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Influence of reinforcement arrangement on the local reinforcement stresses in composite materials
Journal of the Mechanics and Physics of Solids 52 (2004) 1355 – 1377
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In&uence of reinforcement arrangement on the local reinforcement stresses in composite materials P. Ganguly, W.J. Poole∗ Department of Metals and Materials Engineering, University of British Columbia, 309-6350 Stores Road, Vancouver, BC, Canada V6T 1Z4 Received 28 May 2003; received in revised form 4 November 2003; accepted 14 November 2003
Abstract The state of stress in and around reinforcements governs a number of physical processes in composite (multi-phase) materials, including the initiation of damage by either reinforcement cracking or interfacial decohesion. The stresses in the reinforcements have been observed to depend on the spatial distribution of the reinforcements, although the exact correlation is unclear. The present work determines the reinforcement stress for di4erent reinforcement arrangements, ranging from a linear array of three uniformly spaced particles, to random and clustered microstructures. The stress calculations for elastic matrices were undertaken using a computationally e5cient iterative technique. The technique was validated by comparing the results to 7nite element models, and the range of validity was determined. For the three-particle arrangements, the maximum reinforcement stress was observed when the particles were close to each other along the line of loading (a vertical arrangement). On the other hand, when the particle arrangement made a large angle with the loading direction, the reinforcement stress was low. Similar observations were recorded for the random and clustered arrangements where the location of the maximum reinforcement stress coincided with a vertical arrangement. The present work also develops a scheme for determining ‘representative volume elements’ for composite micromechanical models, based on the length scales of stress 7eld interactions. These observations can be used to rationalize damage evolution mechanisms in commercial composites, and aid the development of physically based failure models for such materials. ? 2003 Elsevier Ltd. All rights reserved.
∗
Corresponding author. E-mail address: [email protected] (W.J. Poole).
0022-5096/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2003.11.005
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Keywords: A. Stress concentrations; B. Inhomogeneous material; Reinforcement arrangement; C. Finite elements
1. Introduction Multi-phase composites present interesting possibilities for producing tailor-made materials, through suitable choice of the intrinsic phase properties, and proper control of the extrinsic factors like the volume fraction, spatial distribution and geometry (shape) of the di4erent phase arrangements. These materials have de7ned the scope of engineering materials in the past decades, using di4erent phase combinations for di4erent applications. In many structural applications, composite materials made of a relatively soft and ductile matrix, reinforced by hard and brittle reinforcements have been used. Metal matrix composites and polymer matrix composites, both reinforced by hard ceramic reinforcements, are prime examples in this category. The metallic matrix in MMCs deforms elasto-plastically, while the deformation of the polymer matrix is elastic or in some cases viscoelastic. In both cases, the primary contribution of the elastic reinforcements is to enhance the sti4ness and strength of the composite. Predicting the overall (macroscopic) properties of these composites, given the intrinsic and extrinsic contributions, is critical to their applicability. Continuum models that predict macroscopic properties that depend on volume averaged material responses (e.g. elastic modulus, yield strength and strain hardening rate) are well developed. In these cases, the intrinsic phase properties and their respective volume fractions dominate the material behavior, suitably weighed by average values of reinforcement shape and distribution characteristics. Suitable analytical (Eshelby, 1957; Mori and Tanaka, 1973; Pedersen, 1983; Corbin and Wilkinson, 1994) and 7nite element models (Christman et al., 1989; Bao et al., 1991; Shen et al., 1995) have been developed for such responses, both for elastic and elasto-plastic matrices. However, the existing models are less secure in predicting properties which are governed by the extrema of the constituent responses, e.g. failure processes governed by the ‘weakest link’ or the largest &aw. In these cases, the distribution of the reinforcements needs to be explicitly accounted for, and may play a signi7cant role in determining the macroscopic composite property. For example, changing the spatial distribution of reinforcements in cast MMCs by hot extrusion has been observed to cause a 7ve-fold increase in the composite failure strain (Ganguly et al., 2001). Numerical models, which can capture such a dependence of the material ductility on the reinforcement arrangement, are currently unavailable. Damage initiation in metal matrix composites primarily occurs by either cracking of the reinforcements (Lloyd, 1991), or by decohesion at the reinforcement–matrix interface (Manoharan and Lewandowski, 1990), while for polymer matrix composites, decohesion at the matrix–reinforcement interface is an important damage initiation mechanism (Smith, 2000). Particle cracking events can be directly related to the stress state in the reinforcements and the reinforcement fracture strength (Brechet et al., 1991). The decohesion events are governed by the traction across the interface (Needleman, 1987), and can also be related to the reinforcement stress state, given the continuity of
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the radial stress across the reinforcement–matrix interface. Predictive modeling of damage initiation would thus require an accurate determination of the reinforcement stress state, assuming that the intrinsic parameters governing the damage initiation process (reinforcement and interfacial strengths) are known a priori. For elasto-plastic matrices, 7nite element models (Brockenbrough et al., 1992; Ghosh and Murthy, 1998) have examined the local stress distributions for di4erent reinforcement arrangements. The reinforcement arrangements are generally categorized as random (hard-core) and clustered. For a random arrangement, the spatial distribution of the reinforcement centroids follows a random distribution, provided no reinforcements overlap, and all are contained within the given domain. A clustered distribution is characterized by significant non-uniformity in the reinforcement distribution, with one or more regions (the clusters) of pronounced reinforcement proximity. Both Brockenbrough et al. (1992) and Ghosh and Moorthy (1998) observed that the maximum reinforcement stress laid inside a reinforcement cluster. However, the variation of the reinforcement stress between di4erent clusters was signi7cant, and the maximum reinforcement stress in some of the clusters was lower than that in a random distribution (Ghosh and Moorthy, 1998). A signi7cant variability in the cluster stresses has also been observed by Pyrz and Bochenek (1998) for composites with elastic matrices. This indicates that global characterizations such as random or clustered do not completely dictate reinforcement stress patterns, and further investigations on the reinforcement distribution factors that determine the stress are required. Such analysis may yield suitable characterization schemes for reinforcement distributions, and lead to optimal fracture models for composite materials. Further, determining the optimal size of the representative volume element (RVE) for modeling the reinforcement stress distribution (and damage initiation) in composite materials is required. Numerical models considering a small number of reinforcements have been observed to yield non-unique stress characteristics for random and clustered microstructures (Ghosh et al., 1997). On the other hand, large microstructural models are limited by the existing computational capabilities. The present work attempts to determine the critical distribution characteristics, which dominate the local stress patterns and the resulting damage initiation characteristics in composite materials. The approach taken has been to start with simple idealized reinforcement arrangements where the local stresses in the reinforcements can be solved by an iterative elastic approach. Finite element calculations have been used to evaluate the e4ects of inter-particle spacing and matrix plasticity on the stress in the reinforcements. For relevance to commercial composites, AA6061–Al2 O3 metal matrix composite was chosen as a suitable material system, and the micromechanical calculations were conducted assuming the appropriate phase properties. The salient observations for the model particle distributions were veri7ed for large non-uniform microstructures, where stress distributions for 600 reinforcements in random and clustered arrangements were determined using the iterative approach. The analysis also yields the interaction length scales of the reinforcement stress 7elds, and helps develop a basis for estimating the ‘representative volume elements’ in multi-phase materials. Finally, the applicability of the present analysis to commercial composite models is discussed, with reference to fracture processes in metal matrix composites.
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2. Methodology 2.1. Formulation of the iterative model An iterative model was used for determining the stress 7eld due to multiple (elastic) circular reinforcements in an in7nite, elastic matrix subjected to a far-7eld stress of ∞ . The model algorithm was originally developed by Pijaudier-Cabot and Bazant (1991) and has been used previously by Pyrz and co-workers (Axelsen and Pyrz, 1995, 1997; Pyrz and Bochenek, 1998) for matrix and interfacial cracking in elastic composites. The present work validates the iterative model (using 7nite element calculations) as a tool for reinforcement stress determination in elastic composites, and applies it for stress 7eld determination in a variety of random and clustered microstructures. The multi-reinforcement problem can be reduced to determining the stress 7eld in each reinforcement, subjected to the far-7eld stress (∞ ) and a force 7eld re&ecting the reinforcement–reinforcement interactions. The matrix and the reinforcements were elastic, with their respective moduli being Dm and Dc . For any given (ith) reinforcement, the reinforcement stress (i ) and strain (i ) could be related by i = Dc i :
(1)
The composite may be transformed into a homogeneous solid (with an elastic modulus of Dm ) with an eigenstrain K imposed in the reinforcements. i = Dm (i − Ki ):
(2)
Equating the stress in Eq. (2) with that in Eq. (1) gives Ki = (Dm − Dc )Dc−1 i ;
(3)
where Ki (=Dm Ki ) is the unbalanced stress inside the reinforcement. This unbalanced stress 7eld may be balanced by applying tractions (pi ) at the reinforcement boundary pi = −Ki nc ;
(4)
where nc is an outward normal to the reinforcement boundary. Sub-dividing the given reinforcement boundary into a number of segments (each of length ds), the tractions may be replaced by corresponding point forces (F) acting at the center of each boundary segment. Fij = pi dsj ;
(5)
Fij
where is the point force corresponding to the jth segment of the boundary of the ith reinforcement. Each of the point forces would generate a stress 7eld in the homogeneous solid (Muskhelishvili, 1963). All such stress 7elds may be superposed to determine the perturbation stress 7eld (c ) at each point in the homogeneous solid. If ic is the value of the 7eld at the geometrical point corresponding to the centroid of the given reinforcement, then the reinforcement stress 7eld (i ) is given by i = ∞ + ic − Ki :
(6)
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Initially, the reinforcement stress i was assumed to be equal to ∞ . Ki and pi were determined from Eqs. (3) and (4), respectively, and the reinforcement stress (i ) determined at the end of each step (from Eq. (6)) was used as the stress estimate at the next step. This procedure was repeated until the point forces did not change signi7cantly, i.e. for steps k and (k + 1) N j j T s i=1 Nj=1 (F (F ) ) (k+1) i i ¡ 0:001; − 1 (7) N Ns j j T j=1 (Fi (Fi ) )(k) i=1 where Fij (Fij )T yields the square of the magnitude of each point force vector (Fij ), Ns is the number of boundary segments per reinforcement, and N is the number of reinforcements considered in the model. The composite was deformed under plane strain tension and a far-7eld stress of unit magnitude (22 =1, 11 =12 =0 at far-7eld). Both the matrix and the reinforcement were assumed to be linear elastic materials. The reinforcement elastic modulus was 6.7 times that of the matrix. The Poisson’s ratio (elastic) was assumed to be 0.33 for the matrix and 0.34 for the reinforcement. These values are consistent with the properties of the AA6061 matrix (Metal’s Handbook, 1979) and the alumina reinforcement (Simmons and Wang, 1971), respectively. 2.2. Reinforcement distribution generation The reinforcement distributions were computer generated and involved arrangement of equal sized, circular particles on a planar domain (matrix). The simplest distributions involved three reinforcements arranged / such that the inter-particle distances 1 were equal and the particle centroids were collinear (see Fig. 1). For a given reinforcement radius (r), these distributions were characterized by two parameters, the angle with respect to the loading axis () and the center-to-center spacing between the reinforcements (R). The angle was varied from 0◦ to 90◦ . For =0◦ , the reinforcements were above or below each other (vertical arrangement), whereas for = 90◦ , they were side by side (horizontal arrangement). The spacing between the reinforcement centroids was varied from 3 times to 10 times the reinforcement radius. These models will henceforth be referred as the three-particle uniform arrangements. More complicated reinforcement arrangements included computer-generated microstructures with random and clustered reinforcement distributions, as shown in Fig. 2. Each microstructure included 600 reinforcements arranged in a 10 × 10 planar domain, and the nominal area fraction of the reinforcements was 0.1. The random arrangement (see Fig. 2(a)) was generated following a hard-core algorithm. The reinforcement centroids were generated using a (Poisson) random number generator, with two constraints, (a) the minimum distance between the reinforcement centroids was three times the reinforcement radius (see discussion in Section 3.1), and (b) all the reinforcements were completely contained in the domain. Each of the clustered arrangements included a single cluster (marked by the dashed rectangle in Fig. 2(b–d)), and will be identi7ed 1
Center to center distance between reinforcements.
