Volume 21 - Composites Part 7 pptx

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). The regimes have also
been confirmed in MMCs and exhibit distinct differences in failure mechanisms (Ref 53, 54).

Fig. 8 Schematic showing the three regimes of fatigue of fiber-reinforced MMCs

Regime 1, with N
f
typically between 1 and 1000 cycles, is dominated by fiber fractures without any matrix
cracks. At elevated temperatures under isothermal fatigue conditions, fiber fractures appear to be precipitated
by progressive ratcheting of the matrix under viscoplastic conditions. Essentially, matrix viscoplasticity results
in the gradual transformation of the matrix strain range from tension-tension to fully reversed- loading tension-
compression, where R = –1, although the composite may be subjected to only tension-tension loading at an R-
ratio (minimum to maximum stress ratio on the composite) of 0.1, for example. The offloading of the matrix
results in progressively increased loading being experienced by the fibers, as required by Eq 14, causing them
to fail with an increased number of cycles. In this scenario, if the final fiber stress is insufficient to cause any
significant breakage of fibers (well below the rounded region of Fig. 6), progressive fiber failure should be
avoidable. Indeed, this maximum stress approximately delineates the boundary between regimes 1 and 2 under
isothermal conditions.
The frequency of loading becomes an important factor in regime 1, because matrix creep can lower the
maximum matrix stress attainable in both the tension and compression part of the cycle at low frequencies
(<0.01 Hz). The result is an even greater load being carried by the fibers and a consequent poorer fatigue
performance with lower frequency. The increased matrix creep and associated transfer of load to the fiber is
manifested in the strain-ratcheting behavior of the composite, which shows increased ratcheting with reduced
frequency at elevated temperatures. Thus, life prediction in regime 1 requires both a modeling of the composite
response based on a good viscoplastic characterization of the matrix and adequate incorporation of fiber
fracture using Weibull statistics. The CCM model for analysis has already been discussed, and reference has
been made to the Bodner-Partom model for viscoplastic characterization of the matrix (Ref 10, 11, and 21). The
matrix responses have been integrated into the available FIDEP and IDAC codes.
Under in-phase thermomechanical fatigue (IP- TMF), the extent of matrix ratcheting and fiber damage is
observed to be larger (Ref 18, 55, and 56), and simultaneously the IP-TMF life is observed to be shorter than
under isothermal conditions. One problem found with various investigations was that often the frequency of
loading was smaller under IP-TMF than under isothermal conditions. In the extreme case, creep of the matrix
may relax its value to zero at the end of the tension cycle. The result is that the entire applied load would then
be carried only by the fibers, causing their stress to be significantly higher than under faster isothermal
conditions. If all these factors are appropriately taken into account, then results under different test conditions

(isothermal and IP-TMF) in regime 1 can be rationalized in terms of the maximum fiber stress (Ref 57).
However, this does not clarify the entire picture, because the CCM model with Bodner-Partom constants
accurately predicts the isothermal ratcheting response at the highest temperature, but significantly underpredicts
the ratcheting response for IP-TMF at the same frequency (Ref 18). Thus, other factors may be present as well,
and fiber damage due to molybdenum weaves was suggested for a SCS6/Ti-15- 3 system (Ref 18, 55).
However, this explanation may only be valid for panels with molybdenum weaves. Overall, a complete
understanding of IP-TMF has yet to emerge, although current predictions are much closer to experiments.
At room temperature, when viscoplastic conditions are negligible, the previous explanation is inadequate to
explain why fibers should fail after the first cycle. In Ref 53, an alternate mechanism was proposed for the
initiation of fiber cracks. Microstructural observations suggested damage in the coating and in the reaction
zone, leading to cracking of fibers.
Regime 2 is matrix-crack dominated, similar to monolithic alloys, but fiber stress also plays an important role.
In this regime, the fatigue life can either be plotted as the stress range (Δσ) versus N
f
, or as the strain range (Δ )
versus N
f
. A plot of Δσ versus N
f
shows that fatigue life increases with fiber volume fraction and is mainly an
effect associated with load transfer from the matrix to the higher-strength elastic fibers. Such a plot also shows
that composites have fatigue performance superior to the matrix alloy. This behavior is also observed with
discontinuous reinforcements. However, if the fatigue life is plotted as Δ versus N
f
, as is most often done with
monolithic alloys, then the MMC is generally found to have slightly poorer performance than the monolithic
alloy. Fatigue tests with R-ratio of –1 show better performance than with an R- ratio of 0.1, at the same strain
range (Ref 58). Microstructures show a greater density of matrix cracks, but the stress in the fibers are only half
of what would occur under tension-tension loading. Because composite failure requires the breakage of fibers,
life is improved under negative R-ratio compression conditioning. Models have been proposed to include both

