Load transfer in short fibre reinforced metal matrix composites

Acta Materialia 55 (2007) 5389–5400 www.elsevier.com/locate/actamat

Load transfer in short fibre reinforced metal matrix composites Gerardo Garces a, Giovanni Bruno a

b,1

, Alexander Wanner

c,*

Department of Physical Metallurgy, CENIM-CSIC, Av. Gregorio del Amo 8, 28040 Madrid, Spain b Manchester Materials Science Centre, Grosvenor Street, Manchester M1 7HS, UK c Institut fu¨r Werkstoffkunde I, Universita¨t Karlsruhe (TH), D-76128 Karlsruhe, Germany Received 28 March 2007; received in revised form 1 June 2007; accepted 2 June 2007 Available online 7 August 2007

Abstract The internal load transfer and the deformation behaviour of aluminium–matrix composites reinforced with 2D-random alumina (Saffil) short fibres was studied for different loading modes. The evolution of stress in the metallic matrix was measured by neutron diffraction during in situ uniaxial deformation tests. Tensile and compressive tests were performed with loading axis parallel or perpendicular to the 2D-reinforcement plane. The fibre stresses were computed based on force equilibrium considerations. The results are discussed in light of a model recently established by the co-authors for composites with visco-plastic matrix behaviour and extended to the case of plastic deformation in the present study. Based on that model, the evolution of internal stresses and the macroscopic stress–strain were simulated. Comparison between the experimental and computational results shows a qualitative agreement in all relevant aspects.  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Metal matrix composites (MMC); Fibres; Neutron diffraction; Internal stresses; Plastic deformation

1. Introduction Short-fibre reinforced metal matrix composites (SFMMCs) exhibit higher strength and much improved creep resistance compared to the corresponding unreinforced metals and alloys [1]. Their microstructure is characterized by a random distribution of short fibres in a plane (with an aspect ratio 50). Such composites with a relatively low volume fraction (<30%vol.) of randomly oriented short fibres are in many cases more attractive from an application point of view than composites with a high content of particles, aligned whiskers or continuous fibres. They exhibit a more balanced property profile, are less anisotropic and present a better mechanical response under multiaxial loading conditions. The mechanics of MMCs reinforced by particles or by aligned fibres has been widely explored over the past decades and is very well understood (for an overview we refer *

1

Corresponding author. Tel.: +49 721 608 4160; fax: +49 721 608 8044. E-mail address: [email protected] (A. Wanner). Present address: Corning SAS, CETC, BP3, F-77210 Avon, France.

to Ref [2]). In contrast, due to the complexity of their microstructure, the processes governing the behaviour of random fibre-reinforced composites during elastic, plastic or creep deformation are difficult to grasp and it is still a challenge to develop reliable models. It is accepted that plastic deformation of SF-MMCs is strongly affected by two competing mechanisms [3–6]: (i) the load transfer from the soft, compliant matrix to the stiff, hard fibres under an applied load and the (ii) internal damage reducing the loadbearing capacity of the fibres in the form of fibres fragmentation, buckling and de-bonding, as well as void formation at the fibre/matrix interface. Dlouhy et al. [4,5] have established a simplified micromechanical creep model based in three simultaneous processes: (i) loading of fibres through the formation of a work hardened zone (WHZ), (ii) recovery, which decreases the dislocation density in the WHZ, and (iii) fragmentation of the fibres. This model is able to rationalize creep curves obtained experimentally at different temperatures and under uniaxial tension. While this model describes the micromechanics at the level of an individual fibre in a detailed and sophisticated fashion, the multidirectionality

1359-6454/$30.00  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.06.003

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of the reinforcement is not truly incorporated. Instead, the fibres are represented by a unidirectional arrangement of fibres exhibiting an ‘‘effective’’ volume fraction. The model is thus restricted to the specific case of uniaxial tensile loading. For uniaxial compressive loading, the model would predict a complete absence of fibre fragmentation and a much increased creep resistance, which is not experimentally observed. Creep tests and acoustic emission measurements carried out by Bidlingmaier et al. [7] on a short-fibre reinforced Al-based alloy have shown that the compressive creep strength is only moderately higher than the tensile creep strength and that fragmented fibres are also observed in specimens deformed in compression. In the case of compressive deformation, however, damage is mainly observed for fibres oriented transverse to the load axis and appears to occur at greater strains than in the case of tensile deformation. Inspired by these observations, Wanner and Garces [8] developed a model in which the orientation of the fibres is fully taken into account using an effective stiffness tensor and internal damage is included within the frame of the continuous damage mechanism (CDM) [9]. The discontinuous fibres are treated as if they were endless but exhibited a reduced elastic modulus. The experimental observation that the fibres gradually loose their efficiency due to fragmentation is introduced into the model via a nonlinear stress–strain relationship: with increasing elastic strain, the elastic modulus of the fibres is assumed to decrease monotonically to zero. This is a straightforward way to introduce he internal damage not only caused by fibre fragmentation, but also by fibre buckling. The Wanner–Garces model rationalizes the anisotropic creep behaviour observed on composites exhibiting a 2D-random orientation distribution [7,10–12] and it reflects internal evolutions observed experimentally under different loading modes [13,14]. It also shows that the internal load transfer must be treated as three-dimensional since the volume-averaged stress state in the matrix is generally biaxial or triaxial even under simple uniaxial loading conditions. Therefore, experimental studies of load partitioning in such composites must include stress measurements in the matrix in the three principal directions. Neutron diffraction has been shown to be a viable technique for this purpose (e.g. [6,15–19]). Due to their large penetration depth, neutrons allow lattice strain measurements in the bulk of the specimen under investigation, thus avoiding unwanted surface effects. The orientation of the scattering vector with respect to the specimen axes can be changed arbitrarily by simple specimen manipulation, which is a prerequisite for three-dimensional stress analysis. In a first approach, we have studied the residual stresses after plastic pre-deformation for different loading conditions [15,16]. In the next step, we performed a first in situ measurement of the internal strains [17]. A limitation of these studies was the impossibility to obtain absolute principal matrix stresses. However, it was shown that already

