A NEW MODEL FOR PREDICTING LOW CYCLE FATIGUE BEHAVIOR OF DISCONTINUOUSLY REINFORCED METAL MATRIX COMPOSITES

Qian Zhang ANEW MODEL FOR PREDICTING LOW CYCLE FATIGUE BEHAVIOR OF DISCONTINUOUSLY REINFORCED METAL MATRIX COMPOSITES MASc, Mechanical Engineering. Ryerson, Toronto, 2004. An analytical model for predicting the crack initiation life of low cycle fatigue (LCF) of discontinuously reinforced metal matrix composites (DR-MMCs) has been proposed. The effects of the volume fraction Vf, cyclic strain hardening exponent n' and cyclic strength coefficient K' on the LCF crack initiation life of DR-MMCs were analyzed. While both the lower level of the plastic strain amplitude and the lower Vf were found to increase the LCF crack initiation resistance, the effects of n’ and K' were more complicated. By considering the enhanced dislocation density in the matrix and the load bearing effect of particles, a quantitative relationship between the LCF life of DR-MMCs and particle size was also derived. This model showed that a decreasing particle size results in a longer LCF life. The theoretical predictions based on the proposed models were found to be in good agreement with the experimental data reported in the literature.


LIST OF TABLES
and with the experimental data [6 ] in an AI2O3 particulate-  [6 ] and with the experimental data [2] in an AI2O3 particulate-reinforced AA6061 composite  [2] and with the experimental data [2] in an AI2O3 particulate-reinforced AA6061 composite   [7] and Bosi et al. [8 ] reported that fine particles benefited fatigue resistance within 10"^ cycles. On the other hand, Han et al. [9,10] reported that coarse particle reinforced composites exhibited slightly superior LCF resistance. Furthermore, a recent study by Uygur et al. [11] showed that the reinforcement size has no significant effect on the LCF life of DR-MMCs. The different results reported in the literature indicated that the effect of reinforcement particle size on the LCF life has not been fully understood.
Therefore, the principal purposes of this study are:  Hi eh wear resistance -Increased wear resistance is an important virtue of DR-MMCs. By the proper introduction of the reinforcement, it is common for wear rates to be reduced by factors of up to ten. In addition, it is often advantageous to control the distribution of reinforcement to achieve high wear resistance in selected surface areas while other regions are suitably tough, strong, thermally conducting, etc. It is common that the high wear resistance is combined with other properties, such as high stiffness and high thermal conductivity.
Relativelv low densitv -Reduction of density of MMCs depends on the kinds of matrix and reinforcement. For instance, nickel-based MMCs are used in gas turbines to reduce the density of the material. However, in many cases of interest, the addition of the reinforcement raises the density slightly, such as A 1 based SiC or AI2O3 reinforced MMCs. But the increase is usually more than offset by the enhancement of stiffness, strength, etc.
Isotropic -In CR-MMCs, the well orientated long fibres result in better axial properties than transverse properties. Compared with the long fibres, the discontinuous reinforcements in DR-MMCs are not well orientated, and consequently lead to better transverse properties than those in CR-MMCs. To some extent, they can be treated as isotropic. As a result, in material preparation, the separating direction is not as strict as that in CR-MMCs [13].
Low cost -DR-MMCs can be produced by standard techniques which considerably reduce the manufacturing cost and provide a good balance between price and mechanical properties.
In general, in progressing from continuous to whisker to particulate reinforcement there is a significant decrease in the reinforcement cost: the typical costs are $ 1 ,0 0 0 /kg for continuous fibres, $20-40/kg for whiskers and $ 10/kg for particulates [13].
Thermal expansion control -The low thermal expansivity of ceramics can be used to tailor the composite expansivity to match that of different materials [1], as shown in Fig. 2.1. This may be extremely useful for components where distortions must be kept to a minimum when exposed to relatively large changes in temperature. Volume % S1C_ in aluminium With all the above advantages, it is no wonder that DR-MMCs have the following applications.

Application Cases
• D iesel engine p isto n [1] -A l allo y /5% AI2O3 short fibre In 1983 in Toyota Motor Co./Art. Métal Manufact. Co., a 5% AI2O3 short fibre insert was used in the piston ring area in order to prevent seizure of the piston ring with the top ring groove and bore ( Fig.2.2). The weight was reduced by 5-10%. In standard tests, wear was reduced by over four times and seizure stress doubled relative to the unreinforced AI alloy.
This was combined with high thermal stability and conductivity. Production of these pistons in Japan has been increasing steadily over the past several years and now runs to millions of units annually.  Al-/20% AlzOgp drive shaft [1].
• Equipment rack [ 1 ] -A1 alloy/25% SiC particulate The 6061 Al-/25% SiCp composite supplied by DWA Composite Specialties Inc. was chosen to produce aircraft racks for its high stiffness, low density and good electrical conductivity   Most powder metallurgy-based processes involve the following steps: 1) mixing and blending of pre-alloyed metallic powder and reinforcements; 2 ) heating and degassing of the mixture; 3) consolidation of the mixture to a DR-MMC. In general, powder metallurgy requires the availability of powders of the matrix material which can place restrictions on this process. As a result, the powder-based process tends to be more expensive than the liquid-based one, and therefore generally occupies the more specialist high cost markets for DR-MMCs. Against this, several advantages are offered compared to liquid state processing [13]: • The mechanical properties obtained in the final DR-MMCs are often superior, • Shrinkage and other defects associated with solidification are avoided, • A more uniform reinforcement distribution is generally obtained, • Matrix property improvements can be obtained as a result of the deformation processing involved in solid state routes.
Liquid state routes are seen as cheap, simple and effective ways to the production of DR-MMCs, for they involve incorporation of the reinforcement into the liquid metal prior to casting to near net shape or semi-finished product for subsequent hot working. However, many of them failed for following reasons [13]: • Lack of wetting of reinforcements by liquid metals, • Porosity development in final product, especially where high volume fractions of reinforcement are required. 1 2 • Poor bonding between the matrix and reinforcement, • Excessive reinforcement/matrix reaction and reinforcement degradation.
Squeeze casting and infiltration, melt stirring methods and compocasting [13] are practical ways to overcome these shortcomings.
In spray methods, a mixture of liquid metal droplets and filler particles which are sprayed into a protective gas atmosphere and then impacted onto a substrate. Its advantages include [13]: • Avoidance of the need for the blending and degassing of powders involved in powder routes, • Rapid solidification conditions can be established permitting the production of fine grain size and metastable phase formation in the metal matrix, • Additional strengthening of the matrix, • Rapid processing and solidification tend to minimize reactions between the matrix and the reinforcement components.
However, in this process, there is a tendency for some porosity to be generated in the product.
The cast or consolidated billet made by the above methods is usually subjected to secondary processes (extrusion, forging, rolling) to break up reinforcement agglomerates, homogenize their distribution, eliminate porosity, improve metal-ceramic bonding, or simply to achieve the desired final shape.