the composite strain range and the fiber stress for predicting fatigue life in regime 2 (Ref 53, 57, 58, and 59).
In regime 2, matrix cracks can initiate anywhere between 10 and 30% of life, depending on the preparation of
samples. In other words, the majority of life is spent in the fatigue crack growth (FCG) domain. A number of
investigators have addressed FCG (Ref 60, 61, 62, and 63), and the common feature of their models is the
reduction of the stress-intensity factor (shielding) at the matrix crack tip by the bridging fibers:


(Eq 30)
The primary difference between the models is the way in which the shear lag analysis is conducted. Canonical
equations are provided in Ref 64 to simplify calculations of fatigue crack growth. However, a factor that has not
been considered in these models is the shielding of the crack due to higher-modulus fibers ahead of the crack
tip. The effect of this shielding is analyzed in Ref 65, and it is observed that the interface tensile strength can
have a substantial effect on the retardation of matrix crack growth in the low ΔK regime. The interface tensile
strength then constitutes a microstructural variable, in addition to the friction shear stress parameter (τ), that
could be used to control the crack growth kinetics.
Regime 3 represents the fatigue limit, which is the stress level below which the material can be cycled infinitely
without damage. The matrix behavior is elastic in this region.
References cited in this section
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
18. 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
21. 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
33. 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
52. 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
53. R. Talreja, Fatigue of Composite Materials, Technomic Publishing Company, 1987
54. B.S. Majumdar and G.M. Newaz, Constituent Damage Mechanisms in Metal Matrix Composites Under
Fatigue Loading, and Their Effects on Fatigue Life, Mater. Sci. Eng. A, Vol 200, 1995, p 114–129
55. P.K. Brindley and P.A. Bartolotta, Failure Mechanisms During Isothermal Fatigue of SiC/Ti-24Al-11Nb
Composites, Mater. Sci. Eng. A, Vol 200, 1995, p 55–67
56. B.S. Majumdar and G.M. Newaz, Damage Mechanisms Under In-Phase TMF in a SCS-6/Ti-15-3
MMC, Proc. 1994 HITEMP Conf., NASA CP 10146, National Aeronautics Space Administration,
1994, p 41.1–41.13
57. T. Nicholas, An Approach to Fatigue Life Modeling in Titanium Matrix Composites, Mater. Sci. Eng.
A, Vol 200, 1995, p 29–37
58. T. Nicholas, Fatigue and Thermomechanical Fatigue Life Prediction, Titanium Matrix Composites
Mechanical Behavior, S. Mall and T. Nicholas, Ed., Technomic Publishing Co., 1998, p 209–272
59. B.A. Lerch and G. Halford, Effects of Control Mode and R-Ratio on the Fatigue Behavior of a Metal
Matrix Composite, Mater. Sci. Eng A, Vol 200, 1995, p 47–54
60. B. Lerch and G. Halford, “Fatigue Mean Stress Modeling in a [0]32 Titanium Matrix Composite,” Paper
21, HITEMP Review- 1995, Vol II, NASA CP 10178, National Aeronautics and Space Administration,
1995, p 1–10
61. D.B. Marshall, B.N. Cox, and A.G. Evans, The Mechanics of Matrix Cracking in Brittle Matrix Fiber
Composites, Acta Metall. Mater., Vol 33, 1985, p 2013–2021
62. R.M. McMeeking and A.G. Evans, Matrix Fatigue Cracking in Fiber Composites, Mech. Mater., Vol 9,
1990, p 217–227
63. L.N. McCartney, New Theoretical Model of Stress Transfer Between Fiber and Matrix in a Uniaxially
Fiber Reinforced Composite, Proc. R. Soc. (London) A, Vol 425, 1989, p 215
64. J.M. Larsen, J.R. Jira, R. John, and N.E. Ashbaugh, Crack Bridging in Notch Fatigue of SCS-6/Timetal
21S Composite Laminates, ASTM STP 1253, W.S. Johnson, J.M. Larsen, and B.N. Cox, Ed., ASTM,
1995
65. B.N. Cox and C.S. Lo, Simple Approximations for Bridged Cracks in Fibrous Composites, Acta Metal.
Mater., Vol 40, 1992, p 1487–1496

