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A unit cell model for brittle fracture of particles embedded in a ductile matrix
Computational Materials Science 25 (2002) 85–91 www.elsevier.com/locate/commatsci
A unit cell model for brittle fracture of particles embedded in a ductile matrix A. Eckschlager *, W. Han 1, H.J. B€ ohm Christian Doppler Laboratory for Micromechanics of Materials, Institute of Lightweight Structures and Aerospace Engineering, Vienna University of Technology, A-1040 Vienna, Austria
Abstract A finite element based approach for modeling the successive failure of brittle particulates embedded in a ductile matrix is presented. Three-dimensional unit cells describing specific arrangements of spherical particles are employed, which are meshed so that an appropriate fracture surface is predefined for each of the inhomogeneities. Activation of brittle cleavage at these surfaces is controlled deterministically on the basis of Weibull-type fracture probabilities, which are evaluated from the current stress distributions within the particles. As a first step uniaxial tensile loading of a particle reinforced metal matrix composite is considered for which damage due to particle fracture is modeled while the other damage modes, interfacial decohesion and ductile failure of the matrix, are not active. Results covering the consecutive cleavage of a number of particles are presented and discussed in terms of the macroscopic response and of the evolution of the fracture probabilities of the particulates. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 07.05.Tp; 62.20.Mk Keywords: Particle reinforced composites; Metal matrix composites; Particle fracture; Weibull fracture probability; Unit cell models
1. Introduction At the length scale of the reinforcements, called the microscale in the following, particle reinforced metal matrix composites (MMCs) and related materials show three basic mechanisms for the initiation and progress of damage, viz. brittle cleavage of the particles, decohesion at the inter*
Corresponding author. E-mail address: [email protected] (A. Eckschlager). URL: http://ilfb.tuwien.ac.at. 1 Present address: Alcan Mass Transportation Systems, CH8048 Zurich, Switzerland.
face between matrix and reinforcements, and ductile failure of the matrix. Under given loading conditions, the relative importance of and the interactions between these local damage modes, and consequently the overall ductility and damage behavior of the composite, are determined by the strength as well as stiffness properties of all constituents and by their geometrical arrangement. A considerable number of micromechanically based modeling studies have been reported on the initiation and evolution of microscale damage and failure in particle reinforced MMCs, in which analytical [17], numerical [16] as well as combined approaches [6,13] were employed. Unit cell
0927-0256/02/$ - see front matterÓ 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 5 2 - 5
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A. Eckschlager et al. / Computational Materials Science 25 (2002) 85–91
models, in which the local stress and strain fields can be resolved to a high degree, have covered all of the above damage modes for planar and axisymmetric model geometries, see e.g. [2,7,15]. Recent work [3,10], however, has shown that twodimensional models tend to give unsatisfactory predictions for the mechanical response of particle reinforced materials. Plane stress models generally underestimate and plane strain as well as generalized plain strain models overestimate the overall elastic and elastoplastic stiffness of such materials. Compared to three-dimensional unit cells, both groups of planar models have been shown to give rise to considerably different phase averages of the microscale stresses and strains and to underestimate the widths of their distributions [3]. As a consequence, two-dimensional models may not give reliable predictions for damage relevant variables. For example the Weibull fracture probabilities of the particles were found to be markedly underestimated by two-dimensional unit cells at a given reinforcement volume fraction [8]. Axisymmetric cell descriptions fare better in this respect, but by design can handle only highly regular microgeometries, so that local ‘‘hot spots’’ caused by irregular particle spacing can hardly be investigated. Accordingly, it is of considerable interest to study the initiation and progress of microscale damage in particulate reinforced MMCs by threedimensional unit cells that involve a number of particles in irregular arrangements. Experiments have shown that in a number of MMC systems, among them Al2618-T4 reinforced by SiC particulates, brittle cleavage of the ceramic particles tends to be the primary microscale mechanism for initiating damage, see e.g. [13]. Cracks have been found to be preferentially oriented normally to the loading direction [14] and, depending on particle shape and size, tend to run through the inclusions’ centers [16,18]. The work presented in the following was carried out as a first
step towards modeling a damage mechanism of this type via three-dimensional multi-particle unit cells.
