Volume 21 - Composites Part 7 pptx

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.. Sun, Modeling Continu

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

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 σf and σm 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 Vp 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, σc, under

dilute conditions (Vp < 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 σc for a perfectly plastic matrix

reinforced with unit cylinders (loaded perpendicular to the axis of the cylinder) show β to be a function of Vp:

(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, Em is the elastic modulus of the matrix, and σo 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 σN is a reference stress, almost equal to the 0.2% yield stress of the composite σN 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 Vp 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 Vp 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 Em 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 (Pf) of a

fiber of length L is expressed as:

(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 σch is:

(Eq 28) where fRes 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 (σfar-field) multiplied by a stress-concentration

factor (k) less the residual radial stress (σrresidual) at the interface Stated mathematically:

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 (Nf) 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 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 R-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 Nf, or as the strain range (Δ )

versus Nf A plot of Δσ versus Nf 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 Nf, 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, Pult, for a center-cracked panel of width W and crack length 2a (Ref 70) and possessing an unnotched strength, σunnotched:

(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, E1 is the modulus in the fiber direction, E2 in the transverse

direction, G12 is the shear modulus, ν12 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 (JIc

is obtained (Ref 71):

(Eq 33) where

(Eq 34)

and βn is approximately 2 Here d is the diameter of the fiber, Vf is the volume fraction of fibers, and σY 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/m2 (4900 ft · lbf/ft2) was reported for a SiC-Ti alloy, which may be compared with a typical toughness of 40 kJ/m2 (2700 ft · lbf/ft2), based on KIc = 70 MPa (64 ksi ) for monolithic Ti-6Al-4V alloy Using Eq 33 with σY = 1040 MPa (150 ksi), Vf= 0.32, and d = 100 μm, a toughness

value of 63 kJ/m2 (4.3 ft · lbf/ft2) is estimated, which compares reasonably well with the experimental data

It is useful to note that JIc predicted by Eq 33 and 34 is quite strongly dependent on the volume fraction of

fibers Thus, (1 – Vf) · (1 – )/ decreases from approximately 0.59 to 0.12 on increasing Vf 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 LD 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:

The main problem with Eq 35 is that it shows an increasing toughness with yield strength, σy, 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 KIc 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 viscoplastic analysis based on FEM procedure, with emphasis on multiaxial loading of laminas

Elastic-plastic-Engineering Mechanics and Analysis of Metal-Matrix Composites

Bhaskar S Majumdar, New Mexico Institute of Mining and Technology

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

35 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

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

45 J.R Rice and D.M Tracey, J Mech Phys Solids, Vol 17, 1969, p 201–217

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

52 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

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,

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

66 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

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 KIc 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

Fracture Analysis of Fiber-Reinforced Matrix Composites

Ceramic-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

Failure strains The magnitude of the total

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

(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, σN,

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, kσ 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, σN, normalized by the respective unnotched tensile strength, σ0 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/kσ, where kσ 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 Γb 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 scale bridging (SSB) conditions and hence extract in a direct manner the intrinsic fracture resistance curve In

small-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

Fig 4 Fracture resistance curve for a Nicalon/LAS (Li2O-Al2O3-SiO2) composite that had been subjected to an embrittling heat treatment in air of 16 h at 800 °C (1470 °F) Upon heat treatment, the carbon coating on the fibers is replaced with an oxide layer that inhibits interfacial debonding and sliding The solid line shows a LSB prediction, based on

a linear softening traction law (Eq 4) The bridging parameters are: τ = 270 MPa (39 ksi),

E = 150 GPa (22 × 106 psi), ν = 0.25, h = 11 μm (4.33 × 10– 6 in.), f0 = 0.22, and R = 7 μm

(2.75 × 10–6 in.); the initiation toughness is K0 = 2 MPa The dashed line shows the corresponding SSB prediction for the same bridging parameters Source: Adapted from Ref 10

Class II behavior is characterized by the formation of multiple matrix cracks in both 0°/90° and ±45°

orientations, initially with negligible fiber fracture The cracks are manifested macroscopically as inelastic strain Some results for a Nicalon/SiC composite are shown in Fig 5(a) The inelasticity serves to redistribute the stresses in notched specimens, such that the peak stress concentration is reduced from its initial (elastic)

value Once fiber fracture begins, the subsequent pullout leads to R-curve behavior in essentially the same

manner as in class I materials; the main difference is that the extent of pullout in class II materials is greater than it is in class I materials, because the interfaces are weaker and hence the lengths over which debonding and sliding occur are greater The combined effects of plastic shielding and fiber pullout lead to highly damage-tolerant and notch-insensitive fracture behavior Large-scale bridging models can be used to simulate the fracture resistance behavior, although complications exist with regard to the coupled effects of the bridging and the inelastic straining

