a design-centered approach in developing al-si-based light

DOI 10.1007/s10820-005-3172-3Journal of Computer-Aided Materials Design (2004) 11: 139–161 © Springer 2005

A design-centered approach in developing Al-Si-based light-weightalloys with enhanced fatigue life and strength

JINGHONG FANa,b,∗ and SU HAOc

aMaterials Mechanics Research Center, ChongQing University, Chongqing 400044, ChinabSchool of Engineering, Alfred University, Alfred, NY 14802, USAcDepartment of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

Received 19 February 2004; Accepted 11 August 2004

Abstract. Material heterogeneities and discontinuities such as porosity, second phase particles, andother defects at meso/micro/nano scales, determine fatigue life, strength, and fracture behavior of alu-minum castings. In order to achieve better performance of these alloys, a design-centered computer-aided renovative approach is proposed. Here, the term “design-centered” is used to distinguish thenew approach from the traditional trial-and-error design approach by formulating a clear objective,offering a scientific foundation, and developing a computer-aided effective tool for the alloy develop-ment. A criterion for tailoring “child” microstructure, obtained by “parent” microstructure throughstatistical correlation, is proposed for the fatigue design at the initial stage. A dislocations pileupmodel has been developed. This dislocation model, combined with an optimization analysis, providesan analytical-based solution on a small scale for silicon particles and dendrite cells to enhance bothfatigue performance and strength for pore-controlled castings. It can also be used to further tailormicrostructures. In addition, a conceptual damage sensitivity map for fatigue life design is proposed.In this map there are critical pore sizes, above which fatigue life is controlled by pores; otherwise itis controlled by other mechanisms such as silicon particles and dendrite cells. In the latter case, theproposed criteria and the dislocation model are the foundations of a guideline in the design-centeredapproach to maximize both the fatigue life and strength of Al-Si-based light-weight alloy.

Keywords: damage sensitivity map, dislocations pileup, fatigue, microstructure, optimization criteria,strength

1. Introduction

Al-Si-based alloys are widely used in the automobile and aerospace industry due totheir outstanding mechanical properties, corrosion resistance, light weight, and cor-responding low energy consumption. The casting aluminum alloys, e.g. A356/A357Al alloy, are well-known for providing economic and near-net-shape parts for manyuses. However, as compared with wrought alloys, this class of casting alloy has poorstrength, ductility, and fatigue properties, which restrict its application because anespecially conservative safety factor is required for a cast aluminum alloy component.

The reasons for the poor mechanical properties are due to the manufacturing processand the resultant microstructure for this class of casting alloys. During manufacturing,aluminum and alloying additions are melted together. Grain refiner and precipitatorelements are added, which form particles in the melt which act as nucleation sites foraluminum grains during solidification. This leads to a finer grain structure.∗To whom correspondence should be addressed. E-mail: [email protected]

140 Jinghong Fan and Su Hao

Direct Chill (DC) cast is used widely to produce a solid ingot. DC casting is usedto give a fast cooling rate so as to obtain small grain size and uniform ingot struc-ture. Although the subsequent metallurgical processing such as homogenization, hotrolling, solution treatment and age hardening are designed for improving materialproperties, after the processing the alloying elements are still very unevenly distrib-uted. For A356/A357 Al alloy, their microstructure usually presents the pro-eutecticaluminum dendrites separated by a boundary eutectic region with several types ofinclusions and micro defects, such as segregated silicon particles and micro porosities.

In recent years the A356-T6 Al alloy has been intensively studied both experi-mentally and analytically [1–4]. The experimental results [1–4] reveal that both thesize of dendrite cells and segregated silicon particles have strong effects on the alloy’smechanical behavior. It is found that fatigue cracking usually commences along theboundary between large silicon particles and alloy matrix. The larger the dendritecell, the lower the fatigue life. On the other hand, the result presented in Refs. [5, 6]demonstrates that the smaller secondary dendrite arm spacing, or smaller size ofdendrite cell, may reduce the strength significantly [5]. This abnormal observation iscalled an inverse size effect for the cast aluminum alloy. More specifically, for A356Al alloy it is found experimentally that the yield strength of specimens with dendritesizes of 20–30 µm is remarkably lower than that of specimens with dendrite sizes of80–100 µm.

In order to make safe and affordable machine components, a primary task in thedesign of Al-Si-based alloys is to establish the optimized relationships among the fivefundamental elements. These elements are alloy selection, process design, microstruc-ture design, properties optimization, and performance [7–9]. The traditional designof Al-Si-based alloy is a highly empirical, time-consuming and expensive process.Often the results of such activities are poorly understood and suffer failure duringscale-up or in the field because they aren’t robust. The challenge in this work is tolay a primary foundation for developing a design-centered computer-aided renovativeapproach by combining numerical simulation, dislocation analysis and traditionalempirically based design for this class of alloys. Here, the term “design-centered”is used to distinguish the new approach from the traditional trial-and-error designapproach by formulating a clear objective, offering a scientific foundation and devel-oping a computer-aided effective tool for the new alloy development.