2. Three-dimensional multi-inclusion unit cell models 2.1. Modeling assumptions In order to keep the study reasonably simple and in accordance with the aims stated above, idealized models of particle reinforced MMCs were generated and studied. Simplifications in terms of the constituent material response took the form of prescribing isotropically elastic material behavior for the particles, assuming perfect bonding between matrix and particles, and describing the matrix by a J2 continuum plasticity model with a modified Ludwik hardening law of the type n
ry ¼ ry;0 þ hðeeq;p Þ :
ð1Þ
Here ry is the actual flow stress, ry;0 the initial yield stress, and eeq;p the accumulated equivalent plastic strain, while h and n stand for the hardening coefficient and the hardening exponent, respectively. The material parameters used in the analyses, which closely follow the data given in [13], are listed in Table 1, where E denotes the Young’s modulus and m the Poisson’s ratio. All analyses were carried out for initially stress free and virgin materials, i.e. the residual stress states that are typically present in MMCs [9] were neglected. Brittle failure of the particles under uniaxial tensile loading was modeled as ‘‘instantaneous total cleavage’’ along predefined ‘‘fracture surfaces’’, which in the undeformed state are assumed to be planar, to be oriented normal to the macroscopic loading direction and to pass through the particles’ centers. These assumptions, while not
Table 1 Material parameters used for the elastoplastic Al2618-T4 matrix (modified Ludwik hardening law) and the elastic SiC reinforcements
Al2618-T4 matrix SiC reinforcement
E (GPa)
m
ry;0 (MPa)
h (MPa)
n
m
rf (GPa)
70 450
0.30 0.17
184 –
722.7 –
0.49 –
– 3.0
– 1.0
A. Eckschlager et al. / Computational Materials Science 25 (2002) 85–91
unreasonable, neglect perturbations to the local stress fields caused by both intact and fractured neighboring inclusions, which may influence the position and orientation of cracks within the inclusions. While always leading to total splitting of the involved particles, the cracks were assumed to be arrested immediately after starting to penetrate into the matrix, and crack growth by interfacial decohesion or ductile matrix damage was not provided for. This modeling approach is closely related to the ones proposed in [7,16]. It limits the evolution of damage to the sequential failure of particles and clearly constitutes a major simplification compared to the complexities of microscale damage in actual particle reinforced MMCs. It is, however, very suitable for testing simulation models and for studying ‘‘what-if’’ scenarios. Weibull-type [19] fracture probabilities were evaluated for each particle j at each load increment by using the expression m Z 1 r1 ðxÞ ðjÞ Pfr ¼ 1 exp dV ; V0 V ðjÞ : r1 >0 rf ð2Þ where r1 ðxÞ describes for the distribution of the maximum principal stress within the jth particle and V ðjÞ : r1 > 0 is that part of the particle’s volume in which r1 is tensile. The reference volume V0 was chosen to be equal to the volume of a single particle. Values for the parameters m and rf , which stand for the Weibull modulus and the characteristic strength of SiC particulates, respectively, are given in Table 1. The use of Weibull fracture probabilities for modeling the brittle failure of particles embedded in ductile matrices can account for effects of the absolute size of the inclusions and is well established in the literature, see e.g. [1,4,11,12]. A detailed discussion of the strengths and weaknesses of such approaches can be found in [18]. For simplicity, in the present study a deterministic (Rankine) failure criterion was used, in which a particle is assumed to fail when its fracture probability according to Eq. (2) exceeds values of Pfrcr;1 ¼ 0:593 or Pfrcr;2 ¼ 0:632. In the case of highly dilute inclusions (in which the stresses and strains are uniform [5]), these choices correspond to failure occurring when the maxi-
87
mum principal stress in the particle reaches 90% or 100% of the characteristic strength rf , respectively. A further simplification was introduced by approximating the microgeometry of an actual composite by an infinite periodic arrangement of particles described by three-dimensional unit cells that contain a limited number of identical inclusions. Within this modeling strategy, the failure of a single inclusion in the unit cell corresponds to a material in which a periodically repeating pattern of local damage occurs. In addition, instead of aiming for unit cell geometries that approach statistically isotropic particle arrangements as closely as possible, as was done e.g. in [3], a simpler scheme geared towards reduced complexity of meshing was employed. It is based on partitioning the cube-shaped unit cell into 64 (i.e. 4 4 4) subvolumes, 15 of which were randomly selected to contain spherical particles of equal size that are embedded in matrix material, whereas the other 49 subvolumes were wholly assigned to the matrix phase. A typical phase arrangement resulting from this procedure is shown in Fig. 1(a), for which the particle volume fraction takes a value of n ¼ 6:3%. Even though considerable irregularity is evident in this microgeometry, interparticle distances are constrained to remain relatively large and particle positions in Fig. 1 clearly show patterns of layering and alignment. 2.2. Finite element modeling The microscale stress and strain fields in the unit cells were evaluated with the finite element code ABAQUS/Standard V.5.8 (Hibbitt, Karlsson and Sorensen Inc., Pawtucket, RI, 1998) using modified tetrahedral elements with quadratic shape functions (3D10M). Meshing was carried out with the preprocessor PATRAN V.8.0 (MacNeal–Schwendler Corp., Los Angeles, CA, 1998), a fracture surface as described above being provided for each particle. Periodic boundary conditions were used throughout and displacement controlled geometrically non-linear analyses describing loading by uniaxial tension were carried out. Typical element counts for a model as shown in Fig. 1 were about 40,000 10-node tetrahedra.