Fig 5 Stress–strain behaviors of 2-D woven composites in 0°/90° and ±45° orientations (a) Nicalon/SiC (plain weave fabric) Source: Ref 11 (b) Carbon-carbon (plain weave) Source: Ref 13 (c) Al2O3/mullite (eight-harness satin weave) Source: Ref 14

Class III behavior materials exhibit essentially linear behavior up to fracture in the 0°/90° orientation, but significant inelasticity in the ±45° orientation This behavior is obtained in CFCCs in which the matrix modulus

is much lower than that of the fibers In 0°/90° orientations, the contribution of the matrix to the initial elastic modulus of the composite is small, and hence any subsequent matrix damage has little effect on the composite response In some cases, the low matrix modulus arises from the presence of a high level of matrix porosity, introduced intentionally to impart high damage tolerance in the absence of weak fiber coatings (Ref 12) In others, it is a consequence of the high density of matrix cracks and fine-scale porosity resulting from the constrained pyrolysis and densification of the matrix precursor material (Ref 13) In the ±45° orientation, the matrix modulus plays a more significant role in the initial elastic response, and hence the matrix damage that occurs as a consequence of loading is manifested in significant inelasticity An example of this anisotropic

behavior in a carbon-carbon (C-C) and an all-oxide CFCC are shown in Fig 5(b) and (c) In notched geometries

in the 0°/90° orientation, the inelasticity occurs in the form of long, slender shear bands aligned parallel to the loading direction This deformation can reduce the stress concentration by a modest amount in some cases, although the efficacy of this deformation is considerably lower than that in class II materials, as demonstrated

in the section “Stress Distributions in Notched Specimens” in this article In other cases, the deformation has

the effect of increasing the stress concentration (Ref 15) Fiber pullout and the associated R-curve are again

obtained during fracture

References cited in this section

9 F.E Heredia, S.M Spearing, T.J Mackin, M.Y He, A.G Evans, P Mosher, and P Brønsted, Notch

Effects in Carbon Matrix Composites, J Am Ceram Soc., Vol 77 (No 11), 1994, p 2817–2827

10 F Zok and C.L Hom, Large Scale Bridging in Brittle Matrix Composites, Acta Metall Mater., Vol 38,

1990, p 1895–1904

11 F Zok, O Sbaizero, C.L Hom, and A.G Evans, The Mode I Fracture Resistance of a Laminated Fiber

Reinforced Ceramic, J Am Ceram Soc., Vol 74, 1991, p 187–193

12 C Cady, F.E Heredia, and A.G Evans, In- Plane Mechanical Properties of Several Ceramic Matrix

Composites, J Am Ceram Soc., Vol 78, 1995, p 2065–2078

13 C.G Levi, J.Y Yang, B.J Dalgleish, F.W Zok, and A.G Evans, Processing and Performance of an

All-Oxide Ceramic Composite, J Am Ceram Soc., Vol 81 (No 8), 1998, p 2077–2086

14 K.R Turner, J.S Speck, and A.G Evans, Mechanisms of Deformation and Failure in Carbon-Matrix

Composites Subject to Tensile and Shear Loading, J Am Ceram Soc., Vol 78, 1995, p 1841–1848

15 J.A Heathcote, X.-Y Gong, J Yang, U Ramamurty, and F.W Zok, In-Plane Mechanical Properties of

an All-Oxide Ceramic Composite, J Am Ceram Soc., Vol 82 (No 10), 1999, p 2721–2730

1 G.M Genin and J.W Hutchinson, Composite Laminates in Plane Stress: Constitutive Modeling and

Stress Redistribution Due to Matrix Cracking, J Am Ceram Soc., Vol 80 (No 5), 1997, p 1245–1255

Fracture Analysis of Fiber-Reinforced Ceramic-Matrix Composites

F.W Zok, University of California at Santa Barbara

Constitutive Laws for Inelastic Straining

The micromechanics of matrix cracking, interface debonding and sliding, and their roles in the macroscopic stress–strain response are very well understood Indeed, there exists a vast scientific literature dealing with this class of problems (see, for example, Ref 16, 17, and 18) Despite their large collective volume, most papers are restricted to unidirectionally-reinforced composites, subject to uniaxial loadings parallel to the fibers Some have dealt with 2-D cross-ply and woven architectures, but again, subject to uniaxial loading along one of the fiber axes Although these studies have yielded important insights into the mechanisms and mechanics of failure in CFCCs, they have not lead directly to the development of tools that are amenable for use in structural analysis, especially when off- axis and/or multitiaxial stresses are present