37. S.G. Warrier and B.S. Majumdar, Elastic Shielding During Fatigue Crack Growth of Titanium Matrix
Composites, Metall. Trans. A, Vol 30, 1999, p 277–286

Engineering Mechanics and Analysis of Metal-Matrix Composites
Bhaskar S. Majumdar, New Mexico Institute of Mining and Technology

Fracture Toughness
Fiber-Reinforced Composites. In ductile matrix systems, a number of different behaviors have been observed,
depending on the strength of the matrix, the fiber-matrix bond strength, and the volume fraction of fibers.
When the matrix strength is low, matrix-dominated shear deformation occurs prior to any fiber fracture. Indeed,
in boron-fiber-reinforced aluminum composites, a shear deformation mode parallel to the fibers is observed
(Ref 66, 67, and 68). Here, an intense slip zone develops over a plastic zone of length L perpendicular to the
crack plane. At a critical load, when L is on the order of 3 to 17 times the crack length, the damage zone stops
propagating and is replaced by failure of fibers at the crack tip. This in turn leads to catastrophic fracture of the
composite along the original notch plane. Most notable is the fact that an H-shaped shear zone is created prior
to crack extension, which is absent during propagation of the crack. The crack extension is not self-similar. A
similar type of damage zone and crack extension has also been observed in glass- fiber-reinforced epoxy
composites (Ref 69).
In the previous type of deformation mode, the effective toughening is extremely high, because the blunted,
deflected crack tip attenuates the stress ahead of the crack tip over a distance of the order of the crack tip
opening displacement. For crack lengths of a few millimeters or longer, toughness values of 100 MPa (91
ksi ) have been realized for the boron aluminum system. The effect is reduced for a smaller crack length
and approaches approximately 30 MPa (27 ksi ) at a crack length less than 0.5 mm (0.02 in.). Thus,
fracture toughness may not be the appropriate parameter for predicting failure in these systems, which do not
obey self-similar crack growth. In Ref 66, an attempt was made to estimate the onset of fracture with H-shaped
cracks, based on the attainment of a critical strain in the fiber direction a distance of over two fiber diameter
(2a) ahead of the crack tip. It was found that the local strain in the representative volume element for specimens
with different notch lengths all fell in the error band for the failure strain of unnotched composites. The
following set of equations may be used to estimate the failure load, P
ult

, for a center-cracked panel of width W
and crack length 2a (Ref 70) and possessing an unnotched strength, σ
unnotched
:


(Eq 31)
where


(Eq 32)
and where + is for and – is for , and


Here, d is the fiber diameter, c is the volume fraction of fibers, τ* is the in-plane shear strength of the
unidirectional composite in the fiber direction, E
1
is the modulus in the fiber direction, E
2
in the transverse
direction, G
12
is the shear modulus, ν
12
is the Poisson's ratio, and b is of the order of distance between two
adjacent fibers.
When the matrix strength or the fiber volume fraction is high, the dominant damage mode is fiber
fragmentation in the zone of intense matrix plastic deformation near the crack tip and ultimately, composite
fracture. In order to predict the fracture toughness, some estimate of the flow stress and the critical
displacement is needed. This was modeled in Ref 71 and 72 by considering that the periphery of fractured fiber

tips acts as the nucleation center from which intense matrix plasticity develops. This is a form of macroscopic
void growth, at a length scale that is significantly larger than the distance between intermetallic particles, which
act as the void nucleation site for fracture of monolithic metallic alloys. The following estimated toughness (J
Ic

is obtained (Ref 71):


