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Simulation of the yield strength of wire drawn Cu-based in-situ composites
COMPUTATIONAL MATERIALS SCIENCE ELSEVIER
Computational Materials Science 5 (1996) 195-202
Simulation of the yield strength of wire drawn Cu-based in-situ composites D. Raabe
*,
U. Hangen
Institut fir Metallkunde und Metallphysik, RWTH Aachen, Kopernikusstrasse 14, D-52056 Aachen, Germany Received 15 July 1995; accepted 31 August 1995
Abstract The yield strength of wire drawn in-situ composites consisting of a face centered cubic (fee) metal matrix (Cu) and 5-30 mass% of a body centered cubic (bee) transition metal (Nb) is modelled by using a modified linear rule of mixtures (MROM). This approach analytically describes the yield strength of the composite as the sum of the volumetric weighted average of the experimentally observed yield strengths of the individual pure phases and a Hall-Petch type contribution arising from the impact of internal boundaries. The latter portion is described in terms of dislocation pile-ups in the fee matrix and the activation of dislocation movement in the bee filaments.
1. Introduction
Cu and most body centered cubic (bee) transition metals like Nb, Ta, MO, V, W, Fe and Cr have negligible mutual solubility in the solid state [ 1,2]. Fiber reinforced micro-or even nanoscaled composites can hence be manufactured by large strain wire drawing of a cast ingot. Such materials are often referred to as in-situ processed metal matrix composites (MMCs). Owing to their extremely high strength and their good electrical conductivity, alloys consisting of Cu and 5-30 mass% Nb have been under thorough investigation for the past 15 years [3-251.
Corresponding author. Tel: +49-241-80-6866/W, fax: +49-241-8888-301; e-mail: [email protected]. l
The strength of heavily wire drawn Cu-based MMCs is much greater than expected from the linear rule of mixtures (ROM) [5,6,9-111. Several models have been proposed to explain the strength anomaly observed. The phase barrier model by Spitzig et al. [3,6] attributes the strength to the difficulty of- propagating plastic flow through the internal fee-bee interfaces (fee = face centered cubic). Funkenbusch and Courtney 1111 interpret the strength in terms of geometrically necessary dislocations owing to the incompatibility of plastic deformation of the fee and bee phases. Both approaches yield a good description of the strength observed. However, their application is limited since they depend on a number of fitting parameters. Recently, Raabe and Hangen [14] have outlined a physical model which analytically accounts for the filament morphology [3-91,
0927-0256/96/$15.00 8 1996 Elsevier Science B.V. All rights reserved SSDI 0927-0256(95)00072-O
D. Raabe, U. Hangen /Computational
196
for the dislocation arrangements observed at internal phase boundaries and for the crystallographic orientation distributions [15-171 of both phases. The current study is primarily concerned with the application of this microstructural model [14]. It is aimed to describe the yield strength of in-situ processed MMCs consisting of Cu and 5-30 mass% Nb.
2. The modified linear rule of mixtures (MROM) In the present model the yield strength of the composite is analytically described in terms of a modified linear rule of mixtures (MROM). It regards the yield strength of the MMC as the sum of the volumetric weighted average of the yield strengths of the individual phases (ROM), uRoM, and a Hall-Petch type contribution which is attributed to the influence of the Cu-Nb phase boundaries, chlr,,,c [14]. Whilst vTRoMis computed on the basis of experimentally observed yield strengths, gMVIMC is derived theoretically and then calculated by using microstructural data. Both contributions are linearly decomposed into the load carried by the Cu phase, a&,,,, a&,, and by the Nb phase, ~,2~, upMc: cu vF’0,2 = vROM
VCu + a&,&&
+ d%lC
‘Nb
9
+ a~,&, (1)
where I$., and VNb are the volume fractions of Cu and Nb. Since no reliable experimental yield strengths were available for heavily wire drawn pure Cu (a;&) and Nb 4) but not in case of low strains (7 < 4), where the UTS exceeds the yield strength by about 20-30%. The true strain of the MMC is defined as qMviMc = ln(ktO/A MMC),where A0 and AMMC are the initial and the actual cross-sections after deformation. The true strains of Nb and Cu are denoted by qm= ln(t’/tI and 7cu = ln(h’/A), where t is the thickness and A the interfilament spacing. From experimental data the ratio of the UTS of pure Nb wires related to that of pure Cu wires,
Materials Science 5 (1996) 195-202
R, is derived as a function of strain, R(q) = fl&(~)/a&(77) = VR~M(~>/V~~M(~X It is stipulated that R also holds for the ratio of at$& and azM, (Eq. 2), (Fig. 1).