With this recognition, two independent groups recently have developed engineering approaches to describe the inelasticity of CFCCs, specifically for use in structural analysis The key papers were published almost simultaneously in 1997 (Ref 15, 19) Although the details of the two approaches differ considerably, both contain three key features:

• They faithfully (albeit approximately) represent the material behavior under multiaxial stress states

• They can be implemented readily into finite- element codes that are used commonly in engineering design

• They can be calibrated using a small number of standard mechanical tests

Both approaches are based on a homogenized representation of the composite without an explicit dependence

on the fiber architecture or the size scale of the microstructure, except insofar as these features influence the macroscopic properties obtained from the mechanical tests used for calibration and the macroscopic material symmetry Preliminary assessments of their capabilities and use in engineering design have been encouraging Selection of one over the other will be based ultimately on trade-offs between the level of complexity and the level of effort required for their calibration and implementation The extent to which they explicitly incorporate the damage mechanisms and associated micromechanics also may have some bearing on the selection process, although the criticality of this feature in engineering design has yet to be established

BHL Approach The first approach, developed by Burr, Hild, and Leckie (Ref 19) and subsequently referred to

as BHL, is couched in terms of continuum damage mechanics To begin, the physical internal variables that characterize the damage state are identified For 2-D laminates or weaves under biaxial stressing, ten internal variables are needed to fully characterize the extent of matrix cracking, interfacial sliding, and fiber damage in the two-ply types State potentials are then derived in terms of the state variables The derivations of these potentials are guided closely by micromechanical models of matrix cracking and interface sliding The potentials are then used in a framework of irreversible thermodynamics to obtain the forces driving each of the damage mechanisms The growth laws for each of the state variables are calibrated by performing uniaxial tension tests in both the 0°/90° and ±45° orientations along with periodic unloading-reloading excursions to measure hysteresis The behavior of the composite in other loading directions or under multiaxial loading is obtained through an interpolation procedure The behavior upon unloading also can be obtained Once calibrated, the constitutive law can be implemented into a finite element code as a user material subroutine

GH Approach The second approach, developed by Genin and Hutchinson (Ref 15) and subsequently referred

to as GH, is based on a purely phenomenological representation of the stress- strain response of a CFCC along the directions that coincide with the material symmetry: 0°/90° and ±45° for balanced 2-D laminates and weaves It begins by partitioning the two in- plane strains for principal stressing in the 0°/90° orientation into two functions, with each function being dependent on only one of the two principal stresses Similar partitioning is invoked for principal stressing in the ±45° orientation, yielding two additional strain functions Because the principal axes are indeterminate for equibiaxial loading, the strain functions obtained from principal stressing in the 0°/90° must match those obtained for principal stressing in the ±45° orientation The latter requirement reduces the number of independent functions to three The nonlinearity in these functions is couched in terms of "stress deficits," defined as the difference between the elastic stress that would exist at a prescribed strain and the one actually obtained For other stress states, where the principal stress directions are

at an arbitrary angle to the fiber axes, the stress deficits are obtained using an interpolation procedure A summary of this procedure is given in the Appendix of this chapter The constitutive law is calibrated through monotonic tension tests in the 0°/90° and ±45° orientations; no unload-reload excursions are required The calibrated constitutive law is implemented into a finite-element code as a user material subroutine, similar to that of the BHL constitutive law The GH law has the advantage of being somewhat easier to calibrate and implement; for balanced 2-D laminates, it requires only three independent strain functions, whereas the BHL law requires calibration of ten state functions

Validation of the BHL and GH constitutive laws has been accomplished by performing tests other than those used for their calibrations and comparing the measured responses with the predicted ones Two test types have been used The first is the Iosipescu test, which is used to measure the response in pure shear parallel to the fibers Comparisons between the BHL law and measurements made on a woven Nicalon/SiC composite are shown in Fig 6 The agreement between theory and experiment is good (within about 5%) Similarly good agreement has been obtained between the predictions of the GH law and pure shear measurements made on a 0°/90° Nicalon/CAS (CaO-Al2O3-SiO2) laminate (Ref 15) The second test geometry is an open-hole tensile specimen Strains are measured at select points around the hole using small, 0.7 mm (0.025 in.), strain gages Comparisons between these types of measurements on a 0°/90° Nicalon/MAS (MgO-Al2O3-SiO2) laminate and those predicted by the GH law are presented in Fig 7 Again, the agreement is very good, thereby providing additional confidence in the predictive capability of the constitutive law A related feature that emerges from the latter experiments is that the local strain at the hole edge (in the region of maximum stress concentration) can attain values that are considerably larger than the unnotched tensile failure strain This difference highlights the importance of size scale and stress gradient effects in the onset of failure This issue is addressed further in the section “Fracture Initiation” in this article