(Eq 33)
where


(Eq 34)
and β
n
is approximately 2. Here d is the diameter of the fiber, V
f
is the volume fraction of fibers, and σ
Y
is the
yield stress of the matrix.
Very few experiments have been conducted on the fracture toughness of titanium-matrix composites. In Ref 73,
a toughness of approximately 71 kJ/m
2
(4900 ft · lbf/ft
2
) was reported for a SiC-Ti alloy, which may be
compared with a typical toughness of 40 kJ/m
2
(2700 ft · lbf/ft

2
), based on K
Ic
= 70 MPa (64 ksi ) for
monolithic Ti-6Al-4V alloy. Using Eq 33 with σ
Y
= 1040 MPa (150 ksi), V
f
= 0.32, and d = 100 μm, a toughness
value of 63 kJ/m
2
(4.3 ft · lbf/ft
2
) is estimated, which compares reasonably well with the experimental data.
It is useful to note that J
Ic
predicted by Eq 33 and 34 is quite strongly dependent on the volume fraction of
fibers. Thus, (1 – V
f
) · (1 – )/ decreases from approximately 0.59 to 0.12 on increasing V
f
from 0.3 to
0.6. High-volume-fraction alumina/aluminum composites are currently being developed for a variety of
applications, such as high-tension electrical cables and piston rods. Because of the lower strength of the alloy
and the high volume fraction of alumina fibers, toughness values significantly less than titanium-matrix
composites are anticipated.
A final note is in order regarding the role of the fiber-matrix interface. Equation 33 shows that the toughness is
an increasing function of L
D
. Weak interfaces would permit greater fiber- matrix sliding, thereby increasing the

fracture toughness. This effect was elegantly used in Ref 74 to toughen aluminum-based composites while
maintaining acceptable transverse strengths. In SiC-reinforced titanium-matrix composites pullout lengths are
typically less than one fiber diameter, even for weak carbon-based interfaces. This is largely because of the
high radial compressive stress that is generated at the fiber-matrix interface at the tip of a cracked fiber. For
strong interfaces, such as those formed without a carbon layer on the SiC monofilaments, the pullout length will
be even shorter. However, the effect of a smaller L
D
may be balanced by a higher flow stress associated with
constrained yielding of the matrix in the fragmentation zone. Tensile tests on unnotched SiC-Ti-matrix
composites with different interfaces indicate correlated fiber fractures, independent of the type of interface. Slip
band observations and ultrasonic images of fiber breaks confirm that correlated fractures are a result of slip
band interactions. A slip band impinging on a fiber localizes sufficient strain to fracture that fiber (Ref 49).
Similar experiments have to be conducted with notched composites to provide an assessment of the role of
interface strength on the toughness of composites with high matrix strength.
Discontinuously Reinforced Composites. Hahn and Rosenfield's ductile fracture model (Ref 75) is by far the
most commonly used model for particulate MMCs. Assuming that crack growth will occur if the extent of
heavily deformed zone ahead of crack tip becomes comparable to the width of the unbroken ligaments
separating cracked particles, the fracture toughness can be expressed as:


(Eq 35)
where K
Ic
is the critical value of the stress-intensity factor, V
p
is the volume fraction of particulates, d is the
particle diameter, E is the composite modulus, and σ
y
is the yield strength of the composite. A key validation
point for the model was the V

p
–1/6
dependence observed in a number of monolithic aluminum alloys. A slight
modification of the model was made in Ref 76 to account for the observed effect of specimen thickness on the
fracture toughness of the material.
The main problem with Eq 35 is that it shows an increasing toughness with yield strength, σ
y
, whereas the
reverse is normally observed. A summary of toughness data with comparisons to models can be found in Ref
77. From a microstructural perspective, a higher strength is usually accompanied by concentrated and localized
slip bands, which accelerates the initiation of microvoids in those bands. Mechanically, a higher strength is
accompanied with a reduction in the work-hardening exponent, N. This effect was accounted for in the model
of Garrett and Knott (Ref 78), which was essentially based on an earlier paper of Hahn and Rosenfield (Ref 79).
The following equation was obtained in Ref 78:


