Composites Science and Technology 55 (1995) 57-61

0 1995 Elsevier Science Limited

Printed in Northern Ireland. All rights reserved ELSEVIER 0266-3538(95)00094-l 0266.3538/95/$09.50

INTRODUCTION OF A MODIFIED LINEAR RULE OF MIXTURES FOR THE MODELLING OF THE YIELD STRENGTH OF HEAVILY WIRE DRAWN IN SITU

COMPOSITES

Dierk Raabe & Ude Hangen

Institut fiir Metallkunde und Metallphysik, Kopernikusstrasse 14, Rheinisch-Westfiilische Technische Hochschule Aachen, 52056 Aachen, Germany

(Received 1 February 1995; accepted 8 June 1995)

Abstract A modified linear rule of mixtures for the calculation of the yield strength of heavily wire drawn in situ processed metal-matrix composites which consist of a face centred cubic (fee) phase (matrix component) and a body centred cubic (bee) phase (minor component) is suggested. The approach 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 presence of fee-bee phase boundaries. The model is tested by using data of a Cu-20% Nb composite. The predictions are in good accord with experiment.

Keywords: linear rule of mixtures, yield strength, wire drawn composites, modelling

1 INTRODUCTION

In situ formed metal-matrix composites (MMCs) which consist of a face centred cubic (fee) matrix phase and a body centred cubic (bee) strengthening phase with negligible mutual solubility in the solid state can be manufactured by large strain wire drawing of a cast ingot. Typically, the fee phase contributes 70-99% and the bee phase l-30% to the total mass of the MMC. Common to such micro- or nanocomposites is their very high yield strength which is much greater than predicted by the linear rule of mixtures (ROM).‘”

The manufacturing, microstructure and resulting mechanical and electrical properties of such MMCs have been the subject of various studies.7-‘4 In recent investigations the high strength of fee-bee MMCs was attributed to the difficulty of propagating plastic flow through the internal phase boundaries* or to the

generation of geometrically necessary dislocations.15 Although yielding a good description of the strength in the case of a Cu-20% Nb alloy, both models depend on various fitting parameters which are often not available for related composites. A rigorous theoretical description of the high yield strength of such MMCs has not yet been attained. The present paper is hence concerned with the introduction of a new microstructural model which describes the strength of fee-bee MMCs in terms of dislocation arrangements typically observed close to internal phase boundaries, crystallographic textures and the morphology of both phases.

2 MODIFIED LINEAR RULE OF MIXTURES

For the description of the yield strength, ~Rpo.2, of fee-bee MMCs, a modified linear rule of mixtures (MROM) is suggested. It describes the yield strength, gRoM, of the composite in terms of the volumetric weighted average of the yield strengths of the individual pure phases (ROM), and a Hall-Petch type contribution, eMMC, arising from the presence of internal phase boundaries. Whereas CRoM is calcu- lated from experimental data, cMMC is derived theoretically:

(T RPO.2 = uROM + (TMMC (1)

Both portions, @ROM and UMMC, are computed as volumetric weighted averages of the strength con- tributions of the individual pure phases:

(+ROM = &&M&x + &$MVbcc

u MMC = (Tf%MCvfcc + &%Cvbcc (2)

where V,, and V,,, are the volume fractions, (T&,, and &&,, are the contributions to the ROM and fl$Mc and && are the corresponding contribu-

57

58

Fig. 1. Schematic representation of the dislocation arrange- ments at the internal phase boundaries as incorporated into

the model for the calculation of gMMP

Raahe, Ude Hangen

by about 20-30%. The true strain of the MMC is

defined as qMMC = In (A’IA,,,) where A0 and AMcc are the initial and the actual cross-section, respec- tively, after deformation. The true strains of the fee matrix and the bee filaments are denoted by 7rcc = In (A”/h) and nbcc = In (to/t), where t is the thickness and A the interfilament spacing.

From experimental data the ratio of the UTS of pure bee wires related to that of pure fee wires can

be derived as a function of strain: R(T) = a&&(~)l&&(rl) = &&,(T)/&~(T). In the pre- sent study it is stipulated that R also holds for the

relationship of a&&. and a&&. (Fig. 2):

tions of both phases resulting from the internal phase boundaries (Fig. 1). Since often no reliable ex- perimental data of the yield strengths of the pure wire drawn constituents are available for the fee ((~?o~) and bee phase (IT&) the linear ROM is calculated from the corresponding ultimate tensile strengths (UTS). This approximation is admissible in the case of large strains (7 > 4) but not in the case of low strains (7 < 4), where the UTS of most pure fee and bee metals which are of interest for the production of micro- and nanocomposites exceed the yield strength

3

_ 2.8

s 2.6 9

& 2.4

d 2.2 . - 2 5! E 1.8

% 1.6

t 1.4

u 1.2

1 0 2 4 6 8 10 12

true strain q

Fig. 2. Ratio of the yield strengths of pure wire drawn CU and Nb specimens as a function of strain, R(I)) 3

&%17)/&:“,(-rl) = c6&.47)/&&(77) (9 (3)).