Fig 6 Comparisons of measured and predicted shear responses of a Nicalon/SiC CFCC The prediction is based on a finite-element calculation using the BHL constitutive law Source: Adapted from Ref 19

Fig 7 Comparisons of measured and predicted strains at two locations in an open-hole tensile specimen of a [0°/90°]3s Nicalon/MAS CFCC The predictions are based on finite- element calculations using the GH constitutive law Source: Adapted from Ref 2

Binary Model One of the deficiencies of both the BHL and GH constitutive laws is the absence of an explicit dependence on the fiber architecture and the nonuniformity of the damage at the scale of the fibers and the fiber tows This deficiency is expected to be important in cases where the stress gradients exist over distances that are comparable to or smaller than the length scales associated with the microstructure An alternate modeling approach that provides a more rigorous numerical representation of the behavior of the fibers and/or the fiber tows and the intervening matrix is the so-called binary model (Ref 20, 21) The model can be implemented in two ways In the first, one-dimensional spring elements are used to represent the individual fibers within a tow, and the surrounding matrix is represented by effective medium elements with the appropriate nonlinear behavior In this implementation, the model is capable of simulating the interactions between fiber fracture

events at the microscale, thereby providing a critical theoretical link between the properties of the individual fibers and the properties of the fiber tows In the second implementation, spring elements are used to represent the fiber tows (rather than the individual fibers) and the surrounding matrix and transverse fiber tows by

effective medium elements In this implementation, the model represents the material behavior at the

mesoscale, with information from the microscale computations integrated accordingly The tow properties may

encompass nonlinear deformation caused by fiber failure, matrix cracking, and interface sliding within the tow The binary model has numerous attractive features:

• It does not rely on periodicity in the fiber architecture

• It can readily deal with large spatial variations in the stress

• It can incorporate any arbitrary nonlinear response in the fibers and the matrix as well as statistical strength distributions

• It can compute stress and strain distributions in the matrix and fiber elements

It is ideal for capturing the interactions between stress concentration sites, fiber weave patterns, and multiaxial three-dimensional stress states Generally, however, the model is more computationally intensive than the homogeneous model and does not lend itself to the simulation of full-scale subelements or components Its role

in structural analysis is expected to be limited to the representation and simulation of those material elements within a component that are most critically stressed and are likely to be the sites of component failure In this context, the binary model would be combined with an appropriate homogenized constitutive law (such as the one of GH or BHL) to perform global/ local analyses, closely analogous to those for textile polymer-matrix composites (For global/ local approaches to polymer composite structures, see, for example, papers by J.D Whitcomb, A Tabiei, and their coworkers) The implementation of this approach awaits further developments

in the numerical codes and an assessment of the use of the codes in damage and failure prediction

References cited in this section

15 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

16 G.M Genin and J.W Hutchinson, Composite Laminates in Plane Stress: Constitutive Modeling and

Stress Redistribution Due to Matrix Cracking, J Am Ceram Soc., Vol 80 (No 5), 1997, p 1245–1255

17 D.B Marshall, B.N Cox, and A.G Evans, The Mechanics of Matrix Cracking in Brittle Matrix

Composites, Acta Metall., Vol 33 (No 11) 1985, p 2013–2021

18 J.W Hutchinson and H.M Jensen, Models of Fiber Debonding and Pullout in Brittle Matrix

Composites, Mech Mater., Vol 9 (No 2), 1990, p 139–163

19 18 A.G Evans, J.M Domergue, and E Vagaggini, Methodology for Relating the Tensile Constitutive

Behavior of Ceramic Composites to Constituent Properties, J Am Ceram Soc., Vol 77, 1994, p 1425–

1435

20 A Burr, F Hild, and F.A Leckie, Continuum Description of Damage in Ceramic- Matrix Composites,

Eur J Mech A-Solids, Vol 16 (No 1), 1997, p 53–78

21 M.A McGlockton, B.N Cox, and R.M McMeeking, “A Binary Model of Textile Composites: High Failure Strain and Work of Fracture in 3D Weaves,” to be published, 2001

1 M.A McGlockton, R.M McMeeking, and B.N Cox, “A Model for the Axial Strength of Unidirectional Ceramic Matrix Fiber Composites,” to be published, 2001

