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Structure analysis of dispersion strengthening
Scripta Memllurgica
et
Pergamon
STRUCTURE
ANALYSIS
OF D I S P E R S I O N
Materialia, Vol. 30, No. 9, pp. 1145-1150, 1994 Copyright © 1994 Elsevier Science Lid Printed in the USA. All rights reserved 0956-716X/94 $6.00 + 00
STRENGTHENING
M. Besterci Institute of Materials Research of Slovak Academy of Sciences, Watsonova 47, 043 53 Ko~ice, Slovak Republic (Received November 2, 1993) (Revised January 7, 1994) Introduction In a recent paper [i] Hazzledin discussed the effects of direct and indirect strengthening in dispersion strengthened systems containing non-coherent particles of a second phase at low temperatures and low stress. Theoretical analysis confirmed that indirect strengthening (interaction of dislocations with grain boundaries according to Hall-Petch) always dominates over direct strengthening (interaction of dislocations with dispersion particles according to Orowan). The objective of this study is to point out that this conception has not been confirmed for real dispersion strengthened materials and, moreover, is conditional upon structure, i.e. dependent on distribution of particles throughout the matrix volume, as well as on their size distribution. Both these factors depend considerably on the method of preparation of systems via powder metallurgy, as well as on final compaction methods. The study will analyse the yield point RD0.2 of the system Cu-AI203 on the basis of an additive character of strengthening contributions for materials prepared by various methods. Experimental Material and M e t h o d s Experiments were carried out using 4 types of materials Cu-Al203, prepared by methods of mechanical homogenization (A, C and D) and reaction milling (B). Final compaction of mixtures was achieved by various ways modifying distribution of particles throughout the volume of the final product. The material A, containing 2.1 - 7.3 vol. % AI203 and material B with 1.3 - 15.4 vol. % AI203 were subjected to final compaction by hot extrusion due to which the AI203 particles displayed a linear arrangement (Fig.l).The final structure of material C with 3 - I0 vol. % A1203, which underwent final compaction by hot extrusion and subsequent forging, and of material D with 2.1 - 7.3 vol. % AI20~, subjected to final compaction by cold and hot isostatic pressing, was nomogeneous from the viewpoint of particle arrangement (Fig.2), allowing the application of the Poisson's stochastic model of spatial distribution. It should be noted that according to the realized metallographic analysis, the microstructure is characterised by large, from the viewpoint of strength, ineffective particles. Extraction carbon replicas were prepared for the purpose of distribution determination of medium size effective particles which decrease the mobility of matrix dislocations (approx. <200 nm). The mean value was obtained by statistical processing of a set of approximately 2000 particles. The obtained results were compared with a smaller set of particles present in thin foils, on which also the mean diameter of the subgrains was measured (Fig.3). Transmission electron microscopy revealed that the matrix consists of small regions (0.4 - 0.9 ~m) which, despite their occasional mutual large-angle orientation, are referred to as subgrains, due to the dominance of polygonization processes in the formation of the final structure. 1145
1146
DISPERSION STRENGTHENING
Fig.l Line arrangement of particles in the system Cu-AI203 with 5 vol.% AI203 after extrusion
Fig.3
Vol. 30, No. 9
Fig.2 Uniform arrangement of particles in the system Cu-AI203 with 5 vol.% AI203 after isostatic pressing
Particles and subgrains in the foil from the Cu-Al203 material containing 2 vol.% AI203 after extrusion Results and Discussion
Our previous studies have been devoted to analysis of the yield point in dispersion strengthened systems Ag-MgO [2], Cu-Al203 [3] and AI-AI4C 3 [4]. First of all, the existing strengthening models were verified, according to Table 1. Regression analysis has shown that of the 4 categories of models (expressing R_0.2 as a function of structural parameters) the 4th, which takes i~to account the volume fraction and radius of dispersion particles as structural parameters is the most optimal.