(2) Internal fee-bee phase boundaries are impenetrable by dislocations (Fig. 2). To deduce cMMC it is hence suggested that dislocations pile up in the Cu matrix in front of the phase boundaries (Fig. 2). Helping the externally imposed load, the internal stresses induced by such pile-ups contribute to the activation of dislocation movement or multiplication inside the filaments [14]. Following Sevillano [26] the critical stress for dislocation movement between two impenetrable walls is given by A = 1.2,
(3)
where S is the phase boundary spacing, G the shear modulus, b the Burgers vector and A a constant valid for mixed dislocations. To describe the critical stress of a dislocation mill which is constrained by two parallel impenetrable walls S/2 must be used instead of S. The yield strength of the MMC is reached when both phases deform plastically. According to Eq. (3) hence the critical stress required for dislocation movement in the Nb phase defines the yield
1
2
3
4
5
6
7
8
9
10
11
12
True Strain, Wlre Drawn
Fig. 1. Ratio of the yield strengths of pure wire drawn Cu and Nb specimens as a function of strain, R(T) = u~~(TJ)/~$‘(~) = O&JO)/(r&;M(~) (EkL(2N.
D. Raabe, U. Hangen /Computational Materials Science 5 (19%) 195-202
strength of the MMC. However, it is assumed that prior to massive plastic deformation of the MMC in the Cu matrix dislocations pile up in front of the phase boundaries, causing an accumulated strain not exceeding 0.2%. At the tips of the pile-ups shear stresses are generated. Owing to this contribution the effective shear stress on the slip planes in the Nb filaments is increased:
(4) where r$ is the shear stress acting at the tip of the dislocation pile-up, nplp the number of dislocations assembled in the pile-up, m the orientation factor between the slip systems of Cu and Nb and M, the Taylor factor of the Nb filaments. The critical shear stress for dislocation movement in the Nb filament, T,,,, is calculated according to
197
Eq. (3). The number of dislocations accumulated in a double ended pile-up between two interfaces is given by 1271:
where A+= h/m,, is the filament spacing normalized by the slip geometry, i.e. A is the distance measured perpendicular to the phase boundary. The Hall-Petch type contribution, uMMMC, and the shear stress on the Nb slip system, 7, are related according to uMMMc = MT. Using Eqs. (4) and (5) one thus obtains
Fig. 2. TEM micrograph showing the Cu matrix (bright) and elongated Nb filaments (dark, thin). In the Cu matrix dislocation pile-ups can be observed in front of the phase boundaries.
D. Raabe, U. Hangen /Computational
198
The expression can be written as f cu
uMMC
=
RW?“G,“b,” 2h+( 1 - ~c”)M,,m
-
Since negative stresses do not make sense in this context the positive sign applies. Using Eqs. (1) and (2) the contribution of the phase boundaries to the yield strength can be written as: (TMMC = (‘cu
+ ‘id)
3. Incorporation
(8)
+&4C.
of experimental
Material Science 5 (1996) 195-202
3.67 ({llO}(lll) slip systems) or M~~s=3.18 ((llO](lll), (112](111) and {123](111) slip systems), respectively. In wire drawn Cu-Nb composites the Nb filaments reveal a curled morphology [30], indicating that FC conditions are not fulfilled locally. Correspondingly, the Taylor factors for ‘relaxed constraints’ conditions (RC, relaxation of all shear strains) (see e.g. [31,321) have to be considered leading to Mg* = 2.45 and WZ”” = 2.15. If single slip is considered the orientation factor for Cu is equal to 0.27 and that for Nb equal to 0.4. The orientation factor between the Cu and the Nb slip systems at the phase boundary then amounts to m = 0.98
4. Results and discussion Incorporating experimental data, uMMCcan be computed as a function of the volume fractions V,-, and VNb and of the true strain:
data
The present model incorporates the filament geometry of wire drawn Cu-20 mass% Nb as published by Verhoeven et al. [28]. The following expressions for the filament thickness, t, and spacing, A, were derived by fitting.
4.25MPa vMMC=
(‘Cu
+ ‘NbR)
M
R Nb
+ t = to exp( A = A0 eXP(
-Ad?MMc) -&$?MMc).
(9)
The true strains of both phases related to that of the MMC, A, and Anb, can be written as
(10) The ratios were found to be ANb = AC” = A = 0.5. As starting values to = 1.2 pm and A0= 6 pm were extrapolated from the transmission electron microscope (TEM) data [281. To incorporate the slip geometry at the phase boundaries, the crystallographic textures and the Taylor factors of both phases were considered. As was reported by Raabe et al. [15,161 and Heringhaus et al. [171 the Cu phase in the MMC reveals a (111) and the Nb phase a (110) fiber texture. Under ‘full constraints’ conditions (FC) [29], the corresponding Taylor factor for Cu amounts to MF”” = 3.16 and for Nb to Mr$t2 =
+5.9MPa*(9.25
- qj]‘;
exP( 4ij.
