##### Document Text Contents

Page 2

Fundamentals of Metal-Matrix Composites

Page 175

Figure 9.8. Tensile stress-strain

curves for composites reinforced

by aligned disc-shaped cylindrical

particles (ulb = 5 and f= 0.2).The

matrix material has the

Ramberg-Osgood stress-strain

curves shown. The solid line

curves for the composite were

computed using a cell model,

while the dashed line curves were

obtained using the estimation

scheme described in the text.

Figure 9.9. Comparison of experimental

data for composites of an AVMg matrix

material reinforced by Sic particles with

stress-strain curves predicted by the

estimation scheme described in the text.

(From Yang et al. 1991.) (a) Uniaxial tensile

curves for matrices reinforced by equiaxed

particles. (b) Uniaxial compression curves for

matrices reinforced by randomly oriented

platelets.

Page 176

168 MICROMECHANICS AND MECHANICS OF DEFORh4ATION

matrix as well as for composites reinforced by three

volume fractions of equiaxed particles whose average

size was 9 microns. The estimation scheme described

above was applied by fitting the Ramberg-Osgood curve

(Equation 9.4) to the matrix curve to obtain II and uo (a

was taken to be 3/7). The value of E in (Equation 9.5)

was taken from the experimental curve (which in turn

was shown to agree well with self-consistent predic-

tions), and ON was determined from Equation 9.8, using

the results for Eo from Figure 9.3 for the unit cylindrical

particles. The dashed-line curves in Figure 9.9(a) are the

result of the estimation procedure. The same procedure

was applied to the uniaxial compression data in Figure

9.9(b) for composites reinforced by randomly oriented

platelets whose average maximum diameter was 25

microns. In this case, the platelets are taken to have a

10: 1 aspect ratio and the results for 50 for the randomly

oriented ellipsoidal platelets in Figure 9.5 were em-

ployed to estimate ON.

9.3 The Influence of Residual Stress

on Composite Yielding

The results presented so far in this chapter are for

materials initially free of residual stress in the matrix

and in the reinforcements. Most metal-matrix compos-

ites are processed at high temperatures and, upon cool-

ing, develop residual stresses as a result of thermal

expansion mismatch between the matrix and the rein-

forcements. Although the residual stresses have no effect

on the purely elastic response of the composite, it is of

interest to determine the effect on the yielding of rein-

forced materials. Such effects can occur because the

residual stress can have a deviatoric component and can

thus influence the process of yielding in the matrix. The

effect has been considered by several investigators in-

cluding Povirk et a]. (1991). However, Zahl and Mc-

Meeking (1991), have provided a series of results for

strongly bonded elastic reinforcements in perfectly plas-

tic matrices showing the influence of the thermal strain

mismatch relative to the volume fraction of reinforce-

ments and the yield strain of the matrix.

The results of Zahl and McMeeking (1991) were

obtained by the unit cell method with finite elements

used for the analysis. The residual stresses were first

generated by cooling the material while the matrix was

permitted to respond elastoplastically. Thereafter, loads

were applied to cause macroscopic deformation. The

magnitude of the residual stresses generated were con-

trolled by the parameter A~ATIEo, where A a is the

thermal expansion coefficient of the reinforcement mi-

nus the thermal expansion coefficient of the matrix, and

AT is the current temperature minus the temperature at

which the composite material is free of residual stress in

both the matrix and the reinforcement. The parameter

EO is, as before, the yield strain in tension of the matrix

material. The calculations were carried out with an

elastic modulus for the reinforcement that is 6.62 times

the elastic modulus of the matrix.

Figure 9.10 shows the stress-strain curves for spher-

ical reinforcements in a perfectly plastic matrix with

A ~ T / E o = 1. This case corresponds to Sic particles in

an Al alloy matrix 256C below the stress-free temper-

ature. A softening of the composite response results at

strains comparable to the matrix yield strain for both

tension and compression with the effect much more

pronounced in the compressive cases. The compressive

stress-strain curve is up to 30% below the tensile curve

in terms of strength at the same strain magnitude.