Microstructural Changes Induced by Reinforcements
In DR-MMCs, the average size of the reinforcing ceramic particles (normally AI2O3 or SiC) is in the range of 3 to 30 pm and they are approximately equiaxed (a typical aspect ratio is 2), with irregular shape and sharp comers, although spherical reinforcements are also found.
Most of the matrix is very close to the ceramic reinforcements, whose thermal and mechanical properties are different from those of the matrix. As a result, the microstructural development of the matrix during primary and secondary processing is significantly modified by the dispersion of the ceramic reinforcements, and this influences the mechanisms of cyclic deformation and the overall fatigue resistance of DR-MMCs. The most notable changes induced by the reinforcements are briefly described below [14].

Load Bearing Effect
In the context of MMCs, one of the objectives might be to combine the excellent ductility and formability of the matrix with the stiffness and load-bearing capacity of the reinforcement. Central to an understanding of the mechanical behavior of a composite is the concept of load bearing effect between the matrix and the reinforcing phase. At equilibrium, if the bonding strength is strong enough, the external load must equal the sum of volume-average loads home by the matrix and the reinforcement. Thus, for simple two-constituent DR-MMCs under a given applied load, a certain proportion of that load will be carried by the reinforcement and the remainder by the matrix. The reinforcement may be regarded as acting efficiently if it earries a relatively high proportion of the externally applied load. The load bearing effect represents an important characteristic of the composite and depends on the volume fraction, shape and orientation of the reinforcement and the elastic properties of both constituents. 14 The most widely used model describing the effect of load bearing effect is the so-called shear lag model, originally proposed by Cox [14] and subsequently developed by others [14].
This model is based on the consideration of the transfer of tensile stress from the matrix to the reinforcement by means of interfacial shear stresses, and the calculation is based on the radial variation of shear stress in the matrix and at the interface. However, the main drawback of this model lies in the neglect of load transfer across the fibre end which is unacceptable in DR-MMCs. Several attempts have been made to introduce corrections, and the modified shear lag model is one of them and will be discussed later.

Enhaneed Disloeation Density
As the manufacturing of DR-MMCs is always carried out at high temperatures and the thermal expansion eoefficient of the ceramie reinforcements is lower than that of the metal, this mismatch will induce thermal residual strains (tensile in the metal and compressive in the ceramic) upon cooling from the processing temperature. It has been shown that the large residual strains generated in the matrix lead to stress relaxation by plastic deformation [14].
As a result, disloeation nucléation was especially intense around the ends of whiskers or at the angular comers of particles, where the computed residual stresses were maximum.
Depending on the deformation characteristics and flow stress of the matrix as well as on the presence of precipitates and inclusions, the dislocation structure near the reinforcement may vary very much from one material to another.
The average dislocation densities in the composite matrix were approximately one order of magnitude higher than those in the unreinforced counterpart under the same thermo-mechanical treatment. Various theoretical models were developed to determine the dislocation density as a function of the distance to the interface and there was a broad agreement that the dislocations were able to move over distances of the order of the particle 15 or whisker diameter, and this may be increased by a factor of 2-3 around the ends of whiskers [14]. The enhanced dislocation density model is one of these theories and will be presented in Chapter four.

Grain Structure and Texture
Cold deformation of the composite induces the generation of dislocations near the interface to accommodate the strain incompatibility between the reinforcement and the matrix. The plastic relaxation, as mentioned above, results in a region (often called the deformation zone) close to the reinforcement. The amount of matrix affected by the deformed zone near the reinforcements was proportional to their volume fraction, and this significantly weakens the grain texture induced by cold deformation in commercial DR-MMCs, as compared to the unreinforced counterparts.
It has been shown that ceramic reinforcements with diameters over 1 pm may stimulate the formation of recrystallization nuclei upon annealing in the deformed region formed during cold or hot deformation. As the composite reinforcements are usually much larger than 1 pm, particle stimulated nucléation is dominant and large particles often nucleate more than one grain. After recrystallization, commercial DR-MMCs exhibit an equiaxed grain structure with an average grain size in the range of a few microns to a few tens of microns, significantly smaller than that of the unreinforced materials processed under the same nominal conditions [14].