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θ
r
R
x2
x1
Fig. 1. Arrangement of three reinforcements in an in7nite matrix. The arrangement is characterized by its orientation w.r.t. the loading axis (), and the inter-reinforcement distance R. The radius of the reinforcements is r.
as the square-cluster, the long-cluster or the large-cluster, depending on the cluster shape and the number of particles in the cluster. For the square-cluster distribution (see Fig. 2(b)), the reinforcements were separated into two sub-domains, (a) a cluster of 108 particles in a 3 × 3 area (reinforcement area-fraction in the cluster = 0:2), and (b) 492 particles in the rest of the area. The cluster for the long-cluster distribution (see Fig. 2(c)) contained the same number of particles in each sub-domain (as in the square-cluster distribution), but the cluster sub-domain was de7ned on a 1:5 × 6 area (i.e. the area of the cluster was similar to the square-cluster, but the cluster aspect ratio was 4, compared to 1 for the square-cluster). The cluster for the large-cluster distribution (see Fig. 2(d)) contained 300 particles in a 5 × 5 area (reinforcement area-fraction in the cluster = 0:2), while the remaining 300 particles were distributed in the rest of the area. The distribution of the particles in each of the sub-domains followed a hard-core pattern. The reinforcement arrangement in each microstructure was a4ected by the starting value of the Poisson random number generator (the generating seed). In order to study the e4ect of the variability in the reinforcement stress due to such differences in reinforcement arrangements, 10 di4erent microstructures were generated for each distribution and the reinforcement stress was calculated for each microstructure. In the simulations described above, the reinforcement stresses were computed by assuming elastic interactions among all reinforcements (i.e. all the reinforcements interacted with each other), and will be henceforth called the full-scale iterative model. The reinforcement stresses for the large-cluster distribution were also calculated by limiting the elastic interactions within the neighborhood reinforcements. The near-neighbors (i.e. all the reinforcements which surrounded any given reinforcement) were 7rst identi7ed by Voronoi tessellation (Green and Sibson, 1978). The neighborhoods for the reinforcements were then assembled layer-by-layer. For a one-layer model, only the
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Fig. 2. Reinforcement arrangements for (a) random, (b) small-cluster and (c) long-cluster and (d) large-cluster distributions, respectively. All the distributions contain 600 reinforcements and the nominal reinforcement volume fraction is 0.1 in all the cases.
near-neighbors were assumed to interact with any given reinforcement. For a two-layer model, the near-neighbors of these reinforcements were also included in the interacting domain. Similarly, the three-layer model also included the near neighbors of these reinforcements. Fig. 3 shows typical neighborhoods for a given reinforcement for models with di4erent number of layers. 2.3. Formulation of the 8nite element models Finite element models were constructed to (a) verify the estimates of the elastic iterative models, and (b) determine the e4ect of matrix plasticity on reinforcement stresses. A plane-strain deformation was assumed, and was modeled using the two-dimensional dynamic 7nite element package LS-DYNA2D (Hallquist, 1990). Fig. 4 shows the geometries used for the two models. Symmetry conditions at x1 = 0 and x2 = 0 were
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(a)
(b)
(c)
Fig. 3. (a) One-layer, (b) two-layer and (c) three-layer models for the shaded reinforcement.
x2 = b
x2
x1 = 0
(a)
x2 = 0
x1
x2 = b
x1 = 0
(b)
x2 = 0
Fig. 4. Schematic representation of the FEM models for (a) veri7cation of the iterative models, and (b) reinforcement stress analysis for di4erent reinforcement arrangements in elasto-plastic matrices.
imposed to reduce the model to 14 of the domain. A constant velocity boundary condition (strain rate ∼0:1 s−1 ) was imposed at the far-7eld (x2 = b). Typical properties corresponding to alumina (single crystal sapphire) and AA6061 aluminium alloy were assigned for the reinforcements and the matrix, respectively. The Al2 O3 properties were obtained from Simmons and Wang (1971). The elastic properties of the AA6061 matrix were obtained from the Metal’s Handbook (1979). The plastic deformation of the matrix was represented by a power law equation = Kn ;
(8)
where and are the e4ective stress and strain, respectively, K is the strength coe5cient, and n is the strain hardening exponent. The values of K and n were estimated from an experimentally determined uniaxial compression curve for the AA6061 matrix.
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Table 1 Properties of the AA6061 matrix and the Al2 O3 reinforcement used in 7nite element simulations Constituent
Eel (GPa)
el
Yield stress (MPa)
K (MPa)
N
AA6061 Al2 O3
70 470
0.33 0.34
54
303
0.24
The various input parameters for the two constituents used in the model are summarized in Table 1. Note that the elastic properties of the matrix and the reinforcement are consistent with those assumed in the iterative models (see Section 2.1). • Veri8cation of iterative models: The models were developed to verify the estimates of the iterative models for a vertical arrangement 2 of three reinforcements, for R=r = 2:5, 3 and 5 (see Fig. 4(a)). The maximum far-7eld imposed strain was ∼5×10−5 . The matrix and the reinforcements were assumed to be elastic. The maximum, minimum and the mean stress (22 averaged over the entire reinforcement) for the central reinforcement were determined from the FEM simulations and divided by the far-7eld stress (from the elements at x2 = b) to yield the respective stress ratios. • Modeling elasto-plastic deformation: The elasto-plastic model consisted of a (a) vertical arrangement of three reinforcements, (b) horizontal arrangement of three reinforcements and (c) square arrangement of nine reinforcements (see Fig. 4(b)). The inter-particle distance was 3 times the reinforcement radius for all the three arrangements. The maximum far-7eld strain was 0.1. The reinforcements were assumed elastic, while the matrix was assumed to be elasto-plastic. 3. Stress distribution for uniform reinforcement arrangements 3.1. Elastic model for three-particle arrangements Fig. 5 shows the variation in the maximum normal stress (22 ) in the central reinforcement with change in the normalized inter-particle distance (R=r) and orientation () of the three-particle arrangement in an elastic matrix. The stress for a single particle in an in7nite matrix (where the 22 was 1.37 times the far-7eld stress (Muskhelishvili, 1963)) has been shown for comparison. The reinforcement stress was dependent on both the inter-particle distance and the angle of orientation. For a low and constant , the reinforcement stress increased with decrease in the inter-particle distance. High reinforcement stresses were observed for low at constant R=r. In the range of study, the highest reinforcement stress was observed for a vertical arrangement of reinforcements when the inter-particle distance was 3 times the reinforcement radius. On the other hand, the reinforcement stress was low for large even when the inter-particle 2 The vertical arrangement was chosen for iterative model veri7cation because it exhibited the largest e4ect of neighborhood reinforcements on the stress 7eld of a given reinforcement (see Section 3.1)
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1.8 Reinforcement stress (σ22)
R = 10r
1.7
R = 5r R = 3r
1.6
single reinf.
1.5 1.4 1.3 1.2 1.1 1
0
10 20 30 40 50 60 70 80 90 Angle of orientation (θ)
Fig. 5. Stress (22 ) in the central reinforcement for the three-particle arrangement, with changing inter-particle distance and the orientation of the arrangement w.r.t. the loading direction. The reinforcement stress for a single reinforcement in an in7nite matrix has been shown for comparison.
distance was small. For ¿ 35◦ , the reinforcement stress was always lower than that in a single reinforcement in an in7nite matrix. The results are qualitatively similar to those obtained by Pyrz and Bochenek (1998) for a di4erent material system. The e4ect of inter-particle distance (R=r) and the orientation of the three-particle arrangement () on the reinforcement stress can be qualitatively explained from the stress distribution around an isolated elastic inclusion in an in7nite elastic matrix (see Fig. 6). High stress regions can be observed above and below the reinforcement ‘a’, while low stress regions can be observed on the side. The stress perturbations range over distances of the order of the reinforcement radius. Consequently, the reinforcement stress in Fig. 5 is relatively insensitive to the presence of the second particle at large inter-particle distances (position ‘b’ in Fig. 6). On the other hand, if the second particle is placed in close vertical proximity to ‘a’ (e.g. ‘c’ in Fig. 6), the overlap of high stress regions around each particle would elevate their stresses, while the horizontal proximity of the two particles (‘d’ in Fig. 6) would lower their stresses. The high reinforcement stress for the vertical and the low stress for the horizontal arrangement (or for any arrangement where ¿ 35◦ ) is evident in Fig. 5. The high reinforcement stress observed for the vertical arrangement is consistent with the observations by several investigators (Watt et al., 1996; Al-Ostaz and Jasiuk, 1997; Pyrz and Bochenek, 1998). A natural consequence of the stress interactions between neighboring particles is that the stress 7elds may not be uniform in the reinforcement. On the other hand, the iterative model assumes the uniformity of the reinforcement stresses. This de7nes the bound of validity of the iterative model: it can only be applied to situations yielding approximately uniform reinforcement stresses. Table 2 shows the FEM predictions of the maximum, mean and minimum stresses (22 ) in reinforcements for uniform vertical arrangements in an elastic matrix, with inter-particle spacing (R) being equal to 5, 3
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1.5
b
1.3 c
1.1 0.9
a
d
b 0.5 0.0
x2
x1
Fig. 6. Stress (22 ) distribution in and around a single elastic reinforcement in an in7nite elastic matrix for a unit far-7eld stress. Elastic moduli of reinforcement = 6:7 times that of the matrix, Poisson’s ratio of matrix = 0:33, Poisson’s ratio of reinforcement = 0:34.
Table 2 Variation of the maximum, mean and the minimum reinforcement stress (22 ) ratios for a vertical arrangement of reinforcements for various R=r ratios; the stress ratio from the iterative solution is also included for comparison R=r
Maximum stress ratio (FEM)
Mean stress ratio (FEM)
Minimum stress ratio (FEM)
Stress ratio (iterative)
5.0 3.0 2.5
1.48 1.79 2.09
1.45 1.62 1.73
1.38 1.37 1.35
1.49 1.72 1.91
and 2.5 times the reinforcement radius (r). The variation in the stress at di4erent points in the reinforcement increased with decreasing inter-particle distance. When the distance between the reinforcement centroids was 5 times the reinforcement radius, the reinforcement stress variation was small. On the other hand, the reinforcement stress variation was signi7cant (∼ ± 21% from the mean value) at an R=r of 2.5. The iterative predictions of reinforcement stress for all the three R=r values lay above the mean value. This was dictated by the stress distribution in the reinforcements, with the center of the reinforcement (where the iterative model calculated the particle stress) being a relatively high stress region (see Fig. 7). The di4erence between the iterative model predictions and the FEM predicted maximum reinforcement stress was minimal (less than 5%) at R=r of 5 and 3, while the iterative prediction was signi7cantly lower
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x2
x1 Fig. 7. FEM estimate of stress (22 ) distribution in the reinforcement and the matrix for a 3-particle vertical arrangement (R = 2:5r) at an imposed strain of 0.00003. All stress values are in Pa.
than the maximum stress at R=r of 2.5. The di4erence between the iterative model predictions and the FEM predicted mean reinforcement stress was also low for R=r of 5 and 3, and relatively high for R=r of 2.5. Assuming that the maximum reinforcement stress determined the propensity of damage initiation, and the mean stress governed the reinforcement’s impact on the stress 7elds of neighboring reinforcements, the iterative model could be expected to be accurate for present purposes at R=r greater than or equal to 3. The reinforcement stress estimation for non-uniform structures was thus limited to distributions where the minimum inter-particle distance was three times the reinforcement radius. 3.2. Elasto-plastic models for uniform composite microstructure The maximum reinforcement normal stress (22 ) at an imposed strain of 0.095 for the vertical, horizontal and square arrangements embedded in an elasto-plastic matrix is shown in Fig. 8. The stress distribution was very inhomogeneous with severe stress concentrations in the square and the vertical reinforcement arrangements. Fig. 9 shows the variation of the maximum reinforcement normal stress ratio in the reinforcements
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x2
x1
Fig. 8. Stress (22 ) distribution in the model composite at an imposed strain of 0.095. The 7bers are arranged in a square, a vertical or a horizontal arrangement. The inter-particle spacing for all the three arrangements was 3 times the reinforcement radius. All the stress values are shown in Pa.
