8 TT g (e^) p" (%) and n - 2= x (ej. (11) If, therefore, the s t r e s s - s t r a i n curve of a p a r t i c u l a r metal
can be approximated by an equation of the form (7) and the Meyer constants are evaluated from the experimental r e s u l t s from ( 1) o r (10), then n must always decrease and 6 must always increase with i n c r e a s i n g amounts of cold work. This conclusion i s not only e s s e n t i a l l y connected with a s t r e s s - s t r a i n curve of the type ( 7 ) , but a necessary feature of the property exhibited by annealed metals, namely an i n i t i a l rapid r a t e of work hardening followed by decreasing r a t e s ,
The above conclusion t h a t 6 i n c r e a s e s and n decreases with i n c r e a s i n g amounts of cold work i s indeed the case for cubic m e t a l s . However, t h i s i s not in accordance with the work of F i n n i s t o n , Jones and Madsen' on non-cubic m e t a l s . They a l s o found t h a t (1) was obeyed by the non-cubic metals i n v e s t i g a t e d . This means t h a t the s t r e s s - s t r a i n curve for p o l y c r y s t a l l i n e non-cubic metals can be approximated by a function s i m i l a r to (7). Since (1) can be derived from ( 7 ) , one would expect, accor-ding to the above d i s c u s s i o n , t h a t 6 will i n c r e a s e and n w i l l decrease with p l a s t i c deformation,
Experiments* were c a r r i e d out on samples of well annealed (finnealed for about 10 hours a t 180° C and slowly cooled to room temperature) t i n , zinc, and cadmium. Four d i f f e r e n t loads (62J^, 100, 150, 167/4 kg) were used in d e t e r m i n i n g b/rr and n from (1) by p l o t t i n g log / L a g a i n s t log a/r. Vickers hardness measurements were also made. In t o t a l about 10 inpressions with various loads have been made on a specimen itrmediately a f t e r i t has been subjected to cold r o l l i n g . The specimens were seperately r o l l e d to the required percentages. The load was automatically applied.
I t was found t h a t a l l t h e s e m e t a l s e x h i b i t e d a d e f i n i t e continuous flow a f t e r the load had been a p p l i e d . The time of loading as well as the b a l l diameters are given in Table I , I t was found t h a t 6 i n c r e a s e d and n decreased with i n c r e a s i n g amounts of cold work (Table I I ) .
9. H.M. Finniston, E.R. W.Jones, and P. E.Madsen, Nature 164,1128, 1949.
* The measurements to be described in t h i s t h e s i s were all car-ried out at the Metallographisch Laboratorium.The author wi sh-es to exprsh-ess h i s g r a t i tude to Prof.Dr I r W. F.Brandsma for the h o s p i t a l i t y he has experienced at t h i s laboratory,
Table I , Time of loading and b a l l diameter used for the d i f f e r e n t metals
Metal Ball diameter Time of loading Tin Zinc Cadmium 10 run 3.18 run 5 nm 2 minutes 2 minutes 1 minute
Table I I , Variation of Ultimate Meyer hardness and n with p l a s t i c deformation*
Metal Cd ^ Zn % deformation 0 3,5 6.5 8.4 9.8 18.0 30.3 0 2.3 5.5 10.6 18.0 0 4.3 7.6 13 n = x + 2 2.31 ± 0.02 2.17 ± 0 . 0 3 2.23 ± 0.03 2.21 ± 0.03 2.14 ± 0.03 2.12 ± 0.03 2.10 ± 0.03 2.22 ± 0.02 2.20 ± 0.01 2.09 ± 0.03 2.11 ± 0.03 2.15 ± 0.02 2.25 + 0.02 2.21 ± 0,07 2.20 ± 0.04 2.19 ± 0.04 kg/mn^ 53 ± 2 21 ± 2 29 ± 2 28 ± 2 27 ± 2 27 ± 2 28 ± 2 13.0 ± 0 . 3 14.1 ± 0 . 3 13.0 ± 0 . 3 13.0 ± 0 . 3 14.5 ± 0 . 3 60 ± 1 63 ± 2 64 ± 2 67 ± 2 Vickers hardness kff/mm^ 24 ± 1 \ 2B ±2 29 ± 2 2S' ± 2 28 ± 2 29 ± 2 31 ± 2 15 ± 1 15 ± 1 15 ± 1 16 ± 1 16 ± 1 53 ± 1 62 ± 2 66 ± 2 66 ± 2 I t must be p o i n t e d out here t h a t a Meyer a n a l y s i s i s very s a i s i t i v e to the cold-worked s t a t e in which a metal i s (compare for example ƒƒ„, n and the Vickers hardness in Table I I ) . In the case of the softer metals which recover at room temperature the b and n corresponding to deformations greater than approximately 20 per cent cannot be r e l i e d upon too s e r i o u s l y since recovery e f f e c t s may be q u i t e marked.
* The author i s indebted to Mr K.J.Blok van Laer for carrying out these measurements.
Another i n t e r e s t i n g point here i s the fact t h a t the Vickers hardness, which according to Tabor i s given by
Hy= 3.0 f (e^ + 0.08),
i s greater than the Ultimate Meyer hardness "u = 2.« ƒ (^o + 0.20).
Since ƒ (e) i n c r e a s e s with i n c r e a s i n g e, t h i s was not to be expected. However, as already pointed out, the time of loading i s very inportant i n . t h e hardness t e s t i n g of soft metals. Since the time of loading in the Vickers t e s t (10 s e c . ) i s s m a l l e r than the times used here t h i s r e s u l t i s not s u r p r i z i n g . As i s seen, t h i s effect i s more pronounced for t i n and cadmium as was to be expected.
Therefore in considering these results, one has to d i s t i n g u i s h between the following e f f e c t s :
( i) work-hardening,
( li) recovery e f f e c t s , i . e . the decrease of hardness a f t e r (or during) r o l l i n g , and
(lii) time e f f e c t s in the hardness t e s t i t s e l f and the e f f e c t of p l a s t i c deformation on t h i s .
I t i s i n t e r e s t i n g to d i s c u s s (9) in g r e a t e r d e t a i l . As a l -ready mentioned, i f the i n i t i a l p l a s t i c deformation e i s taken i n t o account, the c o n s t a n t s g and x a r e independent of e . Equation (9) t h e r e f o r e allows one to c o r r e l a t e the d i f f e r e n t hardness numbers for d i f f e r e n t known amounts of cold work and loads from a knowledge of the constants * (or n) and b for an-nealed metals. I t i s e a s i l y seen, by comparing (9) and (1) for
the case of annealed m e t a l s and s u b s t i t u t i n g the r e s u l t s in Table I I , t h a t
H^i = 24 (e^ + p a/r)^'^^/p^'^^ for cadmium
n^ = 13 (e^ + p a / r j ^ ' - ^ V p " - ^ ^ for t i n (12) % = 60 (e^ + p a/r)^'^^/p°'^^ for zinc
These equations are only v a l i d i f no recovery has taken p l a c e and provided the e f f e c t of time of l o a d i n g does not g r e a t l y a l t e r the conditions of v a l i d i t y of these equations. I f //^ i s p l o t t e d as a function of e + p a / r , the hardness numbers cor-responding to d i f f e r e n t loads ( d i f f e r e n t a ' s ) and degrees of cold work should all l i e on the same curve (12) ( f i g , 1).