The critical stress for dislocation movement between two impenetrable walls is given by:16

AGb S r m =Gln b ; As1.2

i!

where S is the distance of the phase boundaries, G is the shear modulus, b is the Burgers vector and A is a constant valid for mixed dislocations. Figure 1 shows the arrangement of dislocations at the phase boundary as assumed in the present approach. When dislocation multiplication rather than movement is considered, S has to be replaced by S/2 (eqn (4)).

Since the model accounts for heavily deformed MMCs rather than for as-cast materials it is assumed that the yield strength of the composite is reached when both phases deform plastically. Following eqn (4) it appears likely that the critical stress for dislocation movement required in the filament phase (bee) exceeds the stress required in the matrix phase (fee). The stress in the bee phase hence dominates the yield strength of the MMC. It is assumed that prior to plastic deformation of the composite, in the fee matrix phase dislocations pile up in front of the internal phase boundaries (Fig. l), causing an accumulated strain not exceeding 0.2%. At the tips of the pile-ups shear stresses are produced. Consequently, the effective shear stress on the slip planes in the bee filaments is increased. According to the linear approach outlined in eqn (2) and the configuration shown in Fig. 1 the following holds:

ml -__

(5)

where r$ is the shear stress at the tip of the dislocation pile-up, ylPIP is the number of dislocations in the pile-up, m is the orientation factor between the bee and fee glide systems and Mhcc is the Taylor factor of the bee filament phase. M is described by M = ~Z,Y’/E, 1 where Z-yi is the sum of the shear which is contributed by the active glide systems and &I1 is a

Introduction of a modified linear rule of mixtures 59

small step of the macroscopic strain imposed. Multiplied with the critical resolved shear stress the Taylor factor represents an orientation-dependent measure for the stored energy of a grain. It can be predicted for arbitrary modes of deformation and combinations of slip systems. The critical shear stress for dislocation movement in the bee filament, z,, is calculated according to eqn (4). The number of dislocations in a double ended pile-up between two interfaces is given by:”

where At = hlmfC, is the filament spacing normalised by the slip geometry, i.e. h is the distance measured perpendicular to the phase boundary and Y is Poisson’s ratio. The Hall-Petch type contribution, cMMC, and the shear stress on the bee slip system, z, are related according to rMMC = Mz. From eqns (3), (5) and (6) it follows that:

From eqn (7) one obtains:

Since negative stresses are not pertinent in this context the positive sign applies. According to eqns (2) and (3) the contribution of the phase boundaries to the yield strength then amounts to:

UMMC = (v,c + V&R) + d&c (9)

3 COMPARISON OF THE MODEL PREDICTIONS WITH EXPERIMENTAL RESULTS

The ratios were found to be A,,= A,,= A = 0.5. As starting values to = 1.2 pm and ho = 6 pm were extrapolated from transmisson electron microscope (TEM) data.” The initial filament thickness extrapo- lated deviates from the primary dendrite diameter

3.1 The Cu-20 mass% Nh in situ composite 2,400

In the following discussion the predictions of the MROM will be compared to experiment. For this purpose data of a heavily wire drawn Cu-20% Nb in situ composite were used which has been thoroughly investigated in the past.‘-lo Competitive binary systems of Cu with Ta, Cr, MO or V show a similar thermodynamical behaviour, i.e. a negligible mutual solubility in the solid state, but exhibit considerable shortcomings as compared to Cu-Nb. The density of Ta, Cr, MO and V strongly deviate from that of Cu so that gravitational segregation during solidification and the increase of the specific density of the final alloy reduce its potential for technological applications. Moreover, the melting temperatures of Ta and MO are even higher than that of Nb which complicates the

fl =) 1,600

g 1,600 P f 1,400 u) $J 1,200

e 2 1,000

‘al

ii 600

E 3

600

Fig. 3. Ultimate tensile strength versus international annealed Cu standard, IACS, of various &-based in situ

composites and conventional alloys.

melting process. Cu and Nb reveal nearly equal densities (8890 and 8580 kg/m3). Therefore gravita- tional segregation does not occur. Increased Nb content leads to a decrease of ductility, i.e. of elongation to fracture, and to a degradation of electrical conductivity.5-8 Thus, an alloy containing 20% Nb appears to combine optimum electrical and mechanical properties (Fig. 3).