(11) The total yield strength of the MMC can then be calculated according to Eq. (1). In (Fig. 3) the yield strengths [61 of the samples containing 20 mass% Nb, the microstructure of which was studied by Verhoeven et al. [28], are depicted together with the simulation results. Since in the original figures of Spitzig et al. [3,6] the UTS is shown, the yield strengths, Q~, had to be extracted from the true stress-true strain curves [3,6]. As was reported elsewhere [14] the best agreement between the experimental results [6] and the model (Fig. 3) is yielded for the simulation which is based on ME* = 2.15 and on the assumption of dislocation movement rather than
D. Raabe, U. Hangen / Computational Materials Science 5 (19%) 195-202
multiplication. Both presumptions are reasonable. First, the low Taylor factor is attributed to the relaxation of local geometrical constraints which are usually imposed by neighboring crystals. This assumption is vindicated by the curled morphology of the Nb filaments. Furthermore, it is consistent with Taylor type simulations of crystallographic cold rolling textures of pure polycrystalline Nb which yield a good agreement with experiment if the RC approach (relaxation of transverse and longitudinal shear) as well as 48 potential slip systems are considered [15,16,33,34]. Second, since the yield strength rather than the UTS is simulated it seems reasonable that in the first place movement rather than multiplication of dislocations takes place. Furthermore, it was frequently observed that the filament thickness is not homogeneous [8,14,19,20]. It is thus likely
1,600
:
1
1.400
:
: ‘A
that by reaching the yield strength dislocation multiplication starts as well in regions having maximum thickness (Eq. (3)). At low strains (71 < 4) the prediction shows a considerable deviation from experiment. This is attributed to the fact that the UTS, a$, (+$& rather than the yield strength of the pure wire drawn constituents was used for the computation Of uROMe Furthermore, in this strain regime the phases are not yet aligned parallel. At large strains (7 > 4) the simulation and the experimental data are in very good accordance (Fig. 3). However, the data predicted by the model slightly overestimate the true yield strengths. This systematic deviation is attributed to the occurrence of dynamic recovery and recrystallization in the Cu matrix (Fig. 4). In Fig. 5 the simulated yield strengths of four Cu-based alloys with a Nb content of 5-30 mass% are shown as a function of strain. As suggested by the present approach, the yield strength of the composite considerably increases with the Nb content.
5. Comparing the MROM proaches
o-,““““““’ 0
1
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4
5
6
7
3
9
10
11
12
True Strain, Wire Drawn Cu-Nb Fig. 3. Results of the simulation
199
with previous ap-
The Hall-Petch type simulation of Spitzig [3,6] and the MROM employed here bear a certain resemblance. Both models attribute the strength to the difficulty of propagating plastic flow through the fee-bee interfaces. Whereas the model of Spitzig [3,6] is derived by fitting of experimental data, the MROM is based on a physical approach. Since it also leads to a HallPetch type relationship, the present model, in particular the contribution gMMC, can be regarded as a suitable supplementary physical derivation of the barrier model introduced by Spitzig et al. [3,61. However, there is less resemblance between the MROM and the model of Funkenbusch and Courtney [ll]. The latter approach is based on a work hardening mechanism, namely on the generation of geometrically necessary dislocations owing to the incompatibility of plastic deformation of the fee and bee phase. At first sight this
200
D. Raabe, U. Hangen /Computational
mod ;el is opposed by the experimental fact that the ratio between the yield and the tensile strewlgth of the MMC is 80% even for heavily
Materials Science 5 (1996) 195-202
wire drawn samples. If the strengthening me:chanism was based on such a type of work harde :ning [ll], however, one would for heavily defer .med
Fig. 4. Formation of small recrystallized grains in the Cu matrix (arrow) of a heavily wire drawn Q-20 mass% Nb MMC. Alth ‘ough from 1 this micrograph it cannot be decided whether static or dynamic recrystallization has taken place, it is conceivable that such sses lead to a decrease of the yield strength.
D. Raabe, U. Hangen /Computational Materials Science 5 (19%) 195-202
201
and a Hall-Petch type contribution which results from the impact of internal phase boundaries. The latter portion is described in terms of dislocation pile-ups in the Cu matrix and movement of dislocations in the filaments. The crystallographic texture and filament geometry of both phases was considered. The model was used for predicting the yield strengths of alloys containing 5-30 mass% Nb. The predictions reveal a very good agreement with experimental data.