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Fig. 1. (a) Unit cell model of an SiC/Al2618-T4 MMC with 15 identical spherical particles of total volume fraction n ¼ 6:3% and (b) undeformed as well as deformed unit cell showing the status after seven particles have failed by cleavage under loading in the 1-direction (Pfrcr ¼ 0:593, dark shade and solid lines mark initial state, light shade and dotted lines mark deformed state).
Particle failure on the predefined fracture surfaces was modeled by a node release technique and was deterministically controlled by the Weibull fracture probabilities of the individual particles as discussed above. The actual implementation used the ABAQUS user subroutines MPC, UVARM and UEXTERNALDB to identify particles fulfilling the failure conditions, to instantaneously disconnect all degrees of freedom across the fracture surface within such a particle, and to subsequently reestablish equilibrium at a constant applied strain. Despite the numerical difficulties associated with the reduction in overall stiffness and the marked rearrangement of local stresses upon the instantaneous release of a considerable number of degrees of freedom, the above procedure was found to be highly reliable in giving rise to convergence provided only one particle fails at a given applied strain. In the present model, the actual loads at which particle cleavage occurs evidently depend on the local stress distributions and thus on the chosen arrangement of particles. Accordingly, an important point in the implementation was the development of a fully automatic algorithm, in which no user intervention is required during a run involving successive fracture of a number of particles. The calculation of the inclusions’ Weibull fracture probabilities made use of a feature of ABAQUS that allows the approximate evaluation
of volume integrals of some function f over some subvolume, in this case particle j, by a weighted sum of the type Z 1 ðjÞ hf i ¼ ðjÞ f ðxÞ dV V V ðjÞ
N ðjÞ 1 X
V ðjÞ
fl Vl
with
l¼1
N ðjÞ X
Vl ¼ V ðjÞ :
ð3Þ
l¼1
Here fl stands for the value of the function f ðxÞ at integration point l, which is weighted by the volume Vl associated with this integration point. N ðjÞ is the total number of integration points within particle j.
3. Discussion of results Uniaxial tensile loading up to overall nominal strains of some 20% was simulated for the phase arrangement shown in Fig. 1(a). For loading in the 1-direction and using critical Weibull fracture probabilities of Pfrcr;1 ¼ 0:593 and Pfrcr;2 ¼ 0:632 the model predicted failure by brittle cleavage of seven particles, the sequence of fracture being INC08, INC03, INC07, INC13, INC10, INC09 and INC14 in both cases. The end state of the deformation process is displayed in Fig. 1(b), where the fractured particles can be easily discerned.
A. Eckschlager et al. / Computational Materials Science 25 (2002) 85–91
Fig. 2. Overall uniaxial stress–strain curves predicted by the 15particle unit cell model of an SiC/Al2618-T4 MMC (n ¼ 6:3%) shown in Fig. 1 using two different critical Weibull fracture probabilities of Pfrcr;1 ¼ 0:593 and Pfrcr;2 ¼ 0:632 to trigger particle cracking.
The corresponding overall uniaxial stress vs. strain relationship obtained by homogenizing the responses of the unit cell using the above two critical Weibull fracture probabilities is given in Fig. 2. Each of the curves shows sudden reductions of the overall stiffness at five different strain values, the first three of which correspond to failure of a single inclusion, whereas the other two were each caused by the fracture of two particles in quick succession. These marked drops in stiffness, which are not found in experimentally obtained stress vs. strain curves, are a direct consequence of using a periodically repeating microgeometry that contains only 15 particles per unit cell––failure of a single inclusion in the unit cell accordingly corresponds to the simultaneous fracture of approximately 6.7% of all particles in the composite. Comparison between the two stress–strain relations shows that for given material properties and a given geometrical arrangement of the spherical particles within the unit cell only the different critical Weibull fracture probabilities control the occurrence of particle cracking at certain overall strains. In Fig. 2 the sequence of particle cracking is the same for both values of Pfrcr , which indicates that for the material properties used here the nonlinear behavior of the matrix does not give rise to major stress redistribution in the load range where
89
particle fracture takes place. As expected, the maximum load carrying capacity of the model composite under uniaxial loading in the 1-direction depends on the critical Weibull fracture probability; for the present case it is coincident with the failure of the third particle at nominal overall strains of approximately 16.6% and 17.8%. Because a unit cell description was used, the evolution of damage related variables could be followed in some detail in the present model. Fig. 3 shows the evolution of the Weibull fracture probabilities of five selected particles, INC03, INC07, INC08, INC09 and INC15 (numbering as in Fig. 1), the same abscissa scaling being used as in Fig. 2. The first four of these particles fail within the load sequence under study, whereas the last one is not subjected to sufficiently high loads. In interpreting Fig. 3 it should be noted that the highest fracture probabilities shown are those obtained in the last increment before failure occurred and thus the curves do not reach the critical value of Pfrcr ¼ 0:593. Generally, the Weibull fracture probabilities of the particles depend non-linearly on the overall ðjÞ strain (there are rapid increases of the Pfr in terms of the overall stress once the matrix has yielded), which, however, depend markedly on the position of a given inclusion. Load redistribution effects upon fracture of a particle are predicted to be fairly complex. The first failure of a reinforcement
Fig. 3. Predicted evolution of Weibull fracture probabilities under uniaxial tensile loading for selected particles (Pfrcr ¼ 0:593; designations follow Fig. 1). Note that curves stop upon the failure of a particle.