Vol.30,No. 9
DISPERSION STRENGTHENING
The evaluation of the y i e l d p o i n t of c o n t r i b u t i o n s in the equation: Rp0.2 where RpN Rs RD RLC Rp
1147
was b a s e d on the
additive
character
= R p N + R s + R D + RLC + Rp
(i)
is P e i e r l s - N a b a r r o strengthening, c o n t r i b u t i o n of s u b s t i t u t i o n strengthening, c o n t r i b u t i o n of d i s l o c a t i o n s t r e n g t h e n i n g , contribution of s u b g r a i n s t r e n g t h e n i n g a c c o r d i n g to L a n g f o r d - C o h e n , direct c o n t r i b u t i o n of s t r e n g t h e n i n g by p a r t i c l e s .
In the case of the e v a l u a t e d s y s t e m Cu-AI~O3, we consider the structural state characterized by a certaln deformation degree a relevant dislocation density. The individual contributions d e t e r m i n e d as follows:
real with were
The Peierls-Nabarro friction stress required for the m o v e m e n t of dislocations in the d e f e c t - f r e e lattice is low and w a s e s t i m a t e d to be equal to 1 MPa. It can be c o m p u t e d from the following e q u a t i o n
. . . .(l-V) . 2G exp I . . . .2~w .b /
RpN where
Q w G b
is is is is
the the the the
(2)
P o i s s o n ' s r a t i o (= 0.33), w i d t h of d i s l o c a t i o n , m o d u l u s of e l a s t i c i t y in s~ear = 27 000 B u r g e r s v e c t o r = 2.5 . i0 v cm.
N / m m 2,
The c o n t r i b u t i o n of s u b s t i t u t i o n s t r e n g t h e n i n g RS, r e s u l t i n g in local distortion of the m a t r i x lattice by atoms of the d i s s o l v e d metal, was e s t i m a t e d to be equal to 5 MPa. It can be c a l c u l a t e d f r o m the equation: RS = wh e r e
n ~ ki.x i
(3)
k i is the s t r e n g t h e n i n g coefficient, x i is c o n c e n t r a t i o n of the d i s s o l v e d The d i s l o c a t i o n
strengthening
metal
atoms.
R D was d e t e r m i n e d
f r o m the e q u a t i o n
R D = 2~Gb~ where
a is the s t r e n g t h is the density processing).
(4)
c o e f f i c i e n t of the d i s l o c a t i o n of d i s l o c a t i o n s in the final
S t a t i s t i c a l e v a l u a t i o n in thin foils provided ~ = s t i t u t i o n in the e q u a t i o n (4) we o b t a i n R D = 35 MPa.-The s u b g r a i n
strengthening
contribution
RLC = kLC
. d -I
n e t w o r k = 0.2, s t a t e (after thermal 6 . 1 0 9 / c m 2. By sub-
was d e t e r m i n e d
from (5)
wh e r e the c o n s t a n t kLC = 6 Gb and d is the mean d i m e n s i o n of subgrains. The subgrain size was small (less than 1 ~m) and was on the d e c r e a s e with increasing volume fraction of A I 2 0 3 and h o m o g e n e i t y of p a r t i c l e distribution. A f t e r s u b s t i t u t i o n into e q u a £ i o n (5) we o b t a i n R C = 84 MPa for the m a t e r i a l w i t h the l i n e a r a r r a n g e m e n t (A) and RLC = 120 M ~ a for the m a t e r i a l w i t h h o m o g e n e o u s m i c r o s t r u c t u r e (D). D i r e c t c o n t r i b u t i o n of d i s p e r s i o n p a r t i c l e s for the g r o u p of m a t e r i a l s A and B with the l i n e a r m i c r o s t r u c t u r e was determined using a modified F i s h e r - H a r t - P r y [5] in the form: f3/4 Rp = 120 + 170 (6) r
1148
DISPERSION STRENGTHENING
Vol.30,No. 9
Analysis of materials with homogeneous distribution of particles to an equation, according to Ashby [6], in the form:
leads
f
Rp = 165 + 85(---) 1/2 r
(7)