0
2
4
6
6
10
12
References
True Strain, Wire Drawn Cu-Nb (Slmulatlon)
Fig. 5. Simulated yield strength of four &-based alloys with a Nb content of S-30 mass% as a function of strain. As suggested by the present approach, the yield strength of the composite considerably increases with the Nb content.
specimens expect a smaller difference between the yield and the tensile strength. As possible explanation of this contradiction, the occurrence of dynamic recovery and recrystallization (Fig. 4) is conceivable. The model of Sevillano [26] overestimates the yield strength of Cu-Nb when tested with the data of Verhoeven et al. [e.g. 281. However, the mechanisms proposed as being critical, namely the movement or multiplication of dislocations in the filaments, seem to be relevant and were hence used in the present approach. It might be a shortcoming of the Sevillanos [32] model that not the local but only the externally imposed load is considered, i.e. dislocation pile-ups in the Cu matrix which increase the shear stress in the Nb filaments [14] are not taken into account.
6. Conclusions A modified linear rule of mixtures (MROM) for the description of the yield strength of wire drawn Cu-based in-situ composites was suggested. It regards the yield strength of the MMC as the sum of the volumetric weighted average of the yield strengths of the individual pure phases
[l] D.J. Chakrabati and D.E. Laughlin, Bull. Alloy Phase Diagrams 2 (1982) 936. [2] G.I. Terekhov and L.N. Aleksandrova, Izv. Akad. Nauk SSR. Metally 4 (19841 210. [3] W.A. Spitzig, Acta Metall. 39 (1991) 1085. [4] K.R. Karasek and J. Bevk, J. Appl. Phys. 52 (1981) 1370. [5] J. Bevk, J.P. Harbison and J.L. Bell, J. Appl. Phys. 49 (1978) 6031. [6] W.A. Spitzig, A.R. Pelton and F.C. Laabs, Acta Metall. 35 (1987) 2427. [7] F. ,Heringhaus, D. Raabe, L. Kaul and G. Gottstein, Metal1 6 (1993) 558. [8] F. Heringhaus, D. Raabe and G. Gottstein, Metall 48 (19941287. 191 L.S. Chumbley, H.L. Downing, W.A. Spitzig and J.D. Verhoeven, Mater. Sci. Eng. All7 (1989) 59. [lOI T.H. Courtney, in: New Developments and Applications in Composites, eds. D. Kuhlmann-Wilsdorf and W.C. Harrigan Jr., TMS-AIME 1 (1978). [ill P.D. Funkenbusch and T.H. Courtney, Acta Metall. 33 (1985) 913. WI G. Wassermann, Verbundwerkstoffe, DGM-Verlag (1981) p. 63. [131 J.D. Embury, M.A. Hill, W.A. Spitzig and Y. Sakai, MRS Bull. 8 (1993) 57. 1141 D. Raabe and U. Hangen, Proc. 15th Rise Int. Symp., eds. S.I. Andersen, J.B. Bilde-Sorensen, T. Lorentzen O.B. Pedersen and N.J. Sorensen. Riso, Denmark (1994) p, 487. D51 D. Raabe, J. Ball and G. Gottstein Scripta Metall. 27 (1992) 211. 1161D. Raabe and G. Gottstein, J. Phys. (Paris) IV, ~011.C7, supp. J. Phys. (Paris) III 3 (19931 1727. 1171F. Heringhaus, D. Raabe, U. Hangen and G. Gottstein, Proc. ICOTOM 10, Mater. Sci. Forum 157-162 (19941 709. 1181 F. Habbal and J. Bevk, J. Appl. Phys. 54 (1983) 6543. iI91 D. Raabe and F. Heringhaus, Phys. Stat. Sol (al, 142 (1994) 473.
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[20] F. Heringhaus, D. Raabe and G. Gottstein, Acta Metall.
[21] [22] [23] [24] [25] [26]
43 (1995) 1467. F. Herlach, IEEE Trans. Magn. MAG-24 (1988) 1094. H.-J. Schneider-Muntau, IEEE Trans. Magn. MAG-18 (1982) 32. J.D. Verhoeven, W.A. Spitzig, F.A. Schmidt, P.D. Krotz and E.D. Gibson, J. Mater. Sci. 24 (19891 1015. P.D. Funkenbusch, T.H. Courtney and D.G. Kubisch, Scipta Metall. 18 (1984) 1099. A.R. Pelton, F.C. Laabs, W.A. Spitzig and C.C. Cheng, Ultramicrosc. 22 (1987) 251. J.G. Sevillano, J. Phys. (Paris) III 6 (19901 967.
Materials Science 5 (1996) 195-202
[27] G. Leibfried, Z. Phys. 130 (19511 214. [28] J.D. Verhoeven, L.S. Chumbley, F.C. Laabs and W.A. Spitzig, Acta Metall. 39 (1991) 2825. [29] G.I. Taylor, J. Inst. Met. 62 (1938) 307. [30] W.F. Hosford Jr., Trans. TMS AIME 230 (1964) 12. [31] H. Honneff and H. Mecking, in: Proc. ICOTOM 6, ed. S. Nagashima ISIJ (1981) p. 347. [32] U.F. Kocks and H. Chandra, Acta Metall. 30 (1982) 695. 1331 D. Raabe and K. Liicke, Z. Metallk. 85 (1994) 302. 1341D. Raabe and K. Liicke, Proc. ICOTOM 10, Mater. Sci. Forum 157-162 (1994) 1469.
