the associated low strength poses a major limitation to the application of fiber-reinforced MMCs. Attempts are underway to improve this strength with minimal loss of longitudinal properties. Fig. 4 A typical stress-strain curve under transverse loading when the interface bond strength is weak. Debonding initiates at a fairly low stress at B, and is accompanied with small-scale plasticity around the debonded fibers. Large-scale plasticity ensues at C, and failure occurs at D. Source: Ref 6, 7 The residual radial stress at the interface has a strong influence on the stress corresponding to point B, because the local radial stress is simply the sum of the residual clamping stress and the local stress due to far-field transverse loading. Equations have been provided previously for calculating the residual stress as well as the stress due to a transverse applied load. There is also a need to model the postdebonded region, BC, when the material is primarily elastic. Reference 9 does provide equations for calculating displacements for a slipping fiber (similar to Eq 13 shown previously), and they may be used to calculate the postdebonded stress-strain behavior in region BC. Stress-Strain Response and Stress Distribution Under Elastic-Plastic and Elastic-Viscoplastic Conditions Longitudinal and Thermal Loading. Axisymmetric conditions are maintained under longitudinal loading, and the CCM model is particularly advantageous. However, plasticity rules must be invoked. A method for incorporating plasticity and viscoplasticity into the CCM analysis is indicated in Ref 10 and 11. The calculations are based on the successive approximation approach of plasticity (Ref 12), as well as the elastic- plastic calculations performed earlier using a CCM model (Ref 13, 14). The matrix cylinder of Fig. 1 is divided into a series of thin concentric cylinders, and a finite-difference scheme is used to integrate Eq 2 and 4. Any arbitrary strain-hardening behavior can be modeled using the CCM formalism. A simpler, but less accurate, method is to simply use a one-dimensional isostrain model. Essentially, the composite stress is expressed as: (Eq 14) where σ f and σ m are the stresses in the fiber and matrix, respectively. At any given strain, the stresses in the fiber and the matrix can be obtained from the respective stress-strain data, and the results summed according to Eq 14. This approach cannot account for the triaxial stress state around the fiber, but does provide a reasonably good estimate of the stress-strain plot. A typical stress-strain plot for a longitudinally loaded composite is illustrated in Fig. 5. The onset of nonlinearity of the stress-strain curve is associated with yielding of the matrix, as confirmed by observation of slip bands and using transmission electron microscopy (Ref 6, 15). The yielding of the matrix is influenced by the residual axial stress in the matrix, which is usually tensile, and the yield strength of the matrix. If the stress- strain behavior of the fiber-free “neat” material is known, then the residual axial stress in the matrix can be estimated from the knee, as shown in Ref 6. Fig. 5 Typical stress-strain curve for a longitudinally loaded MMC The postyield domain of the stress-strain plot is matrix-plasticity-dominated. However, toward the end of region BC in Fig. 5, fiber cracks start occurring, so that there is combination of plasticity and damage. Here, the statistical fiber-fracture model in Ref 16 and 17 can be used to incorporate the effects of fiber failure. Essentially, the fiber fracture model is used to determine an effective nonlinear stress response of the fiber (see subsequent equations), as indicated in Fig. 6. The effective stress-strain behavior of the damaged fibers can then be used either in the elastic-viscoplastic CCM model, as was done in Ref 18, or in a simple one-dimensional representation of the composite longitudinal response. Fig. 6 Schematic of the effective stress-strain response for damaging brittle fibers, based on the statistical model of Ref 16, 17 For time-dependent loading, viscoplastic or creep models have to be used. Among them, Bodner-Partom's viscoplastic model with directional hardening (Ref 19, 20) has been used extensively in the finite difference code for elastic- plastic