In order to gain both high strength and high fatigue life with an optimized bal-ance between these two properties, a design-centered approach is necessary. Here, therelationship among microstructures and properties are emphasized so that a guidelinecan be developed for microstructure design and consequently for the alloy selectionand process design. The key to a truly useful fatigue and strength design approachis the explicit presence of microstructural parameters in the approach. In the exist-ing fatigue and strength design approaches, these microstructure parameters are eitherignored or ambiguously taken into account through the dependence of fatigue andstrength constants on microstructure. This is of little use to engineers who must spec-ify material conditions such as sizes and distributions of the second phase, dendritecell, and large pores, etc. for alloy selection and process design.

The substantial difficulty in reaching this goal is how to get useful informationrelated to fatigue life and strength from complicated microstructures. The first chal-lenge is to develop an effective tool to get necessary relevant information that should

A design-centered approach 141

be helpful for the model development. The second challenge is to develop micro-structure-based criteria. This paper describes how we can face these challenges anddevelop a design-centered approach to reach this goal for Al-Si-based light-weightalloy. In the next section, a computational approach and an operational criterion fortailoring microstructure to reach high fatigue strength are proposed. In Section 3, afatigue damage sensitivity map (FDS) is proposed for fatigue optimization design.In Section 4, a dislocation model is proposed for optimizing both fatigue life andstrength. A summary for the design-centered approach is presented in Section 5.

2. Tailoring microstructure for high fatigue strength

Recently, there was essential progress in microstructure design of materials under cer-tain controlled conditions [9–12]. Specifically, various statistical correlation functionswere extracted from a 2-dimensional micrograph of an experimental “parent” micro-structure and these correlation functions are then used in a digital “microstructurereconstruction” method to generate “child” microstructures that are statistically sim-ilar to the parent structure. This procedure yields a family of microstructures that canbe analyzed to extract similarities and differences among these child microstructuresthat are identical to the chosen level of statistical correlation. Results show the insen-sitivity of elastic modulus and work hardening to high-order correlation functions,but quite sensitivity of the stress/strain concentration because of “the onset of local-ization around particular microstructural features in each child” [12]. This indicatesthat the “hot spots” at which high stresses or strains occur in different child micro-structure are quite different. One core issue related to this significant development isthat which one of these child microstructures is the best in fatigue strength. In thiswork, we propose a criterion for tailoring the child microstructure by computationalmicromechanics [3, 4] to reach high fatigue strength for fatigue crack initiation. Theproposed maximum critical stress (MCS) criterion can be described as follows:

Among the different children of microstructures of Al-Si-based alloys, the higherthe critical effective stress that a child microstructure has, the higher the lowerbound of fatigue strength. The child material with the highest lower bound is thegood candidate for fatigue-based microstructure design.

Here, the critical stress (σCR) as defined in Ref. [4] is the maximum asymptoticapplied effective stress at which the calculated microscopic plastic strain at the dis-continuities, i.e., pores and particles, is zero [4]. This criterion tells us that amongdifferent child microstructures we should choose the one that has a high value of themaximum stress at which no microscopic plastic strain is found at “hot spots”. Thiscritical stress for each child microstructure is calculated by microstructural finite ele-ment methods under the same cyclic loading conditions [4]. In the following, we willdescribe the basic principle of this criterion and how to use this criterion in practice.

The basic principle for the proposed criterion is a generally accepted understand-ing that fatigue crack initiates from local plasticity (or from dislocation pile-ups)due to stress concentration or microscopic defects. More accurately, if there is nolocal plasticity (or no dislocation pile-ups), a fatigue crack has a low probability of


Figure 1. Distribution of effective plastic strain, εpeff , within a realistic microstructure with intact par-

ticle/matrix interface for a cast A356-T6 Al alloy (after Fan et al. [3]).

occurring. In this section, we will describe this cracking mechanism from the viewpoint of local plasticity or microplastic strain; in Section 4 we will describe it fromthe view point of dislocation.