However, as the strain increases beyond EO = AaAT,

the compressive and the tensile stress-strain curves con-

verge toward the curve for the material without initial

residual stress. The limit strength is thus the same

whether or not there are initial residual stresses. Because

the compressive stress-strain curve also represents ten-

sion applied to a material with ACXATIEO = - 1 (the sign

of the residual stresses reversed), the limit strength is

unaffected by whether the residual stresses in the matrix

are tensile or compressive. Because the limit strength of

reinforced materials is independent of the initial resid-

ual stresses, the behavior of the composite material

when the strain greatly exceeds EO = AaAT is corre-

spondingly independent of them too. However, the

limited ductility of particulate composites means that

such large strains are rarely achieved in tension and are

unusual in compression unless accompanied by internal

damage. As a consequence, tension-compression asym-

metries in the yielding of particulate composites are to

be expected, and will generally persist until fracture of

the material occurs.

The degree of yielding caused by thermal expansion

mismatch between the matrix and the reinforcements

depends on the magnitude of A~ATIEo. When this

parameter equals 1, as in the case discussed above,

approximately 50% of the matrix around spherical

particles is yielded. When AaA7'/~0 = 2, 60% of the

matrix has yielded. When A~AT/Eo = 5, the entire

matrix has deformed plastically upon cooling. The ef-

fect of these different degrees of yielding on a material

with 20% of elastic spheres on the compression and

tension stress-strain curves is shown in Figures 9.1 1 and

9.12. It can be seen that in both tension and compres-

sion the greater thermal expansion misfit causes a softer

response. Because the matrix is fully yielded when

A ~ T / E o = 5, any magnitude of AaAT/&o larger than 5

will give rise to the same tension and compression

stress-strain curves as occur for haAT/~o = 5. It is of

Page 349

Index 341

characteristics of, variation in, effects of, 206,208

of continuously reinforced composites, 204-206

of discontinuous reinforcements, 204,206-210

microstructural damage during, 205

in presence of applied strcss, 207-208, 208f-209f

and residual stresses, 69

summary of, 210-211

Thermal expansion coefficient. See Coefficient of thermal

expansion

Thermal expansion tensor, 74

Thermal management composites, 317,319f

Thermal mismatch strains, 40,209

Thermal ratcheting, prediction of, 204-205

Thermal residual stresses, 68-69, 69f, 120

calculation of, 74-75

in continuous fiber-reinforced composites, 184-187

plastic relaxation of, 124-125, 130

yielding caused by, 168-169, 169f-171f

Thermomechanical processing, 3 1-34,32f-34f

microstructure after, 33f, 33-34, 36, 36f

in polycrystalline composite model, 153

Thermomechanical variables, in aging response, 127-13 1

Thin-film cracking, 220, 220f

Thixomolding, 11

Thixotropic nature, of composite slurries, 12

Three-dimensional network, for preform fabrication, 17, 18f

Three-phase damage model, of elastic deformation in

particle fracture, 241-242, 242f

Threshold. See also Near-threshold behavior

intrinsic, 290

Tire studs, 3 14

Titanium alloys

aerospace applications of, 307, 309-310, 310t, 312

applications of, 320

properties of, 3 10t

Tool guide, for hole drilling, 66-67, 67f

Toughness measurements

of fiber-reinforced composites and unreinforced matrix

of particle- and whisker-reinforced composites, 252t

alloys, 252t

Toyota Motor Corporation, 3, 298

Transition bands, 11 1-1 12

Translation state, 104-106, 106f

Transmission electron microscopy

of aging response, 121, 122t-l23t, 126, 129, 131-132,

of chemical processes at metaVceramic interfaces, 100

conventional, 83-84, 84f

of crack propagation, 263, 263f-264f

of dislocation density, 124f, 124-126

of matrix failure, 234

objective lens of, geometric beam path through, 84, 85f

subgrain size measurement with, 116

of thermal cycle strain, 210,210f

132f

Transmission function, 84

Transport phenomena

in infiltration processes, 5-8

in spray processes, 14

after deposition, 14-15