Computational Materials Science 5 (1996) 195-202
Simulation of the yield strength of wire drawn Cu-based in-situ composites D. Raabe
*,
U. Hangen
Institut fir Metallkunde und Metallphysik, RWTH Aachen, Kopernikusstrasse 14, D-52056 Aachen, Germany Received 15 July 1995; accepted 31 August 1995
Abstract The yield strength of wire drawn in-situ composites consisting of a face centered cubic (fee) metal matrix (Cu) and 5-30 mass% of a body centered cubic (bee) transition metal (Nb) is modelled by using a modified linear rule of mixtures (MROM). This approach analytically describes the yield strength of the composite as the sum of the volumetric weighted average of the experimentally observed yield strengths of the individual pure phases and a Hall-Petch type contribution arising from the impact of internal boundaries. The latter portion is described in terms of dislocation pile-ups in the fee matrix and the activation of dislocation movement in the bee filaments.
1. Introduction
Cu and most body centered cubic (bee) transition metals like Nb, Ta, MO, V, W, Fe and Cr have negligible mutual solubility in the solid state [ 1,2]. Fiber reinforced micro-or even nanoscaled composites can hence be manufactured by large strain wire drawing of a cast ingot. Such materials are often referred to as in-situ processed metal matrix composites (MMCs). Owing to their extremely high strength and their good electrical conductivity, alloys consisting of Cu and 5-30 mass% Nb have been under thorough investigation for the past 15 years [3-251.
Corresponding author. Tel: +49-241-80-6866/W, fax: +49-241-8888-301; e-mail: [email protected]. l
The strength of heavily wire drawn Cu-based MMCs is much greater than expected from the linear rule of mixtures (ROM) [5,6,9-111. Several models have been proposed to explain the strength anomaly observed. The phase barrier model by Spitzig et al. [3,6] attributes the strength to the difficulty of- propagating plastic flow through the internal fee-bee interfaces (fee = face centered cubic). Funkenbusch and Courtney 1111 interpret the strength in terms of geometrically necessary dislocations owing to the incompatibility of plastic deformation of the fee and bee phases. Both approaches yield a good description of the strength observed. However, their application is limited since they depend on a number of fitting parameters. Recently, Raabe and Hangen [14] have outlined a physical model which analytically accounts for the filament morphology [3-91,
0927-0256/96/$15.00 8 1996 Elsevier Science B.V. All rights reserved SSDI 0927-0256(95)00072-O
D. Raabe, U. Hangen /Computational
196
for the dislocation arrangements observed at internal phase boundaries and for the crystallographic orientation distributions [15-171 of both phases. The current study is primarily concerned with the application of this microstructural model [14]. It is aimed to describe the yield strength of in-situ processed MMCs consisting of Cu and 5-30 mass% Nb.
2. The modified linear rule of mixtures (MROM) In the present model the yield strength of the composite is analytically described in terms of a modified linear rule of mixtures (MROM). It regards the yield strength of the MMC as the sum of the volumetric weighted average of the yield strengths of the individual phases (ROM), uRoM, and a Hall-Petch type contribution which is attributed to the influence of the Cu-Nb phase boundaries, chlr,,,c [14]. Whilst vTRoMis computed on the basis of experimentally observed yield strengths, gMVIMC is derived theoretically and then calculated by using microstructural data. Both contributions are linearly decomposed into the load carried by the Cu phase, a&,,,, a&,, and by the Nb phase, ~,2~, upMc: cu vF’0,2 = vROM
VCu + a&,&&
+ d%lC
‘Nb
9
+ a~,&, (1)
where I$., and VNb are the volume fractions of Cu and Nb. Since no reliable experimental yield strengths were available for heavily wire drawn pure Cu (a;&) and Nb
Materials Science 5 (1996) 195-202
R, is derived as a function of strain, R(q) = fl&(~)/a&(77) = VR~M(~>/V~~M(~X It is stipulated that R also holds for the ratio of at$& and azM, (Eq. 2), (Fig. 1).