the principal stress differences (deviatoric stresses) offered substantial new insight into the mechanics of SF-MMCs. In particular, correlating the evolution of the deviatoric internal stresses to the macroscopic stress–strain curve during an in situ experiment allowed explaining the strengthening mechanisms in SF-MMCs during uniaxial compression in a direction parallel to the fibre plane. It was possible to divide the compression curve into three regimes: an elastic regime, a high strain-hardening rate regime and high damage region, where fibres are heavily damaged by fragmentation and buckling and compromise their capacity to store elastic strain energy. A simple Eshelby-based model [20] was successfully used to rationalise this scenario. This model did not contain any matrix strain hardening and assumed elastic-perfectly plastic matrix behaviour. The aim of the present work is twofold: 1. To enhance the understanding of the mechanical behaviour and internal damage in SF-MMCs by in situ neutron diffraction analysis of two further modes of loading: uniaxial tension parallel to the fibre plane (TP mode) and uniaxial compression normal to the fibre plane (CN mode). As mentioned above, the deformation and damage characteristics SF-MMCs depend strongly on the loading mode. 2. To seek a further generalization of the Wanner–Garces model described above: In order to understand the damage during plastic deformation in different loading modes, the Wanner–Garces model, which introduced damage under the CDM theory under creep condition, is modified to account for matrix plasticity rather than matrix creep.

2. Experimental procedures 2.1. Specimen material and sample preparation The composite material investigated in the present study was prepared via squeeze casting at Mahle GmbH, Stuttgart, Germany. They consisted in an aluminium matrix reinforced with 10% or 15%vol. Al2O3 fibres. Fibre preforms with dimensions 55 · 73 · 20 mm3 were positioned and supported in the centre of the cylindrical casting mould with height 75 mm and diameter 150 mm. The fibre performs consisted of discontinuous Al2O3 fibres (Saffil by ICI), and the fibre volume fractions in the final composites were 10% and 15%. Fig. 1 shows the microstructure of the Al–15%vol. of Saffil composite as well as the co-ordinate system definition. The fibres exhibit a random-planar orientation distribution. No voids were found in the material by light-optical inspection of polished cross-sections. According to the preform specifications, the average fibre length and diameter are 150 lm and 3 lm, respectively. Tensile and compressive samples were produced for testing in different loading modes (see photo in Fig. 2) for both in situ neutron diffraction as well as ex situ testing. Both

G. Garces et al. / Acta Materialia 55 (2007) 5389–5400

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specimens was always parallel to the fibre plane. For the TN mode samples had a cross-section of 4 · 2 mm2, a gauge length of 6.5 mm and a curvature radius of 3 mm. The axial direction of these specimens was always perpendicular to the fibre plane. For tensile testing cylinder with an initial length of 200 mm and a diameter of 9 mm were cut from the composite initial ingot. The diameter of this cylinder was machined down to 8 mm. For CP and CN modes samples were simple cylinders with a length of 18 mm and a diameter of 9 mm. Fig. 3 shows a sketch of the geometry of the tensile and compressive samples for in situ experiments. 2.2. Uniaxial tests

Fig. 1. Orthogonal cross-sections illustrating the three-dimensional reinforcement architecture of the composite (in the example: Al–15%vol. of Saffil). The material coordinate system is also represented. The axis X2 is normal to the fibre plane.

The specimens were deformed uniaxially at a nominal strain rate of 104 s1 (in both tension and compression) using a screw-driven electromechanical test rig (Instron). In the case of the compression tests, in order to reduce the friction between the specimen and the punches in compression tests, fine boron nitride powder slurry was used as lubricant. 2.3. In situ neutron diffraction measurements The in situ neutron diffraction measurements were carried out in the neutron beamline SALSA [21] at the reactor source of the Institut Laue-Langevin (ILL), Grenoble, France. Fig. 4 shows a sketch of the experimental set-up, with the samples mounted on the test rig in the longitudinal and transverse directions. The specimens were deformed uniaxially using a servo-hydraulic Instron unit, capable of 50 kN maximum load. The rig was placed on the sample manipulator, a Stewart platform [22], as shown in Fig. 5 (see also [17]). ˚ Monochromatic neutrons with wavelength k = 1.77 A were used. A gauge volume of approximately (due to the beam divergence) 4 · 4 · 4 mm3 was defined by primary

Fig. 2. Photo of the tensile/compressive test samples investigated in this study. The original billet is also shown.

X2

CP

10%, 15%

X3

tensile and compression specimens were produced with axes parallel to the fibre plane (TP and CP modes, respectively). However, due to the limited preform height, for the loading direction perpendicular to the fibre plane only compression samples could be produced (CN mode). The samples were labelled according to the loading mode and the reinforcement content (in volume percent). Therefore, for example, TP10 indicates the tensile samples with fibres parallel to the loading axis, containing 10 vol.% of Saffil fibres. For the TP mode cylindrical samples with a gauge length of 28.5 mm and a diameter of 5 mm were machined from the initial batch (Fig. 2). The axial direction of these

X1 CN

X3 15%

X2 X1 TP

X2 10%

X3

X1

Fig. 3. Sketch of the specimens used for in situ experiments. The coordinate systems (for CN, CP, TP) are also included, as defined in Fig. 1.

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Primary slit q5

Primary slit

PSD

q3

q1

q4

q2

q6 PSD Sample

Secondary slit

Sample

Secondary slit

Fig. 4. Sketch of diffraction geometries for two perpendicular arrangements with respect to the load axis: (a) transversal, (b) parallel. The scattering vectors qi and the tilt angles wi are also indicated.