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(INC08) leads to marked unloading of INC07, which is a direct neighbor in loading direction, and causes a very small increase of the failure probability of INC03, which also is a close neighbor due to periodicity, but is somewhat offset. As a consequence, initial trends in fracture probability are changed and INC03 fails considerably earlier than INC07. Failure of both INC08 and INC03 gives rise to small decreases of the fracture probability for particle INC09, but increases for particle INC15, both of which are situated relatively far away from the fractured inclusions. The latter inclusion does not become critical for the loads considered here.
4. Conclusions and outlook A finite element based three-dimensional multiinclusion unit cell model of a particle reinforced MMC was discussed, which allows studying the successive failure of a number of particulates due to brittle cleavage. The fracture of any given particle was controlled deterministically via a Weibull-type fracture probability, was assumed to take place instantaneously along a predefined fracture surface, and was implemented via a node release technique. This modeling strategy allows to account for load redistribution effects upon the failure of any particle in three-dimensional microgeometries. Results from computational runs on a simplified particle arrangement were presented as proof of concept. To keep the complexity of the model at a reasonable level, a number of simplifications were introduced with respect to microgeometry, activation of particle cleavage by a Weibull-based Rankine fracture criterion, and location as well as orientation of crack planes. Work currently in progress is aimed at improvements with respect to these simplifications, while retaining the overall modeling strategy. Phase arrangements based on random sequential adsorption algorithms in analogy to [3] allow considerably more realistic descriptions of statistically isotropic microgeometries, even though in the short term the number of particles per unit cell will remain limited to 15–20 on account of computational requirements. A fully
statistical cleavage model for the particles can be implemented by triggering cleavage by MonteCarlo-type procedures based on the Weibull fracture probabilities of each inclusion, compare [16]. Performing a number of runs of this type for a given microgeometry and evaluating ensemble averaged stress vs. strain diagrams results in a higher number of less prominent stiffness drops compared to Fig. 2. The main practical difficulty with such an approach lies in its high computational costs. The modeling strategy presented here in principle can also be extended in terms of improving the selection of the positions and orientations of the crack planes in the reinforcements as damage progresses. Because the microscale stresses and strains in the particles can be noticeably influenced by load redistribution effects due to the failure of other inclusions, in such a case the use of predefined fracture surfaces, however, would have to be abandoned in favor of remeshing right before a particle fails.
Acknowledgements The authors wish to acknowledge the financial support of the Christian Doppler Research Society and helpful discussions with Prof. F.G. Rammerstorfer.
References [1] T. Antretter, F.D. Fischer, Particle cleavage and ductile crack growth in a two-phase composite on a microscale, Comput. Mater. Sci 13 (1998) 1–7. [2] H. Berns, A. Melander, D. Weichert, N. Asnafi, C. Broeckmann, A. Gross-Weege, A new material for cold forging tools, Comput. Mater. Sci 11 (1998) 166–188. [3] H.J. B€ ohm, W. Han, Comparisons between three-dimensional and two-dimensional multi-particle unit cell models for particle reinforced metal matrix composites, Modell. Simul. Mater. Sci. Engng. 9 (2001) 47–65. [4] Y.J.M. Brechet, J.D. Embury, S. Tao, L. Luo, Damage initiation in metal matrix composites, Acta Metall. Mater. 39 (1991) 1781–1786. [5] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. R. Soc. London A 241 (1957) 376–396. [6] R. Estevez, E. Maire, P. Franciosi, D.S. Wilkinson, Effect of particle clustering on the strengthening versus damage
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rivalry in particulate reinforced elastic plastic materials: A 3-D analysis from a self-consistent modelling, Eur. J. Mech. A/Solids 18 (1999) 785–804. S. Ghosh, S. Moorthy, Particle fracture simulation in nonuniform microstructures of metal-matrix composites, Acta Mater. 46 (1998) 965–982. W. Han, A. Eckschlager, H.J. B€ ohm, The effects of threedimensional multi-particle arrangements on the mechanical behavior and damage initiation of particle-reinforced MMCs, Compos. Sci. Technol. 61 (2001) 1581–1590. S. Ho, E.J. Lavernia, Thermal residual stresses in metal matrix composites: A review, Appl. Compos. Mater. 2 (1995) 1–30. T. Iung, M. Grange, Mechanical behavior of two-phase materials investigated by the finite element method: necessity of three-dimensional modeling, Mater. Sci. Engng. A 201 (1995) L8–L11. C.A. Lewis, P.J. Withers, Weibull modeling of particle cracking in metal matrix composites, Acta Metall. Mater. 43 (1995) 3685–3699. J. LLorca, J.L. Martınez, M. Elices, Reinforcement fracture and tensile ductility in sphere-reinforced metal-matrix