Effects of Microstructural Variables on Fatigue Behavior
The mechanical behavior of DR-MMCs can be quite sensitive to the microstructural details, such as reinforcement shape, size, distribution, volume fraction, property mismatch, aging 16 condition, bonding strength, whisker orientation, etc. The sensitivity covers the reinforcement, the interface, and the matrix. A brief review is given below.

Reinforcement Shape
From the viewpoint of stress concentration, the rates of void initiation and growth are clearly lower for spheres than for angular particles, although sometimes not so dramatically.
Aligned whisker material (6061 Al-20% SiC*) [15,16,17] has been shown to offer a greater resistance to crack propagation when the crack grows transverse to the whisker direction.
Fatigue threshold {AKo) has been found to be the highest in the partieulate-reinforced material, but upon eorrecting for eraek closure, its effective fatigue threshold (AKth,ef) was lower than that for an axial crack in the whisker-reinforced material [1 ].

Reinforcement Size
Experimentally, the effect of particle size on particle cracking is well documented for the Al-SiC system [18,19,20]. In the situation with a large scatter in size, it is the largest particles which tend to fracture [21]. Consequently, in material containing coarse particles, particle fracture is commonplace, whereas for fine particulate, particle cracking is rare. In fatigue, threshold values have been shown to increase with increasing particle size in Al-SiCp composites [22]. This is attributed to the surface roughness-induced closure effect, which was found to be nearly three times greater for coarse particle reinforced systems compared with fine ones [23].
Particle size also had a significant influence on fatigue strength. In high cycle fatigue (HCF), an increased fatigue strength was observed with decreasing particle size [24,25,26]. Tokaji et al. [27] reported that the composite with 5 pm SiC particles showed a higher fatigue strength than the unreinforced alloy. The incorporation of 20 pm SiC particles led to an increase in fatigue strength at a high stress level, but the improvement diminished with decreasing stress level, and a slightly decreased fatigue strength was observed at low stress level, as compared with the unreinforced alloy. The composite with 60 pm SiC particles exhibited a considerable decrease in fatigue strength. On the other hand, in low cycle fatigue (LCF), different results were reported by different investigators, which will be discussed in chapter 4.

Volume Fraction of Reinforcement
In the high cycle fatigue of DR-MMCs, the presence of particulate reinforcements in the metal matrix increases the fatigue limit of the material, and an increasing reinforcement volume fraction makes the fatigue limit even higher. However, in LCF of DR-MMCs, because of the high stress or strain amplitude level, reinforcement cracking occurs very quickly and frequently during the cyclic loading. As a result, the LCF strength of DR-MMCs is inferior to the unreinforced counterparts and a higher volume fraction leads to a lower fatigue strength. This point will be discussed in the later part of this thesis.

Interface strength
The interface between the reinforcement and the matrix is an incoherent interface having a high energy level, and thus is a good site for precipitate nucléation and growth which can influence the bonding strength of DR-MMCs. Another factor which influences the bonding strength is the property mismatch. In most MMCs a stiff reinforcement is incorporated into a weaker, less stiff matrix. The mismatch in properties is thus an essential feature of the composite system. In an attempt to see how closely the interfacial strength was related to fatigue and crack growth in DR-MMCs, an experiment was performed in which SiC particles with naturally oxidized and heavily oxidized interfaces [28]  In terms of fatigue crack growth behavior in DR-MMCs, stronger interfaces and higher void nucléation strains are likely to be beneficial. However, interfacial strength estimates in Al/SiC systems [29,30] suggest that the interfacial strength lies in the range of 1650-3000 MPa. SiC particle fracture is a function of particle size and is also likely to fall around 3000 MPa. Therefore, for the highest strength interfaces with increasing particle size, interfacial firacture will be replaced by particle fracture as the void nucléation mechanism [1 ].

Reinforcement Distribution
Experimental evidence indicates the preferential nucléation of cracks in regions of locally high volume fraction of particles. During crack growth, clustered regions can behave in one of two distinct ways. If the particles move independently of one another under the stresses at the crack tip, then the constraint gives rise to very large plastic strains and the crack displays a preference for passing through clustered regions [31,32]. This is encouraged by incompletely infiltrated regions, and voids are often present within such clusters. However, if the cluster moves collectively, then it behaves as a single large particle, the region between the particles is less strained and the cluster may deflect the crack.

2.4 Basic Fatigue Theories
Fatigue is a process which causes premature failure or damage of components subjected to repeated or cyclic loading. Fatigue failure is a common mode of failure observed in various mechanical components and structures. Proper prediction of life of these components/structures is very important, and any underestimation can cause catastrophic failure. To accurately describe and model the fatigue process on the microscopic level involves a complicated metallurgical process. Despite this complexity, fatigue analysis methods have been developed, which will be briefly described below.

Fatigue Process
It has been generally accepted that the fatigue process can be divided into four phases: (1) Crack initiation; (2)  and/or softening, respectively. After cyclic hardening/softening is completed, the cycled material exhibits a steady-state behavior which is known as a saturation state of the material.
In this state, stress and strain amplitudes have reached their saturated values, and there are no further changes in the hysteresis loop shape and area. If a series of either stress-or strain-controlled tests is performed at different amplitudes, then a set of stable, saturated hysteresis loop shape and area will be reached.
It is necessary to stress that there is no well defined border between the stages. For example, there is no well defined crack length to separate fatigue nucléation and propagation. It is normally dependent on the techniques to detect the crack size. However, fatigue analysis 2 1 methods which involve both crack initiation and propagation have been developed.