Fig. 5. Photo of the stress rig mounted on the Hexapod stage of beamline SALSA at the ILL, Grenoble, France.

and secondary slits. The gauge volume was always positioned in the centre of the specimen under investigation. Volume-averaged lattice strain measurements were performed on the matrix of the composite using the 311 peak of aluminium, which was found at a Bragg diffraction angle around 2h  91. The counting time for each peak was around 3 min. The Saffil (Al2O3) reinforcement phase was previously observed to be essentially amorphous by

X-rays [15] and therefore no peak was searched. In both the tensile and compressive tests, the applied load was increased stepwise. At each loading step, the stress was held constant while the 311 peak was acquired at six different specimen tilt angles w. The tilt angle w was defined as the angle between the specimen long axis (direction X3 for CP and TP or X2 for CN, see Fig. 1) and the scattering vector q (bisecting the angle between the incident and the diffracted beam). Thus, at w = 0 the scattering vector was parallel to the axial direction of the specimen, while at w = 90 the scattering vector was oriented in a radial direction (i.e. transverse to load). The tilt angles actually accessed in the experiments were: for the compressive tests w1 = 1.0, w2 = 7.5, w3 = 10.5, and w4 = 88, w5 = 79, and w6 = 101, for the tensile tests w1 = 1, w2 = 6.0, w3 = 84, and w4 = 94 or w1 = 0, w2 = 94, i.e. three down to one tilt angles, grouped around w = 0 and w = 90. Each different tilt wi corresponds to a different scattering vector qi (see Fig. 4). The diffraction peaks were fitted with a Gaussian function. As it will be seen in the result section, the evolution of Al-2h311 was monitored for each tilt w. However, in order to increase grain statistics, an average of the three tilts around the axial direction and of those around the radial direction was done for each measurement to calculate stresses. Two independent experiments in both the tensile and the compressive tests were performed on different samples from the same batch. In the first test, the radial direction of the w tilt was oriented parallel to the 2D-random fibre plane (‘‘in-plane’’ measurements), in the second test the radial direction was oriented perpendicular to the fibre plane (‘‘out of plane’’ measurements). By combining the results of the two tests it was possible to monitor the evolution of elastic matrix strain in all three principal axes (see Fig. 1). In the case of the CN sample, only one orientation was investigated for symmetry reasons. 2.4. Stress analysis As outlined in Ref. [17], the principal stress differences are obtained from the pair-differences between the Bragg angles h1, h2 and h3:

G. Garces et al. / Acta Materialia 55 (2007) 5389–5400

r1  r3 ¼ 

E ðh1  h3 Þctgh ð1 þ mÞ

r2  r3 ¼ 

E ðh2  h3 Þctgh ð1 þ mÞ

r1  r2 ¼ 

E ½ðh1  h3 Þ  ðh2  h3 Þctgh ð1 þ mÞ

ð1Þ

Here h1 is the Bragg angle measured when the scattering vector is oriented parallel to the principal direction i. Hence, also the von Mises equivalent stress qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 rvM ¼ pffiffiffi ðr1  r2 Þ þ ðr2  r3 Þ þ ðr3  r1 Þ ð2Þ 2 is readily obtained from the measured quantities (h1h3) and (h2h3). In our case the latter quantities correspond directly to the average values of the measured 2h around w = 0 (h3) and w = 90 (h1 or h2 for the in-plane or the out-of-plane tilts, respectively). To compute the stresses, we used the plane-specific Young’s modulus E311 = 69.3 GPa and Poisson’s ratio m311 = 0.353, as calculated using a program by Wern [23] based on a Kro¨ner model. 3. Experimental results In Fig. 6 the tensile and compressive stress–strain curves for the fibre-reinforced composite in a direction parallel (P) and normal (N) to the fibre plane are shown for both the composites with 10%vol. and the 15%vol. of Saffil reinforcement. For comparison, also the unreinforced matrix material compression curve is displayed. Although the composites show different plastic behaviour depending on the stress mode, there are some similarities in all cases considered: Right after yielding, a pronounced strain hardening is observed, where the flow stress increases rapidly. However, at a certain strain, the strain hardening decreases

350

Stress (MPa)

300 250 200

TP15 TN15 CP15 CN15 CP10 Matrix

150 100 50 0 0.00

0.02

0.04

0.06

0.08

0.10

Strain Fig. 6. Experimental stress–strain curves for CP/TP15, CN/TN15, CP10, and the unreinforced material. The latter two are represented with thick lines.

5393

more or less abruptly. The stress and the strain at which this is observed depend on the deformation mode and we will see that it is related to the internal damage during deformation. The composite in the CP mode yields at about 110 MPa and exhibits pronounced strain hardening in the strain range from 0 to about 0.02. Here the flow stress reaches about 300 MPa. In the further course of compressive deformation, the strain hardening is much less pronounced and at 0.09 strain a flow stress of 350 MPa is reached. This behaviour is similar to that of TN, although in the latter case a lower maximum stress is reached (about 230 MPa) and the sample breaks at a strain of about 0.03. The CN and TP specimens exhibit a more sudden transition from the high to the low strain hardening behaviour and this transition occurs already at a strain of about 0.003. The ultimate tensile strength is 185 MPa for TP. For CN a compressive stress of 220 MPa is reached at a strain of 0.07. In fact, the composites in the CN and TN modes are placed between the CP and TP modes. In the case of the TN mode the change in the work hardening rate was not observed and it is expected that it will appear at higher strain. However, fracture occurs before that. The CP composite reinforced with 10%vol. of Saffil fibre follows the same tendency but fibre damage occurs at lower applied stress. Moreover, strain hardening is less pronounced in both regions mentioned above. Fig. 7a–g shows the Al-311 peak positions (2h) plotted against the applied stresses for all tilt angles w measured in the ‘‘in-plane’’ and ‘‘out of plane’’ sample orientations. In the compression cases, a peak shift to higher 2h values is observed with increasing compressive stress for the three w tilts near to 0 (axial direction), while the peak position remains roughly constant for the three w tilts near to 90 (radial directions). In the tension case, the 2h values are shifted to lower values with increasing applied stress for the w angles near to 0 (axial direction). Similar to the compression case, the peak positions for the three w tilts near to 90 (radial directions) remain roughly constant. It must be noted that the agreement of the three measurements (at different tilts w) within one group is rather weak. This is caused by the different families of grains sampled at the different w tilts and by the fact that the grain size is very coarse, as typical in cast aluminium. In order to diminish the effects of this kind of scatter on the stress results in the following, the average peak positions of the three tilts grouped around each principal direction were always taken. In Fig. 8a–c, the evolutions of the principal stress differences (Eq. (1)) and of the von Mises equivalent stress (Eq. (2)) are plotted against the applied stress for the experiments carried out in the present study as well as for the CP10 data published earlier [17]. We report here that we have detected a computation error in our earlier work which we have accounted for by multiplying the CP10 results shown in [17] by a factor of 2. As we will see later,