Fracture Analysis of Fiber-Reinforced Ceramic-Matrix Composites

F.W Zok, University of California at Santa Barbara

Stress Distributions in Notched Specimens

The homogeneous nonlinear constitutive laws can be used to calculate the spatial extent of inelastic straining around notches and holes and to assess the role of this inelasticity in mitigating stress concentrations The results of some finite- element calculations of the crack-tip stresses and the inelastic zones under SSY conditions are plotted in Fig 8 and 9 The examples are selected to be representative of the behaviors of class II and class III materials All calculations are based on the GH model For the purpose of these calculations, the

in-plane properties of the class II material are assumed to be isotropic The tensile stress–strain response is

taken to bilinear, with the initial moduli E0 = E45 The change in slope occurs at a critical cracking stress, σc,

and the tangent moduli beyond cracking are E0′ = E45′ = 0.2E0 Similar properties are assumed for the class III material, with the exception that the response in the fiber direction is taken to be linear for all stress levels Two pertinent features emerge:

• The shape of the inelastic zone for the class III material is distinctly elongated along the loading

direction, because of the strong anisotropy of the inelasticity The length of this zone is ≈0.1(K/σc)2 Furthermore, there is essentially no inelasticity directly ahead of the crack tip along the incipient fracture plane By contrast, the inelastic zone in the class II material is essentially equiaxed and of a size

≈0.1(K/σc)2

• The tensile stresses ahead of the crack in the class II material are reduced considerably to about half of the elastic values, over distances that are comparable to the inelastic zone size Only small reductions in stresses are obtained in the class III material, largely because of the absence of inelasticity ahead of the crack tip Clearly, the efficacy of the inelasticity in stress redistribution is superior in class II materials

Fig 8 Inelastic zones around a crack tip under SSY yielding conditions for (a) class II

and (b) class III materials The coordinates x and y are measured horizontally and

vertically, respectively, from the crack tip, and the stresses are applied remotely along the

y-direction The effective plastic strain is denoted p The normalizing distance is (K/σc)2,

where K is the applied mode I stress-intensity factor and σc is the matrix cracking stress Courtesy M.Y He, University of California at Santa Barbara

Fig 9 Stress distribution along the incipient fracture plane, for class II and class III materials as well as in an elastic, isotropic material The stresses are normalized by the matrix cracking stress, σc The corresponding inelastic zones are plotted in Fig 8 Courtesy M.Y He, University of California at Santa Barbara

The reduction in the peak stress concentration due to inelasticity in notched specimens can be determined in an

approximate way using the method developed by Neuber (Ref 22) Neuber's rule states that the stress and strain concentration factors, kσ and k , following the onset of local nonlinear straining are related to the elastic stress concentration factor, ke, through the relation Neuber demonstrated this relation to be strictly valid for metals subject to antiplane shear loading (Ref 22) Subsequently, the law has been used extensively in predicting stress concentrations in metals for a variety of notch and loading configurations (Ref 23) A graphical representation of Neuber's rule is shown in Fig 10 From an operational viewpoint, the law is implemented by finding the intersection point between the tensile stress-strain curve and a hyperbola described

by σ = where σ and ∊ are the local (maximum) stress and strain, and σA and A are the

corresponding applied values The stress concentration factor, kσ, is then the ratio of the stress at the intersection point to the applied stress In applying Neuber's rule to CFCCs subject to remote loading in the 0°/90° orientation, the relevant stress-strain curve is taken to be the one measured in that same orientation

Fig 10 The procedure used to implement the Neuber law in calculating the stress concentration factor in an elastic-plastic material

Figure 11 shows the stress concentration factors obtained from finite-element calculations for a class II material and those from Neuber's rule for some typical notched geometries The comparisons indicate that Neuber's rule

provides an accurate description of kσ over a wide range of applied stress and notch shape The results can be used to estimate the notched strength by combining the stress concentration factor with the measured unnotched tensile strength, assuming the failure criterion to be deterministic However, as demonstrated in the subsequent section, this approach yields overly conservative estimates of the notched strength and fails to predict size-scale effects (Similarly conservative estimates are obtained when the stress concentration factors obtained through Neuber's rule are used to predict fatigue lives in notched metal components; empirical methods have been developed to account for these size-scale effects and have found use in fatigue lifing [Ref 23]) Nevertheless, when a conservative design is necessarily required, or as a first step in the design process, this approach is expected to provide some useful guidance

Fig 11 Effects of inelasticity and notch shape on the stress concentration factors in a class

II CFCC The finite-element results are based on the GH constitutive law, assuming bilinear stress–strain behavior with the parameters shown in the inset of Fig 8(a) Also shown for comparison are the predictions of the Neuber law Courtesy X.-Y Gong, University of California at Santa Barbara