High Cycle Fatigue and Low Cycle Fatigue
Two different regimes have been identified in fatigue behavior, namely high cycle fatigue (HCF) and low cycle fatigue (LCF). As the resistance to fatigue depends essentially on a number of factors, such as stress concentration, surface roughness, frequency of loading, loading history, residual stress-strain fields, temperature, environmental condition, etc., the dividing line between HCF and LCF changes with the material being considered, but usually falls between 1 0 "^ and 1 0^ cycles.
In HCF, the strains are predominantly elastic and the maximum applied stresses are well below the tensile and compressive yield stress. Therefore, plastic deformation develops slowly in the material to nucleate a crack which propagates until failure, and most of the fatigue life in this regime is spent in the eraek initiation. The fatigue limit, Gc defined as the maximum stress amplitude which may be sustained infinitely without causing failure (see where is the fatigue-ductility coeflScient given by the extrapolation of ttie Coffin-Manson [14] curve to the first half cycle (2iV/=l) and c is the fatigue ductility exponent, which is in the range of -0 . 5 to -0 . 7 for most metals.  that the response of the material at the crack tip is due to the very high stress concentration of a plastic type. It is the cyclic plastic strain, rather than the total or elastic cyclic strain, that controls the crack behavior. Nevertheless, the elastic solution has produced many important results, which also hold to a good approximation for real fatigue cracks. the potential energy is [35] n = const - Let -d nri da, then the critical stress for crack propagation Gc can be expressed Irwin suggested an elementary approach to include the plastic deformation in the concept of linear elastic fracture mechanics. A size of plastic zone was added to the actual crack, and the modified stress intensity factor was introduced as, The crack driving force can be approximately expressed as [35],

(2.9)
Such an approach is valid only when the correction for the plastic effect is comparatively small. In principle, nonlinear fracture mechanics ought to be based on the theory of plasticity in which fatigue is considered as cyclic plastic deformation taking into account hysteretic phenomena, strain hardening and/or softening.

27
A simplified approach is the model of a thin plastic zone suggested by Leonov with Panasyuk and Dugdale [35], see Fig.2.12. In this model, all the plastic effects are concentrated in the plastic zone Dc, while the basic part of the material remains elastic. A crack begins to grow when the crack tip opening displacement ô reaches a critical magnitude, 6c, which is treated as a fracture toughness parameter given by formula: where is the yield strength of the material.
Linear and nonlinear fracture mechanics may be connected tlirough the use of the so-called path-independent integral technique. J-integral is one of them. It is concluded that the J-integral taken along any unclosed contour between crack surfaces is in fact path independent. Thus, even though considerable plasticity may occur in the vicinity of the crack tip, any path sufficiently far from the crack tip can be selected to be conveniently analyzed. where C and m are constants which can be determined for a given material and environment.
Hence, a plot of crack growth rate against AKi, in log scales, should give a straight line in the Paris regime, with a gradient equal to m, see Fig.2.13. At low stress intensities, there exists a threshold, AKth, below which no crack growth occur, while the crack growth rate usually accelerates as the level for fast fracture, K[c, is approached. The threshold region was identified when the average crack increment per cycle was equivalent to or smaller than one atomic lattice spacing [14]. The crack growth rate in this regime is very sensitive to the microstructure, environment, and stress ratio.

Fatigue of DR-MMCs
Fatigue failure of metallic materials is induced by the nucléation of one or several 31 microcracks which propagate slowly during cyclic loading until one of them reaches the critical size and catastrophic failure occurs. In DR-MMCs, the changes in the composite response upon cyclic loading are primarily induced by the plastic deformation of the matrix.
It was found that the damage was initially generated randomly by reinforcement fracture, decohesion at the matrix/reinforcement interface and the formation of matrix cracks. This homogeneous damage condition continued until a dominant microcrack was nucleated and damage was then rapidly localized around the propagating crack.

HCF and LCF in DR-MMCs
The division of HCF and LCF in metallic materials is also valid for DR-MMCs. It is well known that the nucléation phase takes up to 90% of the fatigue life in the HCF regime and, on the contrary, up to 80% of fatigue life is spent on fatigue crack propagation in the LCF region.
In the LCF of DR-MMCs, the composite fatigue life is normally inferior to that of the matrix if they are compared in terms of the cyclic strain amplitude (either total or plastic). However, this result is based on the fact that the unreinforced matrix carries a significantly lower stress at equivalent strain levels owing to its lower stiffness and strain hardening capacity. In fact, opposite result would occur if they are compared in terms of the cyclic stress amplitude rather than the cyclic strain amplitude [14].
The overall HCF performance as well as the fatigue limit of DR-MMCs improved with increasing reinforcement volume fraction due to an increase in the number of cycles to initiate a crack. The benefits in the crack nucléation time have to be added to those of the lower crack propagation rates in the near-threshold region at low stress ratios, because the increases in stiffiiess and strength with reinforcement volume fraction reduced the crack 32 opening levels and promoted higher levels of fatigue crack closure. In the mean time, the ceramic reinforcements reduced the average inclusion size as well as the average stresses acting on the matrix and on the inclusions which increases the crack nucléation time.
Therefore, a higher fatigue limit was reported for the composites [14].
It should be noted that if the composite is correctly processed, the fatigue limit and the fatigue strength can be improved. Porosity, particle clusters and other microstructural defects On the other hand, several typical sites for crack nucléation in HCF were observed [36]. The first type was imperfections associated with cast materials, such as porosity and large reinforcement clusters formed by particle segregation during solidification. The second group of nucléation sites was related to the reinforcing particles. In cast composites, cracks were initiated by decohesion at the interface in materials reinforced with particles, whiskers, and short fibers. On the contrary, fatigue crack initiation at the reinforcements in powder-metallurgy composites was normally due to particle fracture. These broken particles 33 were well above the average particle size or were found in particle clusters with a high local volume fraction of reinforcement. Thirdly, a large lumber of fatigue cracks nucleated at large intermetallic inclusions in the matrix in all materials.