(Eq 36)
Typical values of the parameters were C = 0.025 m and = 0.1.
This form does indeed show the correct inverse dependence of toughness on strength, because N generally
decreases sharply with increasing strength. However, the problem with the analysis is that the results of Ref 80
illustrate that local strains along any orientation θ (measured with respect to crack plane) are extremely
insensitive to the material parameters, rather than the strong N dependence that was assumed in Ref 78 and 79.
A recent discussion on these models, in the context of the micromechanisms of fracture, is provided in Ref 37.
This reference also presents an alternate model, based on localized slip, to explain the large decrease in
toughness in the peak-aged condition of DRAs. Although reasonably good agreement was found with limited
experimental data, further validation of the model is necessary.
References cited in this section
49. 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
66. 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
67. G.J. Dvora, Y.A. Bahei-El-Din, and L.C. Bank, Eng. Fract. Mech., Vol 34 (No. 1), 1989, p 87–104 and
p 105–123
68. J. Awerbuch and G.T. Hahn, J. Compos. Mater., Vol 13, 1979, p 82–107
69. E.D. Reedy, Analysis of Center-Notched Monolayers with Application to Boron/Aluminum
Composites, J. Mech. Phys. Solids, Vol 28, 1980, p 265–286
70. J. Tirosh, J. Appl. Mech. (Trans. ASME), Vol 40, 1973, p 785–790
71. J.F. Zarzour and A.J. Paul, J. Mater. Eng. Perform., Vol 1 (No. 5), 1992, p 659–668
72. J.B. Friler, A.S. Argon, and J.A. Cornie, Mater. Sci. Eng., Vol A162, 1993, p 143–152
73. A.S. Argon, Comprehensive Composite Materials, A. Kelly and C. Zweben, Ed., Vol 1, Pergamon
Press, Oxford, 2000
74. S.J. Connell, F.W. Zok, Z.Z. Du, and Z. Suo, Acta Metall., Vol 42 (No. 10), 1994, p 3451–3461
75. A.S. Argon, M.L. Seleznev, C.F. Shih, and X.H. Liu, Int. J. Fract., Vol 93, 1998, p 351–371
76. G.T. Hahn and A.R. Rosenfield, Metallurgical Factors Affecting Fracture Toughness of Aluminum
Alloys, Metall. Trans. A, Vol 6,1975, p 653–670
77. A.B. Pandey, B.S. Majumdar, and D.B. Miracle, Effects of Thickness and Precracking on the Fracture
Toughness of Particle Reinforced Al-Alloy Composites, Metall. Trans., A, Vol 29,(No. 4), 1998, p
1237–1243
78. J.J. Lewandowski, Fracture and Fatigue of Particulate Composites, Comprehensive Composite
Materials, A. Kelly and C. Zweben, Ed., Vol 3, Metal Matrix Composites, T.W. Clyne, Ed., Elsevier,
2000, p 151–187
79. G.G. Garrett and J.F. Knott, The Influence of Composition and Microstructural Variations on the
Mechanism of Static Fracture in Aluminum Alloys, Metall. Trans. A, Vol 9,1978, p 1187–1201
80. G.T. Hahn and A.R. Rosenfield, Sources of Fracture Toughness: The Relation Between K
Ic
and the
Ordinary Tensile Properties of Metals, Applications Related Phenomena in Titanium Alloys, ASTM STP
432, ASTM, 1968, p 5–32
1. R.M. McMeeking, Finite Deformation Analysis of Crack-Tip Opening in Elastic- Plastic Materials and
Implications for Fracture, J. Mech. Phys. Solids, Vol 25,1977, p 357–381

Engineering Mechanics and Analysis of Metal-Matrix Composites
Bhaskar S. Majumdar, New Mexico Institute of Mining and Technology

Software
The following software programs are available for MMC analysis:
• NDSANDS, developed at Air Force Research Laboratory, Materials Directorate. Elastic analysis
parallel and perpendicular to fiber axis, laminates
• FIDEP, developed at Air Force Research Laboratory, Materials Directorate. Elastic-plastic- viscoplastic
concentric cylinder model
• VISCOPLY, developed at National Aeronautics and Space Administration (NASA) Langley Research
Center. Elastic-viscoplastic model based on the vanishing fiber diameter analysis
• IDAC, developed at Research Applications Inc., San Diego, under Air Force contract. Elastic-plastic-
viscoplastic analysis based on FEM procedure, with emphasis on multiaxial loading of laminas