To make it more specific, we will take the microstructure shown in Figure 1to describe the idea of the criterion. This microstructure was obtained from a2-dimensional optical image analysis on a multi-axial fatigue specimen with lowporosity and then meshed for the plane strain finite element calculation. The micro-structural finite element technique developed in Refs. [3, 4] has been used in carryingout calculations of maximum effective plastic strain for this realistic microstructureunder cyclic applied stress. More details of the finite element modeling techniqueand boundary conditions can be found in Refs. [3, 4]. Figure 1 cited from Ref. [3]shows the distribution of effective plastic strain in particle clusters under completelyreversed remote cyclic tension-compression strain amplitude of 0.2% along the x-direction. Since particles were assumed as linear elastic, all particles domains appearas white, i. e., zero plastic strain. From this figure, we see that all particles wereintact. It was found that the local microplastic strain concentration occurred near theparticle tips (site A and B), micro-crack (site D), or between particles (site C). Themaximum local microplastic strain was found as high as 247 µε for applied, com-pletely reversed total strain amplitude of 0.2%. For this strain amplitude, the overallspecimen response was essentially in the elastic range, and the corresponding stressamplitude is approximately 133 MPa.

Furthermore, numerical simulations were conducted for applied remote strainamplitudes of 0.1, 0.125, 0.15, 0.25, and 0.3% for the same microstructure of Fig-ure 1. For each case, the corresponding maximum local microplastic strain was deter-mined. However, for an applied strain amplitude of 0.1% (stress amplitude of about74 MPa), no microplastic strain was found for this low porosity microstructure.The calculated results obtained for all these simulations are given in Figure 2. The

plot of maximum local effective plastic strain (εpeff =

√2/3ε

pij ε

pij ) versus applied stress

shows that when the applied strain amplitude decreased, the maximum local effective


Figure 2. Maximum local effective plastic strain as a function of applied stress from finite elementsimulations of a representative realistic microstructure of a cast A356-T6 alloy (after Fan et al. [3]).

Figure 3. Experimentally measured multi-axial fatigue life versus effective strain amplitude (after Fanet al. [3]).

microplastic strain decreased. By extrapolation, the local maximum microplasticstrain approached zero for applied stain amplitude of 0.122%.

This result was compared with the ε–N curve shown in Figure 3. This curve wasobtained by a multi-axial fatigue test. The test takes the fatigue specimen whosemicrostructure image was used for the finite element analysis. From Figures 2 and3, it merits note that the maximum predicted strain amplitude of 0.122% (or stress90 MPa), below which local yielding within the microstructure does not occur, is closein value of 0.125% (or the stress 92 MPa) which is determined experimentally for afatigue life of 107 cycles (see Figure 3). This stress value of 90 MPa is, therefore, theestimated fatigue strength at a fatigue life of 107 cycles. From this comparison, itappears that there is a critical stress amplitude σCR below which there is insufficientlocal plasticity within the microstructure to drive fatigue crack formation or to prop-agate pre-existing small crack-like defects within a lifetime of practical interest. Thiscriterion stress σCR can be easily determined by finite element analysis and there isno necessary to carry on any time-consuming fatigue testing. Therefore, this criterionis effective in the initial stage of tailoring microstructure of cast alloys for enhancedfatigue performance.


Figure 4. A schematic of geometry and boundary conditions for finite element analysis.

3. A fatigue damage sensitivity map for fatigue life design

One of the most harmful flaws in a cast alloy is a large pore. In Ref. [4], effects ofpores on crack incubation were examined using both realistic pore microstructuresand idealized geometries. Most pores were assumed to reside in the bulk with negli-gible interaction with the free surface. Finite element analysis has been performed forthe parametric study which included effects of size, spacing, local curvature and clus-tering of pores on fatigue crack incubation. Figure 4 is a schematic of the geometricarrangement of the clusters of largest pores for finite element calculation. This calcu-lation is for developing fatigue defect sensitivity (FDS) map. This map is designed toestablish the relationship between the size of largest pores, the nearest pore distance,stress amplitude and fatigue life.

The basic idea of the fatigue model for cast alloys is described in detail in Refs.[2, 4]. Here, we will use this basic idea for a fatigue model which can be used fordeveloping FDS map. It merits notice that a pore itself is not a crack but a notchwith a stress concentration at the blunted tip. This situation will be substantiallychanged if a tiny crack is formulated at the tip of the pore. To illustrate this situa-tion, suppose the pore maximum dimension, Dmax, is 300 µm and the cracks formu-lated at the two ends of pore are 5 µm in length. After these tiny cracks were formed,the pore together with the tip tiny cracks formulated a long crack with a length of310 µm. This indicates that the crack length from 5 to 310 µm was developed imme-diately after the tiny fatigue cracks were formulated. Under this mechanism, it isassumed that the stages of microscopically and physically small crack propagation [2,4] were passed as soon as the coalescence of the pores in the cluster with the largestpores was completed. In other words, the total fatigue life can be calculated by thesum of Ninc, Ncluster, and Nlong, i.e.,

NT =Ninc +Ncluster +Nlong, (1)

where the members of the right part with subscripts “inc”, “cluster” and “long” referto the loading cycles, respectively, spent on micro-crack incubation, crack coalescence


in the cluster with largest pores, and macro-crack growth until the critical length offatigue failure of the specimen.