Crack Propagation Characteristics of DR-MMCs
The relationship between the crack growth rate da/dN and the variation in the stress intensity factor AK (Fig.2.12), which is the theoretical framework to analyze the kinetics of crack propagation in metals, can be extrapolated to DR-MMCs. However, the interaction between the fatigue crack and the ceramic reinforcements during fatigue crack propagation increases the AKth and the exponent m, while the fracture toughness of the composite was reduced, see   (2) and (3)). In the DR-MMCs, the preferred locations for the fatigue crack initiation are broken reinforcement particles or interfaces between the particles and matrix [5,8,14]. It would be reasonable to regard the formation of a dominant microcrack as the crack initiation, and then fatigue damage propagates along this crack until the catastrophic failure occurs. The relative proportion of the life in the crack initiation and propagation depends on the loading conditions. For the HCF, the crack initiation takes up about 90% of the total fatigue life, while in the LCF it accounts for less than 20% because of the high stress level.
Ding et al. [2] proposed a LCF life prediction model based on the fatigue crack propagation process, but a LCF life prediction model for the DR-MMCs, which takes into account both the crack initiation and the crack propagation, has not been reported so far. Although the crack initiation would occur very quickly in the strain-controlled LCF, to make a more accurate prediction, it should not be ignored, especially for the relatively low stress/strain level. This part of the thesis is, therefore, aimed at developing an analytical LCF crack initiation model, which is based on Gibbs free energy change, and consequently, predicting the whole fatigue life of DR-MMCs. Several factors on the crack initiation, such as, strain level, volume fraction, cyclic strain hardening exponent, cyclic strength coefficient, etc., will be discussed. The results obtained from the crack initiation model in conjunction with Ding et al.'s crack propagation model [2 ], which forms the total fatigue life prediction, will be compared with the experimental data reported in the literature.

Ding et al.'s Model for Fatigue Life Prediction in DR-MMCs
A LCF life prediction model for DR-MMCs was presented by Ding et al. in 2002 [2]. In their model, the LCF behavior of particulate-reinforced MMCs is treated as a localized damage development phenomenon activated by the applied cyclic loading. The localized cyclic stress and strain concentration and fatigue damage evolution of microstructural elements within the fatigue damaged zone ahead of the crack tip are considered to dominate the whole low-cycle fatigue processes. In high-strain low-cycle fatigue conditions, the fatigue-damaged zone is described as the region in which the local cyclic stress level approaches the ultimate tensile 36 strength of the composite and within which the actual degradation of the composite material takes place. The fatigue crack growth rate is directly correlated to the range of the crack tip opening displacement. The following is a brief introduction of this model [2], which will form the basis of our model.
It is well established that the reinforcement of hard particles in a soft metallic matrix produces composites with a higher yield strength compared to that of the matrix, and the strengthening effect can be expressed as It is well known that there is a cyclic plastic zone or fatigue damaged zone (FDZ) in front of the crack tip (see Fig.3

Gibbs Free Energy Change and Crack Initiation
Bhat and Fine [38] reported a fatigue crack nucléation model for iron and high strength low alloy steel, according to Gibbs free energy change. In thermodynamics, the Gibbs free energy is a state function By assuming that in the low strain cycling, cracks often nucleate in the persistent slip bands (PSBs) and the shape of the new created crack is approximately that of a half of a penny with radius, a*, an equation proposed to predict the number of cycles (A^,) required to create a fatigue crack was developed [38]; where TV , is the number of cycles to create a fatigue crack, E is Young's modulus, y is the surface energy density, o is the peak stress or the stress amplitude within a cycle, v is Poisson's ratio, Ea and are the stress and plastic strain ranges in the cycle, / is the 41 ! I defect-energy-absorbing efficiency factor, t is the total thickness of the regions on each side of the crack drained of defects when the crack forms, f t may be regarded as an indicator of the energy that is stored from cycle to cycle, and a* is the critical crack radius [38].
Experimental data from commercial purity iron and a high strength low alloy steel were used to verify the model. A reasonable agreement with the experimental data was reported.
However, because the fatigue mechanism of DR-MMCs is different firom that of pure iron or steel, this model can not be directly applied to DR-MMCs. A new model has to be developed based on the presented model.

A New Life Prediction Model for DR-MMCs
The above model is basically applicable for the crack initiation occurred at PSBs. In the LCF of DR-MMCs the locations of crack initiation are not normally at the PSBs, as reported by several researchers. Llorca [14] pointed out that the stiff ceramic reinforcements acted as preferential sites for the crack nucléation, and were ultimately responsible for the poor LCF properties of the DR-MMCs. Fractographic studies by Bosi et al. [8 ] showed that the crack initiation occurred at the specimen surface and was associated with the microstructural defects such as clusters of particles, coarse particles or other inclusions. Levin and Karlsson [5] reported that cracks initiated either near the particle-matrix interfaces in regions of high volume fraction of particles or by the fracturing of individual particles. However, it would be reasonable to consider that the crack initiation process of DR-MMCs is similar to that of purity iron and high strength low alloy steel based upon the energy point of view, irrespective of the crack initiation site or the type of materials. That is, the formation of a crack leads to the reduction of elastic strain energy and the generation of surface energy. In the case of particle/matrix decohesion in the DR-MMCs, two different kinds of surface energy for the matrix surface and particle surface should be taken into consideration.