Transverse creep, 193

Transverse loading, matrix stresses and composite creep rate

Transverse shear modulus, 177, 181, 182t

Transverse strengthening, of continuous fiber-reinforced

composites, 159, 161-163, 162f

under, prediction of, 193-194

Transverse Young’s modulus, 176, 180-181, 182t

Tresca-type yield criterion, 199

Triangle packing model, 179f, 183

Tungsten

applications of, 136f, 309t, 315-316

characteristics of, 299t

Turbine blades, hollow, fabrication of, with monotapq 24-25,27f

Tyranno, 300

Ultrasonics, measurement of residual stresses with, 62

Uniaxial creep, 191-193

Uniform strain

in elastic deformation, 175

rate of, 164

Unit cell model

of continuous fiber composites, 178-179, 179f-l81f,

of interfacial decohesion, 247-248, 248f

of particle fracture, 242-243

of polycrystalline particulate composites, 144-145, 145f

of residual stress, 168

of single crystal composites, 143-144, 144f

182t, 183

Unit density, strength and stiffness per, 23

United States, research and development programs in, 325

Unloading solution, elastic, 73

Unreinforced matrix alloys, fracture properties of, 251,252t

UTS, 252-253

Vacuum-driven infiltration, 5

V-blender, 26

Vertical clustering, of reinforcement, and stress-strain curve

Vibrational damping composite steels, 3 19

Vickers diamond pyramid indentor microhardness, aging

wetting during, promotion of, 5 1

and ductility, 236f, 236-239

response represented by, 121, 122t-l23t, 127

as function of aging time, 127-129, 128f-129f

Viscosity, of liquid metals, 11-12, 12f

Void nucleation, growth, and coalescence, 239-240, 244f,

244-245

in long fiber reinforced composites, 258-262, 259f-262f

and plastic constraint, 239, 239f

Void volume fraction, 236, 237f-238f

Voigt-type approximation, 176

Volume fraction

and crack closure levels, 286, 286f

and fatigue crack growth, 264

and fatigue life behavior, 280-282, 281f-282f

and near-threshold behavior, 288-289, 288f-289f

Vortex method, 10

Wagner’s first-order interaction parameter, 48-49

W-based alloys, 300

Weak-beam imaging, of misfit dislocations, 92-93

Wear-resistant composites, 23, 298, 314f, 314-316, 316f

Weibull distribution, 202, 202f

Wetting

enhancement of, 51-52

nonreactive, 45-48

reactive, 48-50

Whisker(s), 299t, 300. See also specific type

cracked, 244-245, 245f

deformation zones around, 112, 112f

macroscopic effect of, 112

textural weakening caused by, 114

Page 350

342 INDEX

Whisker-reinforced composites

fatigue life behavior of, 281-282

microstructural evolution in, 109-118

strengthening in, and plastic constraint, 184

stress-strain behavior of, 114-116, 115f-116f

toughness measurements of 252t

Williams’s singularity, 221-224

Work-hardened interface, strengthening effect of, 201

Work hardening, cyclic, 272

Work of adhesion, 43, 8 1

determination of, 82

experimental values of, for nonreactive metals, 45-46,

45t-47t

influence of alloying additions on, 46, 47f

Work of immersion, 43

XD synthesis, 15, 17, 310, 313

X-ray(& penetration depth of, for difference materials, 64t

X-ray diffraction, 62, 64-65

X-ray reflectivity curves, 99,99f

Yield criterion

Mises, 72, 74, 159, 182

Tresca-type, 199

Yielding, composite, influence of residual stress on, 168,

Yield strength

169f-171f

effect of CTE mismatch on, 204

in elastic cell, 230, 230f

steady-state shielding ratio as function of 228, 228f

step-function decay in, model system with, 230, 230f

Yield zone, development of, in matrix punched by glass

sphere, 125, 125f

Young-Dupr6 equation, 43,82,82f

Young’s modulus, 72,165,230,240-242

axial, 176, 181, 182t, 182

transverse, 176, 180-181, 182t

Z-axis surface scattering spectrometer, 95

Zener’s pinning effect theory, 33

Zero clearance piston, 305

Zero-degree square packing, 179

Fundamentals of Metal-Matrix Composites

Page 175

Figure 9.8. Tensile stress-strain

curves for composites reinforced

by aligned disc-shaped cylindrical

particles (ulb = 5 and f= 0.2).The

matrix material has the

Ramberg-Osgood stress-strain

curves shown. The solid line

curves for the composite were

computed using a cell model,

while the dashed line curves were

obtained using the estimation

scheme described in the text.