(2) Internal fee-bee phase boundaries are impenetrable by dislocations (Fig. 2). To deduce cMMC it is hence suggested that dislocations pile up in the Cu matrix in front of the phase boundaries (Fig. 2). Helping the externally imposed load, the internal stresses induced by such pile-ups contribute to the activation of dislocation movement or multiplication inside the filaments [14]. Following Sevillano [26] the critical stress for dislocation movement between two impenetrable walls is given by A = 1.2,
(3)
where S is the phase boundary spacing, G the shear modulus, b the Burgers vector and A a constant valid for mixed dislocations. To describe the critical stress of a dislocation mill which is constrained by two parallel impenetrable walls S/2 must be used instead of S. The yield strength of the MMC is reached when both phases deform plastically. According to Eq. (3) hence the critical stress required for dislocation movement in the Nb phase defines the yield
1
2
3
4
5
6
7
8
9
10
11
12
True Strain, Wlre Drawn
Fig. 1. Ratio of the yield strengths of pure wire drawn Cu and Nb specimens as a function of strain, R(T) = u~~(TJ)/~$‘(~) = O&JO)/(r&;M(~) (EkL(2N.
D. Raabe, U. Hangen /Computational Materials Science 5 (19%) 195-202
strength of the MMC. However, it is assumed that prior to massive plastic deformation of the MMC in the Cu matrix dislocations pile up in front of the phase boundaries, causing an accumulated strain not exceeding 0.2%. At the tips of the pile-ups shear stresses are generated. Owing to this contribution the effective shear stress on the slip planes in the Nb filaments is increased:
(4) where r$ is the shear stress acting at the tip of the dislocation pile-up, nplp the number of dislocations assembled in the pile-up, m the orientation factor between the slip systems of Cu and Nb and M, the Taylor factor of the Nb filaments. The critical shear stress for dislocation movement in the Nb filament, T,,,, is calculated according to
197
Eq. (3). The number of dislocations accumulated in a double ended pile-up between two interfaces is given by 1271:
where A+= h/m,, is the filament spacing normalized by the slip geometry, i.e. A is the distance measured perpendicular to the phase boundary. The Hall-Petch type contribution, uMMMC, and the shear stress on the Nb slip system, 7, are related according to uMMMc = MT. Using Eqs. (4) and (5) one thus obtains
Fig. 2. TEM micrograph showing the Cu matrix (bright) and elongated Nb filaments (dark, thin). In the Cu matrix dislocation pile-ups can be observed in front of the phase boundaries.
D. Raabe, U. Hangen /Computational
198
The expression can be written as f cu
uMMC
=
RW?“G,“b,” 2h+( 1 - ~c”)M,,m
-
Since negative stresses do not make sense in this context the positive sign applies. Using Eqs. (1) and (2) the contribution of the phase boundaries to the yield strength can be written as: (TMMC = (‘cu
+ ‘id)
3. Incorporation
(8)
+&4C.
of experimental
Material Science 5 (1996) 195-202
3.67 ({llO}(lll) slip systems) or M~~s=3.18 ((llO](lll), (112](111) and {123](111) slip systems), respectively. In wire drawn Cu-Nb composites the Nb filaments reveal a curled morphology [30], indicating that FC conditions are not fulfilled locally. Correspondingly, the Taylor factors for ‘relaxed constraints’ conditions (RC, relaxation of all shear strains) (see e.g. [31,321) have to be considered leading to Mg* = 2.45 and WZ”” = 2.15. If single slip is considered the orientation factor for Cu is equal to 0.27 and that for Nb equal to 0.4. The orientation factor between the Cu and the Nb slip systems at the phase boundary then amounts to m = 0.98
4. Results and discussion Incorporating experimental data, uMMCcan be computed as a function of the volume fractions V,-, and VNb and of the true strain:
data
The present model incorporates the filament geometry of wire drawn Cu-20 mass% Nb as published by Verhoeven et al. [28]. The following expressions for the filament thickness, t, and spacing, A, were derived by fitting.
4.25MPa vMMC=
(‘Cu
+ ‘NbR)
M
R Nb
+ t = to exp( A = A0 eXP(
-Ad?MMc) -&$?MMc).
(9)
The true strains of both phases related to that of the MMC, A, and Anb, can be written as
(10) The ratios were found to be ANb = AC” = A = 0.5. As starting values to = 1.2 pm and A0= 6 pm were extrapolated from the transmission electron microscope (TEM) data [281. To incorporate the slip geometry at the phase boundaries, the crystallographic textures and the Taylor factors of both phases were considered. As was reported by Raabe et al. [15,161 and Heringhaus et al. [171 the Cu phase in the MMC reveals a (111) and the Nb phase a (110) fiber texture. Under ‘full constraints’ conditions (FC) [29], the corresponding Taylor factor for Cu amounts to MF”” = 3.16 and for Nb to Mr$t2 =
+5.9MPa*(9.25
- qj]‘;
exP( 4ij.