42
Suppose that the first crack occurs at the interface between the matrix and the particles, then the total change in the surface energy per unit area of the crack consists of two parts: (1) the energy increased by the new generated particle surface and matrix surface ); (2 ) the energy decreased by the release of adhesion work (-Wad) between the particle and the matrix. Dhindaw used Firifalco-Good relationship (Adamson 1982) [39] to calculate Wad as: where ysp and ysm are the surface energy density of the reinforced particle and matrix, respectively, and Wad is the adhesion work between them. The coefficient 0 has a value of 0.24 [39]. By assuming a perfect contact between the matrix and particle, the total change in the surface energy per unit area of a crack can be expressed as:

r^rsp+rsn,-^■^^^|rsprsn, ■ (3-H)
If the crack initiation occurs in the matrix, ysp in Eq.(3.11) could be replaced by ysm, resulting in: On the other hand, if the crack initiation mechanism is associated with the particle fracture, ysm in Eq.(3.11) could be replaced by ysp, giving rise to: According to references [40,41], ysp and ysm could be assumed to be approximately 1 Nm .
Therefore, Eq. (3.11) will be used in the following derivation.
In the LCF, the peak stress or the stress amplitude, a, can be expressed as [2]: where Aey2 is the externally applied/measured plastic strain amplitude. Han et al. [42] considered in the LCF of DR-MMCs, due to the high constraint effect of the reinforcement on the matrix, the external strain measured in the composites is lower than the "real" or "actual" strain in the matrix near the reinforcement. As the crack initiation is a localized phenomenon, the real plastic strain around the crack is apparently higher than the external measured one. However, the localized real plastic strain is a complicated parameter which is influenced by a number of micro structural parameters, such as the size, shape, volume fraction and clustering of particles. As a first approximation, it may be reasonable to use the "average real plastic strain in matrix" to replace the "localized real plastic strain". The "average real plastic strain in matrix", s,.p, proposed by Han et al. [42], can be expressed as follows, (3.13) where Vf\s the volume fraction of the reinforcement particles. Substituting Eqs. (3.14) As mentioned above, jsp and jsm could be assumed to be 1 Nm \ and a conservative value of 250 nm is used for a*, which was detected by Bhat and Fine [38]. Given a group of data of 44 the plastic strain amplitude Afip/2 and the related critical number of cycles to initiate a fatigue crack Ni, together with the material parameters Ec, n\ K', Vf and Vc, an average value ÎOX f t -the indicator of the energy that is stored from cycle to cycle -can be obtained. Then Eq. (3.14) can be used to further predict the fatigue crack initiation life Ni, in conjunction with the effects of a number of factors.

Effect of the Plastic Strain Level and Volume Fraction
The volume fraction of the reinforcement particles can affect various properties of  Also, the effect of the plastic strain amplitude level, Ae/2, on the LCF crack initiation life can be seen from this figure. If a combined effect of both K' and «' (see the values given in Table 3.1) is considered, as demonstrated in Fig.3.8, the monotonie decreasing trend with increasing K' value disappears. It is replaced by the occurrence of a maximum A, value at K' = 1839 MPa. That is, the LCF crack initiation life first increases and then decreases with increasing cyclic strength coefficient, K', for a given plastic strain amplitude, which will be discussed later.   In the following calculations, Q =0.2,5 = 1 .7 , u/=2 mm [1], a,=2a*-2x250 nm -500 nm [38], and the values of V/^ Vm, n' and K' for different composite materials are listed in Table 3 It should be noted that the effect of particle size on the fatigue life was not directly reflected in the above model. As described in Chapter two, the particle size is an important parameter related to the fatigue property of DR-MMCs. It is thus necessary to further model the effect of the particle size on the fatigue life.      parameters, such as the volume fraction, size, shape and distribution of the reinforcing particles, bonding force between the matrix and particles and age condition, etc. For the effect of particle size, Chawla et al. [7] studied an AA2080 reinforced with 10, 20, 30% SiC particles and showed that fine particles benefited fatigue resistance within 1 0 ' cycles, and Bosi et al. [8 ] reported a similar result on AAôOôl/ALOgp composites. This was explained by the fact that coarse particles had a lower strength and also brought more microcracks to the composites due to more self-defects compared to finer ones. On the other hand, Han et al. [9,10] reported that coarse particle reinforced composites exhibited slightly superior LCF resistance based on their studies on SiCp/Al composites with a constant reinforcement volume fraction but different particle sizes at ambient and elevated temperatures. This was considered to be due to a larger number of microcracks arising from more particles in the fine particle reinforced composites. Furthermore, a recent study on AA2124/SiCp composites with different particle sizes by Uygur et al. to the enhanced strength of the matrix due to the increase in the dislocation density in the matrix. While the strengthening effect in the former case was reported to be constant [45], the enhanced dislocation density effect was revealed to be associated with the relaxation of the thermal residual stress created during manufacturing and the diameter of the particles [1 ].
On the basis of these two models, the yield strength of the composite may be expressed as [45], or,. = /,)(! + /;),

04.2)
where fd andfi are the improvement factors associated with the dislocation strengthening of the matrix and the load bearing effeet of the reinforcement respectively. According to Ramakrishnan [45], for the particulate-reinforced composites,// = 0 .5 0 .was considered to be a direct consequence of the increase in the dislocation density due to the residual plastic strain caused by the difference in the coefficient of thermal expansion (ACTE) between the reinforcement particles and the matrix during the post-fabrication cooling. It was also reported by Ramakrishnan [45], say, A£p/2=4.66xl0'^, but this effect becomes weaker at the higher plastic strain amplitude level, especially for d>5 pm. This is mainly due to the breakage of particles during the cyclic deformation, which will be discussed later. constant (15 pm) and J^is changed firom 1% to 30%. An unusual but weak trend is observed: 6 1 as the volume fraction increases, the LCF life becomes slightly longer. The explanation for this will be given in Chapter five.     [2] and with the experimental data [2] in an AI2O3 particulate-reinforced AA6061 composite material (f/=20%, J=12.8pm) tested at -100°C.