3.2 Incorporation of the filament geometry In the as-cast state the Nb dendrites are oriented randomly.‘1-‘4 With increasing degree of wire deformation the dendrites align parallel to the drawing axis.14,15 For q> 4 the composite consists of elongated Nb filaments embedded parallel in the Cu matrix (Fig. 4). The present model is valid only for a well-aligned and homogenous morphology, i.e. for 7 > 4, and uniaxial strain. In order to check the model by use of experimental data the filament geometry of wire drawn Cu-20% Nb as measured previously18 was incorporated. The following expressions for the filament thickness, t, and spacing, A, were derived by fitting:

t = t’exp ( - &~~TIMM~)

h = h’exp ( - &“771~1~) (IO)

The true strains of both phases related to that of the MMC, IL-~ and hNb are denoted by:

A,“== 77~Mc

&&2= VMMC

40 50 60 70 60

Intern. Anneal. Cu Stand., IACS

60 Dierk Raabe, Ude Hangen

Fig. 4. TEM micrograph showing a longitudinal section of a heavily wire drawn Cu-10% Nb composite with parallel Nb filaments (dark) embedded in the Cu matrix (bright),

?J = 10.

observed by use of a scanning electron microscope (SEM) and optical microscopy (tOden = 6.2 pm),‘* indicating that the curve fitted from the TEM data is valid only for true strains n > 4.

3.3 Contribution of the ROM The model regards the yield strength of the MMC as the sum of the volumetric weighted average of the yield strengths of the individual pure constituents

(ROM), (TRoM and a Hall-Petch type contribu- tion attributed to the presence of internal phase boundaries, crMMc, i.e. (TROM is calculated from experimental data of pure Cu and Nb wires. After heavy deformation pure Nb wires reveal a higher UTS, &k,(n), than pure Cu wires, cr$s(n). Figure 2 shows the ratio R(n)=&&(q)/&&(n) = (T~&~(~)/(T~~~(~) as a function of strain. It is

/ “.\ \ L__. .i

/i -,‘_.+ /i//l-J@ !,

1”” 110

Fig. 5. Texture of the Nb filaments in the wire drawn MMC, 7 = 10. Measurement by the use of X-ray diffraction.

stipulated that this relationship also holds for aMMc

(eqn (3)).

3.4 Consideration of the slip geometry and the Taylor factors After heavy wire drawing of the MMC both phases reveal different crystallographic textures.“m’4 The orientation distribution of the Nb filaments is characterised by a crystallographic (110) direction parallel to the wire axis (Fig. 5). The Cu matrix reveals a (111) direction parallel to the drawing direction (Fig. 6). Accounting for full constraints Taylor type conditions (FC)” the corresponding Taylor factor for Cu amounts to MEA* = 3.16 and for Nb to ME;* = 3.67 ((llO}(lll) slip systems) or M f$* = 3.18 ({110}(111), {112}(111) and {123}(111) slip systems), respectively. In heavily wire drawn cu-20% Nb the filaments reveal a curled morphology’ indicating that FC conditions are not fulfilled locally. Correspondingly, the Taylor factors for relaxed constraints conditions (RC, relaxation of all shear strains)*’ were additionally considered, i.e. M’,“A’= 2.45 and MKz8 = 2.15. For single slip the orientation factors for Cu and Nb are 0.27 and 0.4, repectively. The orientation factor between the Cu and the Nb slip systems at the phase boundary is m = 0.98.

3.5 Comparison of simulation and experiment Using the experimental input, @MM<: (eqn (8)) can be computed as a function of the true strain:

uMMC = (0.8 + 0.2R)( - 4.2;;a R

4.25 MPa +

MNh Rr + 5.9 MPa’(9.25 - z))exp (z)

(12) As is evident from Fig. 7 the best agreement between the experimental results8,21 and the model is yielded for the simulation which is based on MEt8 = 2.15 and on dislocation movement rather than on multiplication (eqn (4)). Whereas for large strains (7 > 4) the accord between the MROM and the experimental data is very satisfying, for low strains (n < 4) the prediction shows a considerable deviation from experiment. This is attributed to the fact that the UTS, a&, &‘&,

Fig. 6. Texture of the Cu matrix in the wire drawn MMC, 77 = 10. Measurement by the use of X-ray diffraction.

Introduction of a modified linear rule of mixtures 61

1,600 c 1

-z

1,000

5 5 600 P it * 600

0 2 4 6 6 10 12

true strain q Fig. 7. Results of the simulation (MGT= 2.15, dislocation movement) compared to experimental results.‘.21 Both contributions to the total yield strength, i.e. aRoM and oMMC are shown separately. It becomes apparent that the deviation at low strains (7 < 4) is entirely attributed to uROM which was calculated from the UTS rather than from the

yield strength of the pure constituents.

rather than the yield strengths of the pure wire drawn constituents was employed for the computation of UROM. Furthermore, in this strain regime (77 < 4) the phases are not yet aligned parallel.

4 CONCLUSIONS

A MROM for the description of the yield strength of an in situ processed metal-matrix composite which consists of a fee phase (matrix component) and a bee phase (minor component) was suggested. The approach describes the yield strength of the composite in terms of the volumetric weighted average of the yield strengths of the individual pure phases and a Hall-Petch type contribution which is attributed to the presence of the internal fee-bee phase boundaries. The latter term is described in terms of dislocation

pile-ups in the fee matrix phase and the movement or multiplication of dislocations in the bee filament phase. The crystallographic texture and the morphol- ogy of both phases was considered. The model was tested using data of a Cu-20% Nb in situ composite. The predictions are in good accord with experiment.

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