Ninc can be calculated by the Coffin–Manson law as follows [3]:

Ninc =(

C ′

��

)1/α′

, (2)

where

�� =�γ Pmax

[1+k

σ maxn

σy

], (3)

where �γ P∗max and σ max

n were introduced in [13] and their values can be obtained fromfinite element computation [4].

Nlong is calculated by Paris Law as follows:

da

dN= cm(�k)n, (4)

where �k indicates the range of stress intensity factor which has the relationship withthe range of cyclic stress �σ and crack length, a, through a geometric factor α asfollows [14]:

�k =α�σ√

πa. (5)

Substituting (5) into (4) we have

da

dN= c′

m(�σ)nan/2, (6)

where c′m = 1.7369 × 10−11 m/cycle and n = 5.018 for A356-T6 Al alloy. These values

are obtained based on the fatigue testing data of R =0.1 from Westmoreland Testingand Research, Inc.

Ncluster is actually the cycle number spent on joining the pores with the nearestpore in a largest pore cluster. The fatigue crack propagation rate during this coales-cence process is proposed as follows:

da

dN=Cηβ(a)n1, (7)

where “a” is the crack length that equals a half of the pore size (Dmax/2) plus thelength of a short crack measured from the tip of the pore. In (7) η is the damagefactor proposed in Ref. [15] whose explicit expression is as follows:

η=∫ εp

0exp

(1.5σm

σe

)dεp, (8)

where σm denotes the mean stress, σe the deviatoric stress, and the ratio of σm/σe isthe stress triaxiality. η can be calculated by finite element analysis.

Based on the assumption that fatigue life is controlled by the cluster with largestpores, an exemplifying pore-controlled FDS map shown in Figure 5 was developedunder the idealized pore configuration shown in Figure 4. Where the cluster consistsof four pores which are assumed to be ellipses with Dmin/Dmax = 1/2; the spacing


Figure 5. An exemplifying pore-controlled FDS map which shows relationships between fatigue life,the largest pore size, stress amplitude, nearest pore spacing and critical pore size for R =−1.

between horizontal pores is B and that between vertical pores is L. The value of theratio B/L is one. It is assumed that the crack will be formed at the tip of the Dmax,then passing through 0.5 B to join the crack initiated from the other side of a hori-zontal pore. The pore spacing B is assumed to be 120, 280 and 720 µm. The calcu-lated results for different remote stress, pore spacing and pore size under symmetriccyclic loading is shown in Figure 5 of the FDS map by discreted points.


In this map, the fatigue life and largest pore size are expressed by the vertical andhorizontal axis. The curves of fatigue life versus the largest pore sizes are functions ofthe stress amplitude. This figure includes curves corresponding to fits of calculationresults (discrete points in the Figure 5) at strain (stress) amplitude levels of 0.15%(110 MPa) and 0.25% (165 MPa) for completely reversed (R=−1) loading, along withthe 0.2% amplitude level (133 MPa) reported in Ref. [3]. The other two curves atstress levels of 60 and 85 MPa are only schematics at this time; they are not obtainedthrough calculations. It is seen that the larger the applied stress amplitude the lowerthe fatigue life.

The most important things in the FDS map are the critical pore sizes atdifferent stress amplitudes which are shown at points of A, B and C. The criticalpore size defined by Couper et al. [16] is the size below which the fatigue life is notcontrolled by pores, but by other factors, such as debonding or fracture of siliconparticles. Thus, critical pore sizes are important for microstructure design because afurther reduction of pore sizes below these critical sizes is useless to increase fatiguelife. Critical pore sizes are related to a competition between different fatigue failuremechanisms. They are functions of the amplitude of applied stress because differentcompetition mechanisms have different kind of dependence on stress amplitude.

In this work, critical pore sizes are operationally defined based on cases where thefatigue life is no longer observed to be controlled by pore size but rather by siliconparticles and/or by oxide films, intermetallics, etc. These lives are determined by thefatigue cyclic numbers of the specimens that were not failure by pores. Specifically,the following data were used: 2 × 106 cycles for the strain amplitude level of 0.15%;1.7×105 for 0.2%; and 4.1×104 for 0.25%. Only the second case (0.2%) is verified byexperimental observation (see Appendix A and [17]). The lives roughly correspond tothe inflection point of the computed fatigue life versus largest pore size plot for thestrain amplitude of 0.2% and 0.25%. For 0.15%, the live of 2 × 106 corresponds tothe cycle number at the starting point, A, of the fast slop increase region.