Fig. 9. Variation of the ratio of the maximum stress (22 ) in the reinforcements to the far-7eld stress at different imposed strains for a square, a vertical and a horizontal reinforcement arrangement in an elasto-plastic matrix (open symbols). The iterative solution for the stress ratio for the vertical and horizontal arrangements in an elastic matrix (7lled symbols) has also been shown for comparison. The inter-particle spacing for all the arrangements was 3 times the reinforcement radius.
(i.e. maximum 22 in the reinforcement divided by the far-7eld 22 in the matrix) as a function of strain for the square, vertical and horizontal arrangements. The stress ratio for the vertical and the horizontal three-particle arrangements from the iterative models (involving elastic matrices and in7nitesimal strains) are also shown in the 7gure. The reinforcement stress ratio was higher for the vertical arrangement than the horizontal arrangement, both for elasto-plastic and elastic matrices. However, the di4erence between the two stress ratios decreased with the plastic deformation of the matrix. The stress ratio for the vertical reinforcement arrangement was also higher than that for the square arrangement, although the di4erence was relatively small. The reinforcement stresses obtained from the elasto-plastic simulations are consistent with the results from the elastic (iterative) model, with the high stresses
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coinciding with the vertical reinforcement arrangement in both. The primary in&uence of plasticity is to reduce the di4erence in the reinforcement stresses observed for the vertical and the horizontal arrangement. In the following section, the e4ect of reinforcement arrangement on the reinforcement stress in many-particle composites is investigated. The reinforcement arrangement was represented by random and clustered distributions, with 600 reinforcements in each microstructure (see Section 2.2). Since the construction of elasto-plastic, 7nite element models for such large non-uniform microstructures was computationally prohibitive, the stress distributions were determined by the iterative models. Given the above consistency in elastic and elasto-plastic models, the stress distributions were expected to reveal the general trends of the microstructure–stress correlation in composites at low imposed strains. 4. Stress distributions for non-uniform reinforcement arrangements The maximum normal reinforcement stress (22 ) distribution for the representative random and clustered microstructures is shown in Fig. 10. As mentioned earlier, the minimum inter-particle distance allowed for the microstructures was three times the reinforcement radius. The largest 22 value observed in the random, square-cluster, long-cluster and large-cluster distributions were 1.63, 1.63, 1.66 and 1.63 times the far-7eld stress, respectively. For all distributions, proximity of reinforcements along the loading direction resulted in high reinforcement stresses. On the other hand, proximity along the X1 (horizontal) direction resulted in lower stresses. This is consistent with the earlier observations on uniform arrangements in elastic and elasto-plastic matrices (see Figs. 5 and 9). The maximum reinforcement stress for a clustered distribution was observed inside the cluster for large-cluster and long-cluster distributions, while it was outside the cluster for the square-cluster arrangement. For all the microstructures shown in Fig. 10, the maximum reinforcement stress always coincided with a set of reinforcements aligned along the loading direction (a vertical arrangement 3 ). The occurrence of the maximum reinforcement stress in the cluster depended on the presence of favorable vertical arrangements within it. The probability of such an occurrence increased as more and more particles were added to the cluster, or when the cluster shape favored a vertical arrangement. The occurrence of the maximum reinforcement stress inside the cluster was thus more probable for a large-cluster or for a high aspect ratio cluster aligned along the loading direction (the long-cluster). However, among favorably (vertically) arranged reinforcements, the maximum stress was observed in regions of low horizontal proximity (i.e. when reinforcements at large angles to the loading direction were not present near a favorable arrangement). The clustered distributions yielded more reinforcements with high stresses, as compared to the random distribution. For each microstructure containing 600 particles (see 3
By de7nition, a vertical arrangement indicates a set of reinforcements aligned perfectly along the loading direction. However, in reality, such perfect alignment is seldom observed, and the term hereby signi7es only an approximate reinforcement alignment.
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Fig. 10. Reinforcement stress (22 ) distributions for the (a) random, (b) square-cluster, (c) long-cluster and (d) large-cluster distributions. The reinforcements with maximum stresses are indicated by arrows.
Fig. 10), the number of reinforcements with a stress (22 ) more than 1.5 times the far-7eld stress was 21 for the random distribution, 27 for the square-cluster, 34 for the long-cluster and 29 for the large-cluster. In clustered arrangements, the high stress reinforcements occurred predominantly in the clusters, and thus were spatially close to each other. The square-cluster and the long-cluster contained 19% and 44% of the high stress particles in 9% of the area, respectively, while the large-cluster contained 72% of the high stress particles in 25% of the area. On the other hand, the high stress reinforcements were relatively uniformly distributed for the random microstructure. A recurrent observation for the stress distribution in the clustered microstructures was that the high stress reinforcements (inside the cluster) were predominantly observed near the side-edges of the cluster (see Fig. 10(b–d)). To investigate this phenomenon, clustered microstructures, with uniform (square arrangement) reinforcement distributions inside and outside the cluster, were generated. Fig. 11 shows the stress distribution in reinforcements for the uniform large-cluster (624 reinforcements in total,
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Fig. 11. Reinforcement stress (22 ) distribution for a large-cluster microstructure, given that the reinforcement arrangements both inside and outside the cluster were regular. Note that, for clarity, a slightly di4erent stress range has been used in this 7gure compared to that elsewhere in the paper.
overall reinforcement volume fraction ∼0:1, reinforcement volume fraction in cluster ∼0:2). The reinforcements with the maximum stress were observed near the side-edges for such an arrangement. This was primarily due to the fact that the inter-particle spacing on one side of such reinforcements was dominated by the overall volume fraction (∼0:1) and was thus larger than that inside the cluster. As shown in Figs. 5 and 6, the presence of proximal reinforcements at large angles to the loading direction reduced the reinforcement stress. In this case, the absence of such proximal neighbors on one side of the side-edge reinforcements resulted in their stress being larger compared to the reinforcements near the cluster center. Similarly, the reinforcements (inside the cluster) near the cluster top and bottom edges showed a lower stress as compared to those near the center, due to their vertical neighbors being farther away on one side. These factors may be responsible for the prevalent presence of high stress reinforcements near the side-edges of the random clusters, compared to the cluster top or bottom-edge or the cluster center. 4.1. E9ect of statistical variability of microstructures on the reinforcement stress distributions As mentioned in Section 2.2, a variety of di4erent microstructures may be generated for each reinforcement distribution, by changing the parametric values of the random number generator. The variety in the microstructure (given a particular distribution) is a direct result of the natural randomness, and is also present in commercial composite microstructures. The microstructural variability is expected to increase when a smaller number of reinforcements are considered for each distribution. This may cause non-uniqueness in the stress characteristics of di4erent distributions. Ghosh et al. (1997) observed that the stress distributions for random and clustered
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arrangements (single and triple cluster) were di4erent when the number of particles considered in each microstructure was small (25 particles). However, when larger microstructures were considered (100 particles), the stress distributions were similar to each other. Thus, while discussing trends in microstructure-stress correlation in composite materials, the evidence of the trends for su5ciently large microstructures must be ensured. Axelsen and Pyrz (1997) have studied the variation of the stress characteristics with changes in the microstructure size. They observed that the applied stress corresponding to interfacial cracking of 10% of the particles remained relatively unchanged for microstructures with more than 200 particles. In the present work, each microstructure consisted of 600 particles, and 10 di4erent microstructures were generated for each of the four di4erent reinforcement arrangements, viz. random, square-cluster, long-cluster and large-cluster. The variation in the stress characteristics for a given distribution was small. The highest reinforcement stress was predominantly observed inside the cluster for the long and the large-cluster distributions, and outside the cluster for the square-cluster arrangement. The clustered arrangements yielded a higher number of reinforcements with a high stress (22 ¿ 1:5 times the far-7eld stress), as compared to the random arrangements. For each of the clustered arrangements, the reinforcements with the high stress occurred predominantly in the clusters. The long-cluster arrangement yielded the highest reinforcement stress among all the generated microstructures, the reinforcement stress being 1.73 times the far-7eld stress. The predominance of the high stressed reinforcements (inside the clusters) near the side-edges of the clusters was also evident. Overall, the variation of the maximum reinforcement stress for di4erent microstructures (given a particular arrangement) was less than 7%. 4.2. Length scale of reinforcement–reinforcement interactions Stress (22 ) distributions were computed for the one-layer, two-layer, three-layer and full-scale iterative models for the large-cluster distribution. The stress patterns for the one-layer model did not converge monotonically to the prescribed error level (see Eq. (7)). The stress estimates for the two- and three-layer models were close to the full-scale model estimates. The maximum reinforcement normal stresses (22 ) were 1.62 for the two-layer, 1.63 for the three-layer and 1.63 for the full-scale model respectively, given a unit stress at the far-7eld. The e4ective variation between the di4erent models were estimated through an error function E() N f − i 2 1 22 22 ; (9) E() = i N 22 1 where E() is the root mean sum square of the di4erences between the reinforcef i ment stresses 22 and 22 , and N is the number of reinforcements (600 in the present f case). 22 was the more accurate (7nal) model for the reinforcement stress (i.e. more i neighboring layers considered) as compared to 22 (initial model). Table 3 shows the variation of E() for the di4erent models. The di4erence between the stress estimates for the approximate (two- and three-layer) models and the full-scale model was
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Table 3 E4ect of stress interaction length scale on the error function E() Final model
Initial model
E()
Three-layer Full-scale
Two-layer Three-layer
0.0005 0.0001
E() measures the change in the reinforcement stresses observed by limiting the stress interactions among a given number of neighboring reinforcements.
minimal. This de7nes the length scale of interaction of the reinforcements (given a minimum inter-particle spacing equal to three times the reinforcement radius), and that of the representative volume element in order to e4ectively model reinforcement stress. For example, in order to determine the reinforcement stress in the cluster, an RVE consisting of the cluster surrounded by two to three layers of reinforcements may be e4ective, as the reinforcement distribution outside this neighborhood does not in&uence the cluster reinforcement stress. Fig. 12 compares the (iterative) reinforcement stress calculations for such an RVE (with three layers of neighbors) with the stress distributions from the full-scale iterative model, for the large-cluster microstructure. The stress distributions were similar for the two models. The maximum reinforcement stress was 1.62 for the RVE model and 1.63 for the full-scale model. The error function E() for the reinforcement stress in the large-cluster between the RVE (initial) and the full-scale model (7nal) was 6×10−4 . The RVE shown in Fig. 12(a) may be used for stress computation in elastic composites. Given that the interaction length scales are expected to be smaller for an elasto-plastic matrix (Yip and Wang, 1997), the elastic RVE should be su5cient for stress calculations even in elasto-plastic composites. Thus, the present iterative model may be used to determine the e4ective RVE for a given elasto-plastic composite microstructure, provided that the minimum inter-particle distances and the reinforcement shapes are similar to the iterative model characteristics. 4.3. Summary of results from numerical models on uniform and non-uniform composites The salient observations from the iterative and 7nite element models were • Proximity of reinforcements along (or at small angles to) the loading direction increased reinforcement stress, while proximity at large angles to the loading direction decreased the reinforcement stress. Thus, multi-directional proximity of reinforcements (as generally found in reinforcement clusters) did not guarantee the highest reinforcement stress. The reinforcement stress was maximum when a number of reinforcements were aligned along the loading direction, and other reinforcements at large angles to the loading direction were absent near such an arrangement. • The maximum reinforcement stresses in the clustered microstructures were not signi7cantly higher than that in the random microstructure. However, the number of reinforcements with relatively high stress was higher in the clustered microstructures,
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Fig. 12. Reinforcement stress (22 ) distributions in the large-cluster for computations taking into account interactions between (a) only the reinforcements in the RVE, and (b) all the reinforcements, for an unit far-7eld stress. The reinforcements with the maximum stresses are indicated by arrows.
as compared to the random one. For the clustered microstructures, the higher stress reinforcements also primarily occurred inside the clusters and were thus in spatial proximity to each other. The high stress reinforcements inside the clusters were predominantly observed near the side-edges of the cluster. • The cluster shape had a signi7cant e4ect on the reinforcement stress. Clusters that contained more reinforcements, or were preferentially oriented to the loading direction were more probable to yield reinforcements with the maximum stress. These distributions also generated more reinforcements with relatively high stresses. • For uniform reinforcement arrangements, the e4ect of the arrangement on the reinforcement stress was qualitatively similar for elastic and elasto-plastic matrices. However, plastic deformation of the matrix reduced the stress di4erential between the di4erent arrangements. Note that models with non-uniform reinforcement arrangements in elasto-plastic matrices were not developed due to computational limitations.