References cited in this section

23 H Neuber, Theory of Stress Concentration for Shear-Strained Prismatic Bodies with Arbitrary

Non-Linear Stress-Strain Laws, J Appl Mech., (Trans ASME), Vol E28, 1961, p 544

1 J.A Bannantine, J.J Comer, and J.L Handrock, Fundamentals of Metal Fatigue Analysis, Prentice Hall,

Englewood Cliffs, NJ, p 124–157

Fracture Analysis of Fiber-Reinforced Ceramic-Matrix Composites

F.W Zok, University of California at Santa Barbara

Fracture Initiation

Fracture initiation occurs essentially when the peak tensile stress exceeds the fiber bundle strength and the inelastic strains become highly localized in the region of extensive fiber fracture Experiments on several CFCCs indicate that the conditions for fracture initiation depend on the volume of highly stressed material Direct evidence of these size effects has come from comparisons of the maximum strains that are attained in test specimens with varying stress gradients For instance, in the [0°/90°]3s Nicalon/MAS composite, the unnotched tensile failure strain is 1.0%, the tensile failure strain in four-point flexure is ≈1.4%, and the failure strain at the edge of the hole in one a center-hole tensile specimen is ≈1.6% (Ref 2, 24) These measurements are consistent with the expectation that the failure strain increases with decreasing volume of material in the most heavily stressed region of a structure

The volume-dependence of the fracture initiation condition has been modeled using two approaches The first is

based on the point-stress failure criterion Here, fracture is postulated to initiate when a critical stress (taken to

be the unnotched tensile strength) is attained over a characteristic length, d, ahead of the notch This approach is

analogous to that used to describe cleavage fracture in ferrous alloys containing carbide particles; in the latter case, the characteristic distance correlates with the particle spacing, and the critical stress is dictated by the particle strength (Ref 6) The procedures for its implementation and assessment include:

• Performing nonlinear finite-element calculations of the stresses around the notches

• Calculating the notched tensile strength using several assumed values of the characteristic distance, d

• Making comparisons between the predictions and the experimental data in order to infer the characteristic distance and to check on the consistency in the trends with various structural dimensions and geometric features

Figure 12 shows examples for the Nicalon/ SiC system (Ref 2) The effects of hole diameter, 2a, at a constant value of a/w (Fig 12a) and of specimen width at a constant value of hole diameter (Fig 12b) are captured remarkably well using this approach, when the characteristic distance is selected to be d = 0.75 mm (0.030 in.)

Furthermore, in situ observations indicate that strain localization does not occur prior to the ultimate strength, confirming that fracture coincides with the initiation event Similarly good correlations have been obtained for other CFCCs, including Nicalon/MAS (Ref 2), Sylramic/SiC (Ref 3), and Al2O3/mullite (Ref 14) Interestingly, the inferred characteristic distances for these CFCCs lie in the rather narrow range of ≈0.50 to 0.75 mm (0.020

to 0.030 in.) This correlation suggests a commonalty in the size- dependence of fiber bundle failure and strain

localization; however, quantitative connections between d and the size scales in fiber bundle failure have yet to

be established Nevertheless, the narrowness of the inferred range of d suggests that the approach is quite robust

and probably applicable to other CFCCs of interest From an engineering viewpoint, it has the further advantage

of being relatively easy to implement and use Its main deficiency is in the lack of a sound physical basis

Fig 12 Comparisons of measured strengths with predictions based on the point-stress failure criterion The dashed lines show predictions based on the elastic stress

concentration factor and the unnotched tensile strength Hole is centered d is the

characteristic distance (a) Hole-to-width ratio is constant (b) Hole diameter is constant,

2a= 2.5 mm (0.098 in.), as width varies Source: Adapted from Ref 2

To further illustrate the size effect, Fig 12 also shows notched strength predictions stemming from a

deterministic (size-independent) failure criterion (labeled d = 0) This prediction is based on the assumption that

failure occurs when the maximum tensile stress (at the hole edge) reaches the unnotched tensile strength These predictions strongly underestimate the measured notched strengths and fail to correctly predict the size-scale effects that are found experimentally

An alternate approach to modeling fracture initiation is based on the premise that the composite strength is

probabilistic and follows weakest-link fracture statistics Figure 13 shows the results of a preliminary attempt to

assess the probabilistic approach in failure prediction of a Nicalon/SiC CFCC The predictions are based on finite-element calculations of the stresses in the notched specimens using the GH constitutive law and assuming that the composite strength distribution follows the Weibull function The comparisons suggest that the

composite Weibull modulus is m = 15±5, which is significantly higher than that of the fibers alone (m≈ 3–5)