The finding that the critical pore size is near the inflection point of the curveof fatigue life versus the largest pore size merits notice or the starting point offast slop increasing region. The increase of the slope in the plot to the left of theinflection point is physically associated with a significantly decreased fatique drivingforce which makes the calculated fatigue life increase fast when the pores towardssmaller. It must be emphasized that the calculations in obtaining the curves in Fig-ure 5 include only the influence of pores and do not incorporate the effects of anyother discrete microstructure features such as Si particles or triple points of den-drite cells. The great decrease of the pore driving force after pore size below theinflection point makes the relative importance of other factors. Accordingly it islikely that other life-limiting microstructure features such as Si particle debondingand propagation along intermetallics in the interdendritic regions take a role. Theywill dominate and produce lower fatigue lives than predicted by the pore analy-ses to the left of the inflection points or the starting point of fast slop increasingregion.

The following critical pore sizes are determined from the intersection points of A,B, and C, between horizontal lines at these specific fatigue lives of 2×106, 1.7×105

and 4.1×104, respectively, with the corresponding curves 0.15, 0.2, and 0.25% drawnthrough the mean of our calculations results. The critical pore sizes are 30 µm for


110 MPa, 100 µm for 135 MPa, and 190 µm for 165 MPa. This computed variationof the critical pore size is consistent with the fact that “various critical pore sizeshave been reported from the low values of Couper et al. to the far higher ones ofTing” [18]. Furthermore, these values are close to the corresponding pore sizes givenby the curve in Figure 7 of Ref. [16]. Specifically, their values of critical pore sizes are183 µm for 165 MPa, 107 µm for 133 MPa and 20 µm for 110 MPa. Hence, the cap-ture of the trend is very promising. It merits notice that if a imaging curve passingthrough points of A, B, C and other points denoting critical pore sizes then the FDSmap will be divided into two regions: the top right region in which the fatigue lifeis controlled by pores, and the left bottom region in which it is controlled by othermechanisms such as silicon particles.

It must be emphasized that as our treatment of microstructure-sensitivity analysisfor small fatigue crack initiation and propagation improves, so will the estimateof the FDS map. None-the-less, this example provides some limited experimen-tal validation of the concepts. It is interesting to note that the results are a bitcounter-intuitive. At first, we might speculate that increasing the stress amplitudeshould result in a lower critical pore size, based on an analogy to critical flaw sizein fracture mechanics. However, we must remember that Si particle fracture and deb-onding controls the fatigue crack formation and early growth process below somecritical pore size, leading to a change of mechanism at this bifurcation point. Thissituation differs from classical fracture mechanics of initial defects, and largely moti-vates our multi-scale modeling approach in the first place. Indeed, the FDS mapshows that for a given target fatigue life, the stress amplitude must be decreasedas the pore size is increased; however, the transition pore size is a different mattersince it represents a transition in micromechanical fatigue failure mode. This transi-tion depends on the competition between different mechanisms. More specifically, itdepends on the sensitivity of different mechanisms on stress amplitude.

4. A dislocation model for optimizing fatigue life and strength

In this section, microstructure design is discussed in the region of fatigue cracks incu-bation which is controlled by the impurties such as second phase particles, dendritecells, etc. The decohesion between inclusions and matrix results in nucleation of mi-crovoids, which can be considered as another type of pores. Hence, this mechanismcorresponds to the left-bottom region of an imaginary curve that connecting pointsof A, B, C of the critical pore sizes in Figure 5.

The analysis of Refs. [2–4] demonstrates that the incubation fatigue life is inverselyrelated to the second-phase hard particle size when the amplitude of applied stress islower than macroscopic yielding. Experimental results [1] also show that the largerthe dendrite cell size the lower the fatigue life. On the other hand, the studies inRef. [5] indicates that the stress–strain curves of A356 cast aluminum alloys exhibitan unusual size effect on flow properties: the finer the microstructure, the lower thetensile flow strength. Here, the size effect includes the sizes of dendrite cells and par-ticles [19]. Therefore, a practical issue is to find appropriate dendrite cell and parti-cle sizes which can provide the optimized combination of incubation fatigue life andflow strength. This is crucial for metallurgical process design because the sizes of thesilicon particles and dendrite cells in Al-Si-based casting alloys are controllable by


adding modifier elements such as sodium and calcium under sophisticated designedmelting and solidification process [20]. For this purpose, an analysis of dislocationspile-up around second-phase hard particles is performed in Ref. [21]. Here, we willbriefly introduce the model and its application to the optimization of both strengthand fatigue life for Al-Si-based alloys.