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4.4. Application: fracture analysis in MMCs The above results from model composite microstructures may help rationalize the physical phenomena dominating fracture processes in commercial composites, e.g. metal matrix composites (MMCs). Damage nucleation in metal matrix composites, either by particle cracking or interfacial decohesion, is related to the reinforcement stress state (in this case, the 22 ). The reinforcement stress 22 modeled in the present work may thus dictate MMC damage nucleation. Note that the present work only deals with reinforcement stresses in undamaged composites, and thus may only explain damage initiation events, or the early stages of damage progression (when the damage events are far apart from each other and spatially uncorrelated). As in any continuum model, the iterative model does not include any intrinsic length scale and is thus unable to model localization phenomena (e.g. microscopic necking leading to severe strain localization into shear bands which occurs during void coalescence (Brown and Embury, 1973)). Zahl et al. (1994) have shown that at high reinforcement volume fractions (volume fraction ∼0:39 for square reinforcement arrangement loaded along the square-edge, ∼0:68 for hexagonal arrangement and ∼0:78 for square arrangement loaded along the squarediagonal), considerable stress enhancements may occur due to reinforcement impingement of the plastic &ow planes. Under such situations, the iterative model predictions (which assume an elastic matrix) may be signi7cantly di4erent from the reinforcement stresses observed in an elasto-plastic matrix. However, the iterative model in the present work is restricted to a minimum inter-particle spacing of 3 times the reinforcement radius (see Section 3.1), which translates to a maximum reinforcement volume fraction of 0.35 for the square arrangement (under plane strain deformation). In addition, damage initiation in commercial MMCs may occur at strains as low as 0.01 at room temperature (Lloyd, 1991; Brechet et al., 1991). Even at elevated temperatures, macroscopic necking (the limit for spatially uncorrelated damage evolution) was observed in a AA6061 + 20%Al2 O3 MMC at strains ∼0:04 (Ganguly, 1998). At such macroscopic strains, the trends in the reinforcement arrangement e4ects on the stress 7elds in undamaged composites with elasto-plastic matrices have been shown to be similar to those in undamaged elastic composites (see Fig. 9). Thus, the results from the elastic composite models with non-uniform distribution may be qualitatively valid for damage initiation and early stages of damage evolution in commercial MMCs (with moderate clustering of reinforcements), where the matrix deformation is elasto-plastic. The similarity of the maximum 22 stress for the clustered and random distribution suggests a concurrence of damage initiation (the 7rst damage nucleation event) in the di4erent microstructures. The number of high stress reinforcements observed in the clustered arrangements was higher than that in the random arrangements. However, considering that the percentage of reinforcement with stress (22 ) greater than 1.5 times the far-7eld stress ranged from 3.5% (21 out of 600 reinforcements) for the random arrangement to 5.7% (34 out of 600 reinforcements) for the long-cluster, the early stages of damage evolution (when the damage events are non-interacting) may be generally similar for the random and the clustered arrangements. Experimentally, Ganguly (1998) observed that the necking strains for an as-cast (clustered microstructure) and an extruded (random microstructure) AA6061–20%Al2 O3 MMC were similar, although
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their failure strains (measured from reduction of area at the fracture surface) di4ered by an order of magnitude. This seems to indicate that the damage evolution in random and clustered microstructures may be signi7cantly di4erent in the later stages. The later stages are generally characterized by interactions between the nucleated damage and intact reinforcements, either by a rearrangement of the local stress/strain 7elds or by physical coalescence of the voids. The stress/strain perturbations caused by nucleated damage can change the stress in the neighboring reinforcements. Ganguly and Poole (2002) have shown that a damage event reduces the stress in reinforcements positioned vertically above or below it, while it increases stress in the reinforcements arranged horizontally or along the diagonal for a square arrangement. Thus, while vertical arrangements aid damage nucleation, the propagation of the damage is hindered due to absence of reinforcements at the sides or along the diagonal. This may explain the observed dominance of clusters in the MMC failure process: the clusters contain reinforcements close to each other along all directions, and thus may be optimally suited for both damage nucleation or propagation. Isolated vertical arrangements away from the clusters may be well suited (in some cases, more than the clusters, see Fig. 10(b)) for damage nucleation, but the propagation of the damage in such situations may be hindered. In addition, in the clustered microstructures, the reinforcements with high stresses (which may be considered probable candidates for damage nucleation) were mostly located inside the cluster, while they were relatively uniformly distributed in the random microstructures. Thus, the nucleated damage may be expected to be close to each other (and mostly concentrated in the clusters) for the clustered microstructures, while it may be relatively uniform in random microstructures. High damage densities in MMC reinforcement clusters have been experimentally observed by several investigators (Lloyd, 1991; Lewandowski et al., 1989; Ganguly et al., 2001). The proximity of damage nucleation events for clustered microstructures may aid substantial damage– reinforcement interactions (reorientation of the local stress/strain 7elds) or damage– damage interactions (enhanced coalescence of the voids). Both these processes may lead to catastrophic failure processes and suitably lower failure strains. 5. Conclusions The e4ect of reinforcement distribution on the reinforcement stresses in model composites was studied using micromechanical (iterative and 7nite element) models. Both uniform and non-uniform (random and clustered) reinforcement arrangements in elastic and elasto-plastic matrices were considered. The reinforcement stress calculations for the elastic composites were undertaken using a computationally e5cient iterative algorithm. The validity of the algorithm was demonstrated by comparing the iterative results with those from 7nite element models, and the bounds of validity (i.e. conditions when the iterative results would no longer be valid) were determined. The reinforcement stress was sensitive to both the proximity and the angular orientation (w.r.t. the loading axis) of the neighboring reinforcements. The highest reinforcement stress was observed when the reinforcements were close to each other along the loading
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direction. On the other hand, reinforcement arrangements aligned at a large angle with the loading direction yielded low stresses. The maximum normal reinforcement stress was similar for random and clustered distributions, suggesting a concurrence of damage initiation for di4erent microstructures. However, reinforcements with high stresses were fairly uniformly distributed for the random distribution, but concentrated in the clusters for the clustered distributions. This may lead to signi7cant damage–reinforcement and damage–damage interactions in the clusters and make them the preferred sites for damage evolution. The e4ect of the reinforcement arrangement on the reinforcement stresses was observed to be limited to small length scales. For any given reinforcement, the stresses were relatively insensitive to reinforcement arrangements beyond the immediate neighborhood (2 or 3 layers of neighboring reinforcements), thus de7ning the relevant length scale for micromechanical ‘representative volume elements’. The results of this study could be used to explore experimental observations in more complicated composite systems, e.g. commercial metal matrix composites, and may aid the development of physically relevant models for fracture processes in such materials. Acknowledgements The authors would like to acknowledge the 7nancial support of Natural Sciences and Engineering Research Council of Canada (NSERC) for this work. We would also like to gratefully thank Professor Y. Brechet for useful comments on the manuscript, and an anonymous reviewer for useful suggestions. References Al-Ostaz, A., Jasiuk, I., 1997. The in&uence of interface and arrangement of inclusions on local stresses in composite materials. Acta. Mater. 45, 4131–4143. Axelsen, M.S., Pyrz, R., 1995. Microstructural in&uence of the fracture toughness in transversely loaded unidirectional composites. In: Poursartip, A., Street, K. (Eds.), Proceedings of 10th International Conference on Composite Materials, Vol. 1. Woodhead Pub., Cambridge (UK), pp. 471– 478. Axelsen, M.S., Pyrz, R., 1997. In&uence of disorder on the evolution of interface cracks in unidirectional composites. Sci. Eng. Comp. Mater. 6, 151–158. Bao, G., Hutchinson, J.W., McMeeking, R.M., 1991. Particle reinforcement of ductile matrices against plastic-&ow and creep. Acta Metall. Mater. 39, 1871–1882. Brechet, Y., Embury, J.D., Tao, S., Luo, L., 1991. Damage initiation in metal matrix composites. Acta Metall. Mater. 39, 1781–1786. Brockenbrough, J.R., Hunt, W.H., Richmond, O., 1992. A reinforced material model using actual microstructural geometry. Scripta Metall. Mater. 27, 385–390. Brown, L.M., Embury, J.D., 1973. The initiation and growth of voids at second phase particles. Proceedings of the Third International Conference on Strength of Metals and Alloys. Institute of Metals, London, pp. 164 –169. Christman, T., Needleman, A., Suresh, S., 1989. An experimental and numerical study of deformation in metal ceramic composites. Acta Metall. 37, 3029–3050. Corbin, S.F., Wilkinson, D.S., 1994. The in&uence of particle distribution on the mechanical response of a particulate metal-matrix composite. Acta Metall. Mater. 42, 1311–1318. Eshelby, J.D., 1957. The determination of the elastic 7eld of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. London A 241, 376–396.