(11) The total yield strength of the MMC can then be calculated according to Eq. (1). In (Fig. 3) the yield strengths [61 of the samples containing 20 mass% Nb, the microstructure of which was studied by Verhoeven et al. [28], are depicted together with the simulation results. Since in the original figures of Spitzig et al. [3,6] the UTS is shown, the yield strengths, Q~, had to be extracted from the true stress-true strain curves [3,6]. As was reported elsewhere [14] the best agreement between the experimental results [6] and the model (Fig. 3) is yielded for the simulation which is based on ME* = 2.15 and on the assumption of dislocation movement rather than
D. Raabe, U. Hangen / Computational Materials Science 5 (19%) 195-202
multiplication. Both presumptions are reasonable. First, the low Taylor factor is attributed to the relaxation of local geometrical constraints which are usually imposed by neighboring crystals. This assumption is vindicated by the curled morphology of the Nb filaments. Furthermore, it is consistent with Taylor type simulations of crystallographic cold rolling textures of pure polycrystalline Nb which yield a good agreement with experiment if the RC approach (relaxation of transverse and longitudinal shear) as well as 48 potential slip systems are considered [15,16,33,34]. Second, since the yield strength rather than the UTS is simulated it seems reasonable that in the first place movement rather than multiplication of dislocations takes place. Furthermore, it was frequently observed that the filament thickness is not homogeneous [8,14,19,20]. It is thus likely
1,600
:
1
1.400
:
: ‘A
that by reaching the yield strength dislocation multiplication starts as well in regions having maximum thickness (Eq. (3)). At low strains (71 < 4) the prediction shows a considerable deviation from experiment. This is attributed to the fact that the UTS, a$, (+$& rather than the yield strength of the pure wire drawn constituents was used for the computation Of uROMe Furthermore, in this strain regime the phases are not yet aligned parallel. At large strains (7 > 4) the simulation and the experimental data are in very good accordance (Fig. 3). However, the data predicted by the model slightly overestimate the true yield strengths. This systematic deviation is attributed to the occurrence of dynamic recovery and recrystallization in the Cu matrix (Fig. 4). In Fig. 5 the simulated yield strengths of four Cu-based alloys with a Nb content of 5-30 mass% are shown as a function of strain. As suggested by the present approach, the yield strength of the composite considerably increases with the Nb content.
5. Comparing the MROM proaches
o-,““““““’ 0
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6
7
3
9
10
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True Strain, Wire Drawn Cu-Nb Fig. 3. Results of the simulation
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with previous ap-
The Hall-Petch type simulation of Spitzig [3,6] and the MROM employed here bear a certain resemblance. Both models attribute the strength to the difficulty of propagating plastic flow through the fee-bee interfaces. Whereas the model of Spitzig [3,6] is derived by fitting of experimental data, the MROM is based on a physical approach. Since it also leads to a HallPetch type relationship, the present model, in particular the contribution gMMC, can be regarded as a suitable supplementary physical derivation of the barrier model introduced by Spitzig et al. [3,61. However, there is less resemblance between the MROM and the model of Funkenbusch and Courtney [ll]. The latter approach is based on a work hardening mechanism, namely on the generation of geometrically necessary dislocations owing to the incompatibility of plastic deformation of the fee and bee phase. At first sight this
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D. Raabe, U. Hangen /Computational
mod ;el is opposed by the experimental fact that the ratio between the yield and the tensile strewlgth of the MMC is 80% even for heavily
Materials Science 5 (1996) 195-202
wire drawn samples. If the strengthening me:chanism was based on such a type of work harde :ning [ll], however, one would for heavily defer .med
Fig. 4. Formation of small recrystallized grains in the Cu matrix (arrow) of a heavily wire drawn Q-20 mass% Nb MMC. Alth ‘ough from 1 this micrograph it cannot be decided whether static or dynamic recrystallization has taken place, it is conceivable that such sses lead to a decrease of the yield strength.
D. Raabe, U. Hangen /Computational Materials Science 5 (19%) 195-202
201
and a Hall-Petch type contribution which results from the impact of internal phase boundaries. The latter portion is described in terms of dislocation pile-ups in the Cu matrix and movement of dislocations in the filaments. The crystallographic texture and filament geometry of both phases was considered. The model was used for predicting the yield strengths of alloys containing 5-30 mass% Nb. The predictions reveal a very good agreement with experimental data.
0
2
4
6
6
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12
References
True Strain, Wire Drawn Cu-Nb (Slmulatlon)
Fig. 5. Simulated yield strength of four &-based alloys with a Nb content of S-30 mass% as a function of strain. As suggested by the present approach, the yield strength of the composite considerably increases with the Nb content.
specimens expect a smaller difference between the yield and the tensile strength. As possible explanation of this contradiction, the occurrence of dynamic recovery and recrystallization (Fig. 4) is conceivable. The model of Sevillano [26] overestimates the yield strength of Cu-Nb when tested with the data of Verhoeven et al. [e.g. 281. However, the mechanisms proposed as being critical, namely the movement or multiplication of dislocations in the filaments, seem to be relevant and were hence used in the present approach. It might be a shortcoming of the Sevillanos [32] model that not the local but only the externally imposed load is considered, i.e. dislocation pile-ups in the Cu matrix which increase the shear stress in the Nb filaments [14] are not taken into account.