Engineering Mechanics and Analysis of Metal-Matrix Composites
Bhaskar S. Majumdar, New Mexico Institute of Mining and Technology

References
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
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Fracture Analysis of Fiber-Reinforced Ceramic-
Matrix Composites
F.W. Zok, University of California at Santa Barbara

Introduction
ONE OF THE KEY ATTRIBUTES of continuous fiber-reinforced ceramic composites (CFCCs) is their ability
to undergo inelastic straining upon mechanical loading (Ref 1). The mechanisms for these strains involve
matrix cracking and debonding and sliding along the fiber–matrix interfaces. The inelastic strains impart a high
toughness to CFCCs in essentially the same manner that dislocation plasticity imparts high toughness to
metallic alloys. That is, the inelasticity reduces the local levels of stress around strain-concentrating features,
such as cracks and notches, and hence increases the level of applied stress necessary to initiate fiber fracture at
the crack tip. This phenomenon is referred to as plastic shielding. An additional attribute of CFCCs is the
stochastic nature of fiber fracture within the composite. A consequence is that fiber failure occurs over a range
of locations relative to the macroscopic crack plane. The subsequent pullout of broken fibers leads to additional
shielding of the crack tip.
From a macroscopic viewpoint, the fracture properties of CFCCs differ from those of metals in three important
respects, as listed in Table 1. These differences provide the impetus and the directions for modifying existing
methodologies for damage-tolerant failure prediction (currently used for metallic components) such that they
can be applied to CFCC components.

Table 1 Macroscopic differences in fracture properties of CFCCs and metals
Characteristic CFCC Metal
Consequence
Failure strains The magnitude of the total
straining capacity of CFCCs
is limited to values of ≈1%.
In metallic alloys, failure
strains in the range of 10–
50% are common.
These differences have
implications in the efficacy
of plastic shielding in the
two classes of materials.
Nonlinear
behavior
Because the mechanism of
inelasticity in CFCCs involves
cracking, the process is driven
Plasticity in metals is driven
by the deviatoric component
(a)
of the stress state and is
The constitutive laws for
the mechanical response of
metals and CFCCs in the
largely by normal (tensile)
stresses.
essentially independent of
the hydrostatic stress.
nonlinear regime are

fundamentally different
from one another. A
related feature is the
mechanical anisotropy of
CFCCs with most common
(two dimensional) fiber
architectures.
Fracture
resistance
The increase in fracture
resistance due to fiber
pullout is typically much
greater than the intrinsic
composite toughness (in the
absence of bridging).
Additionally, the amount of
crack growth needed to
attain a steady-state
resistance is typically of the
same order as the
dimensions of CFCC
coupons or components of
interest.
Fracture resistance is
dictated largely by the
behavior of an enclave of
heavily deformed material
at the crack tip. In most
cases of practical interest,
this enclave is very small in

relation to other structural
dimensions, and thus a
small-scale yielding (SSY)
treatment is adequate. The
problem of large-scale
yielding (LSY) arises in
very tough metals,
especially under plane-
stress conditions.
A large-scale bridging
(LSB) mechanics is needed
to describe the structural
response in CFCCs,
including the conditions
associated with crack
stability.
(a) The stress component that is related to the difference in the stress and the mean stress. The hydrostatic stress
is the mean stress. This terminology is used in modeling the observation that plastic deformation is a shear
phenomena and not dependent on hydrostatic stress.
Some trends in the degree of damage tolerance, as manifested in the notch sensitivity of strength, are
highlighted in Fig. 1, based on Ref 2, 3, 4, and 5. Results are presented for the net-section tensile strength, σ
N
,
of open-hole specimens for typical metals, CFCCs, and polymer-matrix composites (PMCs), all with the same
normalized hole radius, a/w = 0.2. The metals listed exhibit no notch sensitivity for hole diameters approaching
10 mm (0.4 in.). This behavior is attributable to the extensive plasticity that occurs across the entire net section
prior to fracture and the effects of this plasticity on reducing the stress concentration at the hole edge. At the
other extreme, PMCs exhibit severe notch sensitivity. The strength diminishes rapidly with increasing hole
diameter and eventually approaches the value predicted from the elastic stress concentration factor, k
σ

. This
notch sensitivity is largely a consequence of the absence of inelastic straining mechanisms in these composites.
Continuous fiber-reinforced ceramic composites with high toughness exhibit intermediate behavior in the sense
that their strength diminishes gradually with hole diameter and appears to saturate at a relatively high value,
typically 70% of the unnotched strength.