where f is the volume fraction of particles and r is their radius. By s u b t r a c t i n g t h e individual strengthening contributions from the measured value of yield point we obtain Rp = 125 MPa for material A and Rp = 163 MPa for material D. Graphical illustration of this situation is shown in Fig. 4. The threshold value R_0.2 for the zero volume f r a c t i o n (i.e. matrix prepared by unconventiona~ way) was in good agreement with the experiment (RD0.2 = 150 MPa for the material B). The above mentioned thus suggests that#the direct strengthening by particles, given by the term Rp, exceeds the indirect one, in our case the strengthening by subgrains according to Langford-Cohen. We should ponder over the different direct strengthening values caused by particles in materials prepared by various ways. Our assumption is that this is caused by the microstructure of materials. Metallographic analyses of "in situ" deformed surfaces [7] of extruded materials determined 3 - 10-times higher mean distances of lines in comparison with the mean distances of particles in the bulk of material having a homogeneous distribution of particles. At the same time, the mean distance of lines is considerably higher at lower volume fractions of particles and is undirectly proportional to the value of Rp. The assumption that the direct contribution of particles (interaction o~ particles with dislocations) is predominantly determined by the distance between lines in heterogeneous structures allows us to state that the mentioned contribution will be lower in the heterogeneous microstructure, in comparison with the uniform one, as it is confirmed by the calculated values. This explains the
4~
z
3oc
200
,B
J/
i,o
100
•",
C
•
D
|
I
1,0
2,0
3,0
f ~ r -t • Fig. 4
The relationship between yield point and microstructural parameters for materials A, B, C and D.
differences in the type of relations for yield point estimation, valid for homogeneous and heterogeneous microstructures, expressed by modified equations according to Ashby and Fisher-Hart-Pry. Important is the fact that dispersion particles, either those introduced into the system from the
Vol. 30, No. 9
DISPERSION STRENGTHENING
1149
outside or developing "in situ" during the p r e p a r a t i o n of systems, have besides their i n t e r a c t i o n effect with dislocations, also a secondary one based on the fact that these particles also modify the m i c r o s t r u c t u r e and substructure formation. Conclusion The low temperature yield point comprised of 5 a d d i t i v e components has two decisive contributions, namely the indirect component of strengthening by subgrains, smaller than the direct one, resulting from the interaction of non-coherent p a r t i c l e s with mobile dislocations. The d i r e c t strengthening contribution is c o n s i d e r a b l y affected by the distribution of strengthening particles in the bulk of the material. This effect is more pronounced at a u n i f o r m d i s t r i b u t i o n of dispersion particles in the matrix.
References i. 2. 3.
M. Hazzledine, Scripta Metall. Mater. 26, 1992, 57. M. Besterci, V. Proch~zka, Neue Hutte 9, 1975, 546. M. ~les~r, G. Jangg, M. Besterci, Zeitschrift f. M e t a l l k u n d e 6, 1981, 423. M. Besterci, M. ~les~r, G. Jangg, Powder Metall. Int. i, 1992, 27. J.C. Fisher, E.W. Hart, R.M. Pry, Acta Metall. i, 1953, 336. M.F. Ashby, Z e i t s c h r i f t f. Metallkunde 5, 1964, 55. M. ~les~r, K.E. Easterling, M. Besterci, B. Bengtsson, Metal materials (in Slovak) 22, 1984, 6, 740.
4. 5. 6. 7.
Table i: Schematic Functional
and functional
relation
relationships
Author
of direct s t r e n g t h e n i n g Schematical
relation
i. R - ebA 2. R~ " - l o g A 3. l~gRp - log A
Unckel Gensamer Show
Rp = gl(~)
4.