O . C . . a X . < . . a025 o , e. . oor? • . « , . 0 0 * 3 A . < , - 0076 T . e. - ao > Zn o . to mO. • . C _ ao» s , <• -OLOH » . « . .aoê4 i^ . 4 , . a o « o , £0 . aw <- , ^ - 0 3 0 > Cd
Fig. 1. Meyer hardness numbers plotted as a function of e + pa/r. The curves were calculated from (12). (N.B. In the text e % ) •
As i s seen, for moderate amounts of p l a s t i c deformation ( s m a l l e r than about 20%), e q u a t i o n s (12) a r e s a t i s f a c t o r i l y s a t i s f i e d . For higher amounts, the values of Ht, l i e below the c u r v e s . This was to be expected in view of recovery e f f e c t s which are known to take pi ace at room tenperature for the softer metals,. I t i s t h e r e f o r e believed t h a t the e f f e c t s observed by Finniston,. Jones and Madsen are not c h a r a c t e r i s t i c of non-cubic metals but due to some recovery of t h e i r specimens. Lead will also show the same e f f e c t .
In view of t h i s , i t i s believed that the Ultimate Meyer hardness invariably increases and that n invariably decreases (provided recovery i s not important) with increasing amounts of plastic deformation and that the interpretation of n - 2 as a measure of the work-hardening capacity of a metal (Tabor and O'Neill) can s t i l l be maintained. For materials which do not worit-harden (plasticine), these quantities will not change with plastic deformation.
5. Rockwell hardness nuabers
(a). Introduction
Rockwell hardness numbers are based on the additional dqjth to which an indenter i s driven into a metal by a heavy load beyond the depth to wliich the same indenter has been driven by a light load, the conditions under which t h i s h^pens being arbitrary but definite.
To put i t more exactly, Rockwell hardness numbers are based on the depth
h m h - h. (13) of an indentation, where h i s the depth to which a hardened
steel ball* i s driven by a load P and h. the depth to w^ich the same ball i s driven by a l i f t e r load P^, The minor load Pj i s f i r s t £pplied and thereafter an additional load Pg - P4 i s applied and removed, leaving the minor load Pj s t i l l applied. In this way a partial elastic recovery of the depth of the in-dentation and indenter i s brouwt about. The author has shown"* that this recovery i s only about one per cent of h and will therefore be neglected in the following.
Unit Rockwell hardness corresponds to h • 2 x 10" nm. The Rockwell hardness tester i s so constructed that h/2 x ICT i s
automatically subtracted from an arbitrary constant, in this case 130. High hardness numbers Hj^, given by
H^ ' 130 - 5 X 10^ h (14) mean a shallow indentation and thus a hard metal.
10. M.A,Meyer, A p p . S c i . R e s . A3, 1 1 , 1951.
* In what f o l l o w s , only b a l l i n d e n t e r s w i l l be c o n s i d e r e d . Harck n e s s numbers o b t a i n e d with b a l l i n d e n t e r s a r e c a l l e d Rockwell b a l l h a r d n e s s numbers.
The minor load i s applied for the i n i t i a l measurement of the depth of the impression and i s s t i l l applied when the hardness number i s read from the d i a l gauge. The e l a s t i c deformation of the frame of the t e s t e r i s t h e r e f o r e the same a t the s t a r t of the t e s t and at the reading and need not be considered further, A l e t t e r designation has been assigned for d i f f e r e n t p o s s i b l e combinations of r and P . not y i e l d i n g n e g a t i v e values for ///j (see Table IV, P^ - 10 k g ) .
Peek and Ingerson' ^ made an analysis of the Rockwell t e s t by studying the r e s u l t s of d i f f e r e n t loads P and ball r a d i i . They found the following empiricsd r e l a t i o n between the d i f f e r e n t i a l depth h <* h ' hj^ at eai i n d e n t a t i o n , the b a l l d i a m e t e r , and major load
h/2r= {q (P^ -P^/l^cry)•)"'' (15)
where a^ i s some constant of the material having the dimensions of s t r e s s , q and m are dim.ensionless c o n s t a n t s . This r e l a t i o n holds when h/2 r i s small ( i , e , < 0 , 1 ) , There are i n d i c a t i o n s t h a t m i s a measure of the s t r a i n - h a r d e n i n g p r o p e r t i e s of a material having the l i m i t i n g value of u n i t y for metals showing l i t t l e or no s t r a i n - h a r d e n i n g . The quantity g i s a function of the work-hardening p r o p e r t i e s of the m a t e r i a l . I f cr^ i s taken to be equal to the t e n s i l e s t r e n g t h , then 0,07 < q < 0,09 for a number of meted.s,
(b) A geometrical theory ignoring elastic recovery.
I t has already been shown'" t h a t the e l a s t i c recovery of the b a l l and depth of the i n d e n t a t i o n which takes p l a c e when the major load i s reduced to P j i s small compared with the depth of
the indentation,
Thus, i f the e l a s t i c deformation of the i n d e n t e r and the e l a s t i c recovery e f f e c t s may be n e g l e c t e d , the depth of the indentation due to the minor load, i s given by ( f i g . 2)
h , ^ r . ( r ^ . a,^f/^
where a;^ i s the r a d i u s of the p r o j e c t e d c o n t a c t a r e : of the ball and indentation when P^ i s applied. The corresponding depth due to the major load i s
/ig = r - (r^ - a^^)^\
11. R.L.Peek and W. E. Ingerson, Proc. Am. Soc. Test. Mat. 39, 1270, W39. 21
IN
2 * ,Fig. 2. The contact areas in the Rockwel1 t e s t ,
Whether the major load i s r e t a i n e d or not i s i r r e l e v a n t , since i t i s assumed t h a t a l l recovery e f f e c t s are n e g l i g i b l e . Hence the incremental depth i s
h . h^ - hk ' (r^ - a^^)^ - Cr2 - a ^ j ! ^ . (16, a)
This equation may be written in a more useful form by using (1). Therefore,
h-r {('i-Pfc2/"/r'^/" b^/")^ - (1 ^ pj/" ^ r^/'' 6^/"/'^} (17, a)
When (a/r) i s smal] conpared with u n i t y , (16, a) reduces to h . fa^2 - a^^) / 2r (16, b) and (17, a) to
h - (P^^ - P ^ 2 / " ; / (2r(^/")-i 6 ^ / " ; . (17, b) Therefore, for i n d e n t a t i o n s vrfiich are not too deep,
Hff « 130 - 250 r P / / " - Pfc^/"; / (rC'/'')'^ ö 2 / " ^ (IS) I f the material constants 6 and n are known for a p a r t i c u l a r metal, i t s Rockwell hardness number corresponding to any p a r t i
-cular combination of r, P , and P, may be c a l c u l a t e d . (c). Comparison with experiment.
