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SiCp metal–matrix composite
Acta mater. 48 (2000) 1563±1573 www.elsevier.com/locate/actamat
VISCOPLASTIC DEFORMATIONS AND COMPRESSIVE DAMAGE IN AN A359/SiCP METAL±MATRIX COMPOSITE Y. LI 1, K. T. RAMESH 1{ and E. S. C. CHIN 2 1
Laboratory for Impact Dynamics & Rheology, Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA and 2Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen Proving Ground, MD 21005, USA (Received 30 July 1999; accepted 9 November 1999)
AbstractÐRecent work by the authors has examined the high-strain-rate compression of a metal±matrix composite consisting of an A359 Al alloy matrix reinforced by 20 vol.% of silicon carbide particulates (SiCp). The work-hardening that is observed in the experiments is much lower than that predicted by analytical and computational models which assume perfect particle±matrix interfaces and undamaged particles. In this work, we show that the discrepancy is a result of particle damage that develops within the A359/ SiCp composite under compression. The evolution of particle damage has been characterized using quantitative microscopy, and is shown to be a function of the applied strain. A simple analytical model that incorporates evolving damage within the composite is proposed, and it is shown that the analytical predictions are consistent with the experimental observations over a wide range of strain rates. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Mechanical properties; Plastic; Composites; Aluminium alloys; Functionally graded materials (FGM)
1. INTRODUCTION
Particle-reinforced metal±matrix composites (MMCs) have signi®cant advantages (including high speci®c modulus, high speci®c strength and high wear resistance) over traditional metal alloys for some structural applications. A set of applications of substantial interest consists of functionally graded metal±ceramic composite structures in which the ceramic particle volume fraction is graded [1, 2]. Graded armor structures represent one speci®c application of this type. Since armor materials are designed to survive impact loading, the eect of microstructural features such as ceramic volume fraction, particle size, particle distribution and particle aspect ratio on the high-strain-rate response of particle-reinforced MMCs must be understood. The literature on the high-strain-rate properties of particle-reinforced metal±matrix composites has recently been reviewed by Li and Ramesh [3]. In summary, it is known that the strain rate hardening of a particle-reinforced MMC may be signi®cantly higher than that of the matrix material, provided the matrix itself is rate-sensitive. Li and Ramesh [3]
{ To whom all correspondence should be addressed.
also studied the eects of particle shape and aspect ratio on the high-strain-rate response of MMCs. They concluded that reinforcement geometry has a strong in¯uence on the ¯ow stress of MMCs at high strain rates. An analytical description of the uniaxial constitutive response of particle-reinforced MMCs was developed from these numerical results [3]. This analytical description was shown to be consistent with the experimental results of other workers on two dierent MMCs over a wide range of strain rates. In a more recent paper, Li et al. [4] present both experimental and computational results on the mechanical behavior of an A359/SiCp MMC in compression over a very wide range of strain rates. This material system is of interest because the matrix is a cast alloy, and thus provides a costeective processing route for the development of graded armor structures. Li et al. [4] show that the constitutive formulation previously used by Li and Ramesh [3] is able to describe the rate-dependence of the yield stress of the MMC, but is unable to capture the observed strain-hardening of this composite. The eective strain hardening of the composite was less than that of the matrix material at all strain rates, apparently because of the development of particle damage (even though the stress states were all predominantly compressive). This paper
1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 4 3 0 - 9
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presents an approach to the incorporation of evolving damage within the rate-dependent constitutive formulation of Li and Ramesh [3] for MMCs. The eects of damage on the overall composite response in tension have been widely studied for low rates of deformation, and several microscopic failure mechanisms have been reported (e.g. [5]). Typically, three kinds of failure (or combinations thereof) can occur in a metal±matrix composite: (i) cracking of the reinforcing particles [5±12]; (ii) partial debonding between the particle and matrix interface, resulting in the nucleation of voids within the matrix [6, 13, 14]; and (iii) the growth and coalescence of voids in the matrix [15, 16]. Often one of these three mechanisms dominates. Which mechanism is dominant will depend on the matrix and on the particle size, shape, distribution and volume fraction. For example, it has been observed that both particle cracking and void nucleation within the matrix tend to originate preferentially in clustered regions of high local volume fraction [6, 15]. The development of damage is a progressive process and the dominant mechanism may change as the damage evolves (e.g. Ref. [17]). Since particle fracture is readily observed in most failed composites, it is often suggested that this is the dominant failure process [18]. A decrease in work hardening under tension is observed with an increase in either particle volume fraction or particle size [7, 8, 11]. This is believed to be due to accelerated damage in the form of particle cracking in the higher volume fraction and/or larger particle size materials. Many experimental investigations [5, 7, 9, 11, 19±21] have been conducted to determine the relationship between particle damage and the average strain or stress. A number of models have also been developed [8, 18, 22±24] to predict the ¯ow stress in tension at low rates of deformation for a composite containing damaged particles. It appears that the incidence of particle damage is related not only to the overall stress or strain level in the composite, but also to the particle size, matrix properties and particle properties [25± 27]. There are few reports in the literature on the development of damage within MMCs under quasistatic and dynamic compressive loadings. Prangnell et al. [28] and Barnes [29] observed damage within SiC particle-reinforced aluminum matrix composites deformed in compression at low strain rates. Particle fracture was identi®ed as the dominant mechanism in regions of low local reinforcement volume fraction. Void formation was observed within the matrix in regions containing strongly clustered particles. On the other hand, Kiser et al. [20] observed no damage after low-rate compression of a SiC particle reinforced aluminum alloy MMC. In the latter study, the composites exhibited the same hardening characteristics as the corresponding matrix material. Several investigators have studied
the properties of MMCs under dynamic compression (see Ref. [3] for a review), but there are relatively few reports of damage as a result of compressive deformations. This paper describes the plastic deformation and resulting damage of an A359/SiCp metal±matrix composite subjected to compression over a very wide range of strain rates (10ÿ4±10+5/s). The experimental techniques and overall mechanical properties have been presented in a previous paper and so are only brie¯y delineated here. This paper focuses on the characterization of the evolving damage, and on the development of a model for the evolving damage based on the analytical model of Li and Ramesh [3]. Finally, this paper presents a comparison between the predictions of this damage model and the experimental results. 2. MATERIALS
The materials used in this study were A359 aluminum alloy and an A359 aluminum alloy reinforced by 20 vol.% of SiC particles (F3S.20S). The A359 aluminum alloy was cast in the form of a bar 1.5 inches in diameter by Alcan International Limited. The composition of the A359 aluminum alloy is as follows in wt%: 0.2 Cu, 0.5±0.7 Mg, 0.1 Mn, 8.5±9.5 Si, 0.20 Fe, 0.10 Zn, 0.20 Ti, 0.20 other, with the balance being aluminum. The composite was also cast in the form of a bar with a diameter of 1.5 inches. Optical micrographs of the as-cast unreinforced alloy and of the as-cast composite are presented in Figs 1(a) and (b), respectively. The unreinforced alloy shows the classic as-cast microstructure with an interdendritic aluminum±silicon eutectic containing ®ne silicon particles. The composite shows that the reinforcing silicon carbide particles are generally distributed along the eutectic phase, and that the particles are faceted, with a slight preferred orientation. Vickers microhardness measurements were made (Table 1) in the unreinforced alloy and in the matrix of the composite. The load was chosen so that the length of the long diagonal of the indent in the matrix was less than half the distance to the nearest particle. The results shown in Table 1 indicate that the hardness of the matrix material is very similar to that of the unreinforced alloy. This, together with the similarity of the microstructures, provides some con®dence that the rate-dependent properties of the unreinforced alloy can be assumed to be the properties of the matrix in the composite for the purposes of theoretical and analytical modeling. Quantitative microscopy was used to determine the statistical characteristics of the reinforcement distribution in the composite. Metallographic specimens were prepared by sectioning the as-cast composite bar along the bar axis and transverse to the bar axis (axial and transverse sections) using a low-
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Fig. 1. Optical micrographs of (a) as-received A359 alloy and (b) the as-received F3S.20S metal±matrix composite (A359 containing 20 vol.% SiC particles, obtained from Alcan).