Fig. 1 Notch sensitivity. Effects of hole diameter on the tensile strength of metals, CFCCs
(Ref 2, 3), and polymer-matrix composites (PMCs) (Ref 4, 5). The data are presented on
the basis of the net-section strength, σ
N
, normalized by the respective unnotched tensile
strength, σ
0
. The composites have two-dimensional (2D) fiber architectures (either
laminated or woven), and the loads are applied parallel to one of the fiber axes. In all
cases, the normalized hole diameter is a/w = 0.2. The lower limit on the notched strength is
given by 1/k
σ
, where k
σ
is the elastic stress concentration factor. Data on metals courtesy
of J.C. McNulty, University of California, Santa Barbara
The objective of this article is to review the mechanics of inelastic deformation and fracture of CFCCs, as
needed for the development of damage-tolerant failure prediction methodologies for use in engineering design.
An underlying theme pertains to the anisotropy in their mechanical properties and its effect on notch sensitivity
of strength. The scope of the article is restricted to CFCCs with balanced 0°/90° fiber architectures, because of
the emphasis on these architectures within the CFCC community.
Many of the concepts and models described here have been adapted from analogous problems in monolithic
materials. Notable examples include: models of stress redistribution around strain concentrations due to
inelastic straining; fracture resistance curves and the conditions associated with crack stability; cohesive or

bridging zone concepts; and effects of material and structural size scales, leading to large scale bridging (LSB)
or large scale yielding (LSY). Such connections are noted where appropriate.
The coverage in this article is organized as follows:
• A general framework for damage-tolerant design with structural materials is outlined. The framework
identifies two broad classes of phenomena that are obtained in such materials: crack-tip inelasticity prior
to the onset of fracture initiation, and crack bridging during crack propagation.
• The common classes of fracture behavior of CFCCs are described. The classification system provides a
rationale for selecting the pertinent features and mechanisms into the failure prediction methodology.
• The constitutive laws needed to describe crack-tip inelasticity are presented.
• The stress distribution section demonstrates the effects of inelasticity on crack-tip stress fields.
• Models for crack initiation are discussed.
• Crack propagation models are derived.
• Environmental degradation effects on damage tolerance are addressed.
References cited in this section
2. A.G. Evans and F.W. Zok, The Physics and Mechanics of Fibre-Reinforced Brittle Matrix Composites,
J. Mater. Sci., Vol 29, 1994, p 3857–3896
3. J.C. McNulty, F.W. Zok, G.M. Genin, and A.G. Evans, Notch-Sensitivity of Fiber-Reinforced Ceramic
Matrix Composites: Effects of Inelastic Straining and Volume-Dependent Strength, J. Am. Ceram. Soc.,
Vol 82 (No. 5), 1999, p 1217–1228
4. J.C. McNulty, M.Y. He, and F.W. Zok, Notch Sensitivity of Fatigue Life in a Sylramic/SiC Composite
at Elevated Temperature, Compos. Sci. Technol., in press2001
5. J.M. Whitney and R.J. Nuismer, Stress Fracture Criteria for Laminated Composites Containing Stress
Concentrations, J. Compos. Mater., Vol 8 (No. 3) 1974, p 253–265
6. J. Awerbuch and M.S. Madhukar, Notched Strength of Composite Laminates: Predictions and
Experiments–A Review, J. Reinf. Plast. Compos., Vol 4 (No. 1), 1985, p 3–159

Fracture Analysis of Fiber-Reinforced Ceramic-Matrix Composites
F.W. Zok, University of California at Santa Barbara