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composites, Fatigue Fract. Engng. Mater. Struct. 20 (1997) 689–702. J. LLorca, C. Gonzalez, Microstructural factors controlling the strength and ductility of particle reinforced metalmatrix composites, J. Mech. Phys. Sol. 46 (1998) 1–28. N.M. Mawsouf, A micromechanical mechanism of fracture initiation in discontinuously reinforced metal matrix composite, Mater. Charact. 44 (2000) 321–327. S.R. Nutt, A. Needleman, Void nucleation at fiber ends in Al–SiC composites, Scripta Metall. 21 (1987) 705–710. R. Pandorf, Beitrag zur FE-Simulation des Kriechens partikelverst€arkter Metallmatrix-Verbundwerkstoffe, VDI Fortschritt-Berichte, Reihe 5, vol. 585, 2000. K. Tohgo, G.J. Weng, A progressive damage mechanics in particle-reinforced metal-matrix composites under high triaxial tension, J. Engng. Mater. Technol. 116 (1994) 414–420. K. Wallin, T. Saario, K. T€ orr€ onen, Fracture of brittle particles in a ductile matrix, Int. J. Fract. 32 (1987) 201– 209. W. Weibull, A statistical distribution function of wide applicability, J. Appl. Mech. 18 (1951) 293–298.
A unit cell model for brittle fracture of particles embedded in a ductile matrix A. Eckschlager *, W. Han 1, H.J. B€ ohm Christian Doppler Laboratory for Micromechanics of Materials, Institute of Lightweight Structures and Aerospace Engineering, Vienna University of Technology, A-1040 Vienna, Austria
Abstract A finite element based approach for modeling the successive failure of brittle particulates embedded in a ductile matrix is presented. Three-dimensional unit cells describing specific arrangements of spherical particles are employed, which are meshed so that an appropriate fracture surface is predefined for each of the inhomogeneities. Activation of brittle cleavage at these surfaces is controlled deterministically on the basis of Weibull-type fracture probabilities, which are evaluated from the current stress distributions within the particles. As a first step uniaxial tensile loading of a particle reinforced metal matrix composite is considered for which damage due to particle fracture is modeled while the other damage modes, interfacial decohesion and ductile failure of the matrix, are not active. Results covering the consecutive cleavage of a number of particles are presented and discussed in terms of the macroscopic response and of the evolution of the fracture probabilities of the particulates. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 07.05.Tp; 62.20.Mk Keywords: Particle reinforced composites; Metal matrix composites; Particle fracture; Weibull fracture probability; Unit cell models
1. Introduction At the length scale of the reinforcements, called the microscale in the following, particle reinforced metal matrix composites (MMCs) and related materials show three basic mechanisms for the initiation and progress of damage, viz. brittle cleavage of the particles, decohesion at the inter*
Corresponding author. E-mail address: [email protected] (A. Eckschlager). URL: http://ilfb.tuwien.ac.at. 1 Present address: Alcan Mass Transportation Systems, CH8048 Zurich, Switzerland.
face between matrix and reinforcements, and ductile failure of the matrix. Under given loading conditions, the relative importance of and the interactions between these local damage modes, and consequently the overall ductility and damage behavior of the composite, are determined by the strength as well as stiffness properties of all constituents and by their geometrical arrangement. A considerable number of micromechanically based modeling studies have been reported on the initiation and evolution of microscale damage and failure in particle reinforced MMCs, in which analytical [17], numerical [16] as well as combined approaches [6,13] were employed. Unit cell
0927-0256/02/$ - see front matterÓ 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 5 2 - 5
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A. Eckschlager et al. / Computational Materials Science 25 (2002) 85–91
models, in which the local stress and strain fields can be resolved to a high degree, have covered all of the above damage modes for planar and axisymmetric model geometries, see e.g. [2,7,15]. Recent work [3,10], however, has shown that twodimensional models tend to give unsatisfactory predictions for the mechanical response of particle reinforced materials. Plane stress models generally underestimate and plane strain as well as generalized plain strain models overestimate the overall elastic and elastoplastic stiffness of such materials. Compared to three-dimensional unit cells, both groups of planar models have been shown to give rise to considerably different phase averages of the microscale stresses and strains and to underestimate the widths of their distributions [3]. As a consequence, two-dimensional models may not give reliable predictions for damage relevant variables. For example the Weibull fracture probabilities of the particles were found to be markedly underestimated by two-dimensional unit cells at a given reinforcement volume fraction [8]. Axisymmetric cell descriptions fare better in this respect, but by design can handle only highly regular microgeometries, so that local ‘‘hot spots’’ caused by irregular particle spacing can hardly be investigated. Accordingly, it is of considerable interest to study the initiation and progress of microscale damage in particulate reinforced MMCs by threedimensional unit cells that involve a number of particles in irregular arrangements. Experiments have shown that in a number of MMC systems, among them Al2618-T4 reinforced by SiC particulates, brittle cleavage of the ceramic particles tends to be the primary microscale mechanism for initiating damage, see e.g. [13]. Cracks have been found to be preferentially oriented normally to the loading direction [14] and, depending on particle shape and size, tend to run through the inclusions’ centers [16,18]. The work presented in the following was carried out as a first
step towards modeling a damage mechanism of this type via three-dimensional multi-particle unit cells.