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Ganguly, P., 1998. High temperature deformation and failure in aluminium-alumina particulate metal matrix composites. M.A.Sc. Thesis, University of British Columbia. Ganguly, P., Poole, W.J., 2002. E4ect of reinforcement distribution on failure mechanism in aluminium based metal matrix composites. Mater. Sci. Forum 396, 1139–1144. Ganguly, P., Poole, W.J., Lloyd, D.J., 2001. Deformation and fracture characteristics of AA6061-Al2 O3 particle reinforced metal matrix composites at elevated temperatures. Scripta Mater. 44, 1099–1105. Ghosh, S., Moorthy, S., 1998. Particle fracture simulation in non-uniform microstructures of metal-matrix composites. Acta Mater. 46, 965–982. Ghosh, S., Nowak, Z., Lee, K., 1997. Quantitative characterization and modeling of composite microstructures by Voronoi cells. Acta Mater. 45, 2215–2234. Green, P.J., Sibson, R., 1978. Computing Dirichlet tessellation in plane. Comput. J. 21, 168–173. Hallquist, J.O., 1990. LS-DYNA2D User’s Manual. Livermore Software Technology Corporation, Livermore, CA. Lewandowski, J.J., Liu, C., Hunt, W.H., 1989. E4ects of matrix microstructure and particle distribution on fracture of an aluminium metal matrix composite. Mater. Sci. Eng. A 107, 241–255. Lloyd, D.J., 1991. Aspects of fracture in particulate reinforced metal matrix composites. Acta Metall. Mater. 39, 59–71. Manoharan, M., Lewandowski, J.J., 1990. Crack initiation and growth toughness of an aluminium metal matrix composite. Acta Metal. Mater. 38, 489–496. Metals Handbook, 1979. Properties and Selection: Nonferrous Alloys and Pure Metals, Vol. 2. ASM International, Metals Park, Ohio, p. 115. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with mis7tting inclusions. Acta Metall. 21, 571–574. Muskhelishvili, N.I., 1963. Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordho4 Ltd., Groningen. Needleman, A., 1987. A continuum model for void nucleation by inclusion debonding. J. Appl. Mech. 54, 525–531. Pedersen, O.B., 1983. Thermoelasticity and plasticity of composites. 1. Mean 7eld theory. Acta Metall. 31, 1795–1808. Pijaudier-Cabot, G., Bazant, Z.P., 1991. Crack interacting with particles or 7bers in composite materials. ASCE J. Eng. Mech. 117, 1611–1630. Pyrz, R., Bochenek, B., 1998. Topological disorder of microstructure and its relation to the stress 7eld. Internat. J. Solids and Structures 35, 2413–2427. Shen, Y-L., Finot, M., Needleman, A., Suresh, S., 1995. E4ective plastic response of 2-phase composites. Acta Metall. Mater. 43, 1701–1722. Simmons, G., Wang, H., 1971. Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook. MIT Press, Cambridge (USA), p. 146. Smith, P.A., 2000. Carbon 7ber reinforced plastics—properties. In: Kelly, A., Zweben, C. (Eds.), Comprehensive Composite Materials, Vol. 2. Elsevier, Oxford (UK), pp. 107–150. Watt, D.F., Xu, X.Q., Lloyd, D.J., 1996. E4ects of particle morphology and spacing on the strain 7elds in a plastically deforming matrix. Acta Mater. 44, 789–799. Yip, T-H., Wang, Z., 1997. E4ect of particle distribution on deformation behaviour of particulate metal matrix composites. Mater. Sci. Technol. 13, 125–134. Zahl, D.B., Schmauder, S., McMeeking, R.M., 1994. Transverse strength of metal matrix composites reinforced with strongly bonded continuous 7bers in regular arrangements. Acta Metall. Mater. 42, 2983–2997.
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In&uence of reinforcement arrangement on the local reinforcement stresses in composite materials P. Ganguly, W.J. Poole∗ Department of Metals and Materials Engineering, University of British Columbia, 309-6350 Stores Road, Vancouver, BC, Canada V6T 1Z4 Received 28 May 2003; received in revised form 4 November 2003; accepted 14 November 2003
Abstract The state of stress in and around reinforcements governs a number of physical processes in composite (multi-phase) materials, including the initiation of damage by either reinforcement cracking or interfacial decohesion. The stresses in the reinforcements have been observed to depend on the spatial distribution of the reinforcements, although the exact correlation is unclear. The present work determines the reinforcement stress for di4erent reinforcement arrangements, ranging from a linear array of three uniformly spaced particles, to random and clustered microstructures. The stress calculations for elastic matrices were undertaken using a computationally e5cient iterative technique. The technique was validated by comparing the results to 7nite element models, and the range of validity was determined. For the three-particle arrangements, the maximum reinforcement stress was observed when the particles were close to each other along the line of loading (a vertical arrangement). On the other hand, when the particle arrangement made a large angle with the loading direction, the reinforcement stress was low. Similar observations were recorded for the random and clustered arrangements where the location of the maximum reinforcement stress coincided with a vertical arrangement. The present work also develops a scheme for determining ‘representative volume elements’ for composite micromechanical models, based on the length scales of stress 7eld interactions. These observations can be used to rationalize damage evolution mechanisms in commercial composites, and aid the development of physically based failure models for such materials. ? 2003 Elsevier Ltd. All rights reserved.
∗
Corresponding author. E-mail address: [email protected] (W.J. Poole).
0022-5096/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2003.11.005
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Keywords: A. Stress concentrations; B. Inhomogeneous material; Reinforcement arrangement; C. Finite elements
1. Introduction Multi-phase composites present interesting possibilities for producing tailor-made materials, through suitable choice of the intrinsic phase properties, and proper control of the extrinsic factors like the volume fraction, spatial distribution and geometry (shape) of the di4erent phase arrangements. These materials have de7ned the scope of engineering materials in the past decades, using di4erent phase combinations for di4erent applications. In many structural applications, composite materials made of a relatively soft and ductile matrix, reinforced by hard and brittle reinforcements have been used. Metal matrix composites and polymer matrix composites, both reinforced by hard ceramic reinforcements, are prime examples in this category. The metallic matrix in MMCs deforms elasto-plastically, while the deformation of the polymer matrix is elastic or in some cases viscoelastic. In both cases, the primary contribution of the elastic reinforcements is to enhance the sti4ness and strength of the composite. Predicting the overall (macroscopic) properties of these composites, given the intrinsic and extrinsic contributions, is critical to their applicability. Continuum models that predict macroscopic properties that depend on volume averaged material responses (e.g. elastic modulus, yield strength and strain hardening rate) are well developed. In these cases, the intrinsic phase properties and their respective volume fractions dominate the material behavior, suitably weighed by average values of reinforcement shape and distribution characteristics. Suitable analytical (Eshelby, 1957; Mori and Tanaka, 1973; Pedersen, 1983; Corbin and Wilkinson, 1994) and 7nite element models (Christman et al., 1989; Bao et al., 1991; Shen et al., 1995) have been developed for such responses, both for elastic and elasto-plastic matrices. However, the existing models are less secure in predicting properties which are governed by the extrema of the constituent responses, e.g. failure processes governed by the ‘weakest link’ or the largest &aw. In these cases, the distribution of the reinforcements needs to be explicitly accounted for, and may play a signi7cant role in determining the macroscopic composite property. For example, changing the spatial distribution of reinforcements in cast MMCs by hot extrusion has been observed to cause a 7ve-fold increase in the composite failure strain (Ganguly et al., 2001). Numerical models, which can capture such a dependence of the material ductility on the reinforcement arrangement, are currently unavailable. Damage initiation in metal matrix composites primarily occurs by either cracking of the reinforcements (Lloyd, 1991), or by decohesion at the reinforcement–matrix interface (Manoharan and Lewandowski, 1990), while for polymer matrix composites, decohesion at the matrix–reinforcement interface is an important damage initiation mechanism (Smith, 2000). Particle cracking events can be directly related to the stress state in the reinforcements and the reinforcement fracture strength (Brechet et al., 1991). The decohesion events are governed by the traction across the interface (Needleman, 1987), and can also be related to the reinforcement stress state, given the continuity of
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the radial stress across the reinforcement–matrix interface. Predictive modeling of damage initiation would thus require an accurate determination of the reinforcement stress state, assuming that the intrinsic parameters governing the damage initiation process (reinforcement and interfacial strengths) are known a priori. For elasto-plastic matrices, 7nite element models (Brockenbrough et al., 1992; Ghosh and Murthy, 1998) have examined the local stress distributions for di4erent reinforcement arrangements. The reinforcement arrangements are generally categorized as random (hard-core) and clustered. For a random arrangement, the spatial distribution of the reinforcement centroids follows a random distribution, provided no reinforcements overlap, and all are contained within the given domain. A clustered distribution is characterized by significant non-uniformity in the reinforcement distribution, with one or more regions (the clusters) of pronounced reinforcement proximity. Both Brockenbrough et al. (1992) and Ghosh and Moorthy (1998) observed that the maximum reinforcement stress laid inside a reinforcement cluster. However, the variation of the reinforcement stress between di4erent clusters was signi7cant, and the maximum reinforcement stress in some of the clusters was lower than that in a random distribution (Ghosh and Moorthy, 1998). A signi7cant variability in the cluster stresses has also been observed by Pyrz and Bochenek (1998) for composites with elastic matrices. This indicates that global characterizations such as random or clustered do not completely dictate reinforcement stress patterns, and further investigations on the reinforcement distribution factors that determine the stress are required. Such analysis may yield suitable characterization schemes for reinforcement distributions, and lead to optimal fracture models for composite materials. Further, determining the optimal size of the representative volume element (RVE) for modeling the reinforcement stress distribution (and damage initiation) in composite materials is required. Numerical models considering a small number of reinforcements have been observed to yield non-unique stress characteristics for random and clustered microstructures (Ghosh et al., 1997). On the other hand, large microstructural models are limited by the existing computational capabilities. The present work attempts to determine the critical distribution characteristics, which dominate the local stress patterns and the resulting damage initiation characteristics in composite materials. The approach taken has been to start with simple idealized reinforcement arrangements where the local stresses in the reinforcements can be solved by an iterative elastic approach. Finite element calculations have been used to evaluate the e4ects of inter-particle spacing and matrix plasticity on the stress in the reinforcements. For relevance to commercial composites, AA6061–Al2 O3 metal matrix composite was chosen as a suitable material system, and the micromechanical calculations were conducted assuming the appropriate phase properties. The salient observations for the model particle distributions were veri7ed for large non-uniform microstructures, where stress distributions for 600 reinforcements in random and clustered arrangements were determined using the iterative approach. The analysis also yields the interaction length scales of the reinforcement stress 7elds, and helps develop a basis for estimating the ‘representative volume elements’ in multi-phase materials. Finally, the applicability of the present analysis to commercial composite models is discussed, with reference to fracture processes in metal matrix composites.
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2. Methodology 2.1. Formulation of the iterative model An iterative model was used for determining the stress 7eld due to multiple (elastic) circular reinforcements in an in7nite, elastic matrix subjected to a far-7eld stress of ∞ . The model algorithm was originally developed by Pijaudier-Cabot and Bazant (1991) and has been used previously by Pyrz and co-workers (Axelsen and Pyrz, 1995, 1997; Pyrz and Bochenek, 1998) for matrix and interfacial cracking in elastic composites. The present work validates the iterative model (using 7nite element calculations) as a tool for reinforcement stress determination in elastic composites, and applies it for stress 7eld determination in a variety of random and clustered microstructures. The multi-reinforcement problem can be reduced to determining the stress 7eld in each reinforcement, subjected to the far-7eld stress (∞ ) and a force 7eld re&ecting the reinforcement–reinforcement interactions. The matrix and the reinforcements were elastic, with their respective moduli being Dm and Dc . For any given (ith) reinforcement, the reinforcement stress (i ) and strain (i ) could be related by i = Dc i :
(1)
The composite may be transformed into a homogeneous solid (with an elastic modulus of Dm ) with an eigenstrain K imposed in the reinforcements. i = Dm (i − Ki ):
(2)
Equating the stress in Eq. (2) with that in Eq. (1) gives Ki = (Dm − Dc )Dc−1 i ;
(3)
where Ki (=Dm Ki ) is the unbalanced stress inside the reinforcement. This unbalanced stress 7eld may be balanced by applying tractions (pi ) at the reinforcement boundary pi = −Ki nc ;
(4)
where nc is an outward normal to the reinforcement boundary. Sub-dividing the given reinforcement boundary into a number of segments (each of length ds), the tractions may be replaced by corresponding point forces (F) acting at the center of each boundary segment. Fij = pi dsj ;
(5)
Fij
where is the point force corresponding to the jth segment of the boundary of the ith reinforcement. Each of the point forces would generate a stress 7eld in the homogeneous solid (Muskhelishvili, 1963). All such stress 7elds may be superposed to determine the perturbation stress 7eld (c ) at each point in the homogeneous solid. If ic is the value of the 7eld at the geometrical point corresponding to the centroid of the given reinforcement, then the reinforcement stress 7eld (i ) is given by i = ∞ + ic − Ki :
(6)
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Initially, the reinforcement stress i was assumed to be equal to ∞ . Ki and pi were determined from Eqs. (3) and (4), respectively, and the reinforcement stress (i ) determined at the end of each step (from Eq. (6)) was used as the stress estimate at the next step. This procedure was repeated until the point forces did not change signi7cantly, i.e. for steps k and (k + 1) N j j T s i=1 Nj=1 (F (F ) ) (k+1) i i ¡ 0:001; − 1 (7) N Ns j j T j=1 (Fi (Fi ) )(k) i=1 where Fij (Fij )T yields the square of the magnitude of each point force vector (Fij ), Ns is the number of boundary segments per reinforcement, and N is the number of reinforcements considered in the model. The composite was deformed under plane strain tension and a far-7eld stress of unit magnitude (22 =1, 11 =12 =0 at far-7eld). Both the matrix and the reinforcement were assumed to be linear elastic materials. The reinforcement elastic modulus was 6.7 times that of the matrix. The Poisson’s ratio (elastic) was assumed to be 0.33 for the matrix and 0.34 for the reinforcement. These values are consistent with the properties of the AA6061 matrix (Metal’s Handbook, 1979) and the alumina reinforcement (Simmons and Wang, 1971), respectively. 2.2. Reinforcement distribution generation The reinforcement distributions were computer generated and involved arrangement of equal sized, circular particles on a planar domain (matrix). The simplest distributions involved three reinforcements arranged / such that the inter-particle distances 1 were equal and the particle centroids were collinear (see Fig. 1). For a given reinforcement radius (r), these distributions were characterized by two parameters, the angle with respect to the loading axis () and the center-to-center spacing between the reinforcements (R). The angle was varied from 0◦ to 90◦ . For =0◦ , the reinforcements were above or below each other (vertical arrangement), whereas for = 90◦ , they were side by side (horizontal arrangement). The spacing between the reinforcement centroids was varied from 3 times to 10 times the reinforcement radius. These models will henceforth be referred as the three-particle uniform arrangements. More complicated reinforcement arrangements included computer-generated microstructures with random and clustered reinforcement distributions, as shown in Fig. 2. Each microstructure included 600 reinforcements arranged in a 10 × 10 planar domain, and the nominal area fraction of the reinforcements was 0.1. The random arrangement (see Fig. 2(a)) was generated following a hard-core algorithm. The reinforcement centroids were generated using a (Poisson) random number generator, with two constraints, (a) the minimum distance between the reinforcement centroids was three times the reinforcement radius (see discussion in Section 3.1), and (b) all the reinforcements were completely contained in the domain. Each of the clustered arrangements included a single cluster (marked by the dashed rectangle in Fig. 2(b–d)), and will be identi7ed 1
Center to center distance between reinforcements.