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In-plane

Out-of-plane 91.2

91.2

CP10

91.1

o

ο

90.8 90.7

ο

2θ ( )

90.9

ψ o

-7.5 o 1 o 10.5 o 79 o 88 o 101

91.0

2θ ( )

-7.5 o 1 o 10.5 o 79 o 88 o 101

91.0

CP10

91.1

ψ

90.9 90.8 90.7

90.6

90.6

90.5 -300

-250

-200

-150

-100

-50

90.5 -300

0

-250

-200

Applied Stress (MPa) 91.2

91.2

91.1

91.1 ψ

90.8

ο

2θ ( )

ο

2θ ( )

90.9

90.6

0

-250

-200

-150

-100

-50

ψ o

-7.5 o 1 o 10.5 o 79 o 88 o 101

90.9 90.8 90.7 90.6

CP15

90.5 -300

90.5 -300

0

-250

-200

-150

-100

-50

0

Applied Stress (MPa)

Applied Stress (MPa) 91.2

91.2

TP10

91.1

ψ

91.0

90.8

ο

90.9

2θ ( )

o

-1 o 5 o 84 o 94

90.9

o

90.7

90.6

90.6

50

100

150

90.5

200

ψ

0 o 94

90.8

90.7

0

TP10

91.1

91.0 ο

-50

91.0

o

-7.5 o 1 o 10.5 o 79 o 88 o 101

90.7

2θ ( )

-100

CP15

91.0

90.5

-150

Applied Stress (MPa)

0

50

100

150

200

Applied Stress (MPa)

Applied stress (MPa) 91.2

CN15

91.1

ψ

ο

2θ ( )

91.0

o

-7.5 o 1 o 10.5 o 79 o 88 o 101

90.9 90.8 90.7 90.6 90.5 -300

-250

-200

-150

-100

-50

0

Applied Stress (MPa) Fig. 7. (a)–(g) The Bragg diffraction angle 2h vs. applied stress for all w tilts. The left column shows the in-plane and the right column the out-of-plane measurements. Note that increasing applied stress is pointing to the left for the compression case.

G. Garces et al. / Acta Materialia 55 (2007) 5389–5400

symmetry with respect to axis X2 in all cases (see Fig. 1). This is in agreement with the rotational symmetry of the 2D random fibre orientation.

100

Matrix stress (MPa)

80

σ1-σ3(in-plane)

σ2-σ3(out-of-plane)

σ2-σ1(out-of-plane)

σVM

60 40 20

3.2. Compressive applied stress (rapp < 0)

TP 10%

0 -20

CP 10%

-40 -60

III

II

I

-80 -300 -250 -200 -150 -100

-50

0

50

100

150

Applied Stress (MPa) 100 80

Matrix Stress (MPa)

5395

60

In the compressive tests, both CP modes (for 10% and 15%vol. of Saffil reinforcement) show a monotonically increasing (r2  r3), while both (r1  r3) and rvM seem to have an inflection point around jrapplj = 120 MPa. Moreover, (r1  r3) and rvM go below (r2  r3) above a certain amount of applied stress. In the case of the CN mode, the stress difference (r2  r3) is less than zero and its amount increases steadily. At high amount of applied stress the increase seems to accelerate, analogously to what happens to CP10 (compare Fig. 6a and c).

40

CP 15% 20

3.3. Tensile applied stress (rapp > 0)

0 -20 -40 -60 -80 -300

σ1-σ3(in-plane)

σ2-σ3(out-of-plane)

σ2-σ1(out-of-plane)

-250

-200

-150

σVM

-100

-50

0

Applied Stress (MPa) 100

In the tensile test, both (r1  r3) and (r2  r3) tend to moderately increase (in absolute value) but at very high applied stress (near to rupture) the curves bend over, while (r2  r1) tends to vanish. The values of the stress differences are in this case smaller than in the compressive case. 4. Modelling

Matrix Stress (MPa)

80 60 40

CN 15%

20 0 -20 -40

σ2-σ1 = σ2-σ3 (out-of-plane) σVM

-60 -80 -300

-250

-200

-150

-100

-50

0

Applied Stress (MPa) Fig. 8. Principal stress differences and von Mises stress plotted against the applied stress for the cases: (a) CP/TP10, (b) CP15, and (c) CN15. In (a) the three region described in the text are indicated.

this does not undermine the conclusions drawn in [17] and in the present work, and even improves the agreement between experiment and modelling results. The following remarks can be made: 3.1. Zero applied stress (rapp = 0) The first measurement during the in situ compression tests was done at a small but non-zero applied stress. This was necessary to hold the sample in place on the rig. By extrapolation to zero applied stress, we find that initially (r1  r3)  0 in almost all the cases and the (r2  r3) values are negative. Moreover, it holds (r2  r3)  (r2  r1). Since the initial RS existing in the matrix is caused by the thermal expansion coefficient (CTE) mismatch between matrix and fibres, it is expectedly deviatoric and exhibits a rotational