speed diamond saw. After the surfaces were polished, scanning electron micrographs of ®ve locations on each section were used to measure the particle aspect ratio and particle size. The measured distributions of particle size and particle aspect ratio within the axial and transverse sections are presented in Fig. 2. The ``frequency'' in Fig. 2 is the percentage of the total number of particles that are present at any given size. Figures 2(c) and (d) show that the mean particle size in both the axial and the transverse sections is in the range of 6±18 mm. The population of particles with a size smaller than 10 mm is greater in the axial section than in the transverse section. There are very few particles larger than 24 mm. Most of the particles have aspect ratios between 1.5 and 2.5 in both axial and transverse sections [Figs 2(b) and (d)]. This suggests that most of the particles are present in the form of small plates.
3. EXPERIMENTAL PROCEDURES
The rate-dependent mechanical behaviors of both the unreinforced alloy and the MMC were studied in compression over strain rates of 10ÿ4±105/s using a combination of quasistatic and dynamic testing techniques. A servohydraulic testing machine was used to conduct compression experiments in the strain rate range of 10ÿ4±100/s. The compression Kolsky bar, or split-Hopkinson pressure bar, developed by Kolsky [30], was used to attain strain rates
of 102±6 103/s. Specimen recovery in the Kolsky bar experiment was achieved by using the recovery modi®cation developed by da Silva and Ramesh [31]. The high-strain-rate pressure-shear plate impact technique [32] was used in order to study the response of both the unreinforced alloy and of the composite at strain rates between 5 104 and 106/s. Detailed descriptions of these experimental techniques are presented by Li et al. [4]. 4. EXPERIMENTAL RESULTS: COMPARISON WITH LI AND RAMESH MODEL
A summary of the rate-dependent mechanical response of the unreinforced A359 alloy is presented in Fig. 3 for a wide range of strain rates between 10ÿ4 and 2 105/s. The strength of the matrix material increases substantially with increasing strain rate, and increases strongly at the higher strain rates developed within pressure±shear plate impact tests. The plate impact test results in Fig. 3 were converted from shear stress±shear strain data to true stress±true strain data using J2-¯ow theory [33], a reasonable assumption for this aluminum alloy. With the mechanical behavior of the matrix material well-characterized (Fig. 3), the numerical and analytical models of Li and Ramesh [3] have been used [4] to predict the rate-dependent response of the composite material. The analytical model of Li and Ramesh [3] for the rate-dependent response of the composite is given by the following equation:
Table 1. Comparison of Vickers microhardness and yield strength of the monolithic alloy and the matrix of the composite material (the yield strength is estimated using VHN/3) Material A359 Al alloy F3S.20S matrix
Mean diagonal (mm)
Vickers microhardness (kg/mm2)
Yield strength (MPa)
35.020.4 34.520.8
75.6 78.0
252 260
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LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
Fig. 2. Distribution of particle size and aspect ratio within the as-received Alcan composite. (a) Particle size distribution in axial section; (b) aspect ratio distribution in axial section; (c) particle size distribution in transverse section; (d) aspect ratio distribution in the transverse section.
Fig. 3. Stress±strain curves obtained on the unreinforced A359 alloy over the full range of strain rates (pressure±shear data incorporated using J2-¯ow approach).
LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
s f, e, e_ s0 eg f 1
e_ e_ 0
m !
m ! e_ 1 f e_ 0
1 where s is the overall ¯ow stress sustained by the composite with reinforcement volume fraction f at an overall strain of e and an overall strain rate of e_ : In equation (1), s0(e ) represents the stress±strain response of the matrix at quasistatic rates of deformation and is obtained directly from experimental data (Fig. 3), and m and e_ 0 are parameters that determine the rate-sensitivity of the matrix material (also determined from Fig. 3). The strengthening function g( f ) represents the dependence of the ¯ow stress ratio on the volume fraction f (at quasistatic rates) as computed using a unit cell model; a polynomial approximation to the numerical results [4] for this composite is g f 1 1:17f 2:28f 2 21:0f 3 : The predictions of this model are compared with experimental data on the composite in Fig. 4 (reproduced from Li et al. [4]). While the observed ¯ow stresses are reasonably well predicted at small strains, the predicted ¯ow stress increasingly overestimates the experimental ¯ow stresses at larger strains. The model does an acceptable job of cap-
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turing the rate-dependence of the composite ¯ow stress, but does not capture the strain hardening of the composite. The model assumes no microstructural damage and therefore predicts a strain hardening at large strains that is essentially the same as that of the matrix. However, a comparison of Figs 3 and 4 shows that the strain hardening of the composite is actually substantially less than that of the matrix alloy. Two pieces of evidence indicate that microstructural damage may have developed within this composite during compressive plastic deformations. First, the strain hardening of the composite is lower than that of the corresponding matrix alloy. Second, the experimentally observed strain hardening of the composite is lower than that predicted by analytical and numerical models that assumed no damage. Direct microstructural examination of the deformed composite (presented in the next section) shows that substantial particle damage occurs within the F3S.20S composite even during compressive deformations. The evolving damage must be incorporated within models of the type developed by Li and Ramesh [3] in order to better capture the overall response of the composite to compression over this range of strain rates.
Fig. 4. Comparison of predicted and observed stress±strain curves for the metal±matrix composite (A359 alloy containing 20 vol.% SiC particles) over the full range of strain rates. Pressure±shear data again incorporated using the J2-¯ow approach. Note that the predictions at the strain rates of 10ÿ4 and 100/s overlap.
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LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
5. MICROSTRUCTURAL CHARACTERIZATION OF EVOLVING DAMAGE
Quantitative microscopy was used to characterize the microstructural damage within this metal± matrix composite after compressive deformations. Deformed cylindrical specimens were sectioned in the longitudinal and transverse directions using a low-speed diamond saw. Each section was carefully polished and examined under the scanning electron microscope (SEM). Four random regions were chosen within each sectioned plane, and SEM micrographs were obtained from each region. Inspection of the micrographs showed little or no particle± matrix debonding and no evidence of matrix void nucleation. Each micrograph was further scanned and then processed using a set of custom imageprocessing routines to obtain particle statistics. Particle statistics and particle fracture statistics were obtained for each micrograph. The statistics were averaged over the four regions for each of the axial and transverse sections. Typically the statistics are obtained over 500 particles. Results are presented for two dierent strain rates: a quasistatic strain rate of 10ÿ4/s and a dynamic strain rate of 2 103 =s: Recovery experiments were performed at each strain rate, with the specimens deformed to various controlled strains in order to determine the evolution of the damage with strain (the corresponding stress±strain curves are shown in Fig. 5). Statistical data is not presented for the pressure± shear experiments because those experiments were not performed in a recovery mode, and so the observed damage cannot be related to the observed stress±strain response. The statistical information acquired from the recovered specimens included particle size, particle aspect ratio, and number of cracked particles. However, this paper presents only the size and number density statistics. A more complete description of the evolving damage will be presented in a forthcoming paper.