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θ
r
R
x2
x1
Fig. 1. Arrangement of three reinforcements in an in7nite matrix. The arrangement is characterized by its orientation w.r.t. the loading axis (), and the inter-reinforcement distance R. The radius of the reinforcements is r.
as the square-cluster, the long-cluster or the large-cluster, depending on the cluster shape and the number of particles in the cluster. For the square-cluster distribution (see Fig. 2(b)), the reinforcements were separated into two sub-domains, (a) a cluster of 108 particles in a 3 × 3 area (reinforcement area-fraction in the cluster = 0:2), and (b) 492 particles in the rest of the area. The cluster for the long-cluster distribution (see Fig. 2(c)) contained the same number of particles in each sub-domain (as in the square-cluster distribution), but the cluster sub-domain was de7ned on a 1:5 × 6 area (i.e. the area of the cluster was similar to the square-cluster, but the cluster aspect ratio was 4, compared to 1 for the square-cluster). The cluster for the large-cluster distribution (see Fig. 2(d)) contained 300 particles in a 5 × 5 area (reinforcement area-fraction in the cluster = 0:2), while the remaining 300 particles were distributed in the rest of the area. The distribution of the particles in each of the sub-domains followed a hard-core pattern. The reinforcement arrangement in each microstructure was a4ected by the starting value of the Poisson random number generator (the generating seed). In order to study the e4ect of the variability in the reinforcement stress due to such differences in reinforcement arrangements, 10 di4erent microstructures were generated for each distribution and the reinforcement stress was calculated for each microstructure. In the simulations described above, the reinforcement stresses were computed by assuming elastic interactions among all reinforcements (i.e. all the reinforcements interacted with each other), and will be henceforth called the full-scale iterative model. The reinforcement stresses for the large-cluster distribution were also calculated by limiting the elastic interactions within the neighborhood reinforcements. The near-neighbors (i.e. all the reinforcements which surrounded any given reinforcement) were 7rst identi7ed by Voronoi tessellation (Green and Sibson, 1978). The neighborhoods for the reinforcements were then assembled layer-by-layer. For a one-layer model, only the
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Fig. 2. Reinforcement arrangements for (a) random, (b) small-cluster and (c) long-cluster and (d) large-cluster distributions, respectively. All the distributions contain 600 reinforcements and the nominal reinforcement volume fraction is 0.1 in all the cases.
near-neighbors were assumed to interact with any given reinforcement. For a two-layer model, the near-neighbors of these reinforcements were also included in the interacting domain. Similarly, the three-layer model also included the near neighbors of these reinforcements. Fig. 3 shows typical neighborhoods for a given reinforcement for models with di4erent number of layers. 2.3. Formulation of the 8nite element models Finite element models were constructed to (a) verify the estimates of the elastic iterative models, and (b) determine the e4ect of matrix plasticity on reinforcement stresses. A plane-strain deformation was assumed, and was modeled using the two-dimensional dynamic 7nite element package LS-DYNA2D (Hallquist, 1990). Fig. 4 shows the geometries used for the two models. Symmetry conditions at x1 = 0 and x2 = 0 were
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(a)
(b)
(c)
Fig. 3. (a) One-layer, (b) two-layer and (c) three-layer models for the shaded reinforcement.
x2 = b
x2
x1 = 0
(a)
x2 = 0
x1
x2 = b
x1 = 0
(b)
x2 = 0
Fig. 4. Schematic representation of the FEM models for (a) veri7cation of the iterative models, and (b) reinforcement stress analysis for di4erent reinforcement arrangements in elasto-plastic matrices.
imposed to reduce the model to 14 of the domain. A constant velocity boundary condition (strain rate ∼0:1 s−1 ) was imposed at the far-7eld (x2 = b). Typical properties corresponding to alumina (single crystal sapphire) and AA6061 aluminium alloy were assigned for the reinforcements and the matrix, respectively. The Al2 O3 properties were obtained from Simmons and Wang (1971). The elastic properties of the AA6061 matrix were obtained from the Metal’s Handbook (1979). The plastic deformation of the matrix was represented by a power law equation = Kn ;
(8)
where and are the e4ective stress and strain, respectively, K is the strength coe5cient, and n is the strain hardening exponent. The values of K and n were estimated from an experimentally determined uniaxial compression curve for the AA6061 matrix.
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Table 1 Properties of the AA6061 matrix and the Al2 O3 reinforcement used in 7nite element simulations Constituent
Eel (GPa)
el
Yield stress (MPa)
K (MPa)
N
AA6061 Al2 O3
70 470
0.33 0.34
54
303
0.24
The various input parameters for the two constituents used in the model are summarized in Table 1. Note that the elastic properties of the matrix and the reinforcement are consistent with those assumed in the iterative models (see Section 2.1). • Veri8cation of iterative models: The models were developed to verify the estimates of the iterative models for a vertical arrangement 2 of three reinforcements, for R=r = 2:5, 3 and 5 (see Fig. 4(a)). The maximum far-7eld imposed strain was ∼5×10−5 . The matrix and the reinforcements were assumed to be elastic. The maximum, minimum and the mean stress (22 averaged over the entire reinforcement) for the central reinforcement were determined from the FEM simulations and divided by the far-7eld stress (from the elements at x2 = b) to yield the respective stress ratios. • Modeling elasto-plastic deformation: The elasto-plastic model consisted of a (a) vertical arrangement of three reinforcements, (b) horizontal arrangement of three reinforcements and (c) square arrangement of nine reinforcements (see Fig. 4(b)). The inter-particle distance was 3 times the reinforcement radius for all the three arrangements. The maximum far-7eld strain was 0.1. The reinforcements were assumed elastic, while the matrix was assumed to be elasto-plastic. 3. Stress distribution for uniform reinforcement arrangements 3.1. Elastic model for three-particle arrangements Fig. 5 shows the variation in the maximum normal stress (22 ) in the central reinforcement with change in the normalized inter-particle distance (R=r) and orientation () of the three-particle arrangement in an elastic matrix. The stress for a single particle in an in7nite matrix (where the 22 was 1.37 times the far-7eld stress (Muskhelishvili, 1963)) has been shown for comparison. The reinforcement stress was dependent on both the inter-particle distance and the angle of orientation. For a low and constant , the reinforcement stress increased with decrease in the inter-particle distance. High reinforcement stresses were observed for low at constant R=r. In the range of study, the highest reinforcement stress was observed for a vertical arrangement of reinforcements when the inter-particle distance was 3 times the reinforcement radius. On the other hand, the reinforcement stress was low for large even when the inter-particle 2 The vertical arrangement was chosen for iterative model veri7cation because it exhibited the largest e4ect of neighborhood reinforcements on the stress 7eld of a given reinforcement (see Section 3.1)
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1.8 Reinforcement stress (σ22)
R = 10r
1.7
R = 5r R = 3r
1.6
single reinf.
1.5 1.4 1.3 1.2 1.1 1
0
10 20 30 40 50 60 70 80 90 Angle of orientation (θ)
Fig. 5. Stress (22 ) in the central reinforcement for the three-particle arrangement, with changing inter-particle distance and the orientation of the arrangement w.r.t. the loading direction. The reinforcement stress for a single reinforcement in an in7nite matrix has been shown for comparison.
distance was small. For ¿ 35◦ , the reinforcement stress was always lower than that in a single reinforcement in an in7nite matrix. The results are qualitatively similar to those obtained by Pyrz and Bochenek (1998) for a di4erent material system. The e4ect of inter-particle distance (R=r) and the orientation of the three-particle arrangement () on the reinforcement stress can be qualitatively explained from the stress distribution around an isolated elastic inclusion in an in7nite elastic matrix (see Fig. 6). High stress regions can be observed above and below the reinforcement ‘a’, while low stress regions can be observed on the side. The stress perturbations range over distances of the order of the reinforcement radius. Consequently, the reinforcement stress in Fig. 5 is relatively insensitive to the presence of the second particle at large inter-particle distances (position ‘b’ in Fig. 6). On the other hand, if the second particle is placed in close vertical proximity to ‘a’ (e.g. ‘c’ in Fig. 6), the overlap of high stress regions around each particle would elevate their stresses, while the horizontal proximity of the two particles (‘d’ in Fig. 6) would lower their stresses. The high reinforcement stress for the vertical and the low stress for the horizontal arrangement (or for any arrangement where ¿ 35◦ ) is evident in Fig. 5. The high reinforcement stress observed for the vertical arrangement is consistent with the observations by several investigators (Watt et al., 1996; Al-Ostaz and Jasiuk, 1997; Pyrz and Bochenek, 1998). A natural consequence of the stress interactions between neighboring particles is that the stress 7elds may not be uniform in the reinforcement. On the other hand, the iterative model assumes the uniformity of the reinforcement stresses. This de7nes the bound of validity of the iterative model: it can only be applied to situations yielding approximately uniform reinforcement stresses. Table 2 shows the FEM predictions of the maximum, mean and minimum stresses (22 ) in reinforcements for uniform vertical arrangements in an elastic matrix, with inter-particle spacing (R) being equal to 5, 3
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1.5
b
1.3 c
1.1 0.9
a
d
b 0.5 0.0
x2
x1
Fig. 6. Stress (22 ) distribution in and around a single elastic reinforcement in an in7nite elastic matrix for a unit far-7eld stress. Elastic moduli of reinforcement = 6:7 times that of the matrix, Poisson’s ratio of matrix = 0:33, Poisson’s ratio of reinforcement = 0:34.