As mentioned before, in recent works, the authors have used an Eshelby approach to rationalize the residual deviatoric stress as derived from neutron diffraction measurements [15,16]. The model assumed that during loading the von Mises equivalent stress (Eq. (5)) controls the onset of plasticity and from the point where rvM = ry on all principal stresses remain constant. Upon unloading, the same approximation was done and the residual stresses could be calculated in the completely unloaded state. This approximation proved to be very robust, because, at least qualitatively, the behaviour of the RS as a function of pre-strain was correctly predicted. Moreover, this model was successfully applied also to in situ experiments [17] and for instance could roughly predict the abrupt change of the slope of (r1  r3) vs. rappl observed for the specimen CP10 (see Fig. 7a). However, the assumptions used in that approach were somewhat crude, and in particular internal damage and matrix strain hardening were not taken into account. Therefore, we have extended Wanner–Garces model [8] outlined in the introduction by introducing matrix plasticity instead of matrix viscoplasticity. This was accomplished using the Levy–von Mises approach as outlined by Dieter [24] and developed further by Flaig [25]. 4.1. Internal damage The Wanner–Garces model [8] for multidirectional fibrereinforced composites is based on the following considerations: When a load is applied to a composite material,

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the stress in the composite, rc, is partitioned between the fibres and the matrix. For simplicity, the fibres are treated as continuous reinforcements and exhibit the same strain as the composite (iso-strain approach). Thus, the relation between the stress and strain in the fibre will be given by: rf ¼ C f  ef

Table 1 Parameters used in the modelling computations Component

Property

Value

Matrix

Young’s modulus, EM Poisson’s ratio, mM Matrix yield strength ryM Thermal expansion coefficient, aM Plasticity law

70 GPa 0.34 50 MPa 2 · 105 K1 50e0.1

Fibres

Young’s modulus, EF Poisson’s ratio, mF Thermal expansion coefficient, aF Volume fraction, f Aspect ratio (length/diameter), s kt kc

280 GPa 0.2 7.7 · 106 K1 0.15/0.1 50 100 60

Composite

Equivalent Eshelby temperature, DTE

200 K

ð3Þ

where rf is the stress borne by the fibres, ef is the elastic strain of the fibres and Cf is the stiffness tensor of a random distribution of fibres. In the following, we use the reduced Voigt notation. The components of this stiffness matrix can be computed as follows: Z p=2 Z 2p Ef f C fij ¼ aðh; uÞF ij ðh; uÞ sin h dh du ð4Þ C 0 0 1

where f is the fibre volume fraction, C is a normalization constant, Ef is the Young’s modulus of a fibre, a(h, u) represents the normalized orientation distribution function of the fibres, and Fij(h, u) is a trigonometric function, which depends on each stiffness tensor component. Damage in the form of fragmentation or buckling of the fibres is introduced by letting the Young’s modulus of each fibre be a function its own axial strain. It is thus assumed that the Young’s modulus Ef depends solely on the (scalar) axial strain of the fibre in its long direction, ef. The fibre elastic properties describing a real case in an optimum fashion are not known. Therefore, based on our previous results [8], the behaviour of the composite is simulated using an heuristic function. A back-to-back exponential was used with different decay ratios, kt and kc, for tension and compression: Ef ðef Þ ¼ Ef0 expðk t ef Þ if Ef ðef Þ ¼ Ef0 expðk c ef Þ if

ef P 0

ð5Þ

ef < 0

where Ef0 is the Young’s modulus of the undamaged fibres. This function is represented in Fig. 9, built by using the parameters listed in Table 1. The axial strain ef depends on the fibre orientation and the strain state of the composite. Therefore, the constant Ef in Eq. (4) must be replaced by a function Ef ðec1 ; ec2 ; ec3 ; h; uÞ and the stiffness tensor C fij 300 250

E

f

200 150 100

must be evaluated for each composite strain state of interest according to: Z p=2 Z 2p f Ef ðec1 ; ec2 ; ec3 ; h; uÞaðh; uÞF ij ðh; uÞ C fij ¼ C 0 0  sin h dh du

ð6Þ

For more details on this we refer to Ref. [8]. 4.2. Constitutive equations The strain in the matrix has two contributions: the elasm tic strain em elastic and the plastic strain eplastic . It is assumed that the matrix strain is identical to the strain of the composite (iso-strain approach). The matrix stress is thus: m c m rm ¼ C m em elastic ¼ C  ðe  eplastic Þ

ð7Þ

The matrix is assumed isotropic and the elastic components of the stiffness tensor are easily related to the Young’s modulus and the Poisson’s ratio of aluminium [26]. Based on force balance considerations, Eqs. (3) and (7) can be combined to: X X f m c rci þ ð1  f Þ Cm ½ð1  f ÞC m ð8Þ ij ej;plastic ¼ ij þ C ij   ej j

j

For plastic deformation, the directional contribution of plastic flow can be obtained from the Levy–von Mises equations [24]:   dem 1 m ep m m m de1 ¼ r1  ðr2 þ r3 Þ 2 re   m de 1 m ep m m m ð9Þ de2 ¼ r2  ðr1 þ r3 Þ 2 re   dem 1 m ep m m dem ðr ¼ r  þ r Þ 3 3 2 2 1 re

50 0 -0.10

-0.05

0.00

0.05

0.10

f

ε

Fig. 9. Fibre effective Young’s modulus function used in the model. The values of the damage rates, kt and kc, are given in Table 1.