analysis (FIDEP) computer code (Ref 10, 11) that is based on the CCM model. The model contains 12 unknown constants that are estimated from tension, fatigue, stress relaxation, and creep tests on the matrix-only “neat” material. Values for a number of titanium alloys are provided in Ref 11 and 21. A number of other models have also been developed to determine the stress-strain response under viscoplastic deformation. These include the vanishing fiber diameter (VFD) model, (Ref 22, 23, and 24), the method of cells (Ref 25), and the generalized method of cells (Ref 26). The computer code VISCOPLY has been developed based on the VFD model and using the viscoplastic model of Ref 27. Results from that code have been compared with experimental data on titanium matrix composites (Ref 28, 29). Comparisons of the different codes with Bodner-Partom's viscoelastic model were conducted in Ref 21 by considering both in-phase and out- of- phase thermomechanical fatigue loading. The models were compared with results from the FEM method. Transverse Loading. The models referenced in the previous paragraph have been used to determine the stress- strain response under transverse loading. One problem in modeling is that at elevated temperatures, the residual clamping stress at the interface is reduced significantly. Combined with the fact that the transverse strength of the interface is maintained quite low to obtain damage tolerance in the fiber direction, interface debonding occurs quite early at elevated temperatures. However, because of the ductility of the matrix, debonding does not lead to failure. Consequently, plasticity and viscoplasticity with debonded fibers must be considered during transverse loading of a unidirectional composite. As indicated earlier, the FEM method may be relied upon, provided the micromechanisms of deformation and damage (such as debonding) are adequately taken into account, and provided the inelastic deformation of the matrix is modeled accurately. However, FEM is not efficient for thermomechanical loading. In recent years, the method of cells has been extended to account for fiber-matrix debonding. Also, the VFD model has been modified to account for a debonded fiber. Details on these issues may be obtained from the references in the previous section. Simplified equations of the stress-strain behavior under elastic-plastic conditions, based on FEM calculations, have been provided in Ref 30. A Ramberg-Osgood power law model is used to represent the matrix plastic behavior, and it is shown that the effective yield strength of a fully bonded composite is increased over that of the matrix material. Further details are presented in the section on discontinuous composites. Multiaxial Loading. For loading other than in the 0° or 90° direction, one may refer to the work in Ref 31 and 32, where the plastically deformed composite is treated as an orthotropic elastic-plastic material. The flow rule here allows for volume change under plastic deformation, unlike the case of monolithic alloys. The approach has the advantage of collapsing data from different lamina on a single curve. However, the method is semiempirical and is not based on the constituent elastic-plastic deformation behavior of the matrix. A more rigorous formulation based on a FEM technique was adopted in Ref 33 and 34. Stand- alone software, called IDAC, is available, such that any multiaxial stress state can be analyzed. Note that off-axis loading is simply a case of multiaxial loading of a unidirectional lamina. The input requirements for the program are the elastic, plastic, and viscoplastic parameters of the matrix and the tensile strength of the fiber- matrix interface. The latter is included because of the propensity for fiber-matrix debonding at low transverse stresses, which strongly influences the post-debond elastic-viscoplastic response of the composite. References cited in this section 2. Z. Hashin and B.W. Rosen, The Elastic Moduli of Fiber Reinforced Materials, J. Appl. Mech. (Trans ASME), Vol 31, 1964, p 223–232 3. N.J. Pagano and G.P. Tandon, Elastic Response of Multidirectional Coated-Fiber Composites, Compos. Sci. Technol., Vol 31, 1988, p 273–293 4. G.P. Tandon, Use of Composite Cylinder Model as Representative Volume Element for Unidirectional