An Alternative Method of Modeling the Particle Size Effect
According to the above analytical model, fine reinforcement particles had a beneficial effect on the LCF life of DR-MMCs. However, the effect was not so apparent, especially at high plastic strain amplitude levels. In the derivation of that model the modified shear lag effect factor was assumed to be a constant of 0.5 [45], which may not be correct according to the experimental data [2]. The experimental data given in [2] showed that the yield strength of In the MSL model, Eq.(4.2) can also be written as [45], As reported in [2] and also listed in Table 3.1, V/, n' m d K' are usually interrelated, leading to a combined effect of these parameters on the fatigue crack initiation. Fig.3.4 illustrates the combined effects of the volume fraction on the LCF crack initiation life, coupled with the variation of Ec, n\ K' and ft. Due to the limited data (  [2], the values off t can be calculated to change from 3.74x10'^ m to 5.91x10'^m.
It is seen from Fig.3.4 that after the unified effect is taken into consideration, the dependence of the LCF crack initiation life on the volume fraction becomes stronger. The larger the volume fraction and the higher the plastic strain amplitude, the lower the number of cycles for the fatigue crack initiation. This is in good agreement with the results reported in the literature. For example, Srivatsan [49] reported the degradation in the LCF resistance with an increase in the volume fraction of SiCp reinforcement in the 2124 matrix. Ding et al. [2] pointed out that in the strain-controlled LCF tests the MMCs with the higher volume fraction of reinforcement particles exhibit shorter fatigue lives than composites with a smaller volume fraction. Hadlanfard and Mai [50] showed that the higher the strain amplitude, the higher the rate of decay and the lower the fatigue life.
The monotonie effect of the cyclic strain hardening exponent, n', on the LCF crack initiation life (Fig.3.5) may be understood as follows. In the cyclic deformation testing at a given plastic strain amplitude, the cyclic strength coefficient K' and cyclic strain hardening exponent n are the decisive factors for the cyclic stress amplitude according to Eq. (3.11).
Since Aep/2<1 and 0.05<n'<0.3 [33], a larger «'corresponds to a lower saturated cyclic stress amplitude level, A<r/2, if K ' and Agp/2 are constant. In the LCF the cyclic deformation test is normally earned out under strain control. Therefore, an increasing «' would lead to a decreasing value of A(t/2, which would result in an increasing LCF crack initiation life. This is also in agreement with the statement that in the LCF "materials with larger values of «' have longer fatigue lives" [37].  (Table 3.1). As shown in Fig.3.8, for a given plastic strain amplitude level, the LCF crack initiation life increases with increasing K'. After reaching a peak value at K ' -1839 MPa, A,decreases with increasing K'. This is associated with the fact that the «' effect offsets the K ' effect on A. Hu and Cao [51] suggested a relationship between «'and K ' as follows.
«'=0.34 V A -0.05, where is the yield stress of the material. Clearly, an increasing K ' leads to an increasing «', which can also be seen from Table 3.1 at a given test temperature. Mathematically, the exponent «'has a stronger effect on Ea!2 than the coefficient K ' does according to Eq. (3.12).
These two points could be used to explain Figs.3.6 and 3.8.

f t -An Indicator of the Energy Stored from Cycle to Cycle
A parameter that deserves further analysis is ft, where / is the defect-energy-absorbing efficiency factor, t is the total thickness of the regions on each side of the crack drained of 70 defects when the crack forms. In the calculation of the crack initiation, f t is considered as a material constant which determines the percentage of the hysteresis loop area stored as the defect energy to be released when a crack initiates. This combined parameter may be influenced by the microstructural parameters and environmental conditions, such as volume fraction, reinforcement particle size, temperature, etc. As mentioned earlier, if a set of experimental data of Np versus Agp/2 is given, the value of ft can be calculated from Eq.  Table 3.1. It is seen that the value off t decreases with increasing volume fraction when Ec, n' and K' remain constant (Fig.5.1).
However, when the effect of the volume fraction on Ec, n' and K' is simultaneously considered, an opposite trend is observed (Fig.5.2). It means that the combination of Ec, n' and K' has a stronger effect on the value of f t than the volume fraction alone. Regarding the effect of test temperatures, the higher the temperature, the stronger the dependence of the value of f t on the volume fraction, as shown in Fig. 5.2. The reason for this is unclear, and further study is needed.

On the Crack Initiation Process
It was reported that damage was initially generated randomly throughout the specimens, and the predominant damage micromechanisms detected at this stage were the fracture of reinforcement particles, decohesion at the matrix/reinforcement interface, and the formation of matrix cracks. This situation of homogeneous damage continued until a predominant microcrack nucleated at the specimen surface, and damage was then rapidly localized around the propagating crack [14]. In this study, the formation of a predominant microcrack was considered as the crack initiation, in which the energy change produced by the decohesion at the matrix/reinforcement interface was taken into account when calculating the free energy change. As the surface energy arises from the unbalance of the force between  inner atoms or molecules and the interface, it is thus sensitive to the surface morphology, chemical composition and the environment. Consequently, the surface energy would vary with matrix and reinforcement particles. However, as a first approximation, the surface energy could be considered to be a constant of about 1 Nm'*, as reported in recent studies [38,40,41]. A quantitative analysis of the change in the surface energy density caused by the surface morphology, composition and environment is possible, but it would make the derived equation more complicated.
There was no clear definition of crack size for the crack initiation, which was normally determined via experiments, e.g., [2,38]. As suggested in [38], a crack initiation size of 500 nm (2x250 nm) was used in this investigation, which was limited by the ability to detect small cracks. As techniques of detecting cracks progress, the size of the crack initiation may be changed accordingly. As a first approximation, -15% of the total fatigue life might be considered as the crack initiation life. While the crack initiation life model was basically validated via the indirect total fatigue life data, the verification of the model using the experimental data involving the direct fatigue crack initiation would be ideal. Such crack initiation life data are thus important. Although the particle shape was specified as a short fiber with varying aspect ratios in Eq. (3.15), and other important parameters, e.g., the volume firaction, cyclic strain hardening exponent, cyclic strength coefficient, were reflected in the model as well, Eq. (3.14) or (3.17) did not include the effect of the particle size. To model the particle size dependence, a slightly modified consideration has been presented elsewhere [52].