In the previous section i t was shown that Rockwell hardness numbers corresponding to p a r t i c u l a r values of Pj^, P , and r , can to a good degree of approximation be c a l c u l a t e d from the m a t e r i a l c o n s t a n t s b and n. The index n occurring in M e y e r ' s
relation is essentially a measure of the work-hardening capacity of a metal. The two quantities 6 and n ( § 4) together determine the stress-strain cliaracteristics of a metal in a satisfactory manner.
(i) Calculation of Rockwell hardness numbers. The material constants n and b were determined for a number of metals. The experimental values for HB together with the calculated Rock-well hardness numbers (calculated from (18)) are given in Table I I I . I t i s seen that the discrepancy between theory and experi-ment i s less than about five per cent, which can be explained as due to experimental errors. The results for copper and nickel are less satisfactory.
(ii) Comparison with the work of Peek and Ingerson. I t follows from (17, b) that
h _ ,P-2/" . p^2/n
w^ich can be rewritten as
P„ - Py 2/"
without introducing a serious error on account of the limited range of the variables. Putting 2/n = 1/m and 9/4 cr^ = l/l/^f^b, the same relation as (15) i s obtained. The dimensionless quan-tity m, in agreement with the work of Peck and Ingerson will be very nearly equal to unity for fully worked metals {n ~ 2).
An estimate df q can also be made. I t can easily be shown that b i s related to the t e n s i l e strength in the following manner. In the ordinary tensile t e s t , stress (cr ) i s taJcen a* force divided by the original cross section. Since the volume of the material remains constant in p l a s t i c deformation the relation between the true stress (cr) and a' i s given by
cr- (i A, e) cr'. (20) Necking sets in when da' /de = 0 or as follows from (20) and
^'^ "^^ e' = x/il - X). (21)
where e' is the uniform elongation in tensile testing. The ten-sile strength therefore, is given by
<^6 = g fe'JVfi + e') = g fi - X) i-rr-S' (22)
Table I I I . The c o n s t a n t s n and b t o g e t h e r with t h e measured and c a l c u l a t e d Rockwell hardness numbers
Metal Armco I r o n Copper ( drawn) Dural Carbon S t e e l 0.6% C N i c k e l ( a n n e a l e d ) 5 a t . % M o - N i a l l o y ( annealed) 5 a t . % Sn - Ni a l l o y ( anneal ed) 5 a t . % S b - Ni a l l o y ( anneal ed) n 2. 16 ± 0 . 0 3 2 . 1 1 ± 0 . 0 1 2.26 ± 0 , 0 2 2.40 ± 0 . 0 3 2 . 5 1 ± 0 . 0 6 2.34 ± 0 . 0 4 2.35 ± 0 . 0 4 2.38 ± 0 . 0 4 6 '369 ± 1 2 28 1 ± 12 290 ± 3 1000 ± 30 317 ± 10 424 ± 10 720 ± 10 685 ± 10. Rockwell h a r d n e s s l e t t e r d e s i g n a t i o n "Rh «Re »Rk % f "fib «Rh «Rh «Re «Rk «Rf «Rb «Rf «Rh "RU «Rf > »Rk «Rf »Rk "Rf > "Rk Rockwell number measured 107 ± 1 89 ± 1 68 ± 1 80 ± 1 33 ± 1 105 ± 1 99 ± 1 77 ± 1 48 ± 1 108 ± 1 91 ± 1 7 4 ± 1 62 ± 2 87 ± 2 20 ± 3 90 ± 2 2 3 ± 2 69 ± 2 103 ± 1 59 ± 1 87 ± 2 100 ± 1 5 3 ± 1 8 4 ± 1 Rockwell number c a l c u l a t e d 10 5 86 65 71 26 98 99 76 48 109 94 76 70 92 37 87 22 68 101 60 90 101 57 85 24
From (11), i t follows that
Therefore
, 2.8 TT p"'^ .3 - n " " 2
,(n/2)-l n-2 , " ' 2 a = 2.8 TT IL-LJ. Lf (IJUL) a,.
If CT^ i s taken to be equal to the tensile strength, as was done by Peck and Ingerson, i t i s seen that
0.45 (3.n) .n_.2j"-^
so that q i s a function of the work-hardening properties of the metal. Furthermore, i t i s seen that for n = 2.5 and 2.0, q takes
the values 0.09 and 0.11 respectively. These values are indeed very near to those obtained by Peek and Ingerson. This agree-moit i s as satisfactory as can be expected,
6. Relation between the d i f f e r e n t hardness numbers
(a) Relation between Rockwell and Meyer hardness From (1) i t follows that
0 n ^^p(2/n).l ^2-{k/n) ^^^
Substituting in (18) and putting P - P and a - a , gives
H^ = , , 0 - ^ ^ [i - r ^ ) ^/"] (24)
TT r Hff e
o r
79.6 P, fi ,„^, ^fl = ^^0 - —nr^ v = 130 -J- (25) where jj. = 79,6 P Tl/r. This ejqsression only holds when (a/r) i s
small compared with unity, i . e , when the indentation;; are not too deep. This -condition i s almost always satisfied and (25) i s a very good ^proximation even when a/r = 0.3.
I t i s then apparant that, in order to convert Meyer hardness numbers to Rockwell hardness numbers, the value of ri. must be known, and that the inpressions made in the determination of H^, must be geometrically similar to those made by P , I t i s incor-rect in principle, to state that a Meyer hardness ƒƒ„ corresponds to a Rockwell hardness ƒƒ„ without specifying the Meyer index,
even i f the indentations were geometrically s i m i l a r .
For a p a r t i c u l a r v a l u e of Pk/Pe, however, the value of rj v a r i e s by only about s i x per cent when n v a r i e s from 2 to 2 , 5 , Approximate conversion formulae can t h e r e f o r e be given. The d i f f e r e n t t h e o r e t i c a l v a l u e s for ju. for d i f f e r e n t v a l u e s of Pf^/P-, r, and n a r e given i n Table IV.