Fiber Composites, J. Compos. Mater., Vol 29 (No. 3), 1995, p 385–409 5. B. Budiansky, J.W. Hutchinson, and A.G. Evans, Matrix Fracture in Fiber-Reinforced Ceramics, J. Mech. Phys. Solids, Vol 34, 1986, p 167–189 6. S.M. Pickard, D.B. Miracle, B.S. Majumdar, K. Kendig, L. Rothenflue, and D. Coker, An Experimental Study of Residual Fiber Strains in Ti-15-3 Continuous Fiber Composites, Acta Metall. Mater., Vol 43 (No. 8), 1995, p 3105–3112 7. B.S. Majumdar and G.M. Newaz, Inelastic Deformation of Metal Matrix Composites: Plasticity and Damage Mechanisms, Philos. Mag., Vol 66 (No. 2), 1992, p 187–212 8. W.S. Johnson, S.J. Lubowinski, and A.L. Highsmith, Mechanical Characterization of Unnotched SCS6/Ti-15-3 MMC at Room Temperature, ASTM STP 1080, ASTM, 1990, p 193–218 9. A.L. Highsmith, D. Shee, and R.A. Naik, Local Stresses in Metal Matrix Composites Subjected to Thermal and Mechanical Loading, ASTM STP 1080, J.M. Kennedy, H.H. Moeller, and W.S. Johnson, Ed., ASTM, 1990, p 3–19 10. N.I. Muskhelisvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff International, Leyden, The Netherlands, 1963 11. D. Coker, N.E. Ashbaugh, and T. Nicholas, Analysis of Thermo-Mechanical Cyclic Behavior of Unidirectional Metal Matrix Composites, ASTM STP 1186, H. Sehitoglu, Ed., 1993, p 50–69 12. D. Coker, N.E. Ashbaugh, and T. Nicholas, Analysis of the Thermo-Mechanical Behavior of [0] and [0/90] SCS-6/Timetal21S Composites, ASME, Vol 34 (No. H00866- 1993), W.F. Jones, Ed., 1993, p 1– 16 13. A. Mendelson, Plasticity Theory and Application, Macmillan, 1968 14. C.H. Hamilton, S.S. Hecker, and L.J. Ebert, Mechanical Behavior of Uniaxially LoadedMultilayered Cylindrical Composites, J. Basic Eng., 1971, p 661–670 15. S.S. Hecker, C.H. Hamilton, and L.J. Ebert, Elasto-Plastic Analysis of Residual Stresses and Axial Loading in Composite Cylinders, J. Mater., Vol 5, 1970, p 868–900 16. B.S. Majumdar, G.M. Newaz, and J.R. Ellis, Evolution of Damage and Plasticity in Metal Matrix Composites, Metall. Trans. A, Vol 24, 1993, p 1597–1610 17. W.A. Curtin, J. Am. Ceram. Soc., Vol 74, 1991, p 2837 18. W.A. Curtin, Ultimate Strengths of Fibre- Reinforced Ceramics and Metals, Composites, Vol 24 (No. 2), 1993, p 98–102 19. B.S. Majumdar and G.M. Newaz, In-Phase TMF of a 0° SiC/Ti-15-3 System: Damage Mechanisms, and Modeling of the TMC Response, Proc. 1995 HITEMP Conf., NASA CP 10178, Vol 2, National Aeronautics and Space Administration, 1995, p 21.1–21.13 20. S.R. Bodner and Y. Partom, Constitutive Equations of Elastic Viscoplastic Strain Hardening Materials, J. Appl. Mech. (Trans. ASME), Vol 42, 1975, p 385–389 21. K.S. Chan and U.S. Lindholm, Inelastic Deformation Under Non-Isothermal Loading, ASME J. Eng. Mater. Technol. (Trans ASME), Vol 112, 1990, p 15–25 22. D. Robertson and S. Mall, Micromechanical Analysis and Modeling, Titanium Matrix Composites Mechanical Behavior, S. Mall and T. Nicholas, Ed., Technomic Publishing Co., 1998, p 397–464 23. G.J. Dvorak and Y.A. Bahei-El-Din, Plasticity Analysis of Fibrous Composites, J. Appl. Mech. (Trans. ASME), Vol 49, 1982, p 193–221 24. G.J. Dvorak and Y.A. Bahei-El-Din, Elastic- Plastic Behavior of Fibrous Composites, J. Mech. Phys. Solids, Vol 27, 1997, p 51–72 25. Y.A. Bahei-El-Din, R.S. Shah, and G.J. Dvorak, Numerical Analysis of Rate-Dependent Behavior of High Temperature Fibrous Composites, Mechanics of Composites at Elevated Temperatures, AMD Vol 118, American Society of Mechanical Engineers, 1991, p 67–78 26. J. Aboudi, A Continuum Theory for Fiber Reinforced Elastic-Viscoplastic Composites, Int. J. Eng. Sci., Vol 20, 1982, p 605–621 27. S.A. Arnold, T.E. Wilt, A.F. Saleeb, and M.G. Castelli, An Investigation of Macro and Micromechanical Approaches for a Model MMC System, Proc. 6th Annual HITEM Conf., NASA Conf. Publ. 19117, Vol II, National Aeronautics and Space Administration (NASA) Lewis, 1995, p 52.1– 52.12 28. M.A. Eisenberg and C.F. Yen, A Theory of Multiaxial Anisotropic Viscoplasticity, J. Appl. Mech. (Trans. ASME), Vol 48, 1991, p 276–284 29. M. Mirdamadi, W.S. Johnson, Y.A. Bahei- El-Din, and M.G. Castelli, Analysis of Thermomechanical Fatigue of Unidirectional TMCs, ASTM STP 1156, W.W. Stinchcomb and N.E. Ashbaugh, Ed., ASTM, 1993, p 591–607 30. W.S. Johnson and M. Mirdamadi, “Modeling and Life