These equations specify the principal matrix strain increments dem i in the direction i if, under a certain stress state re, the equivalent strain is changed by dem ep . In the computation carried out in the present work, this is considered as follows: For each given stress value, if the von Mises equivalent stress in the matrix exceeds the yields stress of the

G. Garces et al. / Acta Materialia 55 (2007) 5389–5400

matrix, a small increment in the equivalent plastic strain m dem ep is assumed and its directional components dei are calculated according to Eq. (9). The plastic strain increments are then added to the matrix strain and both the composite and the matrix stress states are calculated (Eqs. (7) and (8)). If the equivalent matrix stress is still in excess of the yield strength, the equivalent plastic strain is again increased by a small quantity. This procedure is repeated until the equivalent matrix stress falls below the yield stress. At each successive step, the matrix plastic strain is calculated from the current matrix stress and the increment of the matrix plastic strain in the current step is obtained. The step size of applied stress was 0.005 MPa. It was confirmed that this step size is in a regime of negligible step size dependency. The procedure simulates a stress-controlled experiment and thus becomes unstable in regimes where the composite exhibits strain-softening behaviour. The total matrix plastic strain is obtained by adding the incremental plastic strain to the sum of all strain increments from the previous steps. Matrix strain hardening has been taken into account using a typical power-law equation (Table 1). Since a nonlinear stress–strain relationship for the fibres was assumed, the resulting stiffness tensor Cf describing the integral behaviour of a distribution of fibres is a function of the composite strain tensor ec and must be computed at each step of the stress–strain curve simulations. The initial residual stresses stemming from thermal expansion mismatch between fibres and matrix were computed using an Eshelby model [20]. The 2D random distribution of fibres was taken into account following the treatment described in Refs. [6,27]. The values of the residual stress in the matrix are given by the following equation:  rm ¼ f C m ðhSi  IÞ ðC m  C f ÞðhSi  f ðhSi  IÞÞ 1

C m  C f ðam  af ÞDT

ð10Þ

where f is the reinforcement volume fraction, S is the Eshelby matrix, Cf and Cm are the reinforcement and matrix stiffness matrices, and I the identity matrix in the 6-D Voigt space, am and af are the coefficients of thermal expansion of the matrix and the inclusions, respectively, and DTE is the ‘‘effective’’ temperature drop during the cooling process in the composite production step that generates the misfit strain. As expected, all composites present tensile residual stress in the matrix, due to the difference of the coefficient of thermal expansion between the Saffil fibres and the aluminium matrix. 4.3. Modelling results The stress–strain curves computed for fibre contents f = 0.10 and f = 0.15 and for different loading modes are shown in Fig. 10. Also shown is the stress–strain curve of the unreinforced aluminium used in the computations. The composites exhibit a more pronounced strain hardening and a considerably higher ultimate strength than the unreinforced matrix. The highest ultimate strength is

5397

Fig. 10. Simulated global stress–strain curves for CP/TP10, CN/TN15, CP/TP15 compared with the unreinforced aluminium matrix input function. They are all superposed in same quadrant. Calculations are stopped at the maximum stress, to simulate a stress-controlled experiment.

observed for the TN mode, followed by CP and finally CN and TP (the latter two are almost equivalent). As expected, reducing the fibre volume fraction reduces the strengths in all modes without changing the order. Fig. 11 shows the simulated principal stress differences and the von Mises stress vs. applied stress for the cases of CP10 and TP10 (a), CP15 (b), and CN15 (c).

5. Discussion By comparing the simulated stress–strain curves of Fig. 10 with the experimental curves of Fig. 6, it can be seen that there is qualitative agreement in several aspects: The model reflects the experimentally observed initial pronounced strain hardening behaviour of all the composites in general and the fact that the stress level ultimately reached depends clearly on the loading mode. As far as the loading modes CP, CN, and TP are concerned, the model predicts the strength order in the right way. However, for the TN mode, the strength is grossly overestimated. As outlined in Ref. [8], a peculiarity of the TN loading mode is that the stresses in the matrix are strongly tensile hydrostatic, thus promoting the formation and growth of cavities. This damage mechanism and its detrimental effects are not covered by our model. In our earlier study [17] covering the CP10 experiment only (included in Fig. 8a), three regimes were identified in the evolution of the measured principal stress differences as a function of the applied stress: (I) Linear-elastic regime. The stress difference (r2  r1) remained constant and the stress differences (r1  r3) and (r2  r3) exhibited a linear dependence on the applied stress rappl. The slopes of (r1  r3) and (r2  r3) were found to be similar, i.e. the initial offset between these two quantities was conserved.

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Matrix Stress (MPa)

150

I

100

III

II

50

TP 10%

0 -50

σ2-σ1

CP 10%

σ2-σ3

-100 -150

σ1-σ3 σvM

-300 -250 -200 -150 -100

-50

0

50

100

150

Applied Stress (MPa)

Matrix Stress (MPa)

150 100 50 0 -50 -100 -150 -300

σ2-σ1 σ2-σ3

CP 15%

σ1-σ3 σvM

-250

-200

-150

-100

-50

0

Applied Stress (MPa)

Matrix Stress (MPa)

150 100 σ2-σ3 = σ2-σ1

50

σvM

0 -50 CN 15%

-100 -150 -300

-250

-200

-150

-100

-50

0

Applied Stress (MPa) Fig. 11. Simulated Stress differences and von Mises stress vs. applied stress. (a) CP/TP10, (b) CP15, and (c) CN15. The three regions mentioned in the text and shown in Fig. 8a are also displayed for the CP10 case.