This approach is expected to be more robust than the point-stress failure criterion, because it has a stronger mechanistic basis, but is more cumbersome to implement Furthermore, issues pertaining to the nature and the size of strength-limiting flaws in CFCCs and the broad applicability of weakest-link scaling approaches have yet to be addressed For instance, the critical “flaw” that leads to fracture in CFCCs is not preexisting (as it is in monolithic ceramics), but rather “evolves” during the straining process and comprises clusters of numerous broken fibers The size of these clusters dictates the minimum volume that can be used when applying weakest- link scaling laws at the macroscopic level (Ref 25) The binary model, which is described in the section

“Constitutive Laws for Inelastic Straining” in this article, is expected to be well suited to addressing these issues

Fig 13 Comparisons of measured strengths with predictions based on the probabilistic failure model Hole is centered (a) Hole-to-width ratio is constant (b) Hole diameter is

constant 2a = 2.5 mm (0.98 in.), as width varies Source: Adapted from Ref 2

References cited in this section

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

6 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

14 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

24 J.A Heathcote, X.-Y Gong, J Yang, U Ramamurty, and F.W Zok, In-Plane Mechanical Properties of

an All-Oxide Ceramic Composite, J Am Ceram Soc., Vol 82 (No 10), 1999, p 2721–2730

25 J.C McNulty and F.W Zok, Application of Weakest-Link Fracture Statistics to Fiber-Reinforced

Ceramic Matrix Composites, J Am Ceram Soc., Vol 80 (No 6), 1997, p 1535–1543

2 M Ibnabdeljalil and W.A Curtin, Strength and Reliability of Fiber-Reinforced Composites: Localized

Load-Sharing and Associated Size Effects, Int J Solids Struct., Vol 34, 1997, p 2649–2668

Fracture Analysis of Fiber-Reinforced Ceramic-Matrix Composites

F.W Zok, University of California at Santa Barbara

Crack Propagation

The effects of fiber pullout on crack propagation can be modeled using well-established bridging or cohesive zone concepts The general approach to this class of problem is to establish the integral equations that describe the crack-tip stress-intensity factor and the crack-opening displacement profile for the specimen geometry and loading configuration of interest, following standard procedures in the stress analysis of cracks Solutions to these equations (normally obtained through numerical methods) yield results in the form of stress versus crack length In this context, the key material properties are: the bridging law, which describes the relationship between the crack surface tractions, σb, and the crack- opening displacement, u; and the intrinsic crack- tip toughness, K0

Closely analogous problems associated with crack bridging in other materials systems have been studied extensively; the materials include ductile particle-reinforced ceramics, short fiber- reinforced cementitious materials, and monolithic ceramics, such as Al2O3 and Si3N4 (For details on the latter materials and a good treatment of the pertinent mechanics, see the book by B Lawn, Ref 26, and the references therein.)

A useful pedagogical tool for representing the instability conditions for the propagation of a bridged crack is the tangent construction plot, shown in Fig 14 (This construction has been used in the context of metals that

exhibit R-curve behavior; see, for example, the books by D Broek, Ref 27, and Kanninen and Popelar, Ref 28.) The plot contains two types of curves: the fracture resistance, KR, plotted against the crack extension, Δa (beyond the initial notch length); and the stress-intensity factor, KI, due to the applied load, plotted against the

total crack length, Δa + a0 In this context, the right side of the abscissa is interpreted as Δa and the left side as

a0 The fracture resistance curve initiates at the intrinsic fracture toughness, Ko, (at Δa= 0) and subsequently

increases with crack extension as the bridging zone develops Under SSB conditions, once the bridging elements furthest from the crack tip disengage, both the bridging zone length and the fracture resistance reach

steady- state values, independent of additional crack growth Using the J-integral, the steady-state resistance,

Kss, is given by:

(Eq 1) where is the plane-strain composite modulus and Γb is the bridging toughness:

(Eq 2)

The amount of crack extension, Δass, needed to achieve the steady state under SSB conditions is dependent somewhat on the shape of the traction law, but scales with the quantity For typical values of these

parameters (u0≈10 to 100 μm, ≈ 100 GPa, and σ0≈100 MPa), this bridging length is in the range of ≈10 to100

mm The conditions for instability are:

(Eq 3)

These conditions are satisfied when the curve KI (Δa+ a0) is tangent to the curve KR (Δa) The curves shown in Fig 14 illustrate that when the initial flaw is small in relation to Δass, the instability occurs when the crack has grown only a small amount In this case, the bridging plays a negligible role in the strength By contrast, when

the flaws are comparable or larger than Δass, the instability occurs when the crack has extended by ≈Δass In this case, the steady-state toughness is almost fully used in the composite strength

Fig 14 Tangent construction plot used to represent the stability of bridged cracks