Prediction Methodology of TMCs Subjected to Mission Profiles,” NASA TM 109148, National Aeronautics and Space Administration(NASA) Langley, 1994 31. G. Bao, J.W. Hutchinson, and R.M. McMeeking, Particle Reinforcement of Ductile Matrices Against Plastic Flow and Creep, Acta Metall. Mater., Vol 39, 1991, p1871–1882 32. C.T. Sun, J.L. Chen, G.T. Shah, and W.E. Koop, Mechanical Characterization of SCS- 6/Ti-6-4 Metal Matrix Composites, J. Compos. Mater., Vol 29, 1990, p 1029–1059 33. C.T. Sun, Modeling Continuous Fiber Metal Matrix Composite as an Orthotropic Elastic- Plastic Material, ASTM STP 1032, W.S. Johnson, Ed., ASTM, 1989, p 148–160 34. J. Ahmad, S. Chandu, U. Santhosh, and G.M. Newaz, “Nonlinear Multiaxial Stress Analysis of Composites,” Research Applications, Inc. final report to the Air Force Research Laboratory, Materials and Manufacturing Directorate, Contract F33615-96-C-5261, Wright-Patterson Air Force Base, OH, 1999 1. J. Ahmad, G.M. Newaz, and T. Nicholas, Analysis of Characterization Methods for Inelastic Composite Material Deformation Under Multiaxial Stresses, Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P.J. Bonacuse, Ed., ASTM, 2000, p 41–53 Engineering Mechanics and Analysis of Metal-Matrix Composites Bhaskar S. Majumdar, New Mexico Institute of Mining and Technology Micromechanics of Discontinuously Reinforced MMCs The stress-strain response of discontinuously reinforced composites (DRCs) is influenced by the morphology of particles, both in the elastic and elastic-plastic domain. Most of the applications of DRCs have been with discontinuously reinforced aluminum alloys (DRAs). The particle shapes of alumina or SiC reinforcements, employed most often in DRAs, are generally blocky and angular, rather than spherical or cylindrical. Whiskers are generally modeled as cylinders with a high aspect ratio, the ratio of height to diameter. Elastic Deformation. Although the primary effects of particles are their modulus and volume fraction, their shape has influence on the modulus of the composite. The effects of particle shapes are discussed in Ref 35 and 36. Experimental data in Ref 36, 37, 38, and 39 show that the finite- element results of Ref 36 for a unit cylinder with an aspect ratio of unity provide best agreement with experimental data. The Hashin Shtrikman bounds for the elastic moduli (Ref 40) are too wide apart for making an adequate estimate. Rather, Mura's formulation (Ref 41), although developed for spherical particles, appears to match the unit cylinder FEM solution reasonably well up to a fiber volume fraction of 0.25. Beyond that volume fraction the deviation from the FEM result is large, and actual FEM results, such as those in Ref 36 should be used. Note also that the ROM (Eq 1) overestimates the modulus of DRCs and should not be used. The elastic moduli from Mura's analytical solution (Ref 41) are as follows: (Eq 15) and E and ν for the composite are obtained from: Here, the subscripts “m” and “r” refer to the matrix and reinforcement, respectively, and G and K are the shear modulus and bulk modulus of the composite, respectively. V p is the volume fraction of reinforcement. In addition to the FEM approach, one may use Eshelby's technique to determine elastic modulus for various shapes and volume fractions of reinforcements. Such calculations are nicely illustrated in Ref 35, which provides a computer program at the end of the book. It is also relevant to mention that although particle distribution has negligible effect on elastic modulus at low volume fractions, the effect becomes larger at high volume fractions. The distribution effect is largely experienced through a change in the hydrostatic stress distribution in the matrix, and such a change is anticipated to be larger when the volume fraction of the matrix phase is smaller. However, experimental results are not available that can validate this distribution effect. Elastic-Plastic Deformation. Analysis of elastic-plastic deformation with rigid, spherical particles has been considered in Ref 42 for an elastic-perfectly plastic (no strain hardening) matrix. The flow stress, σ c , under dilute conditions (V p < 0.25) may be expressed as: (Eq 16) where β was estimated to be approximately 0.375 for spherical particles. In Ref 30, FEM analysis