(II) Enhanced strain-hardening regime. The out-of-plane stress difference (r2  r3) increased less rapidly, while the in-plane stress difference (r1  r3) substantially flattened and the out-of-plane stress difference (r2  r1) ran through a maximum. (III) Mild strain-hardening regime. The stress difference (r2  r3) remained constant and (r2  r1) dropped clearly, while (r1  r3) increased. The von Mises equivalent stress increased moderately as well, indicating some strain hardening in the matrix. The corresponding stress differences as computed using the model described above are plotted in Fig. 11a. It can be seen that the general shapes and the relative levels of

the three stress differences and the von Mises stress are very well reflected not only for the compression experiment CP10 but also for the tensile case TP10 (compare the experimental results in Fig. 8a). The values of the applied stress where the transitions between the regimes are observed are matched very well, while the magnitudes of the stress differences and the von Mises stress are slightly overestimated. This suggests that in reality the fibres shield the matrix more than in our model, indicating that the fibre damage function assumed is too severe. This is coherent with the observation that the simulated stress–strain curves (Fig. 10) attain a maximum while the experimental curves exhibit a monotonic increase over the whole strain ranges covered in the tests (Fig. 6). The plots in Fig. 11b and c show the modelling results obtained for the other two cases, CP15 and CN15. For each of the stress differences in these plots, the same three regimes as described above can be identified. Comparing Fig. 11b and c to the corresponding experimental data shown in Fig. 8b and c, respectively, a good quantitative agreement is observed, even better than the case of TP10 and CP10. In Fig. 12a–c, the principal stresses obtained from the model calculations are plotted against the applied stress. The values at zero applied stress are the results of the Eshelby model and represent the residual stresses from thermal mismatch. It can be seen that all residual principal stresses are tensile and that r2 is always smaller than r1 = r3. A striking feature is that the evolution of r2 with applied stress is unaffected by the presence of the fibres: for the CP and TP loading modes, r2 simply stays constant, and for the CN mode the applied stress is simply superimposed to the initial value r02 , i.e. r2 ¼ r02 þ rappl . All of our experimental analysis was concentrated on the aluminium matrix because the fibres did not show any useful diffraction peak. Still, it is possible to estimate these stresses making a simplifying assumption: In view the experimental and computational difficulties in determining the initial residual stresses quantitatively, we set the matrix residual stress component r2 to zero for CP and TP and equal to the applied stress r2(applied) for CN. We have to keep in mind that this simplification causes an offset in both the matrix and the fibre stresses. For the cases of CP and TP, we have thus two ways for computing an estimated matrix stress, r3 , from the measured stress differences: r3 ¼ ðr2  r3 Þ r3

and

¼ ½ðr1  r3 Þ þ ðr2  r1 Þ

ð11Þ

Analogously, the other matrix stress component in the fibre plane is estimated as follows: r1 ¼ ðr2  r1 Þ

and

r1 ¼ ½ðr1  r3 Þ  ðr2  r3 Þ

ð12Þ

We can take the average of the two results in both cases. Based on force equilibrium considerations, the stress r3ðfÞ and r1ðfÞ acting in the fibres can thus be estimated according to

G. Garces et al. / Acta Materialia 55 (2007) 5389–5400

σ3 (Applied stress direction)

100

σ1

1000

0

σ *1

CP TP CP TP

500

-50 -100

Al-10% Saffil- CP-TP

-200 -300 -250 -200 -150 -100

-50

0

50

100

150

Applied stress (MPa) 150

Principal stress (MPa)

Saffil

σ2

50

-150

σ1

50

σ2

0

- 500

- 1000

σ3 (Applied stress direction)

100

- 1500

- 2000 0 -300

-50

-200

-100

0

100

200

Applied Stress (MPa) -100 -150

-1500

Al-15% Saffil- CP

σ*3 Al

-200 -300

-250

-200

-150

-100

-50

0

Estimated Phase Stress (MPa)

150

σ3= σ1

100

σ2(Applied stress direction)

50 0 -50 -100 -150

Al-15% Saffil- CN

σ*1 CN CP CN CP

Saffil

-1000

Applied stress (MPa)

Principal stress (MPa)

σ *3 CP TP CP TP

Al

Estimated Phase Stress (MPa)

Principal stress (MPa)

150

5399

CP CP

-500

0

500

1000 -200 -300

-250

-200

-150

-100

-50

0

Applied stress (MPa)

1500

Fig. 12. Simulated principal stresses vs. applied stress for (a) CP/TP10, (b) CN/TN15, and (c) CP/TP15. The residual stresses are calculated with the 2D planar random Eshelby model described in the text.

r3ðfÞ r1ðfÞ

¼ ðr3ðappliedÞ  ð1 

f Þr3 Þ=f

ð13aÞ

f Þr1 =f

¼ ð1  ð13bÞ where f is the volume fraction of the fibres. In the case of CN we can compute the other two principal matrix stresses as: r3 ¼ r1 ¼ r2ðappliedÞ  ðr2  r3 Þ ð14Þ For a fibres volume fraction of f, we obtain the stress r3ðfÞ in the fibres as r3ðfÞ ¼ ð1  f Þr3 =f ð15Þ In Fig. 13a, the evolutions of the estimated matrix and fibre stress components are plotted against the applied stress for the CP and TP experiments on Al–10%vol. of

0

-50

-100

-150

-200

-250

Applied Stress (MPa) Fig. 13. Load transfer as a function of the applied stress: (a) CP/TP10 and (b) CN/CP15. In all cases, the estimated stresses in the fibre plane, r1 and r3 , are shown.