As mentioned in the previous section, the casting A356-T6 is a two-phase Al-Si-Mg alloy with about 7% (wt) silicon and other additions such as magnesium andcopper. In this alloy the majority of silicon elements precipitate as virtually pureparticles in an interdendrite region of Al-Si eutectic which is formed between den-drite cells of the aluminum solid solution containing the rest of 1% silicon. This alu-minum solid solution in dendrite cells is reinforced by the silicon particles and otherhard impurities such as intermetallics and oxides. For simplification, in the followinganalysis we apply the assumption introduced in Refs. [2–4] where the A356 Al alloyis simplified as a two-phase system: the aluminum-rich matrix and the silicon parti-cles. Also only static and quasi-static load conditions are taken into account. In thefollowing analysis, we will refer both silicon particles and dendrite cells with inter-dendrite region as inclusions. The hard interdendritic eutectic region with the siliconparticles surrounding a dendrite works as a hard “egg shell” which can pile disloca-tions up between that shell and the aluminum matrix.

We start at the approximation that a fatigue crack nucleation is the accumulationof dislocation pileups around hard inclusions driven by applied stress. These disloca-tion pileups cause the subsequent interfacial decohesion or inclusion break to prop-agate the crack along the interdendritic region. On the other hand, the strength iscontrolled by the Orowan mechanism. This is because an aluminum usually existsin face central cubic crystallography which is rich in slipping systems with relativelylow dislocation energies barrier. Therefore, the matrix is considered to be intrinsi-cally “ductile” and the alloy is essentially reinforced by the hard particles. Hence, inRef. [21] we introduce two competing mechanisms: particles-cutting and dislocationspileup-induced decohesion; the dominant one between them controls the processes ofyielding and subsequent micro-cracking initiation in this class of alloy, see Figure 6.

Then the failure of the alloys at ultimate flow can be characterized by

τ appl = τ intrinul . (9)

In this relation the left hand side, i.e., the applied stress τ appl is the driving force tofracture; while the right-hand side refers the material intrinsic resistance against fail-

Figure 6. Orowan’s mechanisms of dislocations–inclusion interaction and cutting model.


ure, which is expressed as the following summation:

τ intrinul = τobst + τdis + τdrag, (10)

where τobst is the resistance stress directly exerted by the obstacles; τdis denotes theresistance stress against dislocation motion in the “soft matrix” and the elastic self-interaction stress between dislocations. When dislocation distribution is known, theexact solution τdis can be obtained [22, 23]. τdrag is a viscous drag proportional tothe velocity of the motion of a dislocation line

τdrag = vdB/b, (11)

where vd is the amplitude of normal velocity of a dislocation line, b is the amplitudeof Burger’s vector, and B is a viscosity coefficient.

We consider static and quasi-static load, so among the three terms on the right-hand side of (10) τobst is the dominant term for an alloy that is reinforced by hardparticles. Hence, it is the focus in this analysis. When a slip system just becomes acti-vated, the corresponding resistance reaches a threshold against the dislocation glidingthat trends to move over a macroscopic distance. The difference between the thresh-old in a matrix with obstacles and that without obstacles is defined as τobst. For cast-ing aluminum alloys, the amplitude of τobst is mainly contributed by the interactionbetween silicon particles and dislocations. The analysis of [21] provides the quantita-tive computation of τobst for the above mentioned two competing mechanisms.

Remark. In Ref. [24] Cottrell defined the local balance of the stress field with distrib-uted dislocations as follows:

τ appl + τobst = τinert + τdis + τdrag, (12)

where the τobst has different sign as compared with (10) and the term τinert representsthe effect of inertia.

Figure 7. (a) Ham’s cutting model, (b) proposed dislocation climbing/pile-up model in Ref. [21].


When a dislocation front line advances approaching to an inclusion, the plasticstrain increases rapidly near the inclusion as illustrated in Ref. [4]. These high valuesof plastic strain are caused by dislocation pileups, which cause additional stresses. Asresult, the inclusions will be either cut through or separated from matrix through dec-ohesion. The former occurs when an inclusion under a high applied stress [19] or inthe case that the crystal structure of the particles is similar to the matrix, so it is alsotermed “order strengthening” model. Ham [25] has studied the “order strengthening”phenomenon for dual phase (γ −γ ′) Fe-Ni-based alloys and proposed an “order dis-location cuts” model as shown in Figure 7a. This model presumes that the secondphase particles with an average diameter r are uniformly distributed with a spacing L.When a dislocation line penetrates such a particle, the original crystal symmetry hasbeen destroyed and the dislocation plane forms a new locally symmetric system dueto the periodicity property of the crystal. Therefore the dislocation plane is termed“anti-phase boundary” and the corresponding energy dissipation, denoted as γAPB, istermed “anti-phase boundary energy”. According to Ham’s model, the obstacle stressagainst particle cuts is

τ cobst =

2rγAPB

bL, (13)

where r and L are particles average radius and spacing, respectively; γAPB is the “anti-phase boundary energy” with the dimension J/M2, and the superscript “c” refersto cutting model. Although silicon can be in fcc or bcc structure, at room temper-ature a pure silicon usually exists in a diamond crystallography. A particle cuttingoccurs only when the particle under high localized applied stress [16], hence, thismechanism becomes significant during elasto-plastic fracture. For the fatigue failureof A356-T6, as indicated in Ref. [4], the interfacial debonding between silicon par-ticle and aluminum matrix is the dominant mechanism causing micro-cracking. Thiscan be seen from Figure 8 referred from Ref. [17]. Fatigue crack growth along the