was conducted for different-shaped rigid particles. The σ c for a perfectly plastic matrix reinforced with unit cylinders (loaded perpendicular to the axis of the cylinder) show β to be a function of V p : (Eq 17) When matrix strain hardening is considered, the results in Ref 30 can be used. Essentially, the matrix is represented by the Ramberg-Osgood formulation: (Eq 18) where α = , n is the inverse of the work- hardening exponent, N, of the matrix, E m is the elastic modulus of the matrix, and σ o is a normalizing parameter approximately equal to the yield strength of the matrix. The corresponding stress-strain response of the particulate-reinforced composite, based on FEM calculations (Ref 30), is estimated to be: (Eq 19) where the subscript “c” refers to the composite, “m” is the matrix, and σ N is a reference stress, almost equal to the 0.2% yield stress of the composite. σ N is a function of the volume fraction, work-hardening rate of the matrix, and the particle shape. It is expressed in Ref 37 as: (Eq 20) where V p is the volume fraction of particles, β can be obtained from Eq 17, and κ is a function of the shape and volume fraction of particles and is plotted in Ref 30. ξ is approximately 0.94 at small plastic strains (less than 3 o , where o is the yield strain of the matrix), but ξ becomes unity at large strains. Approximate values of κ are 3.1, 3.5, and 4.25 at V p of 0.1, 0.15, and 0.2, respectively. All these quantities are valid only for unit cylinder particles, and they are considered here because this shape provides best correlation with the experimentally determined elastic modulus of DRAs. For particles of other shapes, one may refer to Ref 30. In summary, Eq 19 provides the entire stress-strain curve for the composite when the parameters E m and n (= ) in Eq 18 are known for the matrix. Results in Ref 37 and 38 for a silicon carbide particle, SiCp, reinforced 7093 aluminum alloy show that the previous estimation formulas provide reasonable correlation with the experimentally determined stress-strain response of the composite. A few remarks are in order here. The formulas can only provide approximate values, and they were based on rigid particles with infinite elastic modulus. Experiments on composites with the same volume fraction of particles in the same matrix, but with different sizes of particles, show that the strength tends to increase with smaller particle size. This effect is not captured by FEM calculations, where the absolute size of particles do not influence the results. Possible effects of particle size include: • The reduction of grain size of the matrix and, hence, an increased strength of the matrix • The punching of dislocations from the particles and the associated strengthening, which would be more effective at small particle sizes • The limitation of standard FEM solution when the size scales become small • The matrix alloy may be affected by reaction with the particle. These issues are not captured by current modeling practice, and hence the predictive equations provided previously should only be used for initial estimation. The ductility of the composite is an important issue in DRCs, unlike fiber-reinforced systems, where debonding fibers can provide damage tolerance when loaded in the fiber direction. Ductility of DRCs can vary anywhere from 10 to 70% of the matrix, with ductility being affected significantly at volume fractions of 0.25 and higher. Recent discussions on these issues are available in Ref 35, 37, 38, and 39. Important damage mechanisms include particle fracture and particle-matrix debonding (see the article “Fracture and Fatigue of DRA Composites” in Fatigue and Fracture, Volume 19 of ASM Handbook). Particle fracture is particularly dominant for high-strength matrices, such as peak or underaged 2xxx and 7xxx aluminum alloys, and is established by observing mirror halves of the fracture surface. Debonding is observed in lower-strength matrices, such as 6xxx aluminum alloys, although it is often difficult to establish whether failure occurred at the interface or whether it initiated in the matrix immediately adjacent to the interface. The latter mode mostly occurs when the bond is strong and the matrix is quite weak, such as aluminum alloys in the overaged condition. Models