Table IV. Values for T) and /x for d i f f e r e n t values of Pjfe/Pg, r, and n . Pj^ = 10 kg
r = 0.794 mm Rockwell l e t t e r desig-nation
V
^Rb "R. Pk/Pg 1/6 1/10 1/15 n 2.0 2,5 2,0 2,5 2,0 2.5 V 0.83 0,76 0.90 0.84 0.94 0.89 M 5000 4600 9060 8400 14000 13008 r = 1.588 mm \ Rockwell hardness desig-nation ^Rh "Re "Rk ^ k / ^V6
1/10 1/15 n 2.0 2,5 2.0 2.5 2,0 2,5 V 0,83 0,76 0,90 0,84 0,94 0.89 i*2500 1
2300 4500 4200 7100 6700 Equation (25) can be w r i t t e n as Hf,^fji/(130 ' Hn) (26) where /U. has the values given in Table IV for a number ofcombi-n a t i o combi-n s of Pk/Pg '"^^ '"• ^ ® experimecombi-ntal cocombi-nversiocombi-n formulae to B r i n e l l hardness numbers are
Hg - 7300/(130 - Hflfc) 40 < ff^jj, < 100 and
Hg m 3710/(130 - HjfJ 30 < //^^ < 100
Since Meyer hardness numbers are somewhat l a r g e r than the cor-responding ( a s far as a/r i s concerned) B r i n e l l hardness num-b e r s , i t was to num-be expected t h a t the experimental conversion c o n s t a n t s should be smaller than the t h e o r e t i c a l values by a few per c e n t .
(b) Relation betieen Vickers and Meyer hardness numbers The r e l a t i o n between Meyer and Vickers hardness numbers i s given by (5) and ( 6 ) . Roth t h e s e e q u a t i o n s are in agreement with the experimental r e s u l t s . I t i s seen t h a t Meyer and Vickers hardness numbers are equal* only when pa/r - 0.08 or a/r = 0,4, Vickers hardness numbers must be divided by a constant which i s approximately equal to 3,0/2.8 in order to obtain values which can be compared with the Meyer hardness numbers (Table V).
As i s e a s i l y seen, Meyer h a r d n e s s numbers will be l a r g e r than Vickers hardness numbers when a/r > 0,4 and smaller when a/r < 0 . 4 . T h i s d i f f e r e n c e w i l l be s m a l l e r the g r e a t e r the amount of cold work.
Table V. Vickers and Meyer hardness numbers for a number of metals. Second and fourth columns must be compared
Metal Armco Iron Copper Düral 0.6% Carbon Steel Nickel Meyer hardness for a/r^ O.i**
103 ± 4 82 ± 4 74 ± 3 198 ± 6 64 ± 2 Vickers hard-ness number 119 ± 5 94 ± 2 76 ± 2 213 ± 7 67 ± 2 Hy/1.1 108 ±5 85 ±2 69 ±2 192 ±7 60 ±2
7. Relation between hardness and s t r u c t u r e s e n s i t i v e p r o p e r t i e s of n e t a l s
(a) The relation between hardness and elastic limit From (7) £md (9) one obtains
H^=2.8( e „ + pa/r' ) ^ (27)
where a i s the e l a s t i c l i m i t of the metal previously deformed by an amount Cg. From t h i s i t i s seen t h a t there i s no unique relaticm between the e l a s t i c l i m i t of a metal and i t s hardness, but t h a t t h i s r e l a t i o n i s determined by the s i z e of the im-p r e s s i o n and the t e n s i l e index of the metal,
* Neglecting the small difference between 2.8 and 3.0,
•* Thiswas obtained by p l o t t i n g Meyer haidness numbers a.s » function of a/r,
This equation can be compared with K i i r t h ' s ^ ^ experimental r e s u l t s for annealed copper. Taking 0.5 for x, one can e a s i l y c a d c u l a t e cr/Hu for v a r i o u s amounts of cold work (Table V I ) ,
Table VI, The r a t i o of the e l a s t i c l i m i t and Meyer hardness for annealed copper, a/rm 0.2 .
% cr kg/nm^ fljf kg/mn^ a/Hff (exp,) a/Hu ( c a l . ) 0 1.25 27.0 0.046 0 0,055 10.8 48,6 0,22 0.28 0.11 16.5 61.4 0.27 0.31 0.20 21.5 7 3 . 1 0.29 0.32 0.40 28.1 87 0.30 0.33
R e l a t i o n s between V i c k e r s and Rockwell h a r d n e s s and t h e e l a s t i c l i m i t of metals can e a s i l y be obtained by means of the previous formulae,
(b) Relation between Meyer hardness and tensile strength. I t i s p o s s i b l e to deduce the t r u e s t r e s s - s t r a i n curve, the nominal s t r e s s - s t r a i n curve and the t e n s i l e strength of a metal
from i t s Meyer hardness numbers. This p o i n t was d i s c u s s e d in d e t a i l by Tabor'3, From (10) and (22), i t follows t h a t
"M - 2,« (j)' ^ (^^r^)\ (28)
where cr, i s the t e n s i l e strength of the metal under c o n s i d e r a t i o nAs i s seen, according to t h i s a n a l y s i s , the r a t i o fi^/cr^ i s independent of the n a t u r e of the m e t a l . I t depends p r i m a r i l y on the degree of work-hardening of the metal, i , e . on the index x. Tabor has shown t h a t for d i f f e r e n t metals t h i s equation i s a very good approximation.
A s i m i l a r a n a l y s i s can a l s o be c a r r i e d out for Vickers sind Rockwel1 hardness numbers.
8 . C o n c l u s i o n
I t has been shown in the previous s e c t i o n s t h a t the Rockwell, Meyer, B r i n e l l and Vickers hardness numbers can be e3q)ressed in
terms of the s t r e s s - s t r a i n c h a r a c t e r i s t i c s . The most important conclusion t h a t can be drawn from the above discussions i s t h a t the actual hardness numbers are only of secondary importance. Far more important are the s t r e s s - s t r a i n c h a r a c t e r i s t i c s of metals which are s a t i s f a c t o r i l y determined when n and 6 are known. I f these constants are known, as well as the representa-t i v e p l a s representa-t i c deformarepresenta-tion of an i n d e n representa-t a representa-t i o n , representa-the four above-mentioned hardness numbers can be calculated uniquely.
12. A.Kürth, Zei t . Ver. Deu t . I n g . 52, 1560, 1908. 13. D.Tabor, J. I n s t . M e t . 79, 1, 1951.
C h a p t e r I I
THE HARDINESS OF PURE METALS
9 . The d e f o r m a t i o n of m e t a l s
In the previous chapter i t was shown t h a t the various hardn e s s hardnumbers cahardn be based very s a t i s f a c t o r i l y ohardn the s t r e s s -s t r a i n c h a r a c t e r i -s t i c -s of m e t a l -s . A -study of hardne-s-s i -s thu-s e s s e n t i a l l y connected with the mechanism of p i a s t i c deformation. The p l a s t i c p r o p e r t i e s of p o l y c r y s t a l s are determined by (a)
the p l a s t i c p r o p e r t i e s of the c o n s t i t u e n t g r a i n s , (b) the i n t e r -action between the g r a i n s , g e n e r a l l y r e f e r r e d to as the grain boundary i n f l u e n c e . These two f a c t o r s w i l l now be d i s c u s s e d s e p a r a t e l y .
(a) The plastic properties of single crystals
Of the various modes of deformation, s l i p i s the most impor-t a n impor-t one. I impor-t i s now generally assumed impor-t h a impor-t p l a s impor-t i c deformaimpor-tion (pr s l i p ) takes place through the motion of d i s l o c a t i o n s . The fol-lowing p r o p e r t i e s of d i s l o c a t i o n s * may be enumerated'*^'^.