Fig. 5. Stress±strain curves obtained from recovery tests on composite specimens at two ®xed strain rates (10ÿ4 and 2 103/s), with each specimen deformed to a pre-determined strain.
The dierences between the initial particle size distribution (presented in Fig. 2) and the particle size distribution after deformation to a strain of 20% at a strain rate of 10ÿ4/s are presented in Fig. 6. The ``frequency change'' is de®ned as the change in the frequency of occurrence (as de®ned for Fig. 2) of particles of a given size as a result of the deformation (for example, a frequency change of ÿ5% means that there are 5% fewer particles of that size in the deformed composite). Figure 6 shows that the frequency of occurrence of large particles within the composite is reduced after compressive deformations in both the axial and transverse sections, whereas the frequency of occurrence of small particles increases. The large particles are evidently broken up into smaller particles during the deformation. It is apparent that most particles that are larger than 10 mm are fractured into pieces less than 10 mm in size. Figure 6 thus presents strong evidence of particle damage developed during quasistatic compressive deformations. Note that the data indicates that the incidence of particle damage and corresponding degradation of properties may be reduced by changing the initial particle size distribution. These observations of damage during compression may be compared with observations of particle damage under tensile loading [7, 8, 11] (note that other damage modes such as void nucleation are also present in tensile loadings). The increased probability of damage in larger particles is usually due to the increased probability of ®nding pre-existing defects within larger particles. It is worth noting that in this composite, those particles larger than 1±3 mm in size are typically polycrystalline, and may be failing along grain boundaries. While particle damage appears to be the predominant damage mechanism for this composite under compression, some degree of particle±matrix debonding may also occur, but not be perceptible with the optical and scanning electron microscopy used in this study. Debonding has been observed after dynamic compressive deformations in an underaged Al±Zn±Mg±Cu alloy reinforced with SiC particles [34]. Though incontrovertible evidence that debonding does not occur within the F3S.20S composite is lacking, there is ancillary evidence that suggests that this is not a signi®cant issue for the deformations considered here. Some degree of debonding has been observed in an 8090 composite [35]. However, the analytical model of equation (1) captured the experimentally observed behavior assuming perfect bonding [3]. This suggests that the eect of debonding on the composite ¯ow stress under compression is small. Typical micrographs showing fractured particles are presented in Figs 7(a) and (b). Some of the larger particles are shattered into several pieces. Mochida et al. [9] have classi®ed particle damage into three dierent types: shattered particles,
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Fig. 6. Distribution of particle sizes showing particle damage within the Alcan composite after compression to a strain of 20% at a strain rate of 10ÿ4/s. (a) Change in frequency of occurrence of particles of a given size in axial section; (b) change in frequency of occurrence of particles of a given size in transverse section.
cracked particles and debonded particles. They concluded that the reduction in the Young's modulus of the composite is greatest in the case of the shattered particle mode. Particle damage of this type has also been observed by Prangnell [28] in a similar metal±matrix composite. The evolution of the particle damage with strain (for quasistatic rates of deformation) is presented in Fig. 8 for the transverse section. The percentage of damaged particles is de®ned as the percentage ratio of damaged particle number to initial total particle number for any given size. This is plotted as a function of particle size for four dierent macroscopic strains: 5, 10, 20 and 30%. It is evident that the number of particles of all sizes that are damaged increases with increasing strain. Generally, the amount of damage increases more rapidly for larger particle sizes. Analytical modeling of the damage should therefore incorporate the evolution of the damage with strain (essentially with plastic strain, since the elastic strains are comparatively small). The evolution of the particle damage with strain rate is much harder to evaluate. Statistical charac-
terization has only been performed at two very dierent strain rates: 10ÿ4 and 2 103/s. Because of the rate-dependence of the matrix ¯ow stress, the stresses experienced by the particles under dynamic loading are expected to be substantially higher than those under static loading. A comparison of the degree of particle damage for the high and low strain rates at a ®xed strain of 20% is presented in Fig. 9. The primary dierence between the two damage distributions in Fig. 9 is that the larger particles (>18 mm) appear to fracture more easily under the higher rate loading. However, Fig. 2 also shows that the actual number of particles larger than 18 mm is comparatively small over the entire populations of particles, and so the dierence in the percentage of damaged particles at these sizes is in¯uenced by the statistics of small numbers. Thus from the high-strain-rate experiments, the change of particle damage with strain is similar to that presented in Fig. 8 for low strain rates. For these reasons, the modeling of the development of particle damage that follows is idealized as dependent on strain, but not on strain rate.
Fig. 7. Optical micrographs showing typical particle damage after quasistatic compressive deformations.
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LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
Fig. 8. Development of particle damage with increasing strain at a strain rate of 10ÿ4/s (all data from transverse sections). 6. A MODEL FOR COMPRESSIVE DAMAGE
There have been many numerical and analytical models developed to describe the mechanical response of metal±matrix composites with evolving damage [8, 18, 22±24]. Most of these models were developed for rate-independent deformations, and most use parameter sets that are dicult to obtain from experiments of the type presented here. This section develops an approach to the modeling of
evolving damage for compressive deformations. The approach is based on our direct measures of the evolution of the damage with the strain (Fig. 8). This statistical information is used together with a simple physical model of the eect of damage to modify the model developed by Li and Ramesh [3] for rate-dependent deformations of metal±matrix composites. The resulting model for the eect of the evolving damage, while valid only for compres-
Fig. 9. Comparison of particle damage at a strain of 20% after compressive deformations at strain rates of (a) 10ÿ4/s and (b) 2 103/s (all data from transverse sections).
LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
sive deformations, is shown to provide reasonably good agreement with experimental results. The statistics of the evolving damage have been presented in Fig. 8. This evolving damage is expected to result in a degradation of the macroscopic properties (in particular the overall ¯ow stress) of the composite. In order to model the eect of the damage on the overall properties, two features are assumed to represent the most signi®cant aspects of the damage. First, the primary evolving damage parameter is assumed to be the overall fraction of particles that are damaged (averaged over all sizes). The evolution of this damage parameter with macroscopic strain is computed from the data of Fig. 8, and the evolution is assumed to be insensitive to strain rate. Second, every damaged particle is assumed to have failed in the shattering mode. This is obviously an idealization, but the microscopic observations typi®ed by Fig. 7 indicate that particle shattering is the dominant failure mode. This may in part be a result of the fact that the silicon carbide particles larger than approximately 3 mm in size are polycrystalline, and almost all of the reinforcing particles in this composite are larger than this (Fig. 2). The critical physical concept that relates the evolving damage to the macroscopic response is this: once shattered, a particle is assumed to no longer provide a signi®cant constraint to the plastic ¯ow of the matrix, but yet acts as a space-®lling medium that can carry the necessary load. The individual pieces within a shattered particle are assumed to rearrange themselves so as to accommodate the plastic deformation of the matrix. As a consequence, the eective reinforcement volume fraction decreases when a particle is damaged. Thus the evolution of the damage with overall strain translates to an eective reduction in the reinforcement volume fraction with increase in the overall strain. This approach to the description of damage is obviously limited to largely compressive deformations (in tensile deformations, a fractured particle typically no longer carries load). The eective reinforcement volume fraction as a result of compressive damage is thus obtained from the following relation: f1 ÿ r egf0 , f e d
Vd e : V0
The damage fraction rd(e ) varies between the minimum value of 0 (no damage) and the maximum value of 1 (corresponding to the situation when all of the particles have been shattered). When the damage fraction has reached its maximum value of 1, the eective reinforcement volume fraction is reduced to zero, and the composite would behave essentially as though it consisted only of the metal matrix. Note, however, that in fact rd(e ) may never reach the maximum value of 1, because the smallest particles in the initial particle size distribution may be single crystals and may not fracture at all, and they are unlikely to shatter in any case. Thus the initial particle size distribution may determine the limiting response of the composite in terms of the evolving damage. The procedure for computing the damage fraction rd(e ) is as follows. First, the total volume V0 of (undamaged) particles before deformation is estimated using particle counting techniques and estimating the mean particle size from statistical data such as that in Fig. 2. Next, the volume of damaged particles Vd(e ) is computed from sections of deformed specimens in a similar manner, but counting only the damaged particles and estimating the mean size of the damaged particles. This procedure is repeated for several axial and transverse sections of the damaged composite at each strain, and the resulting damage fractions are averaged over all sections for each strain to obtain the rd(e ) that is used in equation (3). The particles are assumed to be cylindrical with unit aspect ratio for all of the volume estimates. Damage fraction estimates are obtained for strains of 5, 10, 20 and 30%, all of the strains having been developed at a low strain rate of 10ÿ4/s. The resulting evolution of the damage function with strain is shown in Fig. 10, as is a power-law curve ®t through the data. The corresponding power-law function is rd(e )=0.7e 0.3. The corre-
2
where f0 is the initial reinforcement volume fraction of the undamaged composite, f(e ) is the eective reinforcement volume fraction at the overall strain e, and rd(e ) is a ``damage fraction'', de®ned as the ratio of the volume of damaged particles Vd(e ) to the total volume V0 of (undamaged) particles before deformation: rd e
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3
Fig. 10. Evolution of the damage function with strain as computed from statistical particle data. Also shown is a power-law curve ®t through the data (the corresponding power-law function is rd(e )=0.7e 0.3).
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LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
now predicted using the eective reinforcement volume fraction f(e ) in place of the volume fraction f in the earlier analytical model of Li and Ramesh [3] [equation (1)]. That is, the volume fraction within the analytical model is allowed to degrade according to the prescription of equation (2). A comparison of the predictions of this damage model with the experimental results on the metal±matrix composite at all strain rates is presented in Fig. 12. The ability of the model to capture the observed behavior is quite good, especially considering the coarseness of the statistical approximations made in the assessment of the damage parameters. Fig. 11. Degradation of the eective reinforcement volume fraction [equation (2)] as a function of the applied overall strain. The initial volume fraction is 20% for this composite, but the eective reinforcement volume fraction is only 10.4% at a strain of 30% according to this model. The curve shown is developed using the power-law form for the damage function.
sponding degradation of the eective reinforcement volume fraction [equation (2)] is plotted as a function of the applied overall strain in Fig. 11. The initial volume fraction is 20% for this composite, but the eective reinforcement volume fraction is only 10.4% at a strain of 30% according to this model. The curve shown in Fig. 11 is developed using the power-law form presented earlier in this paragraph. The macroscopic eect of the evolving damage is
7. SUMMARY
A description has been developed of the viscoplastic response and evolving damage within a metal±matrix composite subjected to compressive deformations. The primary conclusions of this work are the following. 1. Particle damage is developed within the F3S.20S (A359 alloy reinforced with SiC particles) metal± matrix composite during compressive deformations at all of the strain rates studied. The predominant damage mode appears to be particle fracture. 2. Larger particles are fractured more easily than smaller particles; the change in the particle size distribution indicates that particles larger than
Fig. 12. Comparison between the predictions of the model including evolving damage and the experimental observations of composite response at various strain rates. Note that the predictions at the strain rates of 10ÿ4 and 100/s overlap.
LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
10 mm in size are relatively easy to fracture. A signi®cant number of the larger particles are shattered. 3. The degree of particle damage increases with increasing strain, but is only weakly dependent on strain rate. 4. The eective reinforcement volume fraction decreases when a particle is damaged, and the degradation is estimated by simply removing the damaged particle from the composite and replacing it with the matrix. 5. A simple modi®cation of the analytical model of [3], based on the previous assumption, is able to provide good predictions of the high-strain-rate response using a statistical assessment of the actual damage evolution at a low strain rate.
AcknowledgementsÐThis work was supported by the U.S. Army Research Oce under Grant No. DAAH04-95-20006 and by the U.S. Army Research Laboratory through Grant No. DAAL01-96-2-0047. The authors wish to acknowledge useful discussions with K. J. Hemker of Johns Hopkins. The authors also wish to express appreciation for the assistance of R. Mills with analysis of the micrographs, and D. R. Chichili, A. M. Lennon and D. Jia for assistance with the experiments. Last but not least, the authors would like to thank Don Doutre of Alcan International Ltd for supplying us with both the unreinforced alloy and the composite material. REFERENCES 1. Suresh, S. and Mortensen, A., Int. Mater. Rev., 1997, 42(3), 85. 2. Mortensen, A. and Suresh, S., Int. Mater. Rev., 1995, 40(6), 239. 3. Li, Y. and Ramesh, K. T., Acta mater., 1998, 46(16), 5633±5646. 4. Li, Y., Ramesh, K. T. and Chin, E. S. C., The compressive viscoplastic response of an A359/SiCp metal± matrix composite and of the A359 aluminium alloy matrix. Int. J. Solids Struct., in press. 5. Lloyd, D. J., Acta metall. mater., 1991, 39(1), 59±71. 6. Lewandowski, J. J., Liu, C. and Hunt, W. H., Mater. Sci. Engng A, 1989, 107, 241±255. 7. Hunt Jr, W. H., Brockenbrough, J. R. and Magnusen, P. E., Scripta metall. meter., 1991, 25, 15± 20. 8. Yang, J., Cady, C., Hu, M. S., Zok, F., Mehrabian,
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R. and Evans, A. G., Acta metall. mater., 1990, 38(12), 2613±2619. Mochida, T., Taya, M. and Lloyd, D. J., Mater. Trans. Japan Inst. Metals, 1991, 32(10), 931±942. Yeh, J.-W. and Liu, W.-P., Metall. Mater. Trans. A, 1996, 27A, 3558±3568. Brechet, Y., Embury, J. D., Tao, S. and Luo, L., Acta metall. mater., 1991, 39(8), 1781±1786. Flom, Y. and Arsenault, R. J., Acta metall., 1989, 37(9), 2413±2423. Christman, T., Needleman, A., Nutt, S. and Suresh, S., Mater. Sci. Engng A, 1989, 107, 49±61. Lou, B. Y. and Huang, J. C., Metall. Mater. Trans. A, 1996, 27A, 3095±3107. Christman, T., Needleman, A. and Suresh, S., Acta metall., 1989, 37(11), 3029±3050. Llorca, J., Needleman, A. and Suresh, S., Acta metall. mater., 1991, 39(10), 2317±2335. Chichili, D. R. and Ramesh, K. T., Int. J. Solids Struct, 1995, 32(17/18), 2609±2626. Lloyd, D. J., Int. Mater. Rev., 1994, 39(1), 1±23. Vedani, M. and Gariboldi, E., Acta mater., 1996, 44(8), 3077±3088. Kiser, M. T., Zok, F. W. and Wilkinson, D. S., Acta mater., 1996, 44(9), 3465±3476. Zong, B. Y. and Derby, B., J. Mater. Sci., 1996, 31, 297±303. Bao, G., Acta metall. mater., 1992, 40, 2547±2555. Nan, C. W. and Clarke, D. R., Acta mater., 1996, 44(9), 3801±3811. Llorca, J. and Gonzalez, C., J. Mech. Phys. Solids, 1998, 46(1), 1±28. Llorca, J., Martin, A., Ruiz, J. and Mlices, M., Metall. Trans. A, 1993, 24A, 1575±1588. Singh, P. M. and Lewandowski, J. J., Metall. Trans. A, 1993, 24A, 2531±2543. Manoharan, M. and Lewandowski, J. J., Mater. Sci. Engng A, 1992, 150(2), 179±186. Prangnell, P. B., Barnes, S. J., Roberts, S. M. and Withers, P. J., Mater. Sci. Engng A, 1996, 220, 41±56. Barnes, S. J., Prangnell, P. B., Roberts, S. M. and Withers, P. J., Scripta metall. mater., 1995, 33(2), 323±329. Kolsky, H., Proc. Phys. Soc., London, 1949, 62B, 676. da Silva, M. and Ramesh, K. T., Int. J. Plast., 1997, 13(67), 587±610. Yadav, S., Chichili, D. R. and Ramesh, K. T., Acta metall. mater., 1995, 43(12), 4453±4464. Khan, A. S. and Huang, S., Continuum Theory of Plasticity. Elsevier, Amsterdam, 1989. Hong, S. L., Gray III, G. T. and Lewandowski, J. J., Acta metall. mater., 1993, 41, 2337±2351. Vaidya, R. U. and Zurek, A. K., J. Mater. Sci., 1995, 30, 2541±2548.