of ductility have been proposed in Ref 37 and 39 to obtain initial estimates of ductility. The model in Ref 39 is based primarily on statistical particle fracture according to Weibull statistics and subsequent specimen instability according to the Considere criterion. (See the article “Uniaxial Compression Testing” in Mechanical Testing and Evaluation, Volume 8 of ASM Handbook, for an introduction to the Considere criterion.) The problem with this approach is that necking is essentially nonexistent in DRCs possessing any appreciable volume fraction of particles. Nevertheless, reasonable agreement was obtained with experiments conducted by the authors. The model in Ref 37 presupposes the existence of particle cracks, and failure is postulated based on rupture of the matrix between cracked particles. Once again, reasonably good agreement is obtained between the predictions of the model and experimental data on DRAs from a wide number of sources. However, the strain prior to particle fracture is neglected. Reference 39 also provides empirical equations for calculating the particle stress in a power-law hardening matrix at different values of imposed plastic strains. These formulas may be used to estimate the extent of damage as a function of applied strain. An alternate simplified methodology is suggested in Ref 37 for calculating particle stress and then determining particle strength based on the fraction of cracked particles. Such analyses suggest a Weibull modulus of approximately 5 and a Weibull strength of 2400 MPa (350 ksi) for 10 μm size SiC particles. The previously mentioned elastic-plastic models assume a uniform distribution of particles. Although clustering may be considered small in well-processed powder-metallurgy-derived composites of volume fractions less than 0.2, nonuniformity and clustering is the rule rather than the exception. A Voronoi cell FEM approach has been developed in Ref 43 to assess elastic-plastic deformation of a multitude of unevenly distributed particles, rather than the uniform distribution assumed in unit cell FEM calculations. The analyses show that particle fractures occur early in regions of clusters, and this is accompanied with large plastic strains and hydrostatic stresses in damaged regions. These regions then become the locations for microvoid initiation, and because void growth is linearly proportional to the plastic strain and exponentially dependent on the level of hydrostatic tensile stress (Ref 44), the voids can rapidly grow to coalescence. A ductility model based on Voronoi cell computations remains to be established, but should provide a more accurate estimate of damage and failure for a nonuniform microstructure. References cited in this section 35. G. Bao, J.W. Hutchinson, and R.M. McMeeking, Particle Reinforcement of Ductile Matrices Against Plastic Flow and Creep, Acta Metall. Mater., Vol 39, 1991, p1871–1882 36. T.W. Clyne and P.J. Withers, An Introduction to Metal Matrix Composites, Cambridge University Press, Cambridge, 1993 37. Y.L. Shen, M. Finot, A. Needleman, and S. Suresh, Effective Elastic Response of Two- Phase Composites, Acta Metall. Mater., Vol 42, 1994, p 77–97 38. B.S. Majumdar and A.B. Pandey, Deformation and Fracture of a Particle Reinforced Aluminum Alloy Composite, Part II: Modeling, Metall. Trans A, Vol 31, 2000, p 937–950 39. B.S. Majumdar and A.B. Pandey, Deformation and Fracture of a Particle Reinforced Aluminum Alloy Composite, Part I: Experiments, Metall. Trans. A, Vol 31, 2000, p 921–936 40. J. Llorca and C. Gonzalez, Microstructural Factors Controlling the Strength and Ductility of Particle Reinforced Metal-Matrix Composites, J. Mech. Phys. Solids, Vol 46, 1998, p 1–28 41. Z. Hashin and S. Shtrikman, J. Mech. Phys. Solids, Vol 11, 1963, p 127 42. T. Mura, Micromechanics of Defects in Solids, 2nd ed., Martinis Nijhoff, The Hague, 1987 43. J. Duva, A Self Consistent Analysis of the Stiffening Effect of Rigid Inclusions on a Power-Law Material, J. Eng. Mater. Struct. (Trans. ASME), Vol 106, 1984, p 317 44. S. Ghosh and S. Moorthy, Elastic-Plastic Analysis of Arbitrary Heterogeneous Materials with the Voronoi Cell Finite Element Method, Comp. Methods Appl. Mech. Eng., Vol 121, 1995, p 373–409 16. J.R. Rice and D.M. Tracey, J. Mech. Phys. Solids, Vol 