Figure 9.9. Comparison of experimental

data for composites of an AVMg matrix

material reinforced by Sic particles with

stress-strain curves predicted by the

estimation scheme described in the text.

(From Yang et al. 1991.) (a) Uniaxial tensile

curves for matrices reinforced by equiaxed

particles. (b) Uniaxial compression curves for

matrices reinforced by randomly oriented

platelets.

Page 176

168 MICROMECHANICS AND MECHANICS OF DEFORh4ATION

matrix as well as for composites reinforced by three

volume fractions of equiaxed particles whose average

size was 9 microns. The estimation scheme described

above was applied by fitting the Ramberg-Osgood curve

(Equation 9.4) to the matrix curve to obtain II and uo (a

was taken to be 3/7). The value of E in (Equation 9.5)

was taken from the experimental curve (which in turn

was shown to agree well with self-consistent predic-

tions), and ON was determined from Equation 9.8, using

the results for Eo from Figure 9.3 for the unit cylindrical

particles. The dashed-line curves in Figure 9.9(a) are the

result of the estimation procedure. The same procedure

was applied to the uniaxial compression data in Figure

9.9(b) for composites reinforced by randomly oriented

platelets whose average maximum diameter was 25

microns. In this case, the platelets are taken to have a

10: 1 aspect ratio and the results for 50 for the randomly

oriented ellipsoidal platelets in Figure 9.5 were em-

ployed to estimate ON.

9.3 The Influence of Residual Stress

on Composite Yielding

The results presented so far in this chapter are for

materials initially free of residual stress in the matrix

and in the reinforcements. Most metal-matrix compos-

ites are processed at high temperatures and, upon cool-

ing, develop residual stresses as a result of thermal

expansion mismatch between the matrix and the rein-

forcements. Although the residual stresses have no effect

on the purely elastic response of the composite, it is of

interest to determine the effect on the yielding of rein-

forced materials. Such effects can occur because the

residual stress can have a deviatoric component and can

thus influence the process of yielding in the matrix. The

effect has been considered by several investigators in-

cluding Povirk et a]. (1991). However, Zahl and Mc-

Meeking (1991), have provided a series of results for

strongly bonded elastic reinforcements in perfectly plas-

tic matrices showing the influence of the thermal strain

mismatch relative to the volume fraction of reinforce-

ments and the yield strain of the matrix.

The results of Zahl and McMeeking (1991) were

obtained by the unit cell method with finite elements

used for the analysis. The residual stresses were first

generated by cooling the material while the matrix was

permitted to respond elastoplastically. Thereafter, loads

were applied to cause macroscopic deformation. The

magnitude of the residual stresses generated were con-

trolled by the parameter A~ATIEo, where A a is the

thermal expansion coefficient of the reinforcement mi-

nus the thermal expansion coefficient of the matrix, and

AT is the current temperature minus the temperature at

which the composite material is free of residual stress in

both the matrix and the reinforcement. The parameter

EO is, as before, the yield strain in tension of the matrix

material. The calculations were carried out with an

elastic modulus for the reinforcement that is 6.62 times

the elastic modulus of the matrix.

Figure 9.10 shows the stress-strain curves for spher-

ical reinforcements in a perfectly plastic matrix with

A ~ T / E o = 1. This case corresponds to Sic particles in

an Al alloy matrix 256C below the stress-free temper-

ature. A softening of the composite response results at

strains comparable to the matrix yield strain for both

tension and compression with the effect much more

pronounced in the compressive cases. The compressive

stress-strain curve is up to 30% below the tensile curve

in terms of strength at the same strain magnitude.