2. Three-dimensional multi-inclusion unit cell models 2.1. Modeling assumptions In order to keep the study reasonably simple and in accordance with the aims stated above, idealized models of particle reinforced MMCs were generated and studied. Simplifications in terms of the constituent material response took the form of prescribing isotropically elastic material behavior for the particles, assuming perfect bonding between matrix and particles, and describing the matrix by a J2 continuum plasticity model with a modified Ludwik hardening law of the type n
ry ¼ ry;0 þ hðeeq;p Þ :
ð1Þ
Here ry is the actual flow stress, ry;0 the initial yield stress, and eeq;p the accumulated equivalent plastic strain, while h and n stand for the hardening coefficient and the hardening exponent, respectively. The material parameters used in the analyses, which closely follow the data given in [13], are listed in Table 1, where E denotes the Young’s modulus and m the Poisson’s ratio. All analyses were carried out for initially stress free and virgin materials, i.e. the residual stress states that are typically present in MMCs [9] were neglected. Brittle failure of the particles under uniaxial tensile loading was modeled as ‘‘instantaneous total cleavage’’ along predefined ‘‘fracture surfaces’’, which in the undeformed state are assumed to be planar, to be oriented normal to the macroscopic loading direction and to pass through the particles’ centers. These assumptions, while not
Table 1 Material parameters used for the elastoplastic Al2618-T4 matrix (modified Ludwik hardening law) and the elastic SiC reinforcements
Al2618-T4 matrix SiC reinforcement
E (GPa)
m
ry;0 (MPa)
h (MPa)
n
m
rf (GPa)
70 450
0.30 0.17
184 –
722.7 –
0.49 –
– 3.0
– 1.0
A. Eckschlager et al. / Computational Materials Science 25 (2002) 85–91
unreasonable, neglect perturbations to the local stress fields caused by both intact and fractured neighboring inclusions, which may influence the position and orientation of cracks within the inclusions. While always leading to total splitting of the involved particles, the cracks were assumed to be arrested immediately after starting to penetrate into the matrix, and crack growth by interfacial decohesion or ductile matrix damage was not provided for. This modeling approach is closely related to the ones proposed in [7,16]. It limits the evolution of damage to the sequential failure of particles and clearly constitutes a major simplification compared to the complexities of microscale damage in actual particle reinforced MMCs. It is, however, very suitable for testing simulation models and for studying ‘‘what-if’’ scenarios. Weibull-type [19] fracture probabilities were evaluated for each particle j at each load increment by using the expression m Z 1 r1 ðxÞ ðjÞ Pfr ¼ 1 exp dV ; V0 V ðjÞ : r1 >0 rf ð2Þ where r1 ðxÞ describes for the distribution of the maximum principal stress within the jth particle and V ðjÞ : r1 > 0 is that part of the particle’s volume in which r1 is tensile. The reference volume V0 was chosen to be equal to the volume of a single particle. Values for the parameters m and rf , which stand for the Weibull modulus and the characteristic strength of SiC particulates, respectively, are given in Table 1. The use of Weibull fracture probabilities for modeling the brittle failure of particles embedded in ductile matrices can account for effects of the absolute size of the inclusions and is well established in the literature, see e.g. [1,4,11,12]. A detailed discussion of the strengths and weaknesses of such approaches can be found in [18]. For simplicity, in the present study a deterministic (Rankine) failure criterion was used, in which a particle is assumed to fail when its fracture probability according to Eq. (2) exceeds values of Pfrcr;1 ¼ 0:593 or Pfrcr;2 ¼ 0:632. In the case of highly dilute inclusions (in which the stresses and strains are uniform [5]), these choices correspond to failure occurring when the maxi-
87