VISCOPLASTIC DEFORMATIONS AND COMPRESSIVE DAMAGE IN AN A359/SiCP METAL±MATRIX COMPOSITE Y. LI 1, K. T. RAMESH 1{ and E. S. C. CHIN 2 1
Laboratory for Impact Dynamics & Rheology, Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA and 2Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen Proving Ground, MD 21005, USA (Received 30 July 1999; accepted 9 November 1999)
AbstractÐRecent work by the authors has examined the high-strain-rate compression of a metal±matrix composite consisting of an A359 Al alloy matrix reinforced by 20 vol.% of silicon carbide particulates (SiCp). The work-hardening that is observed in the experiments is much lower than that predicted by analytical and computational models which assume perfect particle±matrix interfaces and undamaged particles. In this work, we show that the discrepancy is a result of particle damage that develops within the A359/ SiCp composite under compression. The evolution of particle damage has been characterized using quantitative microscopy, and is shown to be a function of the applied strain. A simple analytical model that incorporates evolving damage within the composite is proposed, and it is shown that the analytical predictions are consistent with the experimental observations over a wide range of strain rates. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Mechanical properties; Plastic; Composites; Aluminium alloys; Functionally graded materials (FGM)
1. INTRODUCTION
Particle-reinforced metal±matrix composites (MMCs) have signi®cant advantages (including high speci®c modulus, high speci®c strength and high wear resistance) over traditional metal alloys for some structural applications. A set of applications of substantial interest consists of functionally graded metal±ceramic composite structures in which the ceramic particle volume fraction is graded [1, 2]. Graded armor structures represent one speci®c application of this type. Since armor materials are designed to survive impact loading, the eect of microstructural features such as ceramic volume fraction, particle size, particle distribution and particle aspect ratio on the high-strain-rate response of particle-reinforced MMCs must be understood. The literature on the high-strain-rate properties of particle-reinforced metal±matrix composites has recently been reviewed by Li and Ramesh [3]. In summary, it is known that the strain rate hardening of a particle-reinforced MMC may be signi®cantly higher than that of the matrix material, provided the matrix itself is rate-sensitive. Li and Ramesh [3]
{ To whom all correspondence should be addressed.
also studied the eects of particle shape and aspect ratio on the high-strain-rate response of MMCs. They concluded that reinforcement geometry has a strong in¯uence on the ¯ow stress of MMCs at high strain rates. An analytical description of the uniaxial constitutive response of particle-reinforced MMCs was developed from these numerical results [3]. This analytical description was shown to be consistent with the experimental results of other workers on two dierent MMCs over a wide range of strain rates. In a more recent paper, Li et al. [4] present both experimental and computational results on the mechanical behavior of an A359/SiCp MMC in compression over a very wide range of strain rates. This material system is of interest because the matrix is a cast alloy, and thus provides a costeective processing route for the development of graded armor structures. Li et al. [4] show that the constitutive formulation previously used by Li and Ramesh [3] is able to describe the rate-dependence of the yield stress of the MMC, but is unable to capture the observed strain-hardening of this composite. The eective strain hardening of the composite was less than that of the matrix material at all strain rates, apparently because of the development of particle damage (even though the stress states were all predominantly compressive). This paper
1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 4 3 0 - 9
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presents an approach to the incorporation of evolving damage within the rate-dependent constitutive formulation of Li and Ramesh [3] for MMCs. The eects of damage on the overall composite response in tension have been widely studied for low rates of deformation, and several microscopic failure mechanisms have been reported (e.g. [5]). Typically, three kinds of failure (or combinations thereof) can occur in a metal±matrix composite: (i) cracking of the reinforcing particles [5±12]; (ii) partial debonding between the particle and matrix interface, resulting in the nucleation of voids within the matrix [6, 13, 14]; and (iii) the growth and coalescence of voids in the matrix [15, 16]. Often one of these three mechanisms dominates. Which mechanism is dominant will depend on the matrix and on the particle size, shape, distribution and volume fraction. For example, it has been observed that both particle cracking and void nucleation within the matrix tend to originate preferentially in clustered regions of high local volume fraction [6, 15]. The development of damage is a progressive process and the dominant mechanism may change as the damage evolves (e.g. Ref. [17]). Since particle fracture is readily observed in most failed composites, it is often suggested that this is the dominant failure process [18]. A decrease in work hardening under tension is observed with an increase in either particle volume fraction or particle size [7, 8, 11]. This is believed to be due to accelerated damage in the form of particle cracking in the higher volume fraction and/or larger particle size materials. Many experimental investigations [5, 7, 9, 11, 19±21] have been conducted to determine the relationship between particle damage and the average strain or stress. A number of models have also been developed [8, 18, 22±24] to predict the ¯ow stress in tension at low rates of deformation for a composite containing damaged particles. It appears that the incidence of particle damage is related not only to the overall stress or strain level in the composite, but also to the particle size, matrix properties and particle properties [25± 27]. There are few reports in the literature on the development of damage within MMCs under quasistatic and dynamic compressive loadings. Prangnell et al. [28] and Barnes [29] observed damage within SiC particle-reinforced aluminum matrix composites deformed in compression at low strain rates. Particle fracture was identi®ed as the dominant mechanism in regions of low local reinforcement volume fraction. Void formation was observed within the matrix in regions containing strongly clustered particles. On the other hand, Kiser et al. [20] observed no damage after low-rate compression of a SiC particle reinforced aluminum alloy MMC. In the latter study, the composites exhibited the same hardening characteristics as the corresponding matrix material. Several investigators have studied
the properties of MMCs under dynamic compression (see Ref. [3] for a review), but there are relatively few reports of damage as a result of compressive deformations. This paper describes the plastic deformation and resulting damage of an A359/SiCp metal±matrix composite subjected to compression over a very wide range of strain rates (10ÿ4±10+5/s). The experimental techniques and overall mechanical properties have been presented in a previous paper and so are only brie¯y delineated here. This paper focuses on the characterization of the evolving damage, and on the development of a model for the evolving damage based on the analytical model of Li and Ramesh [3]. Finally, this paper presents a comparison between the predictions of this damage model and the experimental results. 2. MATERIALS
The materials used in this study were A359 aluminum alloy and an A359 aluminum alloy reinforced by 20 vol.% of SiC particles (F3S.20S). The A359 aluminum alloy was cast in the form of a bar 1.5 inches in diameter by Alcan International Limited. The composition of the A359 aluminum alloy is as follows in wt%: 0.2 Cu, 0.5±0.7 Mg, 0.1 Mn, 8.5±9.5 Si, 0.20 Fe, 0.10 Zn, 0.20 Ti, 0.20 other, with the balance being aluminum. The composite was also cast in the form of a bar with a diameter of 1.5 inches. Optical micrographs of the as-cast unreinforced alloy and of the as-cast composite are presented in Figs 1(a) and (b), respectively. The unreinforced alloy shows the classic as-cast microstructure with an interdendritic aluminum±silicon eutectic containing ®ne silicon particles. The composite shows that the reinforcing silicon carbide particles are generally distributed along the eutectic phase, and that the particles are faceted, with a slight preferred orientation. Vickers microhardness measurements were made (Table 1) in the unreinforced alloy and in the matrix of the composite. The load was chosen so that the length of the long diagonal of the indent in the matrix was less than half the distance to the nearest particle. The results shown in Table 1 indicate that the hardness of the matrix material is very similar to that of the unreinforced alloy. This, together with the similarity of the microstructures, provides some con®dence that the rate-dependent properties of the unreinforced alloy can be assumed to be the properties of the matrix in the composite for the purposes of theoretical and analytical modeling. Quantitative microscopy was used to determine the statistical characteristics of the reinforcement distribution in the composite. Metallographic specimens were prepared by sectioning the as-cast composite bar along the bar axis and transverse to the bar axis (axial and transverse sections) using a low-
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Fig. 1. Optical micrographs of (a) as-received A359 alloy and (b) the as-received F3S.20S metal±matrix composite (A359 containing 20 vol.% SiC particles, obtained from Alcan).