Effect of the Particle Size on the Fatigue Life
The present size effect model is actually based upon the load-bearing effect and the reinforcement-particle-strengthening effect on the matrix. High stiffness and high strength are among the main reasons why MMCs are so attractive. However, just as every coin has two faces, those advantages may sometimes go the other way. As LCF tests are usually conducted under total or plastic strain control, to reach a certain amount of strain, a higher stress level is required in MMCs than in their un-reinforced counterparts due to their high stiffness and high strength. This could be the reason why MMCs have a LCF life inferior to the constituent matrix material. One of the strengthening mechanisms in MMCs is the load bearing feature of the hard reinforcement. In the LCF of DR-MMCs, more loads are shared by the reinforcement particles because of the required high stress. By assuming that the fracture of a reinforcement particle is stress controlled (maximum-stress principle) due to its elastic deformation, Srivatsan [49] reported that the "intrinsic strength" of each particle, Op, is inversely proportional to the square root of its characteristic dimension {d)\

(5.2)
where E is Young's modulus, v is Poisson's ratio, and Gpm is the critical strain energy release rate for the propagation of the microcrack initiated at a brittle reinforcement particle.
Consequently, smaller particles do not crack easily, but larger particles tend to break under lower stresses due to their relatively low "intrinsic strength". The broken particles act as the sources of microcracks in the matrix which will coalesce into macrocracks and lead to fatigue failure of the MMCs eventually. In addition, the larger particles are likely to contain larger inherent defects than the smaller ones. These defects would typically be sharp comers or re-entrant angles that act as potential stress concentrators in the ductile matrix [4 9 ].
Furthermore, short crack growth occupies a large proportion of fatigue lives of DR-MMCs.
Therefore, short crack trapping by the microstructure plays an important part in the resistance of the material to fatigue failure. Li & Ellyin [53] reported that the short crack trapping/untrapping is both particle size and volume faction dependent, and their 74 experimental observations indicated that a large particle dramatically reduces a nearby short crack growth rate, but the fractured particle releases the trapped short crack and leads to a considerable increase in its growth rate. Their FEM model also showed that for a given volume fraction, a fine particle composite provided a better resistance to the short crack growth. Since crack growth results from the damage caused by the reversible movement of dislocations in the cyclic plastic zone ahead of the crack tip, the barriers to the movement of dislocations play an important role in crack growth. In the DR-MMCs, the particles are strong bamers to the movement of dislocations, and the high multiaxial internal stress in the area surrounded by the particles impedes the stage 1 growth [53]. Therefore, the average spacing among particles appears to be a dominant material parameter for short crack growth in the DR-MMCs. For a constant reinforcement volume fraction, smaller particles imply more particles embedded in the matrix, i.e., more barriers to the dislocation movement, and consequently, to the short erack growth. Dieter [37] presented a relationship among the inter-particle spacing (/), volume fraction (V/J and particle diameter (dj: Eq.(5.3) indicates that a decreasing particle size d results in a smaller inter-particle spacing and consequently leads to more multiaxial-intemal-stressed areas, giving rise to a lower short crack growth.
All of the above points can be used to understand why larger particles result in a lower LCF life predicted on the basis of Eq.(4.5) or Eq.(4.9), especially at low plastic strain levels ( Fig.4.1 or Fig.4.7). Chawla et al. [7]  However, in LCF with increasing plastic strain amplitudes, not only large particles break during cyclic deformation, some smaller particles get broken or decohered, which can act as sources of cracks and voids. In view of the fact that there are more particles in the small particle reinforced MMCs, at a very large plastic strain amplitude, there would be more cracked or de-cohered particles in MMCs with smaller particles. This may offset some of the advantages of small particles mentioned above in resisting fatigue failure, giving rise to only a weak size effect at high plastic strain amplitude levels shown in Fig.4.1 or Fig.4.7.
However, how these factors affect the fatigue life quantitatively is not clear and further studies are needed.

Effect of Volume Fraction on the Fatigue Life
As shown in Fig.4.2, fatigue life increases slightly with increasing volume fraction of reinforcement particles when the volume fraction is treated as an only variable. However, most investigators [1,2,7,49] pointed out that an increasing volume fraction led to a decreasing LCF life. This difference can be related to the fact that the volume fraction has also an effect on other microstructural parameters, i.e., K \ n', Q, etc. When these effects are taken into consideration, it can be found that the lower the volume fraction, the longer the LCF life. Based on the available data given in [2], where the effect of volume fraction on E, K', n' is considered. Apparently, when the volume fraction changes from 15% to 2 0 %, fatigue life becomes shorter at all four levels of plastic strain amplitudes, in agreement with the findings reported in the literature [1 ,2 ,7 ,4 9 ].