Figure 8. SEM microstructure characterization near crack initiation sites: (a) Exposed silicon particleswith extremely smooth and rounded surface indicating crack growth along the interface, (b) Extremelyfine matrix striations indicating that the crack growth rates in the vicinity of the debonded particlesare relatively small (after Gall et al. [10]).


interface between a particle and the matrix is inferred from the extremely smoothand rounded surface of the exposed silicon particles at high magnification; this isquite different from the fatigue crack propagation passing through dendrites wherefine fatigue striations are shown (Figure 8). To describe this class of phenomona, adislocation-induced decohesion model [21], as illustrated in Figure 7b, is proposed.In this model the interface between an inclusion and matrix is a curved shear sur-face in which a tangential separation can happen that allows the dislocation flowbypassing. Since the kinematically admissible path becomes curved, the additionalresistance causes localized dislocation pileups which cause interfacial decohesion. Anestimate of the corresponding obstacle stress is

τDobst =

8r�F

Lb, (14)

where �F denotes the interfacial adhesion energy between an inclusion and thematrix; the superscript “D” refers to the “dislocation-induced decohesion model”.(14) is obtained based on the Ham’s cutting model but replacing the inclusion cut-ting plane (πr2) by the sphere inclusion surface (4πr2).

In general, for an inclusion-reinforced alloy the obstacle shear resistance can beexpressed as the minimum of cutting resistance and debonding resistance, i.e.,

τobst =min{τH

obst, τDobst

}=min{

2rγAPB

Lb,

8r�F

Lb

}. (15)

When the average inclusion radius r and spacing L are known, first principle-basedcomputation is required for determining the accurate values of γAPB and �F .

For Al-Si-based alloys with intrinsic ductility, an inclusion-matrix interfacial dec-ohesion is essentially the process of interface separation induced by dislocationspileup/plastic deformation accumulation. This process transforms the work done byapplied stress to plastic dissipation and energy required to form new surfaces. Omit-ting the effect of temperature, �F can be expressed as a summation as following:

�F =γparticleF +γ matrix

F −�segE −�

pileupE −�misfit

E , (16)

where γparticleF and γ matrix

F are the new created surface energy of an inclusion and thematrix, respectively, per unit area; �

pileupE is the reduction of adhesive energy caused

by dislocations pileup; �misfitE is the energy reduction caused by lattice misfit and �

segE

is the energy deduction due to impurity elements interfacial segregation which is usu-ally strongly dependent upon environment. In this analysis, the effect of �

segE on the

Al–Si interface is not taken into account.When there is no dislocation pileup and the particle and matrix are perfect match (no

misfit), the summation of the first two terms on the right-hand side of (16) is the adhesiveenergy at a perfectly coherent interface. Table 1 lists the corresponding γAPB, γ

particleF and

γ matrixF for silicon particle in Al-rich matrix, obtained by first-principle-based computation

[21, 26–29] using the methodology, e. g. introduced in Ref. [30].Now we discuss the determination of �

pileupE . Dislocation pileups are considered

as the major cause of strengthening. This is because, when an inclusion does notwork as an obstacle against dislocations motion, the term τobst in (10) disappears soτ intrinul is merely determined by the “soft matrix”. When a pileup of dislocation occurs


around a hard inclusion, an increasing number of mobile dislocations stay on theequilibrium positions determined by the superposition of their self- and mutual-inter-action forces and external applied stress. These dislocation pileups are actually the“geometrical necessary dislocations” introduced in strain gradient theory, as analyzedin Refs. [31, 32] and calculated in Ref. [4]. The correspondent equilibrium state canbe characterized by the system energy that is computed as the summation of pileupdislocation energy [21]. When this energy accumulation reaches a critical level, it willdrive the dislocations away from the equilibrium position which causes debonding,as formulated by (10, 13–16) and illustrated in Figure 7b. The general theory of dis-locations and solution strategies can be found in [22, 23, 33]. Rice first proposedthe energy criterion of dislocation instability [34]. Applying the Peierle force solu-tion obtained in Ref. [23], a stress instability criterion has been proposed in Ref. [35].The solution of dislocation loop pileups in infinite uniform elastic plane has beenobtained by Leibfried [36, 37]. The general discussion of the pileups in bi-materialssystem can be found, e.g. in Ref. [38].