speed diamond saw. After the surfaces were polished, scanning electron micrographs of ®ve locations on each section were used to measure the particle aspect ratio and particle size. The measured distributions of particle size and particle aspect ratio within the axial and transverse sections are presented in Fig. 2. The ``frequency'' in Fig. 2 is the percentage of the total number of particles that are present at any given size. Figures 2(c) and (d) show that the mean particle size in both the axial and the transverse sections is in the range of 6±18 mm. The population of particles with a size smaller than 10 mm is greater in the axial section than in the transverse section. There are very few particles larger than 24 mm. Most of the particles have aspect ratios between 1.5 and 2.5 in both axial and transverse sections [Figs 2(b) and (d)]. This suggests that most of the particles are present in the form of small plates.
3. EXPERIMENTAL PROCEDURES
The rate-dependent mechanical behaviors of both the unreinforced alloy and the MMC were studied in compression over strain rates of 10ÿ4±105/s using a combination of quasistatic and dynamic testing techniques. A servohydraulic testing machine was used to conduct compression experiments in the strain rate range of 10ÿ4±100/s. The compression Kolsky bar, or split-Hopkinson pressure bar, developed by Kolsky [30], was used to attain strain rates
of 102±6 103/s. Specimen recovery in the Kolsky bar experiment was achieved by using the recovery modi®cation developed by da Silva and Ramesh [31]. The high-strain-rate pressure-shear plate impact technique [32] was used in order to study the response of both the unreinforced alloy and of the composite at strain rates between 5 104 and 106/s. Detailed descriptions of these experimental techniques are presented by Li et al. [4]. 4. EXPERIMENTAL RESULTS: COMPARISON WITH LI AND RAMESH MODEL
A summary of the rate-dependent mechanical response of the unreinforced A359 alloy is presented in Fig. 3 for a wide range of strain rates between 10ÿ4 and 2 105/s. The strength of the matrix material increases substantially with increasing strain rate, and increases strongly at the higher strain rates developed within pressure±shear plate impact tests. The plate impact test results in Fig. 3 were converted from shear stress±shear strain data to true stress±true strain data using J2-¯ow theory [33], a reasonable assumption for this aluminum alloy. With the mechanical behavior of the matrix material well-characterized (Fig. 3), the numerical and analytical models of Li and Ramesh [3] have been used [4] to predict the rate-dependent response of the composite material. The analytical model of Li and Ramesh [3] for the rate-dependent response of the composite is given by the following equation:
Table 1. Comparison of Vickers microhardness and yield strength of the monolithic alloy and the matrix of the composite material (the yield strength is estimated using VHN/3) Material A359 Al alloy F3S.20S matrix
Mean diagonal (mm)
Vickers microhardness (kg/mm2)
Yield strength (MPa)
35.020.4 34.520.8
75.6 78.0
252 260
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LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
Fig. 2. Distribution of particle size and aspect ratio within the as-received Alcan composite. (a) Particle size distribution in axial section; (b) aspect ratio distribution in axial section; (c) particle size distribution in transverse section; (d) aspect ratio distribution in the transverse section.
Fig. 3. Stress±strain curves obtained on the unreinforced A359 alloy over the full range of strain rates (pressure±shear data incorporated using J2-¯ow approach).
LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
s f, e, e_ s0 eg f 1
e_ e_ 0
m !
m ! e_ 1 f e_ 0
1 where s is the overall ¯ow stress sustained by the composite with reinforcement volume fraction f at an overall strain of e and an overall strain rate of e_ : In equation (1), s0(e ) represents the stress±strain response of the matrix at quasistatic rates of deformation and is obtained directly from experimental data (Fig. 3), and m and e_ 0 are parameters that determine the rate-sensitivity of the matrix material (also determined from Fig. 3). The strengthening function g( f ) represents the dependence of the ¯ow stress ratio on the volume fraction f (at quasistatic rates) as computed using a unit cell model; a polynomial approximation to the numerical results [4] for this composite is g f 1 1:17f 2:28f 2 21:0f 3 : The predictions of this model are compared with experimental data on the composite in Fig. 4 (reproduced from Li et al. [4]). While the observed ¯ow stresses are reasonably well predicted at small strains, the predicted ¯ow stress increasingly overestimates the experimental ¯ow stresses at larger strains. The model does an acceptable job of cap-
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turing the rate-dependence of the composite ¯ow stress, but does not capture the strain hardening of the composite. The model assumes no microstructural damage and therefore predicts a strain hardening at large strains that is essentially the same as that of the matrix. However, a comparison of Figs 3 and 4 shows that the strain hardening of the composite is actually substantially less than that of the matrix alloy. Two pieces of evidence indicate that microstructural damage may have developed within this composite during compressive plastic deformations. First, the strain hardening of the composite is lower than that of the corresponding matrix alloy. Second, the experimentally observed strain hardening of the composite is lower than that predicted by analytical and numerical models that assumed no damage. Direct microstructural examination of the deformed composite (presented in the next section) shows that substantial particle damage occurs within the F3S.20S composite even during compressive deformations. The evolving damage must be incorporated within models of the type developed by Li and Ramesh [3] in order to better capture the overall response of the composite to compression over this range of strain rates.
Fig. 4. Comparison of predicted and observed stress±strain curves for the metal±matrix composite (A359 alloy containing 20 vol.% SiC particles) over the full range of strain rates. Pressure±shear data again incorporated using the J2-¯ow approach. Note that the predictions at the strain rates of 10ÿ4 and 100/s overlap.