17, 1969, p 201–217 Engineering Mechanics and Analysis of Metal-Matrix Composites Bhaskar S. Majumdar, New Mexico Institute of Mining and Technology Local Failures of Fiber-Reinforced MMCs Longitudinal Loading. Under monotonic tension loading, failure of the composite is determined by fiber fracture. Generally, fiber strengths follow weak-link Weibull statistics, where the probability of failure (P f ) of a fiber of length L is expressed as: (Eq 21) where “m” is the Weibull modulus and σ o is the Weibull (approximately average) strength for a fiber of length L o . For a ductile matrix, the matrix is always yielded, so that in a one-dimensional model, the composite ultimate strength, σ c U , is simply: (Eq 22) where σ f U is the effective strength of the fiber at instability of the composite, and σ m flow is the flow stress in the matrix at that value of composite strain, typically 0.8 to 1%. The value of σ f U depends on the mode of failure. If the interface is so weak that failure of a fiber at any location is equivalent to loss of load-carrying capability of the entire fiber, then σ f U may be equated to the dry bundle strength (σ dbf ) (Ref 45): (Eq 23) where e is the exponential term approximately equal to 2.718. A more-realistic situation is the ability of the broken fiber to recarry the load after a sliding distance, δ, from the fiber break. In this case, one must consider the frictional sliding stress, τ, which can be independently determined from pushout or fragmentation tests. The associated effective fiber strength, according to Curtin's global load-sharing model (Ref 16, 17), is: (Eq 24) where the characteristic fiber strength σ ch is: (Eq 25) In Curtin's model (Ref 16, 17), the fragmenting fibers essentially follow the constitutive law: (Eq 26) where the subscript “f” refers to the fragmenting fibers. At instability, this leads to an effective fiber strain ( f U ): (Eq 27) The total strain in the composite at failure ( c U ) is then simply: (Eq 28) where f Res is the residual strain in the fiber, being predominantly compressive and negative. This model has been found to correlate quite well with the strength of a number of fiber-reinforced titanium alloys (Ref 17, 46, and 47). However, local load-sharing has also been observed (Ref 48, 49, and 50), where the density of fiber cracks was found to be far below those predicted by the global load-sharing model. Reference 50 provides a comparison of different models in the context of failure of an orthorhombic titanium alloy reinforced with SiC fibers. The local load- sharing situation is well captured by the second fiber fracture model of Zweben and Rosen (Ref 51), and the pertinent equations are also provided in Ref 50. The local load-sharing model gives effective fiber strengths that are slightly lower than the global load-sharing model. The lowest bound on the effective fiber strength is obtained from the dry bundle model. Although this may be overly conservative during room- temperature deformation, when there is significant clamping stress between the fibers and the matrix, the dry bundle model may provide a reasonable lower bound at high temperatures. Transverse Loading. Under transverse loading, the onset of nonlinearity is determined by fiber-matrix separation, as discussed earlier. Debonding occurs when the local radial stress is greater than the bond strength of the interface. The local radial stress is simply the far-field stress (σ far-field ) multiplied by a stress-concentration factor (k) less the residual radial stress (σ r residual ) at the interface. Stated mathematically: (Eq 29) Models for determining k and the thermal residual stress have been described earlier, with the single-fiber case being given by the analytical equations (Eq 22). Typical values of k are in the range 1.2 to 1.5. The ultimate strength is governed by matrix rupture. If the fibers are not debonded, then the models described for discontinuous reinforced particles may be used without much loss of accuracy. Thus, Eq 16 with β = 0.375 may be used. When the fibers are debonded, then the strength of the composite is less than that of the matrix. In this case, one usually resorts to FEM analysis. References cited in this section 17. W.A. Curtin, J. Am. Ceram. Soc., Vol 74, 1991, p 2837 