(i) Any real c r y s t a l contains a number of d i s l o c a t i o n s p r e s e n t i n the l a t t i c e . The motion of t h e s e and those p r e s e n t a t the boundaries of the crystal under the action of applied s t r e s s e s , c o n s t i t u t e s p l a s t i c deformation. Once generated**, a dislocation remote from other s t r e s s f i e l d s , will r e q u i r e a c e r t a i n s t r e s s before i t moves. This s t r e s s was f i r s t c a l c u l a t e d by P e i e r l s ^ , l a t e r extended by N a b a r r o ' and recently by Foreman, Jaswon and Wood*. These workers have shown t h a t the weaker the s h e a r i n g s t r e s s between adjacent atoms in the s l i p plane, the wider the d i s l o c a t i o n . The external shear s t r e s s required to move a d i s -location i s extremely s a i s i t i v e to the width of the d i s l o c a t i o n ,
1. A. H , C o t t r e l l , P r o g r e s s in Metal P h y s i c s I (London, 1 9 4 9 ) , p . 7 7 .
2. N. F.Mott, I n t e r n a t i o n a l Conference on the P h y s i c s of Metals (Amsterdam, 12-17 J u l y , 1948).
3. N.F.Mott, J . I n s t . M e t . 72, 367, 1946. 4 . F . P e i e r l s , P r o c . P h y s . S o c . 52, 256, 1947. 5. F.R.N.Nabarro, Proc.Phys. Soc. 59, 256, 1947.
6. A. J.Foreman, M.A.Jaswon, and J.K.Wood, P r o c . P h y s . .Soc. 6 4 ( 2 ) , 156, 1951.
* Only edge d i s l o c a t i o n s w i l l be c o n s i d e r e d h e r e and i n t h e f o l l o w i n g . This c h o i c e does n o t i n f l u e n c e t h e c o n c l u s i o n s . ** The p r o b l e m o i t h e c r e a t i o n of d i s l o c a t i o n s i s . s t i l l very
i n c o m p l e t e . See a l s o N.F.MotL, Proc.Pliy b. Soc. 64< B), 7 2^ , 11 51.
becoming vanishing]y small a t widths above about t h r e e atomic s p a c i n g s . I f the y i e l d s t r e s s of p u r e , annealed metals i s de-termined by the s t r e s s r e q u i r e d t o move a s i n g l e d i s l o c a t i o n which i s s t r a i ^ t and p a r a l l e l to a c r y s t a l lographic a x i s , then the l a t t e r a u t h o r s ' work suggests t h a t the widths of d i s l o c a -ticxis are g r e a t e r than 1,5 a, «iiere a i s the atomic spacing. (ii) In any pure and annealed c r y s t a l , t h e r e are a number of i n t e r n a l s t r e s s regions (vacancies, impurity atoms, o t h e r d i s -l o c a t i o n s , e t c ) h i n d e r i n g the movement of d i s -l o c a t i o n s . Due t o i t s s t r e s s f i e l d , a d i s l o c a t i o n w i l l i n t e r a c t with t h e s e sources of s t r e s s .
(Hi)A d i s l o c a t i o n i s considered as f l e x i b l e . The shape adopted by a f l e x i b l e d i s l o c a t i o n in a region of i n t e r n a l s t r e s s i s '
p m G a/a^ . (1) where p i s t h e l o c a l value of the r a d i u s of c u r v a t u r e of t h e
d i s l o c a t i o n l i n e in the region nrfiere the i n t e r n a l s t r e s s i s o-^. G i s the shear modulus and o the l a t t i c e constant.
I f the average d i s t a n c e D between t h e s e s t r e s s r e g i o n s i s l a r g e compared with p, as one would expect t o be the case i n annealed pure metals or p r e c i p i t a t i o n hardened a l l o y s , the d i s -l o c a t i o n can fo-l-low the contours of the s t r e s s f i e -l d . In t h i s case the d i s l o c a t i o n i s looped i n t o s e c t i o n s of wavelength and a n p l i t u d e of the o r d e r of D. Each loop w i l l a c t more or l e s s independently of the o t h e r s . To produce p l a s t i c deformation, each loop w i l l have to be taken over a p o t e n t i a l h i l l without the help of o t h e r s e c t i o n s of the d i s l o c a t i o n . To acconplish t h i s , the e x t e r n a l s t r e s s m i l have to exceed crj.
{iv'). On the assumption t h a t p l a s t i c flow occurs «iien the ex-t e r n a l s ex-t r e s s a i s almosex-t equal ex-t o a^, ex-t h e a c ex-t i v a ex-t i o n eaergy for moving a d i s l o c a t i o n loop from one v a l l e y to the n e x t i s '
U (a) . 0.15 a. a D^ (1 - a/a.)^/^. (2) With reasonable v a l u e s for the c o n s t a n t s , t h i s i s very l a r g e
compared with kT a t room temperature u n l e s s cr^ d i f f e r s from a by only a few per c e n t .
(v) The frequency with which a d i s l o c a t i o n v i b r a t e s about a mean p o s i t i o n i s given b y '
f.('J!^)^(l.f//2n (3)
a a U * 7 N . F . M o t t , and F . R . N . N a b a r r o , The S t r e n g t h o f S o l i d s ( L o n d o n , P h y s i c a l S o c i e t y , 1 9 4 8 ) , P ' l « 30wliere d i s the density of the m a t e r i a l . I f cr d i f f e r s from cr^ by a few per cent, ƒ i s of the order of 10* - 10' s e c . ' ',
(vi) I t seems h i g h l y p r o b a b l e t h a t one d i s l o c a t i o n , once i t s t a r t s moving, g e n e r a t e s o t h e r s . Frank* suggested a. p o s s i b l e mechanism for t h i s process,
These p r o p e r t i e s , t o g e t h e r with o t h e r s , form an e s s e n t i a l p a r t of most modern t h e o r i e s of the deformation of c r y s t a l s , The f a c t t h a t the r e s i s t a n c e to s l i p i n c r e a s e s with p l a s t i c deformation, e i t h e r mesuis t h a t i t becomes more d i f f i c u l t to produce d i s l o c a t i o n s in the c r y s t a l or t h a t i t becomes more d i f f i c u l t to move them, Mott and N a b a r r o ' pointed out t h a t the i n t e r n a l s t r e s s e s ' which are known to be p r e s e n t in s t r a i n -hardened c r y s t a l s , ought to help the formation of d i s l o c a t i o n s
but hinder t h e i r motion. I t i s t h e r e f o r e p l a u s i b l e to assume t h a t s t r a i n h a r d e n i n g i s due to the d i f f i c u l t y of moving d i s -l o c a t i o n s .