Mott-Nabarro Ansell-Lenel Trefilov-Mojsejev
Rp = g2(L)
Meiklejohn-Skoda Hart Tanaka-Mori Brown-Stoobs
Rp = g3(f)
Rp
" L
5. Rp - I/~L 6. Rp i/~rL ~
7 8 9 i0
Rp Rp Rp Rp
ii. Rp 12 Rp 13. Rp
f f3/4+ f3/2 f.E f.G _ f/r ~t, (f/r) ~i~ 1 r-
3f 1/3_ 3f [(~-) (~-)]
Orowan Ashby Preston-Grant
14. Rp = f3/2/r
Fisher-Hart/Pry
15. Rp = (-~-) 2r
Hirch-Humpreys
Rp = g4(f/r)
et
Pergamon
STRUCTURE
ANALYSIS
OF D I S P E R S I O N
Materialia, Vol. 30, No. 9, pp. 1145-1150, 1994 Copyright © 1994 Elsevier Science Lid Printed in the USA. All rights reserved 0956-716X/94 $6.00 + 00
STRENGTHENING
M. Besterci Institute of Materials Research of Slovak Academy of Sciences, Watsonova 47, 043 53 Ko~ice, Slovak Republic (Received November 2, 1993) (Revised January 7, 1994) Introduction In a recent paper [i] Hazzledin discussed the effects of direct and indirect strengthening in dispersion strengthened systems containing non-coherent particles of a second phase at low temperatures and low stress. Theoretical analysis confirmed that indirect strengthening (interaction of dislocations with grain boundaries according to Hall-Petch) always dominates over direct strengthening (interaction of dislocations with dispersion particles according to Orowan). The objective of this study is to point out that this conception has not been confirmed for real dispersion strengthened materials and, moreover, is conditional upon structure, i.e. dependent on distribution of particles throughout the matrix volume, as well as on their size distribution. Both these factors depend considerably on the method of preparation of systems via powder metallurgy, as well as on final compaction methods. The study will analyse the yield point RD0.2 of the system Cu-AI203 on the basis of an additive character of strengthening contributions for materials prepared by various methods. Experimental Material and M e t h o d s Experiments were carried out using 4 types of materials Cu-Al203, prepared by methods of mechanical homogenization (A, C and D) and reaction milling (B). Final compaction of mixtures was achieved by various ways modifying distribution of particles throughout the volume of the final product. The material A, containing 2.1 - 7.3 vol. % AI203 and material B with 1.3 - 15.4 vol. % AI203 were subjected to final compaction by hot extrusion due to which the AI203 particles displayed a linear arrangement (Fig.l).The final structure of material C with 3 - I0 vol. % A1203, which underwent final compaction by hot extrusion and subsequent forging, and of material D with 2.1 - 7.3 vol. % AI20~, subjected to final compaction by cold and hot isostatic pressing, was nomogeneous from the viewpoint of particle arrangement (Fig.2), allowing the application of the Poisson's stochastic model of spatial distribution. It should be noted that according to the realized metallographic analysis, the microstructure is characterised by large, from the viewpoint of strength, ineffective particles. Extraction carbon replicas were prepared for the purpose of distribution determination of medium size effective particles which decrease the mobility of matrix dislocations (approx. <200 nm). The mean value was obtained by statistical processing of a set of approximately 2000 particles. The obtained results were compared with a smaller set of particles present in thin foils, on which also the mean diameter of the subgrains was measured (Fig.3). Transmission electron microscopy revealed that the matrix consists of small regions (0.4 - 0.9 ~m) which, despite their occasional mutual large-angle orientation, are referred to as subgrains, due to the dominance of polygonization processes in the formation of the final structure. 1145
1146
DISPERSION STRENGTHENING
Fig.l Line arrangement of particles in the system Cu-AI203 with 5 vol.% AI203 after extrusion
Fig.3
Vol. 30, No. 9
Fig.2 Uniform arrangement of particles in the system Cu-AI203 with 5 vol.% AI203 after isostatic pressing
Particles and subgrains in the foil from the Cu-Al203 material containing 2 vol.% AI203 after extrusion Results and Discussion
Our previous studies have been devoted to analysis of the yield point in dispersion strengthened systems Ag-MgO [2], Cu-Al203 [3] and AI-AI4C 3 [4]. First of all, the existing strengthening models were verified, according to Table 1. Regression analysis has shown that of the 4 categories of models (expressing R_0.2 as a function of structural parameters) the 4th, which takes i~to account the volume fraction and radius of dispersion particles as structural parameters is the most optimal.