45. W.A. Curtin, Ultimate Strengths of Fibre- Reinforced Ceramics and Metals, Composites, Vol 24 (No. 2), 1993, p 98–102 46. A. Kelly and N.H. Macmillan, Strong Solids, 3rd ed., Clarendon Press, Oxford, 1986 47. C.H. Weber, X. Chen, S.J. Connell, and F. Zok, On the Tensile Properties of a Fiber Reinforced Titanium Matrix Composite, Part I, Unnotched Behavior, Acta Metall. Mater., Vol 42, 1994, p 3443– 3450 48. C.H. Weber, Z.Z. Du, and F.W. Zok, High Temperature Deformation and Fracture of a Fiber Reinforced Titanium Matrix Composite, Acta Metall. Mater., Vol 44, 1996, p 683–695 49. D.B. Gundel and F.E. Wawner, Experimental and Theoretical Assessment of the Longitudinal Tensile Strength of Unidirectional SiC-Fiber/Titanium-Matrix Composites, Compos. Sci. Technol., Vol 57, 1997, p 471–481 50. B.S. Majumdar, T.E. Matikas, and D.B. Miracle, Experiments and Analysis of Single and Multiple Fiber Fragmentation in SiC/Ti- 6Al-4V MMCs, Compos. B: Eng., Vol 29, 1998, p 131–145 51. C.J. Boehlert, B.S. Majumdar, S. Krishnamurthy, and D.B. Miracle, Role of Matrix Microstructure on RT Tensile Properties and Fiber-Strength Utilization of an Orthorhombic Ti-Alloy Based Composite, Metall. Trans. A, Vol 28, 1997, p 309–323 10. C. Zweben and B.W. Rosen, A Statistical Theory of Material Strength with Application to Composite Materials, J. Mech. Phys. Solids, 1970, p 189–206 Engineering Mechanics and Analysis of Metal-Matrix Composites Bhaskar S. Majumdar, New Mexico Institute of Mining and Technology Macromechanics Strength of Fiber-Reinforced Composites. The cases of tensile loading in the longitudinal and transverse directions have been described earlier. Figure 7 shows measured and predicted stress-strain plots for 0° SCS6/Ti-15-3 composites, where the sudden increase in the predicted strain response is interpreted as failure of the specimen (Ref 18). Modeling was conducted using the FIDEP code with both elastic-plastic and elastic- viscoplastic matrix using the Bodner-Partom model, which was modified to incorporate fiber fracture according to Eq 26. Figure 7 shows good agreement between the predicted stress- strain curves and strengths with experimental data. This type of correlation also was observed at elevated temperatures, when viscoplastic effects became important. Fig. 7 Comparison of predicted and experimental stress-strain behavior of SCS6/Ti-15-3 composites at room temperature for 15% and 30% fiber volume fractions. Both elastic- plastic and elastic-viscoplastic analysis was conducted, and fiber fractures were incorporated into the model. The sudden increase in strain in the predicted curves signifies specimen failure. Source: Ref 18 For off-axis or multiaxial loading, the IDAC (Ref 33) program may be used to compute the stress-strain response of the composite and the local stresses/strains in the constituents. The onset of failure can then be predicted based on the mechanisms, that is, fiber fracture, transverse failure, or shear failure, depending upon which mechanism can operate at the least value of the far-field load. Strength of Discontinuous Reinforced Composites. The stress-strain curve has been covered in an earlier section. The ultimate strength is dependent on the elongation to failure, which is generally much less than the matrix. Models of ductility have been presented earlier. Fatigue of Fiber-Reinforced MMCs. The longitudinal fatigue life of fiber-reinforced MMCs can generally be grouped under three regimes, in a plot of stress or strain range versus the cycles to failure (N f ). They are illustrated in Fig. 8, which was first postulated for polymer- matrix composites (Ref 52). 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Bahei-El-Din, Elastic- Plastic Behavior of Fibrous Composites, J. Mech. Phys. Solids, Vol 27, 19 97, p 51 72 25. Y.A. Bahei-El-Din, R.S. Shah, and G.J. Dvorak, Numerical Analysis of Rate-Dependent Behavior. Fig. 7 Comparison of predicted and experimental stress-strain behavior of SCS6/Ti-1 5-3 composites at room temperature for 15% and 30% fiber volume fractions. Both elastic- plastic and elastic-viscoplastic. [0] and [0/90] SCS-6/Timetal21S Composites, ASME, Vol 34 (No. H0086 6- 1993), W.F. Jones, Ed., 1993, p 1– 16 21. B.S. Majumdar and G.M. Newaz, In-Phase TMF of a 0° SiC/Ti-1 5-3 System: Damage