This discussion leads one to i n t e r p r e t the p l a s t i c p r o p e r t i e s of metals in terms of the motion of d i s l o c a t i o n s , having the above-mentioned p r o p e r t i e s , in a p o t e n t i a l f i e l d V(x), where x i s the p o s i t i o n of a p a r t i c u l a r p a r t of the d i s l o c a t i o n . The
function V(x), does not only change during cold working but i s
probably also changed by the applied s t r e s s and temperature, T a y l o r ' " suggested t h a t , except for the i n i t i a l ones, the ^ d i s l o c a t i o n s do not p a s s completely through the l a t t i c e but become stuck within the c r y s t a l ; t h e i r number therefore increa-s e increa-s with i n c r e a increa-s i n g increa-s t r a i n . Due to t h e i r increa-s t r e increa-s increa-s f i e l d increa-s t h e increa-s e d i s l o c a t i o n s w i l l oppose the motion of o t h e r s . In T a y l o r ' s theory, V(x) i s considered as a r i s i n g from the s t r a i n s due to a l l other d i s l o c a t i o n s p r e s e n t in the c r y s t a l . T a y l o r ' s simple theory leads to a p a r a b o l i c relation,*
cr m g ^ . ( 4 )
between stress and strain, where g is some constant of the ma-terial wiiich is directly proportional to the shear modulus.
This parabolic relation between stress and strain is closely 8. F . C . F r a n k , The S t r e n g t h of S o l i d s (London, P h y s i c a ] S o c i e t y ,
1948), p . 46.
9 . For a d i s c u s s i o n of t h e i n t e r n a l s t r a i n s i n w o r k - h a r dened m e t a l s s e e C . S . B a r r e t t , The S t r u c t u r e of M e t a l s (Mc Graw H i l l , 1 9 4 3 ) . Also B.E.Warren and B.L. A v e r b a c h , P i t t s b u r g h Symposium on P l a s t i c Deformation of C r y s t a l l i n e Solids (Naval Research L a b o r a t o r y , 1950), p. 113.
10. G . I . T a y l o r , P r o c . Roy. Soc. 145, 388, 1945.
• N . F . M o t t ( R e s e a r c h 2. 162, IQtP) lias shown that, t h i s p a r a -b o l i c law i s a c o n s e q u e n c e of mucli more g e n e r a l a s s u m p t i o n s than t h o s e used by T a y l o r .
obeyed by cubic metals.* The fact t h a t the s t r e s s - s t r a i n curves for hexagonal m e t a l s are l i n e a r , as was p o i n t e d out by S e i t z and Read'*, does not c o n t r a d i c t the b a s i c assumptions of Tay-l o r ' s theory, for a very simpTay-le modification of T a y Tay-l o r ' s argu-ments leads to a l i n e a r law,
T a y l o r ' s theory has been c r i t i c i z e d by Orowan'^, and Andrade and R o s c o e ' ^ . For s i n g l e c r y s t a l s of cadmium and low s t r a i n r a t e s , Andrade and Roscoe observed a l i n e a r law of work-harde-ning and found t h a t the number of s l i p bands was independent of t h e s t r a i n . From these r e s u l t s i t might be concluded t h a t the law of work-hardening for each individual s l i p band i s l i n e a r . In aluminium single c r y s t a l s the number of s l i p bands increases l i n e a r l y with the shear s t r e s s . They suggested t h a t t h i s would also explain the p a r a b o l i c law of work-hardening, and t h a t all metals obey a l i n e a r work-hardening law*'',
(b) The interact ion between the grains
Tbkt t h i s factor e x i s t s i s shown by a l a r g e number of expe-riments and p a r t i c u l a r l y by Miller**, and Boas and Hargreaves*^,
I t i s well known t h a t the hardness of p o l y c r y s t a l l i n e metals** increases with decreasing g r a i n - s i z e * ' , P o l y c r y s t a l l i n e hexago-nal metals are appreciably harder than s i n g l e crystals*** since there i s only one s l i p - p l a n e , which, together with the formation of twins, r e s u l t s in considerable shear hardening. The hardness of p o l y c r y s t a l l i n e cubic metals i s about the same as t h a t of single c r y s t a l s ( l a r g e number of p o t e n t i a l s l i p ' p l a n e s ) . Accor-ding to the experimental r e s u l t s the hardness i n c r e a s e s with
1 1 . F , S e i t i and T, A, Read, J . A p p . P h y s . 12, 100, 170, 470, 538, 1 9 4 1 .
12. E.Orowan, Symposium on I n t e r n a l S t r e s s e s (London, I n s t i t u t e of M e t a l s , 1947), p . 47.
13. E.N.de C.Andrade and R.Roscoe, P r o c . P h y s . S o c . 49, 152,1937. 14. See a l s o D.Kuhlraann, P r o c . P h y s . S o c . «4 ( 2 ) , 140, 1951. 15. R . F . M i l l e r , T r a n s . A.I.M.M, E, 111. 135, 1934.
16. W.Boas and M. E. H a r g r e a v e s , P r o c . R o y , Soc. 193, 8 9 , 1948. 17. R.W.Wood, P h i l . M a g . 10, 1073. 1930; J . H . F r y e and
W.HumeR o t h e r y , P r o c . W.HumeRoy. Soc. 181, 1, 1942; C.F.Elam, The D i s t o r -t i o n of Me-tal C r y s -t a l s ( O x f o r d , 1935); H . U n c k e l , Zel-t.Tde- Zelt.Tde-t a l l k . 29, 414, 1937.