6. Conclusions A modified linear rule of mixtures (MROM) for the description of the yield strength of wire drawn Cu-based in-situ composites was suggested. It regards the yield strength of the MMC as the sum of the volumetric weighted average of the yield strengths of the individual pure phases
[l] D.J. Chakrabati and D.E. Laughlin, Bull. Alloy Phase Diagrams 2 (1982) 936. [2] G.I. Terekhov and L.N. Aleksandrova, Izv. Akad. Nauk SSR. Metally 4 (19841 210. [3] W.A. Spitzig, Acta Metall. 39 (1991) 1085. [4] K.R. Karasek and J. Bevk, J. Appl. Phys. 52 (1981) 1370. [5] J. Bevk, J.P. Harbison and J.L. Bell, J. Appl. Phys. 49 (1978) 6031. [6] W.A. Spitzig, A.R. Pelton and F.C. Laabs, Acta Metall. 35 (1987) 2427. [7] F. ,Heringhaus, D. Raabe, L. Kaul and G. Gottstein, Metal1 6 (1993) 558. [8] F. Heringhaus, D. Raabe and G. Gottstein, Metall 48 (19941287. 191 L.S. Chumbley, H.L. Downing, W.A. Spitzig and J.D. Verhoeven, Mater. Sci. Eng. All7 (1989) 59. [lOI T.H. Courtney, in: New Developments and Applications in Composites, eds. D. Kuhlmann-Wilsdorf and W.C. Harrigan Jr., TMS-AIME 1 (1978). [ill P.D. Funkenbusch and T.H. Courtney, Acta Metall. 33 (1985) 913. WI G. Wassermann, Verbundwerkstoffe, DGM-Verlag (1981) p. 63. [131 J.D. Embury, M.A. Hill, W.A. Spitzig and Y. Sakai, MRS Bull. 8 (1993) 57. 1141 D. Raabe and U. Hangen, Proc. 15th Rise Int. Symp., eds. S.I. Andersen, J.B. Bilde-Sorensen, T. Lorentzen O.B. Pedersen and N.J. Sorensen. Riso, Denmark (1994) p, 487. D51 D. Raabe, J. Ball and G. Gottstein Scripta Metall. 27 (1992) 211. 1161D. Raabe and G. Gottstein, J. Phys. (Paris) IV, ~011.C7, supp. J. Phys. (Paris) III 3 (19931 1727. 1171F. Heringhaus, D. Raabe, U. Hangen and G. Gottstein, Proc. ICOTOM 10, Mater. Sci. Forum 157-162 (19941 709. 1181 F. Habbal and J. Bevk, J. Appl. Phys. 54 (1983) 6543. iI91 D. Raabe and F. Heringhaus, Phys. Stat. Sol (al, 142 (1994) 473.
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[20] F. Heringhaus, D. Raabe and G. Gottstein, Acta Metall.
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43 (1995) 1467. F. Herlach, IEEE Trans. Magn. MAG-24 (1988) 1094. H.-J. Schneider-Muntau, IEEE Trans. Magn. MAG-18 (1982) 32. J.D. Verhoeven, W.A. Spitzig, F.A. Schmidt, P.D. Krotz and E.D. Gibson, J. Mater. Sci. 24 (19891 1015. P.D. Funkenbusch, T.H. Courtney and D.G. Kubisch, Scipta Metall. 18 (1984) 1099. A.R. Pelton, F.C. Laabs, W.A. Spitzig and C.C. Cheng, Ultramicrosc. 22 (1987) 251. J.G. Sevillano, J. Phys. (Paris) III 6 (19901 967.
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[27] G. Leibfried, Z. Phys. 130 (19511 214. [28] J.D. Verhoeven, L.S. Chumbley, F.C. Laabs and W.A. Spitzig, Acta Metall. 39 (1991) 2825. [29] G.I. Taylor, J. Inst. Met. 62 (1938) 307. [30] W.F. Hosford Jr., Trans. TMS AIME 230 (1964) 12. [31] H. Honneff and H. Mecking, in: Proc. ICOTOM 6, ed. S. Nagashima ISIJ (1981) p. 347. [32] U.F. Kocks and H. Chandra, Acta Metall. 30 (1982) 695. 1331 D. Raabe and K. Liicke, Z. Metallk. 85 (1994) 302. 1341D. Raabe and K. Liicke, Proc. ICOTOM 10, Mater. Sci. Forum 157-162 (1994) 1469.