Saffil material. This plot illustrates that the stresses in the fibres attain much higher values than in the matrix. The compressive and tensile branches comply with each other very well. Taking into account that small experimental errors on the matrix stress will cause very large uncertainties on the estimated fibre stress (the amplification factor being proportional to (1  f)/f) the CP and TP results are remarkably consistent and can be combined into one continuous curve. The slope of this curve is about 5–6 in the central region and increases at the extremities of the measured region. This implies that all the measurements were

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G. Garces et al. / Acta Materialia 55 (2007) 5389–5400

done in the load regime where load transfer is still progressing and the detrimental effects of fibre damage are not yet dominating. The fibre stress in the transverse direction is small because the matrix can expand in direction X2 without any significant constraint from the fibres. In Fig. 13b, the evolutions of the estimated matrix and fibre stress components are plotted against the applied stress for the CP and CN experiments on Al–15%vol. of Saffil material. Similarly to Fig. 13a the stresses in the fibres are much larger than those in the matrix. The curves for CN and CP are symmetric and the slope is about 4. At large applied stress the two curves bend into opposite directions. This is most probably due to the fact that in the CN case the strain is much higher and therefore fibre damage is indeed occurring. In fact, even if in the CN mode the strain in the transverse direction is roughly half of the applied strain (plastic deformation), Fig. 6 shows that the applied strain in CN mode is 5–10 times larger than that in CP mode in the applied stress interval between 200 and 250 MPa. Analogously to the Al–10%vol. of Saffil material, the transverse stress in CP15 mode is small and does not vary as a function of the applied stress. Comparing the CP10 and CP15 results of Fig. 13a and b, respectively, it can be seen that, under the same applied load; the fibre stresses are larger for the smaller reinforcement volume fraction, as a result of the rule-of-mixtures. 6. Conclusions In the present study, we have for the first time experimentally characterized the evolution of the internal stresses and the internal load transfer in 2D random short fibre reinforced aluminium matrix composites under different external loading modes. The experiments gave insight into the load transfer from matrix to fibre under tensile and compressive stress for loading parallel and perpendicular to the reinforcement plane. The results have been discussed in the light of a composite model established earlier for random fibre-reinforced composites embedded in a viscoplastic matrix, which now has been extended to the case of plastic matrix behaviour. The unconventional approach of the composite model is that the slender but discontinuous fibres are treated as endless fibres carrying loads only along their axial directions and exhibiting a non-linear, strain-softening behaviour. Based on this simplification, the evolution of internal stresses and the macroscopic stress–strain curves can be simulated for arbitrary loading modes including the specific modes treated in the present study. Comparison between the experimental and computational results shows certainly a good qualitative agreement in all relevant aspects, and sometimes even a quantitative match. This demonstrates that the constraint from the fibre ensemble on the matrix flow and the result-

ing composite deformation behaviour are captured by our approach and thus help to rationalize the mechanics of random short-fibre reinforced composites. Acknowledgements The Institut Laue-Langevin (ILL, Grenoble, France) is kindly acknowledged for beamtime on the beamline SALSA. We thank ILL staff and the FaME38 Team at the ILL-ESRF (in particular Darren Hughes) for their technical support. GG thanks the Spanish Ministry of Science and Education for support via the Ramon y Cajal program. References [1] Clyne TW, Withers PJ. An introduction to metal matrix composites. Cambridge: Cambridge University Press; 1993. p. 73. [2] Suresh S, Mortensen A, Needleman A. Fundamentals of metal matrix composites. Stoneham (MA): Butterworth–Heinemann; 1993. [3] Dragone TL, Schlautmann JJ, Nix WD. Metall Trans A 1991;22:1029. [4] Dlouhy A, Merk N, Eggeler G. Acta Metal Mater 1993;41:3245. [5] Dlouhy A, Eggeler G, Merk N. Acta Metal Mater 1995;43:535. [6] Johannesson B, Ogin SL. Acta Metall Mater 1995;43:4337. [7] Bidlingmaier T, Wolf A, Wanner A, Arzt E. In: Proceedings of Werkstoffwoche’98. Symposium 8: metallurgy, symposium 14: simulation, vol. VI. Weinheim: Wiley–VCH; 1999. p. 471. [8] Wanner A, Garces G. Philos Mag 2004;84:3019. [9] Kachanov LM. Izd Akad Nauk SSSR Tekh Nauk 1958;8:26 [Engl. Transl.: Kachanov LM. Int J Fract 1999;97:xi]. [10] Kuchanova K, Horkel T, Dlouhy A. In: Misra RS, Mukherjee AK, Murty L, editors. Creep behaviour of advanced materials for the 21st century. Warrendale (PA): The Minerals, Metals and Materials Society; 1999. p. 127. [11] Yawny A, Kaustra¨ter G, Skrotzki B, Eggeler G. Scripta Mater 2002;46:837. [12] Huang YD, Hort N, Kainer KU. Composite A 2004;35:249. [13] Schu¨rhoff J, Yawny A, Skrotzki B, Eggeler G. Mater Sci Eng A 2004;387–389:896. [14] Tavangar R, Weber L, Mortensen A. Mater Sci Eng A 2005;395:27. [15] Garces G, Bruno G, Wanner A. Mater Sci Eng A 2006;417:73. [16] Garces G, Bruno G, Wanner A. Int J Mater Res 2006;97:1312. [17] Garces G, Bruno G, Wanner A. Scripta Mater 2006;55:163. [18] Withers PJ, Stobbs WM, Pedersen OB. Acta Metall 1989;37:3061. [19] Daymond MR, Lund C, Bourke MAM, Dunand DC. Metall Mater Trans A 1999;30:2989. [20] Eshelby JD. Proc Roy Soc A 1957;291:376. [21] Pirling T, Bruno G, Withers PJ. Mater Sci Eng A 2006;437:139. [22] Hughes DJ, Bruno G, Pirling T, Withers PJ. Neutron News 2006;17:28. [23] Wern H. XEC program. Germany: Hochschule fu¨r Technik und Wirtschaft Saarbru¨cken; 2000. [24] Dieter GE. Mechanical metallurgy. New York: McGraw-Hill; 1988. p. 90. [25] Flaig A. Doctoral Thesis, University of Stuttgart, Germany; 2000. [26] Nye JF. The physical properties of crystals. Oxford: Clarendon Press; 1985. [27] Brown LM, Clarke DR. Acta Metall 1977;25:563.