• I t s h o u l d be o b s e r v e d a t t h i s p o i n t t h a t K o c h e n d ó ' r f e r and Rohm (See U . D e h l i n g e r , P i t t s b u r g h Symposium on P l a s t i c Defor-mation of C r y s t a l l i n e S o l i d s (Naval Research L a b o r a t o r y , 1 9 5 0 ) , p . l 0 3 ) h a v e found e v i d e n c e to show t h a t the s t r e s s - s t r a i n c u r v e
of s i n g l e c r y s t a l s of aluminium i s l i n e a r when t h e specimen i s deformed i n p u r e s h e a r , i n c o n t r a s t to t h e p a r a b o l i c r e l a -t i o n o b -t a i n e d i n -t h e c l a s s i c a l e x p e r i m e n -t s of T a y l o r . See a l s o F. Kdhm and A.Kochend'eJrf e r . Zei t . f . M e t a l l k . 4 1 , 265, 1950. ** I t i s assumed h e r e and i n t h e f o l l o w i n g t h a t t h e g r a i n s a r e
small compared with t h e d i m e n s i o n s of t h e s p e c i m e n . 32
decreasing grain s i z e , slowly at f i r s t and with increasing r a t e the smaller the grain s i z e becomes,
The usual e:)q)l anations ** • *'• ^° of these e f f e c t s (inhomogeneous deformation of grains r e s u l t i n g in l a t t i c e bending, i n h i -b i t i o n of s l i p a t the grain -boundaries r e s u l t i n g from the dif-ference in o r i e n t a t i o n of the g r a i n s ) are n o t e n t i r e l y free from o b j e c t i o n s . S e i t z * ' pointed out t h a t the d i f f e r e n t t r e a t -ments given to the specimens in order to obtain d i f f e r e n t grain
s i z e s might not e n t i r e l y be removed by annealing. As far as the d i s l o c a t i o n theory i s concerned one would not expect ^appreciable hardening unless the grain s i z e i s of the order of magnitude of the length of a d i s l o c a t i o n ( i . e , 10'* - 10**cm),
I t follows from the work by (Jensamer^* and h i s c o l l a b o r a t o r s who studied the r e l a t i o n between the fineness of the microstruc-t u r e s in s microstruc-t e e l and microstruc-the microstruc-t e n s i l e p r o p e r microstruc-t i e s , microstruc-t h a microstruc-t microstruc-the r e s i s microstruc-t a n c e to p l a s t i c deformation i s independent of the shape of the grains* ( f o r grain s i z e s down to about 10'* cm) but depends on t h e i r s i z e , determined by the meeui free path along which s l i p can o c c u r ,
Roughly speaking, the e f f e c t of p l a s t i c deformation i s to produce s t r e s s f i e l d s in the l a t t i c e of which the exact n a t u r e i s s t i l l unknown'. I t i s p o s s i b l e t h a t these s t r e s s f i e l d s are mainly caused by stuck d i s l o c a t i o n s as follows from considera-tions of the energy' stored during cold working of metals'^'^* ^^,
More or l e s s the same arguments can be applied to the effect of p l a s t i c deformation on the p r o p e r t i e s of p o l y c r y s t a l s with due consideration of the p o s s i b l e e f f e c t s of grain-boundaries, In p a r t i c u l a r , the fact t h a t single c r y s t a l s of hexagonal metals work-harden according to a l i n e a r law, t o g e t h e r with Andrade Eind Roscoe's suggestion and the fact t h a t the grains in a crystal deform inhomogeneously, leads one to suspect that poly-c r y s t a l l i n e hexagonal metals will not obey the l i n e a r law of worit-hardening,
18. W.Boas, The Physics of Metals and Alloys (Melbourne Univer-s i t y PreUniver-sUniver-s 1947)
19. F.Seitz, rfe Physics of Metals (Mc Graw H i l l , 1943). 20. H.Carpenter and J.M.Robertson, Metals (Oxford, 1939). 21. M.Gensamer, E.P.Pearsall, W. S.Pel l i n i , J.R.Low, Trans.A.S.M.
30, 383. 1942; M.Gensamer, G.V.Smith, E. B.Pearsa]. 1, Trans. A. S.M. 28, 380. 1942.
22. G.I.Taylor andH,()uinney, Proc. Roy. Soc. 143, 307, 1934; 163, 157 1937.
23. J . s l K o e h l e r , Phys.Rev. 60, 397. 1941. • This i s not true for cast i r o n .
Experimentally, both for annealed cubic and hexagonal metals, the s t r e s s - s t r a i n curve can closely be approximated by*
^ ' g e' (5) w^ere x ~ 0 . 5 for annealed cubic metals and x ~ 0.3 for annealed
hexagonal m e t a l s , A lower index x means a lower r a t e of work-hardening.
According to ( 5 ) , the r a t i o of the Vickers hardness number of a cubic metal c o l d - r o l l e d t o 80%, to t h a t of the annealed metal i s given by ! « . ^
(§1)
' 8^ 3.3. (see §3) Hy (80%) HY ( 0%)whereas the corresponding r a t i o for hexagonal metals i s 2 . 0 , As i s seen from Table V I I , the experimental r e s u l t s for t h i s
r a t i o i s roughly in accordance with the above r e s u l t s ,
Table VII. The hardness of a number of annealed and cold-worked pure metals Metal Ni Ag Al Ga Au Zn 1 Cxi Hv (0) 67 r38 ^31 ^14 ^21 40 ^37 ^25 ^20 43 23 HY (80) 197 r84 ^96 .50 ^40 ƒ120
Hl7
ƒ60 ^67 50 29 Hv im/HY (0) 2.9 <2.2 ^ 3 . 1 ƒ3.5 ^2.0 ƒ3.0 ^ 3 , 1 ^2,4 ^3,3 1.2 1,3 L i t e r a t u r e |"1
<^ ( . -,24 *^•^ 1
1**) Own measurements. 1 P a r t i c u l a r l y i n t e r e s t i n g i s the fact t h a t t h i s sirrpleanaly-s i analy-s analy-showanaly-s t h a t t h i analy-s r a t i o i analy-s independent of the nature of the metals provided they have the same value for the index x (same
24. Metals Handbook (A.S.M. , 1948), p . 1657.
25. E.Raub, Die E d e l m e t a l l e und i h r e Legierungen ( B e r l i n , 1 9 4 2 ) . • Other more c o m p l i c a t e d s t r e s s - s t r a i n r e l a t i o n s have been s u g g e s t e d . However, ( 5 ) w i l l be c o n s i d e r e d because of i t s s i m p l i c i t y and the c l o s e agreement with experiment in the range of i n t e r e s t h e r e .
crystal structure). Further also, according to this discussion, this ratio is independent of the temperature in the range where recovery does not take place because in this range,;»: is practic-ally independent of the temperature.
10. Relation between hardness and structure i n s e n s i t i v e properties of metals
In order to discuss the plastic properties of polycrystalline metals of different crystal structures, one has to take into account the effect of the number of potential slip-planes cha-r a c t e cha-r i s t i c of a pacha-rticulacha-r ccha-rystal stcha-ructucha-re as well as othecha-r phenomena such as twinning on the strain-hardening properties. As already mentioned the stress-strain curve of a polycrystalline cubic metal will not differ appreciably from the s t r e s s -s t r a i n curve-s of -single cry-stal-s^*, wherea-s polycry-stalline hexagonal metals will be considerably harder than single crys-tals ( § 9)„
For cubic metals, therefore, taking into account the fact that within a crystal in a polycrystalline specimen, the defor-mation is inhomogeneous, and the large nvmiber of potential slip-planefs, one might expect that the p l a s t i c behaviour of poly-crystalline cubic metals will be determined mainly by the inter-action forces between dislocations and between dislocations and other stress-fields. Since this interaction is directly propor-tional to the shear modulus of the crystal, one may write (5) in the form
a- m G A e' and therefore the hardness
H-^ 3'GA (e^ff)' (6)
where e ,, is the effective deformation corresponding to the particular hardness measurement. The constant A will presumably depend on the crystal structure of the crystals composing the
aggregate.
The hardness^' at 0° K of a number of pure annealed metals
is plotted against the corresponding shear moduli^* also at 0° K (fig. 3 ) , This reduction to 0° K was done to minimize the
26. G. I.Taylor, J. Inst.Met. 62, 307, 1938.
27. These r e s u l t s were obtained by linear extrapolation of results given by M.J.Druyvesteyn, App.Sci.Res. A 1, 66,1947 28. Derived from results given by W.Köster, Zeit.Metal Ik. 39,
1, 1948.