Vol.30,No. 9
DISPERSION STRENGTHENING
The evaluation of the y i e l d p o i n t of c o n t r i b u t i o n s in the equation: Rp0.2 where RpN Rs RD RLC Rp
1147
was b a s e d on the
additive
character
= R p N + R s + R D + RLC + Rp
(i)
is P e i e r l s - N a b a r r o strengthening, c o n t r i b u t i o n of s u b s t i t u t i o n strengthening, c o n t r i b u t i o n of d i s l o c a t i o n s t r e n g t h e n i n g , contribution of s u b g r a i n s t r e n g t h e n i n g a c c o r d i n g to L a n g f o r d - C o h e n , direct c o n t r i b u t i o n of s t r e n g t h e n i n g by p a r t i c l e s .
In the case of the e v a l u a t e d s y s t e m Cu-AI~O3, we consider the structural state characterized by a certaln deformation degree a relevant dislocation density. The individual contributions d e t e r m i n e d as follows:
real with were
The Peierls-Nabarro friction stress required for the m o v e m e n t of dislocations in the d e f e c t - f r e e lattice is low and w a s e s t i m a t e d to be equal to 1 MPa. It can be c o m p u t e d from the following e q u a t i o n
. . . .(l-V) . 2G exp I . . . .2~w .b /
RpN where
Q w G b
is is is is
the the the the
(2)
P o i s s o n ' s r a t i o (= 0.33), w i d t h of d i s l o c a t i o n , m o d u l u s of e l a s t i c i t y in s~ear = 27 000 B u r g e r s v e c t o r = 2.5 . i0 v cm.
N / m m 2,
The c o n t r i b u t i o n of s u b s t i t u t i o n s t r e n g t h e n i n g RS, r e s u l t i n g in local distortion of the m a t r i x lattice by atoms of the d i s s o l v e d metal, was e s t i m a t e d to be equal to 5 MPa. It can be c a l c u l a t e d f r o m the equation: RS = wh e r e
n ~ ki.x i
(3)
k i is the s t r e n g t h e n i n g coefficient, x i is c o n c e n t r a t i o n of the d i s s o l v e d The d i s l o c a t i o n
strengthening
metal
atoms.
R D was d e t e r m i n e d
f r o m the e q u a t i o n
R D = 2~Gb~ where
a is the s t r e n g t h is the density processing).
(4)
c o e f f i c i e n t of the d i s l o c a t i o n of d i s l o c a t i o n s in the final
S t a t i s t i c a l e v a l u a t i o n in thin foils provided ~ = s t i t u t i o n in the e q u a t i o n (4) we o b t a i n R D = 35 MPa.-The s u b g r a i n
strengthening
contribution
RLC = kLC
. d -I
n e t w o r k = 0.2, s t a t e (after thermal 6 . 1 0 9 / c m 2. By sub-
was d e t e r m i n e d
from (5)
wh e r e the c o n s t a n t kLC = 6 Gb and d is the mean d i m e n s i o n of subgrains. The subgrain size was small (less than 1 ~m) and was on the d e c r e a s e with increasing volume fraction of A I 2 0 3 and h o m o g e n e i t y of p a r t i c l e distribution. A f t e r s u b s t i t u t i o n into e q u a £ i o n (5) we o b t a i n R C = 84 MPa for the m a t e r i a l w i t h the l i n e a r a r r a n g e m e n t (A) and RLC = 120 M ~ a for the m a t e r i a l w i t h h o m o g e n e o u s m i c r o s t r u c t u r e (D). D i r e c t c o n t r i b u t i o n of d i s p e r s i o n p a r t i c l e s for the g r o u p of m a t e r i a l s A and B with the l i n e a r m i c r o s t r u c t u r e was determined using a modified F i s h e r - H a r t - P r y [5] in the form: f3/4 Rp = 120 + 170 (6) r
1148
DISPERSION STRENGTHENING
Vol.30,No. 9
Analysis of materials with homogeneous distribution of particles to an equation, according to Ashby [6], in the form:
leads
f
Rp = 165 + 85(---) 1/2 r
(7)