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5. MICROSTRUCTURAL CHARACTERIZATION OF EVOLVING DAMAGE
Quantitative microscopy was used to characterize the microstructural damage within this metal± matrix composite after compressive deformations. Deformed cylindrical specimens were sectioned in the longitudinal and transverse directions using a low-speed diamond saw. Each section was carefully polished and examined under the scanning electron microscope (SEM). Four random regions were chosen within each sectioned plane, and SEM micrographs were obtained from each region. Inspection of the micrographs showed little or no particle± matrix debonding and no evidence of matrix void nucleation. Each micrograph was further scanned and then processed using a set of custom imageprocessing routines to obtain particle statistics. Particle statistics and particle fracture statistics were obtained for each micrograph. The statistics were averaged over the four regions for each of the axial and transverse sections. Typically the statistics are obtained over 500 particles. Results are presented for two dierent strain rates: a quasistatic strain rate of 10ÿ4/s and a dynamic strain rate of 2 103 =s: Recovery experiments were performed at each strain rate, with the specimens deformed to various controlled strains in order to determine the evolution of the damage with strain (the corresponding stress±strain curves are shown in Fig. 5). Statistical data is not presented for the pressure± shear experiments because those experiments were not performed in a recovery mode, and so the observed damage cannot be related to the observed stress±strain response. The statistical information acquired from the recovered specimens included particle size, particle aspect ratio, and number of cracked particles. However, this paper presents only the size and number density statistics. A more complete description of the evolving damage will be presented in a forthcoming paper.
Fig. 5. Stress±strain curves obtained from recovery tests on composite specimens at two ®xed strain rates (10ÿ4 and 2 103/s), with each specimen deformed to a pre-determined strain.
The dierences between the initial particle size distribution (presented in Fig. 2) and the particle size distribution after deformation to a strain of 20% at a strain rate of 10ÿ4/s are presented in Fig. 6. The ``frequency change'' is de®ned as the change in the frequency of occurrence (as de®ned for Fig. 2) of particles of a given size as a result of the deformation (for example, a frequency change of ÿ5% means that there are 5% fewer particles of that size in the deformed composite). Figure 6 shows that the frequency of occurrence of large particles within the composite is reduced after compressive deformations in both the axial and transverse sections, whereas the frequency of occurrence of small particles increases. The large particles are evidently broken up into smaller particles during the deformation. It is apparent that most particles that are larger than 10 mm are fractured into pieces less than 10 mm in size. Figure 6 thus presents strong evidence of particle damage developed during quasistatic compressive deformations. Note that the data indicates that the incidence of particle damage and corresponding degradation of properties may be reduced by changing the initial particle size distribution. These observations of damage during compression may be compared with observations of particle damage under tensile loading [7, 8, 11] (note that other damage modes such as void nucleation are also present in tensile loadings). The increased probability of damage in larger particles is usually due to the increased probability of ®nding pre-existing defects within larger particles. It is worth noting that in this composite, those particles larger than 1±3 mm in size are typically polycrystalline, and may be failing along grain boundaries. While particle damage appears to be the predominant damage mechanism for this composite under compression, some degree of particle±matrix debonding may also occur, but not be perceptible with the optical and scanning electron microscopy used in this study. Debonding has been observed after dynamic compressive deformations in an underaged Al±Zn±Mg±Cu alloy reinforced with SiC particles [34]. Though incontrovertible evidence that debonding does not occur within the F3S.20S composite is lacking, there is ancillary evidence that suggests that this is not a signi®cant issue for the deformations considered here. Some degree of debonding has been observed in an 8090 composite [35]. However, the analytical model of equation (1) captured the experimentally observed behavior assuming perfect bonding [3]. This suggests that the eect of debonding on the composite ¯ow stress under compression is small. Typical micrographs showing fractured particles are presented in Figs 7(a) and (b). Some of the larger particles are shattered into several pieces. Mochida et al. [9] have classi®ed particle damage into three dierent types: shattered particles,
LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
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Fig. 6. Distribution of particle sizes showing particle damage within the Alcan composite after compression to a strain of 20% at a strain rate of 10ÿ4/s. (a) Change in frequency of occurrence of particles of a given size in axial section; (b) change in frequency of occurrence of particles of a given size in transverse section.
cracked particles and debonded particles. They concluded that the reduction in the Young's modulus of the composite is greatest in the case of the shattered particle mode. Particle damage of this type has also been observed by Prangnell [28] in a similar metal±matrix composite. The evolution of the particle damage with strain (for quasistatic rates of deformation) is presented in Fig. 8 for the transverse section. The percentage of damaged particles is de®ned as the percentage ratio of damaged particle number to initial total particle number for any given size. This is plotted as a function of particle size for four dierent macroscopic strains: 5, 10, 20 and 30%. It is evident that the number of particles of all sizes that are damaged increases with increasing strain. Generally, the amount of damage increases more rapidly for larger particle sizes. Analytical modeling of the damage should therefore incorporate the evolution of the damage with strain (essentially with plastic strain, since the elastic strains are comparatively small). The evolution of the particle damage with strain rate is much harder to evaluate. Statistical charac-
terization has only been performed at two very dierent strain rates: 10ÿ4 and 2 103/s. Because of the rate-dependence of the matrix ¯ow stress, the stresses experienced by the particles under dynamic loading are expected to be substantially higher than those under static loading. A comparison of the degree of particle damage for the high and low strain rates at a ®xed strain of 20% is presented in Fig. 9. The primary dierence between the two damage distributions in Fig. 9 is that the larger particles (>18 mm) appear to fracture more easily under the higher rate loading. However, Fig. 2 also shows that the actual number of particles larger than 18 mm is comparatively small over the entire populations of particles, and so the dierence in the percentage of damaged particles at these sizes is in¯uenced by the statistics of small numbers. Thus from the high-strain-rate experiments, the change of particle damage with strain is similar to that presented in Fig. 8 for low strain rates. For these reasons, the modeling of the development of particle damage that follows is idealized as dependent on strain, but not on strain rate.
Fig. 7. Optical micrographs showing typical particle damage after quasistatic compressive deformations.
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Fig. 8. Development of particle damage with increasing strain at a strain rate of 10ÿ4/s (all data from transverse sections). 6. A MODEL FOR COMPRESSIVE DAMAGE
There have been many numerical and analytical models developed to describe the mechanical response of metal±matrix composites with evolving damage [8, 18, 22±24]. Most of these models were developed for rate-independent deformations, and most use parameter sets that are dicult to obtain from experiments of the type presented here. This section develops an approach to the modeling of
evolving damage for compressive deformations. The approach is based on our direct measures of the evolution of the damage with the strain (Fig. 8). This statistical information is used together with a simple physical model of the eect of damage to modify the model developed by Li and Ramesh [3] for rate-dependent deformations of metal±matrix composites. The resulting model for the eect of the evolving damage, while valid only for compres-
Fig. 9. Comparison of particle damage at a strain of 20% after compressive deformations at strain rates of (a) 10ÿ4/s and (b) 2 103/s (all data from transverse sections).
LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
sive deformations, is shown to provide reasonably good agreement with experimental results. The statistics of the evolving damage have been presented in Fig. 8. This evolving damage is expected to result in a degradation of the macroscopic properties (in particular the overall ¯ow stress) of the composite. In order to model the eect of the damage on the overall properties, two features are assumed to represent the most signi®cant aspects of the damage. First, the primary evolving damage parameter is assumed to be the overall fraction of particles that are damaged (averaged over all sizes). The evolution of this damage parameter with macroscopic strain is computed from the data of Fig. 8, and the evolution is assumed to be insensitive to strain rate. Second, every damaged particle is assumed to have failed in the shattering mode. This is obviously an idealization, but the microscopic observations typi®ed by Fig. 7 indicate that particle shattering is the dominant failure mode. This may in part be a result of the fact that the silicon carbide particles larger than approximately 3 mm in size are polycrystalline, and almost all of the reinforcing particles in this composite are larger than this (Fig. 2). The critical physical concept that relates the evolving damage to the macroscopic response is this: once shattered, a particle is assumed to no longer provide a signi®cant constraint to the plastic ¯ow of the matrix, but yet acts as a space-®lling medium that can carry the necessary load. The individual pieces within a shattered particle are assumed to rearrange themselves so as to accommodate the plastic deformation of the matrix. As a consequence, the eective reinforcement volume fraction decreases when a particle is damaged. Thus the evolution of the damage with overall strain translates to an eective reduction in the reinforcement volume fraction with increase in the overall strain. This approach to the description of damage is obviously limited to largely compressive deformations (in tensile deformations, a fractured particle typically no longer carries load). The eective reinforcement volume fraction as a result of compressive damage is thus obtained from the following relation: f1 ÿ r egf0 , f e d
Vd e : V0
The damage fraction rd(e ) varies between the minimum value of 0 (no damage) and the maximum value of 1 (corresponding to the situation when all of the particles have been shattered). When the damage fraction has reached its maximum value of 1, the eective reinforcement volume fraction is reduced to zero, and the composite would behave essentially as though it consisted only of the metal matrix. Note, however, that in fact rd(e ) may never reach the maximum value of 1, because the smallest particles in the initial particle size distribution may be single crystals and may not fracture at all, and they are unlikely to shatter in any case. Thus the initial particle size distribution may determine the limiting response of the composite in terms of the evolving damage. The procedure for computing the damage fraction rd(e ) is as follows. First, the total volume V0 of (undamaged) particles before deformation is estimated using particle counting techniques and estimating the mean particle size from statistical data such as that in Fig. 2. Next, the volume of damaged particles Vd(e ) is computed from sections of deformed specimens in a similar manner, but counting only the damaged particles and estimating the mean size of the damaged particles. This procedure is repeated for several axial and transverse sections of the damaged composite at each strain, and the resulting damage fractions are averaged over all sections for each strain to obtain the rd(e ) that is used in equation (3). The particles are assumed to be cylindrical with unit aspect ratio for all of the volume estimates. Damage fraction estimates are obtained for strains of 5, 10, 20 and 30%, all of the strains having been developed at a low strain rate of 10ÿ4/s. The resulting evolution of the damage function with strain is shown in Fig. 10, as is a power-law curve ®t through the data. The corresponding power-law function is rd(e )=0.7e 0.3. The corre-
2
where f0 is the initial reinforcement volume fraction of the undamaged composite, f(e ) is the eective reinforcement volume fraction at the overall strain e, and rd(e ) is a ``damage fraction'', de®ned as the ratio of the volume of damaged particles Vd(e ) to the total volume V0 of (undamaged) particles before deformation: rd e
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3
Fig. 10. Evolution of the damage function with strain as computed from statistical particle data. Also shown is a power-law curve ®t through the data (the corresponding power-law function is rd(e )=0.7e 0.3).
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LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
now predicted using the eective reinforcement volume fraction f(e ) in place of the volume fraction f in the earlier analytical model of Li and Ramesh [3] [equation (1)]. That is, the volume fraction within the analytical model is allowed to degrade according to the prescription of equation (2). A comparison of the predictions of this damage model with the experimental results on the metal±matrix composite at all strain rates is presented in Fig. 12. The ability of the model to capture the observed behavior is quite good, especially considering the coarseness of the statistical approximations made in the assessment of the damage parameters. Fig. 11. Degradation of the eective reinforcement volume fraction [equation (2)] as a function of the applied overall strain. The initial volume fraction is 20% for this composite, but the eective reinforcement volume fraction is only 10.4% at a strain of 30% according to this model. The curve shown is developed using the power-law form for the damage function.
sponding degradation of the eective reinforcement volume fraction [equation (2)] is plotted as a function of the applied overall strain in Fig. 11. The initial volume fraction is 20% for this composite, but the eective reinforcement volume fraction is only 10.4% at a strain of 30% according to this model. The curve shown in Fig. 11 is developed using the power-law form presented earlier in this paragraph. The macroscopic eect of the evolving damage is
7. SUMMARY
A description has been developed of the viscoplastic response and evolving damage within a metal±matrix composite subjected to compressive deformations. The primary conclusions of this work are the following. 1. Particle damage is developed within the F3S.20S (A359 alloy reinforced with SiC particles) metal± matrix composite during compressive deformations at all of the strain rates studied. The predominant damage mode appears to be particle fracture. 2. Larger particles are fractured more easily than smaller particles; the change in the particle size distribution indicates that particles larger than
Fig. 12. Comparison between the predictions of the model including evolving damage and the experimental observations of composite response at various strain rates. Note that the predictions at the strain rates of 10ÿ4 and 100/s overlap.
LI et al.: VISCOPLASTIC DEFORMATION AND COMPRESSIVE DAMAGE
10 mm in size are relatively easy to fracture. A signi®cant number of the larger particles are shattered. 3. The degree of particle damage increases with increasing strain, but is only weakly dependent on strain rate. 4. The eective reinforcement volume fraction decreases when a particle is damaged, and the degradation is estimated by simply removing the damaged particle from the composite and replacing it with the matrix. 5. A simple modi®cation of the analytical model of [3], based on the previous assumption, is able to provide good predictions of the high-strain-rate response using a statistical assessment of the actual damage evolution at a low strain rate.
AcknowledgementsÐThis work was supported by the U.S. Army Research Oce under Grant No. DAAH04-95-20006 and by the U.S. Army Research Laboratory through Grant No. DAAL01-96-2-0047. The authors wish to acknowledge useful discussions with K. J. Hemker of Johns Hopkins. The authors also wish to express appreciation for the assistance of R. Mills with analysis of the micrographs, and D. R. Chichili, A. M. Lennon and D. Jia for assistance with the experiments. Last but not least, the authors would like to thank Don Doutre of Alcan International Ltd for supplying us with both the unreinforced alloy and the composite material. REFERENCES 1. Suresh, S. and Mortensen, A., Int. Mater. Rev., 1997, 42(3), 85. 2. Mortensen, A. and Suresh, S., Int. Mater. Rev., 1995, 40(6), 239. 3. Li, Y. and Ramesh, K. T., Acta mater., 1998, 46(16), 5633±5646. 4. Li, Y., Ramesh, K. T. and Chin, E. S. C., The compressive viscoplastic response of an A359/SiCp metal± matrix composite and of the A359 aluminium alloy matrix. Int. J. Solids Struct., in press. 5. Lloyd, D. J., Acta metall. mater., 1991, 39(1), 59±71. 6. Lewandowski, J. J., Liu, C. and Hunt, W. H., Mater. Sci. Engng A, 1989, 107, 241±255. 7. Hunt Jr, W. H., Brockenbrough, J. R. and Magnusen, P. E., Scripta metall. meter., 1991, 25, 15± 20. 8. Yang, J., Cady, C., Hu, M. S., Zok, F., Mehrabian,
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