Table 2 Variation of the maximum, mean and the minimum reinforcement stress (22 ) ratios for a vertical arrangement of reinforcements for various R=r ratios; the stress ratio from the iterative solution is also included for comparison R=r
Maximum stress ratio (FEM)
Mean stress ratio (FEM)
Minimum stress ratio (FEM)
Stress ratio (iterative)
5.0 3.0 2.5
1.48 1.79 2.09
1.45 1.62 1.73
1.38 1.37 1.35
1.49 1.72 1.91
and 2.5 times the reinforcement radius (r). The variation in the stress at di4erent points in the reinforcement increased with decreasing inter-particle distance. When the distance between the reinforcement centroids was 5 times the reinforcement radius, the reinforcement stress variation was small. On the other hand, the reinforcement stress variation was signi7cant (∼ ± 21% from the mean value) at an R=r of 2.5. The iterative predictions of reinforcement stress for all the three R=r values lay above the mean value. This was dictated by the stress distribution in the reinforcements, with the center of the reinforcement (where the iterative model calculated the particle stress) being a relatively high stress region (see Fig. 7). The di4erence between the iterative model predictions and the FEM predicted maximum reinforcement stress was minimal (less than 5%) at R=r of 5 and 3, while the iterative prediction was signi7cantly lower
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x2
x1 Fig. 7. FEM estimate of stress (22 ) distribution in the reinforcement and the matrix for a 3-particle vertical arrangement (R = 2:5r) at an imposed strain of 0.00003. All stress values are in Pa.
than the maximum stress at R=r of 2.5. The di4erence between the iterative model predictions and the FEM predicted mean reinforcement stress was also low for R=r of 5 and 3, and relatively high for R=r of 2.5. Assuming that the maximum reinforcement stress determined the propensity of damage initiation, and the mean stress governed the reinforcement’s impact on the stress 7elds of neighboring reinforcements, the iterative model could be expected to be accurate for present purposes at R=r greater than or equal to 3. The reinforcement stress estimation for non-uniform structures was thus limited to distributions where the minimum inter-particle distance was three times the reinforcement radius. 3.2. Elasto-plastic models for uniform composite microstructure The maximum reinforcement normal stress (22 ) at an imposed strain of 0.095 for the vertical, horizontal and square arrangements embedded in an elasto-plastic matrix is shown in Fig. 8. The stress distribution was very inhomogeneous with severe stress concentrations in the square and the vertical reinforcement arrangements. Fig. 9 shows the variation of the maximum reinforcement normal stress ratio in the reinforcements
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x2
x1
Fig. 8. Stress (22 ) distribution in the model composite at an imposed strain of 0.095. The 7bers are arranged in a square, a vertical or a horizontal arrangement. The inter-particle spacing for all the three arrangements was 3 times the reinforcement radius. All the stress values are shown in Pa.
Fig. 9. Variation of the ratio of the maximum stress (22 ) in the reinforcements to the far-7eld stress at different imposed strains for a square, a vertical and a horizontal reinforcement arrangement in an elasto-plastic matrix (open symbols). The iterative solution for the stress ratio for the vertical and horizontal arrangements in an elastic matrix (7lled symbols) has also been shown for comparison. The inter-particle spacing for all the arrangements was 3 times the reinforcement radius.
(i.e. maximum 22 in the reinforcement divided by the far-7eld 22 in the matrix) as a function of strain for the square, vertical and horizontal arrangements. The stress ratio for the vertical and the horizontal three-particle arrangements from the iterative models (involving elastic matrices and in7nitesimal strains) are also shown in the 7gure. The reinforcement stress ratio was higher for the vertical arrangement than the horizontal arrangement, both for elasto-plastic and elastic matrices. However, the di4erence between the two stress ratios decreased with the plastic deformation of the matrix. The stress ratio for the vertical reinforcement arrangement was also higher than that for the square arrangement, although the di4erence was relatively small. The reinforcement stresses obtained from the elasto-plastic simulations are consistent with the results from the elastic (iterative) model, with the high stresses
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coinciding with the vertical reinforcement arrangement in both. The primary in&uence of plasticity is to reduce the di4erence in the reinforcement stresses observed for the vertical and the horizontal arrangement. In the following section, the e4ect of reinforcement arrangement on the reinforcement stress in many-particle composites is investigated. The reinforcement arrangement was represented by random and clustered distributions, with 600 reinforcements in each microstructure (see Section 2.2). Since the construction of elasto-plastic, 7nite element models for such large non-uniform microstructures was computationally prohibitive, the stress distributions were determined by the iterative models. Given the above consistency in elastic and elasto-plastic models, the stress distributions were expected to reveal the general trends of the microstructure–stress correlation in composites at low imposed strains. 4. Stress distributions for non-uniform reinforcement arrangements The maximum normal reinforcement stress (22 ) distribution for the representative random and clustered microstructures is shown in Fig. 10. As mentioned earlier, the minimum inter-particle distance allowed for the microstructures was three times the reinforcement radius. The largest 22 value observed in the random, square-cluster, long-cluster and large-cluster distributions were 1.63, 1.63, 1.66 and 1.63 times the far-7eld stress, respectively. For all distributions, proximity of reinforcements along the loading direction resulted in high reinforcement stresses. On the other hand, proximity along the X1 (horizontal) direction resulted in lower stresses. This is consistent with the earlier observations on uniform arrangements in elastic and elasto-plastic matrices (see Figs. 5 and 9). The maximum reinforcement stress for a clustered distribution was observed inside the cluster for large-cluster and long-cluster distributions, while it was outside the cluster for the square-cluster arrangement. For all the microstructures shown in Fig. 10, the maximum reinforcement stress always coincided with a set of reinforcements aligned along the loading direction (a vertical arrangement 3 ). The occurrence of the maximum reinforcement stress in the cluster depended on the presence of favorable vertical arrangements within it. The probability of such an occurrence increased as more and more particles were added to the cluster, or when the cluster shape favored a vertical arrangement. The occurrence of the maximum reinforcement stress inside the cluster was thus more probable for a large-cluster or for a high aspect ratio cluster aligned along the loading direction (the long-cluster). However, among favorably (vertically) arranged reinforcements, the maximum stress was observed in regions of low horizontal proximity (i.e. when reinforcements at large angles to the loading direction were not present near a favorable arrangement). The clustered distributions yielded more reinforcements with high stresses, as compared to the random distribution. For each microstructure containing 600 particles (see 3
By de7nition, a vertical arrangement indicates a set of reinforcements aligned perfectly along the loading direction. However, in reality, such perfect alignment is seldom observed, and the term hereby signi7es only an approximate reinforcement alignment.
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Fig. 10. Reinforcement stress (22 ) distributions for the (a) random, (b) square-cluster, (c) long-cluster and (d) large-cluster distributions. The reinforcements with maximum stresses are indicated by arrows.
Fig. 10), the number of reinforcements with a stress (22 ) more than 1.5 times the far-7eld stress was 21 for the random distribution, 27 for the square-cluster, 34 for the long-cluster and 29 for the large-cluster. In clustered arrangements, the high stress reinforcements occurred predominantly in the clusters, and thus were spatially close to each other. The square-cluster and the long-cluster contained 19% and 44% of the high stress particles in 9% of the area, respectively, while the large-cluster contained 72% of the high stress particles in 25% of the area. On the other hand, the high stress reinforcements were relatively uniformly distributed for the random microstructure. A recurrent observation for the stress distribution in the clustered microstructures was that the high stress reinforcements (inside the cluster) were predominantly observed near the side-edges of the cluster (see Fig. 10(b–d)). To investigate this phenomenon, clustered microstructures, with uniform (square arrangement) reinforcement distributions inside and outside the cluster, were generated. Fig. 11 shows the stress distribution in reinforcements for the uniform large-cluster (624 reinforcements in total,
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Fig. 11. Reinforcement stress (22 ) distribution for a large-cluster microstructure, given that the reinforcement arrangements both inside and outside the cluster were regular. Note that, for clarity, a slightly di4erent stress range has been used in this 7gure compared to that elsewhere in the paper.
overall reinforcement volume fraction ∼0:1, reinforcement volume fraction in cluster ∼0:2). The reinforcements with the maximum stress were observed near the side-edges for such an arrangement. This was primarily due to the fact that the inter-particle spacing on one side of such reinforcements was dominated by the overall volume fraction (∼0:1) and was thus larger than that inside the cluster. As shown in Figs. 5 and 6, the presence of proximal reinforcements at large angles to the loading direction reduced the reinforcement stress. In this case, the absence of such proximal neighbors on one side of the side-edge reinforcements resulted in their stress being larger compared to the reinforcements near the cluster center. Similarly, the reinforcements (inside the cluster) near the cluster top and bottom edges showed a lower stress as compared to those near the center, due to their vertical neighbors being farther away on one side. These factors may be responsible for the prevalent presence of high stress reinforcements near the side-edges of the random clusters, compared to the cluster top or bottom-edge or the cluster center. 4.1. E9ect of statistical variability of microstructures on the reinforcement stress distributions As mentioned in Section 2.2, a variety of di4erent microstructures may be generated for each reinforcement distribution, by changing the parametric values of the random number generator. The variety in the microstructure (given a particular distribution) is a direct result of the natural randomness, and is also present in commercial composite microstructures. The microstructural variability is expected to increase when a smaller number of reinforcements are considered for each distribution. This may cause non-uniqueness in the stress characteristics of di4erent distributions. Ghosh et al. (1997) observed that the stress distributions for random and clustered
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arrangements (single and triple cluster) were di4erent when the number of particles considered in each microstructure was small (25 particles). However, when larger microstructures were considered (100 particles), the stress distributions were similar to each other. Thus, while discussing trends in microstructure-stress correlation in composite materials, the evidence of the trends for su5ciently large microstructures must be ensured. Axelsen and Pyrz (1997) have studied the variation of the stress characteristics with changes in the microstructure size. They observed that the applied stress corresponding to interfacial cracking of 10% of the particles remained relatively unchanged for microstructures with more than 200 particles. In the present work, each microstructure consisted of 600 particles, and 10 di4erent microstructures were generated for each of the four di4erent reinforcement arrangements, viz. random, square-cluster, long-cluster and large-cluster. The variation in the stress characteristics for a given distribution was small. The highest reinforcement stress was predominantly observed inside the cluster for the long and the large-cluster distributions, and outside the cluster for the square-cluster arrangement. The clustered arrangements yielded a higher number of reinforcements with a high stress (22 ¿ 1:5 times the far-7eld stress), as compared to the random arrangements. For each of the clustered arrangements, the reinforcements with the high stress occurred predominantly in the clusters. The long-cluster arrangement yielded the highest reinforcement stress among all the generated microstructures, the reinforcement stress being 1.73 times the far-7eld stress. The predominance of the high stressed reinforcements (inside the clusters) near the side-edges of the clusters was also evident. Overall, the variation of the maximum reinforcement stress for di4erent microstructures (given a particular arrangement) was less than 7%. 4.2. Length scale of reinforcement–reinforcement interactions Stress (22 ) distributions were computed for the one-layer, two-layer, three-layer and full-scale iterative models for the large-cluster distribution. The stress patterns for the one-layer model did not converge monotonically to the prescribed error level (see Eq. (7)). The stress estimates for the two- and three-layer models were close to the full-scale model estimates. The maximum reinforcement normal stresses (22 ) were 1.62 for the two-layer, 1.63 for the three-layer and 1.63 for the full-scale model respectively, given a unit stress at the far-7eld. The e4ective variation between the di4erent models were estimated through an error function E() N f − i 2 1 22 22 ; (9) E() = i N 22 1 where E() is the root mean sum square of the di4erences between the reinforcef i ment stresses 22 and 22 , and N is the number of reinforcements (600 in the present f case). 22 was the more accurate (7nal) model for the reinforcement stress (i.e. more i neighboring layers considered) as compared to 22 (initial model). Table 3 shows the variation of E() for the di4erent models. The di4erence between the stress estimates for the approximate (two- and three-layer) models and the full-scale model was
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Table 3 E4ect of stress interaction length scale on the error function E() Final model
Initial model
E()
Three-layer Full-scale
Two-layer Three-layer
0.0005 0.0001
E() measures the change in the reinforcement stresses observed by limiting the stress interactions among a given number of neighboring reinforcements.