800-Bnwn Hirdnts» 700-«» 0"K 11 600- 500- 400- 300- 200-100' O 5000 10000 1SD00 20000 25000 » - SkMT medulu* «t 0*K F i g . 3 . B r i n e l l h a r d n e s s a t 0 ° K p l o t -t e d as a f u n c -t i o n o f -t h e s h e a r modulus at 0° K. p o s s i b l e e f f e c t s of temperature on the m o b i l i t y of d i s l o c a -t i o n s . As i s seen for a l l me-tals having -the same s l i p elemen-ts, t h e r e i s a lineaif r e l a t i o n between hardness and shear modulus ( e ^ e c i a l l y in the case of f . c . c * m e t a l s ) . The deviations from t h i s l i n e a r r e l a t i o n s h i p in the case of the c, p . h. and b, c, c, metals ( e s p e c i a l l y Mo, a-Fe and W), can be r e l a t e d to a l a r g e r number of s l i p elements, I t must be pointed out t h a t the r e l a t
-ive p o s i t i o n s of these l i n e s and d e v i a t i o n s will depend on the temperature since the hardness v a r i e s considerably with temper-a t u r e .
From the eibove, i t follows t h a t ,
(i) The hardness of a p o l y c r y s t a l l i n e metal of a p a r t i c u l a r c r y s t a l s t r u c t u r e , i n c r e a s e s with i n c r e a s i n g - shear modulus, Since the shear modulus of metals i s again r o u ^ l y proportional to the e l a s t i c modulus, heat of sublimation, melting p o i n t £ind bulk modulus, the hardness wrill also i n c r e a s e with these quan-t i quan-t i e s .
The a b b r e v a t i o n s f . c . c , b . c . c . and c . p . h . w i l l be used for
f a c e - c e n t e r e d c u b i c , b o d y - c e n t e r e d c u b i c and c l o s e - p a c k e d hexagonal r e s p e c t i v e l y .
f i i ) F o r a fixed shear modulus at 0° K, b , c , c . metals are harder than f . c . c . metals aid c . p . h . metals are about as hard as b , c , c . m e t a l s . Whether t h i s i s due to the f a c t t h a t the metals are p o l y c r y s t a l l i n e cannot be s t a t e d , but i t seems to be u n l i k e l y . Ebcceptions a r i s e mainly because of additional s l i p systems.
11. The e f f e c t o f temperature
According to the d i s l o c a t i o n theory, s l i p in a s o l i d takes place through the motion of d i s l o c a t i o n s under the i n f l u e n c e of a small applied s t r e s s . This s t r e s s ( § 9 (a, i)) must exceed a c e r t a i n small v a l u e . I t i s a usual procedure in d i s c u s s i n g the deformation of pure m e t a l l i c c r y s t a l s to neglect t h i s s t r e s s (thereby also n e g l e c t i n g the atomic n a t u r e of the l a t t i c e ) and to consider the effect of internal s t r e s s regions in a l a t t i c e , considered as an e l a s t i c i s o t r o p i c continuum, on the mobility. of d i s l o c a t i o n s . If, for example, these regions are large com-p a r e d with the l o c a l r a d i u s of c u r v a t u r e of the d i s l o c a t i o n l i n e , the d i s l o c a t i o n will be looped i n t o sections of wavelength and amplitude of the order of the distance between these s t r e s s regions. To produce g l i d e , each loop will have to be taken over a p o t e n t i a l h i l l without the help of other sections of the d i s -l o c a t i o n , The chance of a d i s -l o c a t i o n -loop jumping through a region of unfavourable s t r e s s in u n i t time can be written as
p u f e xp (- U (ai)/kT) (7) where ƒ i s the frequency of v i b r a t i o n of d i s l o c a t i o n in a
po-t e n po-t i a l valley arid U (a^) po-the a c po-t i v a po-t i o n energy for jump. For the eibove-mentioned case, the a c t i v a t i o n energy which has to be supplied by thermal a c t i v a t i o n in order to move a d i s l o c a t i o n loop over an unstable region, i s very large compared with kT at room temperature unless the i n t e r n a l s t r e s s a^ d i f f e r s from the applied s t r e s s by a few p e r c e n t . Thus l i t t l e temperature
de-pendence of y i e l d i s to be expected on t h i s b a s i s . Actually, Y the y i e l d point and hardness of metals vary considerably with I temperature,
One may expect that the following e f f e c t s mi^it be oi impor-tance in considering the effect of temperature on the hardness of metals,
(i) the effect of temperature on the p r o b a b i l i t y that a d i s -location will be formed,
(ii) new glide p l a n e s , (Hi) recovery,
(iv) the decrease of the shear modulus with temperature,
37
X
(v) e f f e c t o f t e m p e r a t u r e and atomic v i b r a t i o n s on t h e mobi-l i t y o f d i s mobi-l o c a t i o n s , and
(vi) d e t e d l s o f t h e atomic arrangement i n t h e g l i d e p l a n e s and d i r e c t i o n ,
The higji e n e r g y of f o r m a t i o n o f a d i s l o c a t i o n , l e a d s one t o b e l e a v e t h a t d i s l o c a t i o n s can h a r d l y be formed by thermal f l u c t u a t i o n s i n even a h i g h l y s t r e s s e d b u t o t h e r w i s e p e r f e c t c r y s -t a l , The e f f e c -t of s -t r e s s c o n c e n -t r a -t i o n s a -t c e r -t a i n r e g i o n s o f a c r y s t a l d o e s n o t seem t o make ( ' ) an i m p o r t a n t f a c t o r . By r e s t r i c t i n g t h e d i s c u s s i o n t o such t e m p e r a t u r e s where no a d d i -t i o n a l g l i d e p l a n e s and no r e c o v e r y e f f e c -t s a r e e f f e c -t i v e , (H) and (Hi) can be l e f t o u t o f c o n s i d e r a t i o n . The d e c r e a s e o f t h e s h e a r modulus with t e n p e r a t u r e a l o n e c a n n o t e x p l a i n t h e o b s e r v e d d e c r e a s e of y i e l d s t r e n g t h ( s e e f o r e x a n p l e D r u y v e s t e y n ' ^ ' ) , O i e i s t h e r e f o r e l e f t w i t h (v) and (vi). I t t h e r e f o r e seems p r o b a -b l e t h a t an a t t e n p t t o e x p l a i n t h e e f f e c t o f - t e n p e r a t u r e on t h e y i e l d s t r e n g t h o r h a r d n e s s o f m e t a l s must take t h e f a c t o r s (v)
and (vi) o r a s s o c i a t e d e f f e c t s i n t o a c c o u n t ( s e e a l s o L i e b -f r i e d " a n d N a b a r r o ^ O ) ,
29. G . L i e b f r i e d , Z e i t . f . P h y s . 127, 344, 1950. 30. F.R.N.Nabarro, P r o c . Roy. Soc. 209, 278, 1951.