where f is the volume fraction of particles and r is their radius. By s u b t r a c t i n g t h e individual strengthening contributions from the measured value of yield point we obtain Rp = 125 MPa for material A and Rp = 163 MPa for material D. Graphical illustration of this situation is shown in Fig. 4. The threshold value R_0.2 for the zero volume f r a c t i o n (i.e. matrix prepared by unconventiona~ way) was in good agreement with the experiment (RD0.2 = 150 MPa for the material B). The above mentioned thus suggests that#the direct strengthening by particles, given by the term Rp, exceeds the indirect one, in our case the strengthening by subgrains according to Langford-Cohen. We should ponder over the different direct strengthening values caused by particles in materials prepared by various ways. Our assumption is that this is caused by the microstructure of materials. Metallographic analyses of "in situ" deformed surfaces [7] of extruded materials determined 3 - 10-times higher mean distances of lines in comparison with the mean distances of particles in the bulk of material having a homogeneous distribution of particles. At the same time, the mean distance of lines is considerably higher at lower volume fractions of particles and is undirectly proportional to the value of Rp. The assumption that the direct contribution of particles (interaction o~ particles with dislocations) is predominantly determined by the distance between lines in heterogeneous structures allows us to state that the mentioned contribution will be lower in the heterogeneous microstructure, in comparison with the uniform one, as it is confirmed by the calculated values. This explains the
4~
z
3oc
200
,B
J/
i,o
100
•",
C
•
D
|
I
1,0
2,0
3,0
f ~ r -t • Fig. 4
The relationship between yield point and microstructural parameters for materials A, B, C and D.
differences in the type of relations for yield point estimation, valid for homogeneous and heterogeneous microstructures, expressed by modified equations according to Ashby and Fisher-Hart-Pry. Important is the fact that dispersion particles, either those introduced into the system from the
Vol. 30, No. 9
DISPERSION STRENGTHENING
1149
outside or developing "in situ" during the p r e p a r a t i o n of systems, have besides their i n t e r a c t i o n effect with dislocations, also a secondary one based on the fact that these particles also modify the m i c r o s t r u c t u r e and substructure formation. Conclusion The low temperature yield point comprised of 5 a d d i t i v e components has two decisive contributions, namely the indirect component of strengthening by subgrains, smaller than the direct one, resulting from the interaction of non-coherent p a r t i c l e s with mobile dislocations. The d i r e c t strengthening contribution is c o n s i d e r a b l y affected by the distribution of strengthening particles in the bulk of the material. This effect is more pronounced at a u n i f o r m d i s t r i b u t i o n of dispersion particles in the matrix.
References i. 2. 3.
M. Hazzledine, Scripta Metall. Mater. 26, 1992, 57. M. Besterci, V. Proch~zka, Neue Hutte 9, 1975, 546. M. ~les~r, G. Jangg, M. Besterci, Zeitschrift f. M e t a l l k u n d e 6, 1981, 423. M. Besterci, M. ~les~r, G. Jangg, Powder Metall. Int. i, 1992, 27. J.C. Fisher, E.W. Hart, R.M. Pry, Acta Metall. i, 1953, 336. M.F. Ashby, Z e i t s c h r i f t f. Metallkunde 5, 1964, 55. M. ~les~r, K.E. Easterling, M. Besterci, B. Bengtsson, Metal materials (in Slovak) 22, 1984, 6, 740.
4. 5. 6. 7.
Table i: Schematic Functional
and functional
relation
relationships
Author
of direct s t r e n g t h e n i n g Schematical
relation
i. R - ebA 2. R~ " - l o g A 3. l~gRp - log A
Unckel Gensamer Show
Rp = gl(~)
4.
Mott-Nabarro Ansell-Lenel Trefilov-Mojsejev
Rp = g2(L)
Meiklejohn-Skoda Hart Tanaka-Mori Brown-Stoobs
Rp = g3(f)
Rp
" L
5. Rp - I/~L 6. Rp i/~rL ~
7 8 9 i0
Rp Rp Rp Rp
ii. Rp 12 Rp 13. Rp
f f3/4+ f3/2 f.E f.G _ f/r ~t, (f/r) ~i~ 1 r-
3f 1/3_ 3f [(~-) (~-)]
Orowan Ashby Preston-Grant
14. Rp = f3/2/r
Fisher-Hart/Pry
15. Rp = (-~-) 2r
Hirch-Humpreys
Rp = g4(f/r)