Fracture resistance (KR) is plotted against crack extension (Δa) Stress-intensity factor, KI,

is plotted against total crack length

Rather comprehensive solutions exist for the crack propagation stress in uniaxial tension for a variety of traction law shapes, including rectilinear, linear hardening, linear softening, and parabolic (Ref 29, 30, 31, and

32) All are based on the assumption that the adjacent composite material remains elastic; the coupled effects of

inelastic deformation and fiber pullout have yet to be explored Representative results are plotted in Fig 15 for

center-cracked specimens of infinite width (a/w≈ 0) for the linear softening traction law (Ref 32) The latter

traction law is selected here because it is consistent with the prediction from a shear lag analysis of a single bridging fiber The result of this analysis is:

(Eq 4)

where τ is the interfacial sliding stress, f0 is the volume fraction of fibers that are aligned parallel to the loading

direction, h is the fiber pullout length, and R is the fiber diameter The results are presented in terms of the

propagation stress, σp, normalized by the peak stress, σ0, in the traction law and plotted against a nondimensional crack length, α, defined by (Ref 29, 32):

(Eq 5)

where 2a0 is the initial crack length and u0 is the critical crack-opening displacement upon complete

disengagement of the bridging elements (equivalent to h in the shear lag result in Eq 4) The normalizing crack

length, is the characteristic bridging-length scale that dictates Δass The effects of the intrinsic toughness are incorporated into a normalized toughness parameter, λ, defined by (Ref 32):

(Eq 6) where Γ0 is the intrinsic fracture energy (related to the intrinsic fracture toughness, K 0, through the Irwin relation, ), and the normalizing fracture energy, σ0 u0, is proportional to the bridging toughness The results show three main regimes of behavior, governed mainly by the value of α When the initial crack

length is very small (α≤0.01), the propagation stress is dictated by the intrinsic toughness, K0, in accordance with the Griffith relation (written in non- dimensional form):

(Eq 7)

Fig 15 Normalized strength vs nondimensional crack length (α) Effects of flaw size (α) and the toughness (λ) on the crack propagation stress (σp) for a/w << 1 The bridging law

is assumed to be linear softening, with a peak stress, σ0 The dotted lines on the right are the SSB predictions and are applicable when α > 1 The dashed lines on the left side represent the behavior in the absence of crack bridging and are applicable when α < 0.01 Source: Adapted from Ref 32

The predictions of Eq 7 are plotted as a series of dashed lines on the left side of Fig 15 In this regime, crack bridging has negligible effect on the crack stability, as illustrated by the tangent construction plot in Fig 14 At the other extreme, where the initial crack length is very large (α >1), the propagation stress again follows the Griffith relation, except now the pertinent toughness is the steady-state composite toughness; this defines the SSB regime In this regime, the relation between the strength and the initial crack length can be expressed as:

(Eq 8)

These predictions are shown by the dotted lines on the right side of Fig 15 Between these extremes, the strength is predicted by a LSB model The predictions of the LSB model converge with the nonbridging prediction and the SSB prediction when the initial flaw size approaches the respective limiting value, as required Furthermore, because the flaws of interest are generally smaller than the characteristic bridging-length scale, SSB conditions are rarely achieved, and hence the LSB model is needed to describe the notch sensitivity of the propagation stress

The bridging law parameters can be determined in one of several ways The first is based on micromechanical models, such as the shear lag model that leads to the result in Eq 4, coupled with independent measurement of the pertinent material parameters, including the sliding stress and the pullout length The accuracy of the resulting bridging law depends on the accuracy of these measurements, recognizing that some are difficult to make in a manner that accurately reflects the fracture process For instance, the sliding stress that is obtained by common measurement techniques, such as by fiber push-in (Ref 33) or push-through tests (Ref 34), is usually higher than the value that is representative of the pullout process The discrepancy is the result of the difference

in the sign of the Poisson strain; in some cases, it may also be associated with a dependence of the sliding stress

on the extent of interface sliding (because of wear, for example, Ref 35) The accuracy of the bridging law depends also on the fidelity of the micromechanical model upon which the law is based Notwithstanding these complications, micromechanical models are attractive and popular, because they form a critical link between the physics of the pullout process and the associated macroscopic mechanical response

The second approach involves either direct or semidirect mechanical measurements The most direct method is

to perform a tensile test on a deeply notched specimen and measure the variation in the load with the opening displacement in the regime beyond the load maximum (Ref 35, 36) Although direct, this type of test is prone to instabilities near the load maximum, a consequence of the elastic unloading of the material away from the crack plane when the load falls from its peak value Indeed, in cases where the values of the bridging