mum principal stress in the particle reaches 90% or 100% of the characteristic strength rf , respectively. A further simplification was introduced by approximating the microgeometry of an actual composite by an infinite periodic arrangement of particles described by three-dimensional unit cells that contain a limited number of identical inclusions. Within this modeling strategy, the failure of a single inclusion in the unit cell corresponds to a material in which a periodically repeating pattern of local damage occurs. In addition, instead of aiming for unit cell geometries that approach statistically isotropic particle arrangements as closely as possible, as was done e.g. in [3], a simpler scheme geared towards reduced complexity of meshing was employed. It is based on partitioning the cube-shaped unit cell into 64 (i.e. 4 4 4) subvolumes, 15 of which were randomly selected to contain spherical particles of equal size that are embedded in matrix material, whereas the other 49 subvolumes were wholly assigned to the matrix phase. A typical phase arrangement resulting from this procedure is shown in Fig. 1(a), for which the particle volume fraction takes a value of n ¼ 6:3%. Even though considerable irregularity is evident in this microgeometry, interparticle distances are constrained to remain relatively large and particle positions in Fig. 1 clearly show patterns of layering and alignment. 2.2. Finite element modeling The microscale stress and strain fields in the unit cells were evaluated with the finite element code ABAQUS/Standard V.5.8 (Hibbitt, Karlsson and Sorensen Inc., Pawtucket, RI, 1998) using modified tetrahedral elements with quadratic shape functions (3D10M). Meshing was carried out with the preprocessor PATRAN V.8.0 (MacNeal–Schwendler Corp., Los Angeles, CA, 1998), a fracture surface as described above being provided for each particle. Periodic boundary conditions were used throughout and displacement controlled geometrically non-linear analyses describing loading by uniaxial tension were carried out. Typical element counts for a model as shown in Fig. 1 were about 40,000 10-node tetrahedra.
88
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Fig. 1. (a) Unit cell model of an SiC/Al2618-T4 MMC with 15 identical spherical particles of total volume fraction n ¼ 6:3% and (b) undeformed as well as deformed unit cell showing the status after seven particles have failed by cleavage under loading in the 1-direction (Pfrcr ¼ 0:593, dark shade and solid lines mark initial state, light shade and dotted lines mark deformed state).
Particle failure on the predefined fracture surfaces was modeled by a node release technique and was deterministically controlled by the Weibull fracture probabilities of the individual particles as discussed above. The actual implementation used the ABAQUS user subroutines MPC, UVARM and UEXTERNALDB to identify particles fulfilling the failure conditions, to instantaneously disconnect all degrees of freedom across the fracture surface within such a particle, and to subsequently reestablish equilibrium at a constant applied strain. Despite the numerical difficulties associated with the reduction in overall stiffness and the marked rearrangement of local stresses upon the instantaneous release of a considerable number of degrees of freedom, the above procedure was found to be highly reliable in giving rise to convergence provided only one particle fails at a given applied strain. In the present model, the actual loads at which particle cleavage occurs evidently depend on the local stress distributions and thus on the chosen arrangement of particles. Accordingly, an important point in the implementation was the development of a fully automatic algorithm, in which no user intervention is required during a run involving successive fracture of a number of particles. The calculation of the inclusions’ Weibull fracture probabilities made use of a feature of ABAQUS that allows the approximate evaluation
of volume integrals of some function f over some subvolume, in this case particle j, by a weighted sum of the type Z 1 ðjÞ hf i ¼ ðjÞ f ðxÞ dV V V ðjÞ
N ðjÞ 1 X
V ðjÞ
fl Vl
with
l¼1
N ðjÞ X
Vl ¼ V ðjÞ :
ð3Þ
l¼1
Here fl stands for the value of the function f ðxÞ at integration point l, which is weighted by the volume Vl associated with this integration point. N ðjÞ is the total number of integration points within particle j.
3. Discussion of results Uniaxial tensile loading up to overall nominal strains of some 20% was simulated for the phase arrangement shown in Fig. 1(a). For loading in the 1-direction and using critical Weibull fracture probabilities of Pfrcr;1 ¼ 0:593 and Pfrcr;2 ¼ 0:632 the model predicted failure by brittle cleavage of seven particles, the sequence of fracture being INC08, INC03, INC07, INC13, INC10, INC09 and INC14 in both cases. The end state of the deformation process is displayed in Fig. 1(b), where the fractured particles can be easily discerned.
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Fig. 2. Overall uniaxial stress–strain curves predicted by the 15particle unit cell model of an SiC/Al2618-T4 MMC (n ¼ 6:3%) shown in Fig. 1 using two different critical Weibull fracture probabilities of Pfrcr;1 ¼ 0:593 and Pfrcr;2 ¼ 0:632 to trigger particle cracking.