General Framework for Fracture Analysis

It is instructive to begin by outlining a general framework for the description of fracture in structural materials
in the presence of notches and cracks. There are two broad categories of phenomena (Fig. 2). The first involves
local inelastic straining in the material surrounding the crack tip, which reduces the intensity of the stress
singularity. Inelastic straining can occur by one of numerous mechanisms, including dislocation glide in
metallic alloys, distributed microcracking in two-phase ceramics with large internal stresses, stress-induced
phase transformations in stabilized zirconia alloys, and matrix cracking and interface sliding in CFCCs. The
magnitude of the shielding effect can be computed using standard finite-element methods, provided suitable
constitutive laws for the inelasticity are available.
The second category of phenomena pertains to the fracture process zone (FPZ). The FPZ represents the region
directly ahead of the crack within which the strains become highly localized and lead to the initiation and
propagation of a macroscopic crack. Generally, fracture initiation is stochastic when it involves fracture of
brittle constituents. Indeed, stochastic behavior is obtained in CFCCs as well as in fiber-reinforced polymer-
matrix composites. Following fracture initiation, the mechanical response of the material in the crack wake is
characterized by a bridging traction law, as shown schematically in Fig. 2. Of the two steps in the fracture
process, fracture initiation and crack propagation, the one requiring the higher stress dictates the notched
strength.

Fig. 2 The effects of inelasticity on the crack-tip stresses and the characterization of the
fracture process zone (FPZ) through a bridging traction law. Here Γ
b
is the steady-state
toughness of the FPZ, independent of plastic shielding.
This framework for fracture analysis has been used successfully in the context of numerous fracture
mechanisms, including ductile rupture of metals, cleavage of inhomogeneous alloys, such as steels (Ref 6), and
fracture along bimaterial interfaces (Ref 7). It is anticipated that the framework will be implemented by the
CFCC design community once an understanding of the features specific to CFCCs reaches a suitably mature
level. The remainder of this chapter focuses on the features needed for this implementation.
References cited in this section
7. R.O. Ritchie, R.F. Knott, and J.R. Rice, On the Relationship Between Critical Tensile Stress and
Fracture Toughness in Mild Steel, J. Mech. Phys. Solids, Vol 21, 1973, p 395–410

8. J.W. Hutchinson and A.G. Evans, Mechanics of Materials: Top-Down Approaches to Fracture, Acta
Mater., Vol 48 (No. 1), 2000, p 125–135


Fracture Analysis of Fiber-Reinforced Ceramic-Matrix Composites
F.W. Zok, University of California at Santa Barbara

Classes of Material Behavior
Continuous fiber-reinforced ceramic composites exhibit one of three broad classes of deformation and fracture
behaviors, designated class I, II, and III (Ref 8). The key features associated with each are shown schematically
in Fig. 3. In CFCCs with 0°/90° fiber architectures, these behaviors are elicited by performing tension tests both
with and without notches in two orientations: with the loading direction parallel to one of the two fiber axes,
denoted the 0°/90° orientation; and with the loading direction oriented at ± 45° to the fiber axes.

Fig. 3 The three common classes of fracture behavior in CFCCs. Source: Adapted from
Ref 8
Class I behavior is characterized by the propagation of a dominant matrix crack, accompanied by fiber pullout,
but with otherwise negligible inelastic deformation. This behavior is obtained in materials with unusually high
interfacial toughness and/or sliding resistance (as a consequence of oxidation of the fiber–matrix interfaces, for
example) and yields relatively damage-intolerant behavior. Fiber pullout is manifested in the form of a rising
fracture resistance curve (so-called R-curve). The fracture resistance starts at a value comparable to the fracture
toughness of the matrix and increases with crack growth as the bridging zone develops. Once the broken fibers
at the point furthest from the crack tip completely disengage from their respective matrix sockets, a steady-state
fracture resistance is obtained. Because of the large length scales associated with the bridging, the extent of
crack growth needed to attain steady state is rather large, typically approximately 10 to 100 mm (0.4 to 4 in.).
An implication is that extremely large test specimens, on the order of 1 m (40 in.), are needed to obtain small-
scale bridging (SSB) conditions and hence extract in a direct manner the intrinsic fracture resistance curve. In
smaller test specimens and structures, the fracture resistance curve is influenced by structural dimensions. The
connections between the macroscopic structural response and the fundamental parameters dictating the pullout
process are made through a LSB mechanics (Ref 9). Some typical experimental results for the fracture

resistance behavior of a class I material are shown in Fig. 4. For comparison, predictions based on both LSB
and SSB models, described subsequently, are shown also.