However, as the strain increases beyond EO = AaAT,

the compressive and the tensile stress-strain curves con-

verge toward the curve for the material without initial

residual stress. The limit strength is thus the same

whether or not there are initial residual stresses. Because

the compressive stress-strain curve also represents ten-

sion applied to a material with ACXATIEO = - 1 (the sign

of the residual stresses reversed), the limit strength is

unaffected by whether the residual stresses in the matrix

are tensile or compressive. Because the limit strength of

reinforced materials is independent of the initial resid-

ual stresses, the behavior of the composite material

when the strain greatly exceeds EO = AaAT is corre-

spondingly independent of them too. However, the

limited ductility of particulate composites means that

such large strains are rarely achieved in tension and are

unusual in compression unless accompanied by internal

damage. As a consequence, tension-compression asym-

metries in the yielding of particulate composites are to

be expected, and will generally persist until fracture of

the material occurs.

The degree of yielding caused by thermal expansion

mismatch between the matrix and the reinforcements

depends on the magnitude of A~ATIEo. When this

parameter equals 1, as in the case discussed above,

approximately 50% of the matrix around spherical

particles is yielded. When AaA7'/~0 = 2, 60% of the

matrix has yielded. When A~AT/Eo = 5, the entire

matrix has deformed plastically upon cooling. The ef-

fect of these different degrees of yielding on a material

with 20% of elastic spheres on the compression and

tension stress-strain curves is shown in Figures 9.1 1 and

9.12. It can be seen that in both tension and compres-

sion the greater thermal expansion misfit causes a softer

response. Because the matrix is fully yielded when

A ~ T / E o = 5, any magnitude of AaAT/&o larger than 5

will give rise to the same tension and compression

stress-strain curves as occur for haAT/~o = 5. It is of

Page 349

Index 341

characteristics of, variation in, effects of, 206,208

of continuously reinforced composites, 204-206

of discontinuous reinforcements, 204,206-210

microstructural damage during, 205

in presence of applied strcss, 207-208, 208f-209f

and residual stresses, 69

summary of, 210-211

Thermal expansion coefficient. See Coefficient of thermal

expansion

Thermal expansion tensor, 74

Thermal management composites, 317,319f

Thermal mismatch strains, 40,209

Thermal ratcheting, prediction of, 204-205

Thermal residual stresses, 68-69, 69f, 120

calculation of, 74-75

in continuous fiber-reinforced composites, 184-187

plastic relaxation of, 124-125, 130

yielding caused by, 168-169, 169f-171f

Thermomechanical processing, 3 1-34,32f-34f

microstructure after, 33f, 33-34, 36, 36f

in polycrystalline composite model, 153

Thermomechanical variables, in aging response, 127-13 1

Thin-film cracking, 220, 220f

Thixomolding, 11

Thixotropic nature, of composite slurries, 12

Three-dimensional network, for preform fabrication, 17, 18f

Three-phase damage model, of elastic deformation in

particle fracture, 241-242, 242f

Threshold. See also Near-threshold behavior

intrinsic, 290

Tire studs, 3 14

Titanium alloys

aerospace applications of, 307, 309-310, 310t, 312

applications of, 320

properties of, 3 10t

Tool guide, for hole drilling, 66-67, 67f

Toughness measurements

of fiber-reinforced composites and unreinforced matrix

of particle- and whisker-reinforced composites, 252t

alloys, 252t

Toyota Motor Corporation, 3, 298

Transition bands, 11 1-1 12

Translation state, 104-106, 106f

Transmission electron microscopy

of aging response, 121, 122t-l23t, 126, 129, 131-132,

of chemical processes at metaVceramic interfaces, 100

conventional, 83-84, 84f

of crack propagation, 263, 263f-264f

of dislocation density, 124f, 124-126

of matrix failure, 234

objective lens of, geometric beam path through, 84, 85f