The corresponding overall uniaxial stress vs. strain relationship obtained by homogenizing the responses of the unit cell using the above two critical Weibull fracture probabilities is given in Fig. 2. Each of the curves shows sudden reductions of the overall stiffness at five different strain values, the first three of which correspond to failure of a single inclusion, whereas the other two were each caused by the fracture of two particles in quick succession. These marked drops in stiffness, which are not found in experimentally obtained stress vs. strain curves, are a direct consequence of using a periodically repeating microgeometry that contains only 15 particles per unit cell––failure of a single inclusion in the unit cell accordingly corresponds to the simultaneous fracture of approximately 6.7% of all particles in the composite. Comparison between the two stress–strain relations shows that for given material properties and a given geometrical arrangement of the spherical particles within the unit cell only the different critical Weibull fracture probabilities control the occurrence of particle cracking at certain overall strains. In Fig. 2 the sequence of particle cracking is the same for both values of Pfrcr , which indicates that for the material properties used here the nonlinear behavior of the matrix does not give rise to major stress redistribution in the load range where
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particle fracture takes place. As expected, the maximum load carrying capacity of the model composite under uniaxial loading in the 1-direction depends on the critical Weibull fracture probability; for the present case it is coincident with the failure of the third particle at nominal overall strains of approximately 16.6% and 17.8%. Because a unit cell description was used, the evolution of damage related variables could be followed in some detail in the present model. Fig. 3 shows the evolution of the Weibull fracture probabilities of five selected particles, INC03, INC07, INC08, INC09 and INC15 (numbering as in Fig. 1), the same abscissa scaling being used as in Fig. 2. The first four of these particles fail within the load sequence under study, whereas the last one is not subjected to sufficiently high loads. In interpreting Fig. 3 it should be noted that the highest fracture probabilities shown are those obtained in the last increment before failure occurred and thus the curves do not reach the critical value of Pfrcr ¼ 0:593. Generally, the Weibull fracture probabilities of the particles depend non-linearly on the overall ðjÞ strain (there are rapid increases of the Pfr in terms of the overall stress once the matrix has yielded), which, however, depend markedly on the position of a given inclusion. Load redistribution effects upon fracture of a particle are predicted to be fairly complex. The first failure of a reinforcement
Fig. 3. Predicted evolution of Weibull fracture probabilities under uniaxial tensile loading for selected particles (Pfrcr ¼ 0:593; designations follow Fig. 1). Note that curves stop upon the failure of a particle.
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(INC08) leads to marked unloading of INC07, which is a direct neighbor in loading direction, and causes a very small increase of the failure probability of INC03, which also is a close neighbor due to periodicity, but is somewhat offset. As a consequence, initial trends in fracture probability are changed and INC03 fails considerably earlier than INC07. Failure of both INC08 and INC03 gives rise to small decreases of the fracture probability for particle INC09, but increases for particle INC15, both of which are situated relatively far away from the fractured inclusions. The latter inclusion does not become critical for the loads considered here.
4. Conclusions and outlook A finite element based three-dimensional multiinclusion unit cell model of a particle reinforced MMC was discussed, which allows studying the successive failure of a number of particulates due to brittle cleavage. The fracture of any given particle was controlled deterministically via a Weibull-type fracture probability, was assumed to take place instantaneously along a predefined fracture surface, and was implemented via a node release technique. This modeling strategy allows to account for load redistribution effects upon the failure of any particle in three-dimensional microgeometries. Results from computational runs on a simplified particle arrangement were presented as proof of concept. To keep the complexity of the model at a reasonable level, a number of simplifications were introduced with respect to microgeometry, activation of particle cleavage by a Weibull-based Rankine fracture criterion, and location as well as orientation of crack planes. Work currently in progress is aimed at improvements with respect to these simplifications, while retaining the overall modeling strategy. Phase arrangements based on random sequential adsorption algorithms in analogy to [3] allow considerably more realistic descriptions of statistically isotropic microgeometries, even though in the short term the number of particles per unit cell will remain limited to 15–20 on account of computational requirements. A fully
statistical cleavage model for the particles can be implemented by triggering cleavage by MonteCarlo-type procedures based on the Weibull fracture probabilities of each inclusion, compare [16]. Performing a number of runs of this type for a given microgeometry and evaluating ensemble averaged stress vs. strain diagrams results in a higher number of less prominent stiffness drops compared to Fig. 2. The main practical difficulty with such an approach lies in its high computational costs. The modeling strategy presented here in principle can also be extended in terms of improving the selection of the positions and orientations of the crack planes in the reinforcements as damage progresses. Because the microscale stresses and strains in the particles can be noticeably influenced by load redistribution effects due to the failure of other inclusions, in such a case the use of predefined fracture surfaces, however, would have to be abandoned in favor of remeshing right before a particle fails.
Acknowledgements The authors wish to acknowledge the financial support of the Christian Doppler Research Society and helpful discussions with Prof. F.G. Rammerstorfer.
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