minimal. This de7nes the length scale of interaction of the reinforcements (given a minimum inter-particle spacing equal to three times the reinforcement radius), and that of the representative volume element in order to e4ectively model reinforcement stress. For example, in order to determine the reinforcement stress in the cluster, an RVE consisting of the cluster surrounded by two to three layers of reinforcements may be e4ective, as the reinforcement distribution outside this neighborhood does not in&uence the cluster reinforcement stress. Fig. 12 compares the (iterative) reinforcement stress calculations for such an RVE (with three layers of neighbors) with the stress distributions from the full-scale iterative model, for the large-cluster microstructure. The stress distributions were similar for the two models. The maximum reinforcement stress was 1.62 for the RVE model and 1.63 for the full-scale model. The error function E() for the reinforcement stress in the large-cluster between the RVE (initial) and the full-scale model (7nal) was 6×10−4 . The RVE shown in Fig. 12(a) may be used for stress computation in elastic composites. Given that the interaction length scales are expected to be smaller for an elasto-plastic matrix (Yip and Wang, 1997), the elastic RVE should be su5cient for stress calculations even in elasto-plastic composites. Thus, the present iterative model may be used to determine the e4ective RVE for a given elasto-plastic composite microstructure, provided that the minimum inter-particle distances and the reinforcement shapes are similar to the iterative model characteristics. 4.3. Summary of results from numerical models on uniform and non-uniform composites The salient observations from the iterative and 7nite element models were • Proximity of reinforcements along (or at small angles to) the loading direction increased reinforcement stress, while proximity at large angles to the loading direction decreased the reinforcement stress. Thus, multi-directional proximity of reinforcements (as generally found in reinforcement clusters) did not guarantee the highest reinforcement stress. The reinforcement stress was maximum when a number of reinforcements were aligned along the loading direction, and other reinforcements at large angles to the loading direction were absent near such an arrangement. • The maximum reinforcement stresses in the clustered microstructures were not signi7cantly higher than that in the random microstructure. However, the number of reinforcements with relatively high stress was higher in the clustered microstructures,
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Fig. 12. Reinforcement stress (22 ) distributions in the large-cluster for computations taking into account interactions between (a) only the reinforcements in the RVE, and (b) all the reinforcements, for an unit far-7eld stress. The reinforcements with the maximum stresses are indicated by arrows.
as compared to the random one. For the clustered microstructures, the higher stress reinforcements also primarily occurred inside the clusters and were thus in spatial proximity to each other. The high stress reinforcements inside the clusters were predominantly observed near the side-edges of the cluster. • The cluster shape had a signi7cant e4ect on the reinforcement stress. Clusters that contained more reinforcements, or were preferentially oriented to the loading direction were more probable to yield reinforcements with the maximum stress. These distributions also generated more reinforcements with relatively high stresses. • For uniform reinforcement arrangements, the e4ect of the arrangement on the reinforcement stress was qualitatively similar for elastic and elasto-plastic matrices. However, plastic deformation of the matrix reduced the stress di4erential between the di4erent arrangements. Note that models with non-uniform reinforcement arrangements in elasto-plastic matrices were not developed due to computational limitations.
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4.4. Application: fracture analysis in MMCs The above results from model composite microstructures may help rationalize the physical phenomena dominating fracture processes in commercial composites, e.g. metal matrix composites (MMCs). Damage nucleation in metal matrix composites, either by particle cracking or interfacial decohesion, is related to the reinforcement stress state (in this case, the 22 ). The reinforcement stress 22 modeled in the present work may thus dictate MMC damage nucleation. Note that the present work only deals with reinforcement stresses in undamaged composites, and thus may only explain damage initiation events, or the early stages of damage progression (when the damage events are far apart from each other and spatially uncorrelated). As in any continuum model, the iterative model does not include any intrinsic length scale and is thus unable to model localization phenomena (e.g. microscopic necking leading to severe strain localization into shear bands which occurs during void coalescence (Brown and Embury, 1973)). Zahl et al. (1994) have shown that at high reinforcement volume fractions (volume fraction ∼0:39 for square reinforcement arrangement loaded along the square-edge, ∼0:68 for hexagonal arrangement and ∼0:78 for square arrangement loaded along the squarediagonal), considerable stress enhancements may occur due to reinforcement impingement of the plastic &ow planes. Under such situations, the iterative model predictions (which assume an elastic matrix) may be signi7cantly di4erent from the reinforcement stresses observed in an elasto-plastic matrix. However, the iterative model in the present work is restricted to a minimum inter-particle spacing of 3 times the reinforcement radius (see Section 3.1), which translates to a maximum reinforcement volume fraction of 0.35 for the square arrangement (under plane strain deformation). In addition, damage initiation in commercial MMCs may occur at strains as low as 0.01 at room temperature (Lloyd, 1991; Brechet et al., 1991). Even at elevated temperatures, macroscopic necking (the limit for spatially uncorrelated damage evolution) was observed in a AA6061 + 20%Al2 O3 MMC at strains ∼0:04 (Ganguly, 1998). At such macroscopic strains, the trends in the reinforcement arrangement e4ects on the stress 7elds in undamaged composites with elasto-plastic matrices have been shown to be similar to those in undamaged elastic composites (see Fig. 9). Thus, the results from the elastic composite models with non-uniform distribution may be qualitatively valid for damage initiation and early stages of damage evolution in commercial MMCs (with moderate clustering of reinforcements), where the matrix deformation is elasto-plastic. The similarity of the maximum 22 stress for the clustered and random distribution suggests a concurrence of damage initiation (the 7rst damage nucleation event) in the di4erent microstructures. The number of high stress reinforcements observed in the clustered arrangements was higher than that in the random arrangements. However, considering that the percentage of reinforcement with stress (22 ) greater than 1.5 times the far-7eld stress ranged from 3.5% (21 out of 600 reinforcements) for the random arrangement to 5.7% (34 out of 600 reinforcements) for the long-cluster, the early stages of damage evolution (when the damage events are non-interacting) may be generally similar for the random and the clustered arrangements. Experimentally, Ganguly (1998) observed that the necking strains for an as-cast (clustered microstructure) and an extruded (random microstructure) AA6061–20%Al2 O3 MMC were similar, although
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their failure strains (measured from reduction of area at the fracture surface) di4ered by an order of magnitude. This seems to indicate that the damage evolution in random and clustered microstructures may be signi7cantly di4erent in the later stages. The later stages are generally characterized by interactions between the nucleated damage and intact reinforcements, either by a rearrangement of the local stress/strain 7elds or by physical coalescence of the voids. The stress/strain perturbations caused by nucleated damage can change the stress in the neighboring reinforcements. Ganguly and Poole (2002) have shown that a damage event reduces the stress in reinforcements positioned vertically above or below it, while it increases stress in the reinforcements arranged horizontally or along the diagonal for a square arrangement. Thus, while vertical arrangements aid damage nucleation, the propagation of the damage is hindered due to absence of reinforcements at the sides or along the diagonal. This may explain the observed dominance of clusters in the MMC failure process: the clusters contain reinforcements close to each other along all directions, and thus may be optimally suited for both damage nucleation or propagation. Isolated vertical arrangements away from the clusters may be well suited (in some cases, more than the clusters, see Fig. 10(b)) for damage nucleation, but the propagation of the damage in such situations may be hindered. In addition, in the clustered microstructures, the reinforcements with high stresses (which may be considered probable candidates for damage nucleation) were mostly located inside the cluster, while they were relatively uniformly distributed in the random microstructures. Thus, the nucleated damage may be expected to be close to each other (and mostly concentrated in the clusters) for the clustered microstructures, while it may be relatively uniform in random microstructures. High damage densities in MMC reinforcement clusters have been experimentally observed by several investigators (Lloyd, 1991; Lewandowski et al., 1989; Ganguly et al., 2001). The proximity of damage nucleation events for clustered microstructures may aid substantial damage– reinforcement interactions (reorientation of the local stress/strain 7elds) or damage– damage interactions (enhanced coalescence of the voids). Both these processes may lead to catastrophic failure processes and suitably lower failure strains. 5. Conclusions The e4ect of reinforcement distribution on the reinforcement stresses in model composites was studied using micromechanical (iterative and 7nite element) models. Both uniform and non-uniform (random and clustered) reinforcement arrangements in elastic and elasto-plastic matrices were considered. The reinforcement stress calculations for the elastic composites were undertaken using a computationally e5cient iterative algorithm. The validity of the algorithm was demonstrated by comparing the iterative results with those from 7nite element models, and the bounds of validity (i.e. conditions when the iterative results would no longer be valid) were determined. The reinforcement stress was sensitive to both the proximity and the angular orientation (w.r.t. the loading axis) of the neighboring reinforcements. The highest reinforcement stress was observed when the reinforcements were close to each other along the loading
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direction. On the other hand, reinforcement arrangements aligned at a large angle with the loading direction yielded low stresses. The maximum normal reinforcement stress was similar for random and clustered distributions, suggesting a concurrence of damage initiation for di4erent microstructures. However, reinforcements with high stresses were fairly uniformly distributed for the random distribution, but concentrated in the clusters for the clustered distributions. This may lead to signi7cant damage–reinforcement and damage–damage interactions in the clusters and make them the preferred sites for damage evolution. The e4ect of the reinforcement arrangement on the reinforcement stresses was observed to be limited to small length scales. For any given reinforcement, the stresses were relatively insensitive to reinforcement arrangements beyond the immediate neighborhood (2 or 3 layers of neighboring reinforcements), thus de7ning the relevant length scale for micromechanical ‘representative volume elements’. The results of this study could be used to explore experimental observations in more complicated composite systems, e.g. commercial metal matrix composites, and may aid the development of physically relevant models for fracture processes in such materials. Acknowledgements The authors would like to acknowledge the 7nancial support of Natural Sciences and Engineering Research Council of Canada (NSERC) for this work. We would also like to gratefully thank Professor Y. Brechet for useful comments on the manuscript, and an anonymous reviewer for useful suggestions. References Al-Ostaz, A., Jasiuk, I., 1997. The in&uence of interface and arrangement of inclusions on local stresses in composite materials. Acta. Mater. 45, 4131–4143. Axelsen, M.S., Pyrz, R., 1995. Microstructural in&uence of the fracture toughness in transversely loaded unidirectional composites. In: Poursartip, A., Street, K. (Eds.), Proceedings of 10th International Conference on Composite Materials, Vol. 1. Woodhead Pub., Cambridge (UK), pp. 471– 478. Axelsen, M.S., Pyrz, R., 1997. In&uence of disorder on the evolution of interface cracks in unidirectional composites. Sci. Eng. Comp. Mater. 6, 151–158. Bao, G., Hutchinson, J.W., McMeeking, R.M., 1991. Particle reinforcement of ductile matrices against plastic-&ow and creep. Acta Metall. Mater. 39, 1871–1882. Brechet, Y., Embury, J.D., Tao, S., Luo, L., 1991. Damage initiation in metal matrix composites. Acta Metall. Mater. 39, 1781–1786. Brockenbrough, J.R., Hunt, W.H., Richmond, O., 1992. A reinforced material model using actual microstructural geometry. Scripta Metall. Mater. 27, 385–390. Brown, L.M., Embury, J.D., 1973. The initiation and growth of voids at second phase particles. Proceedings of the Third International Conference on Strength of Metals and Alloys. Institute of Metals, London, pp. 164 –169. Christman, T., Needleman, A., Suresh, S., 1989. An experimental and numerical study of deformation in metal ceramic composites. Acta Metall. 37, 3029–3050. Corbin, S.F., Wilkinson, D.S., 1994. The in&uence of particle distribution on the mechanical response of a particulate metal-matrix composite. Acta Metall. Mater. 42, 1311–1318. Eshelby, J.D., 1957. The determination of the elastic 7eld of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. London A 241, 376–396.
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