subgrain size measurement with, 116

of thermal cycle strain, 210,210f

132f

Transmission function, 84

Transport phenomena

in infiltration processes, 5-8

in spray processes, 14

after deposition, 14-15

Transverse creep, 193

Transverse loading, matrix stresses and composite creep rate

Transverse shear modulus, 177, 181, 182t

Transverse strengthening, of continuous fiber-reinforced

composites, 159, 161-163, 162f

under, prediction of, 193-194

Transverse Young’s modulus, 176, 180-181, 182t

Tresca-type yield criterion, 199

Triangle packing model, 179f, 183

Tungsten

applications of, 136f, 309t, 315-316

characteristics of, 299t

Turbine blades, hollow, fabrication of, with monotapq 24-25,27f

Tyranno, 300

Ultrasonics, measurement of residual stresses with, 62

Uniaxial creep, 191-193

Uniform strain

in elastic deformation, 175

rate of, 164

Unit cell model

of continuous fiber composites, 178-179, 179f-l81f,

of interfacial decohesion, 247-248, 248f

of particle fracture, 242-243

of polycrystalline particulate composites, 144-145, 145f

of residual stress, 168

of single crystal composites, 143-144, 144f

182t, 183

Unit density, strength and stiffness per, 23

United States, research and development programs in, 325

Unloading solution, elastic, 73

Unreinforced matrix alloys, fracture properties of, 251,252t

UTS, 252-253

Vacuum-driven infiltration, 5

V-blender, 26

Vertical clustering, of reinforcement, and stress-strain curve

Vibrational damping composite steels, 3 19

Vickers diamond pyramid indentor microhardness, aging

wetting during, promotion of, 5 1

and ductility, 236f, 236-239

response represented by, 121, 122t-l23t, 127

as function of aging time, 127-129, 128f-129f

Viscosity, of liquid metals, 11-12, 12f

Void nucleation, growth, and coalescence, 239-240, 244f,

244-245

in long fiber reinforced composites, 258-262, 259f-262f

and plastic constraint, 239, 239f

Void volume fraction, 236, 237f-238f

Voigt-type approximation, 176

Volume fraction

and crack closure levels, 286, 286f

and fatigue crack growth, 264

and fatigue life behavior, 280-282, 281f-282f

and near-threshold behavior, 288-289, 288f-289f

Vortex method, 10

Wagner’s first-order interaction parameter, 48-49

W-based alloys, 300

Weak-beam imaging, of misfit dislocations, 92-93

Wear-resistant composites, 23, 298, 314f, 314-316, 316f

Weibull distribution, 202, 202f

Wetting

enhancement of, 51-52

nonreactive, 45-48

reactive, 48-50

Whisker(s), 299t, 300. See also specific type

cracked, 244-245, 245f

deformation zones around, 112, 112f

macroscopic effect of, 112

textural weakening caused by, 114

Page 350

342 INDEX

Whisker-reinforced composites

fatigue life behavior of, 281-282

microstructural evolution in, 109-118

strengthening in, and plastic constraint, 184

stress-strain behavior of, 114-116, 115f-116f

toughness measurements of 252t

Williams’s singularity, 221-224

Work-hardened interface, strengthening effect of, 201

Work hardening, cyclic, 272

Work of adhesion, 43, 8 1

determination of, 82

experimental values of, for nonreactive metals, 45-46,

45t-47t

influence of alloying additions on, 46, 47f

Work of immersion, 43

XD synthesis, 15, 17, 310, 313

X-ray(& penetration depth of, for difference materials, 64t

X-ray diffraction, 62, 64-65

X-ray reflectivity curves, 99,99f

Yield criterion

Mises, 72, 74, 159, 182

Tresca-type, 199

Yielding, composite, influence of residual stress on, 168,

Yield strength

169f-171f

effect of CTE mismatch on, 204

in elastic cell, 230, 230f

steady-state shielding ratio as function of 228, 228f

step-function decay in, model system with, 230, 230f

Yield zone, development of, in matrix punched by glass

sphere, 125, 125f

Young-Dupr6 equation, 43,82,82f

Young’s modulus, 72,165,230,240-242

axial, 176, 181, 182t, 182

transverse, 176, 180-181, 182t

Z-axis surface scattering spectrometer, 95

Zener’s pinning effect theory, 33

Zero clearance piston, 305

Zero-degree square packing, 179