Based on the results of Leibfried [36, 37], a solution of circle dislocation loop pile-ups around a rigid inclusion, as shown in Figure 7b, has been developed in Ref. [21].For the system of Figure 7b, under a remote cyclic applied stress with the maximumamplitude τ appl the summation of the dislocation loop self-created strain energy andmutual interaction energy is

Epileup =N ·κ ·4πr4(

1− v2− v

)(τ appl

τMy

)3

γ My τ appl, (17)

where N is the load cycle number and N ≤ NINC, γ My and τM

y are the shear yieldstrain and yield stress of the matrix, respectively, κ is a constant in the order of 102.In (17) it is presuming that the pileup is not reversible; also the high order smallerquantities terms are omitted. It merits notice from (17) that the pileup stored energyincreases with increase of cyclic number N .

The model in Figure 7b and the Eqs. (10) and (13)–(17) introduce a material fail-ure model which can be described as follows: the applied load cycle piles disloca-tion loops up around inclusions, which causes the self-induced stresses and energyaccumulation. When the dislocations energy reaches a certain level, the correspond-ing self-induced stress will be high enough to drive debonding/decohesion thoughthe amplitude of the applied stress is controlled under very low level. According toRice’s dislocation instability criterion [34], a hypothesis has been made in Ref. [21]

Table 1. γAPB, γparticleF and γ matrix

F for Al and Si

γF (J/M2) γAPB(J/M2) Lattice constants Refrences(room temperature) (nm)

Si (100)DIM 1.52 1.83 0.545 [26, 27]Si (111)DIM 1.31 2.51 0.545 [26]Al (100)FCC 0.93 0.509 0.405 [21]Al (110)FCC 0.96 0.687 0.405 [21]Al (111)FCC 0.991 0.18 0.405 [26]


that the interfacial debonding between an inclusion and the matrix will take place ifEpileup/

(4πr2

), the energy caused by dislocations pileup per unit interfacial area, is

equal to the interfacial adhesion energy

Epileup/(4πr2)=γ

particleF +γ matrix

F . (18a)

which refers to the condition that the piled dislocation loops are released throughinterfacial decohesion when the condition of (18a) is satisfied. Substituting (17) intothe left part of (18a) we obtain the following expression for calculation of �

pileupE :

�pileupE =Epileup/(4πr2)=N ·κ · r2

(1− v2− v

)(τ appl

τMy

)3

γ My τ appl. (18b)

Now we discuss the term �misfitE in (16). According to Table 1 one sees that the

fcc Al matrix has different lattice constant as compared with the diamond-structuredSi particles. In a solid solution of an Al-Si-based alloy such as A356, the local cova-lent bonding may be established between the Al atoms and Si atoms at the interface,which causes a certain mount of residual stress around, as illustrated in Figure 9.Being independent of the applied stress field, this residual stress may trigger dislo-cation loops surrounding the silicon particle. Hence, the corresponding strain energycould reduce the interfacial adhesion. This is why a negative sign appears for �misfit

Ein (16).

Mott and Nabarro [39] propose using the lattice misfit factor kM

kM = 2 (aM −aP)

aM +aP. (19)

To characterize the misfit between a spherical particle and matrix; where aM and aP

are the lattice constant of the matrix and the particle, respectively. Using linear iso-tropic elastic theory, the misfit stress field has been solved for spherical particle whichdemonstrates a purely hydrostatic feature. The corresponding misfit energy Emisfit has

Figure 9. An illustration of the lattice misfit.


been obtained, e.g. in Ref. [40]

Emisfit = 8π (1+ v) r3

3 (1− v)f (µM,µP)

[εmp (kM)

]2(20)

and

�misfit = Emisfit

4πr2, (21)

where εmp is the volumetrical strain of the particle which is the function of kM, andf (µM,µP) is a function of the shear Young’s modules µM (matrix) and µP. In thederivation of (20) it is assumed that the matrix and particles have the same Poisson’sratio.

Since in A356-T6 the most Si particles are with cubic diamond structure, so in thisanalysis we leave out the term �misfit

E from (16) because εmp is small in such a harddiamond crystal. We will discuss this issue in details in Ref. [21].

By substituting (17–19) into (16), then into (15), we finally obtain

τDobst =

8Lb

{(γ

particleF +γ matrix

F

)· r − �̄

pileupE · r3

}, (22)

where

�̄pileupE =N ·κ

(1− v2− v

)(τ appl

τMy

)3

γ My τ appl. (23)

By substituting (22) into (15) and then into (9) and (10), we obtain relationsamong applied stress, load cycle, inclusion size, spacing and materials properties suchas those listed in Table 1 at the instance that the piled dislocation loops becomeunstable, which is corresponding to the incubation of fatigue cracking. The resultedequation demonstrates that the material intrinsic strength is proportional to the pro-duction of the interfacial adhesion energy and inclusion radius r minus the pileupenergy times r3, but inversely proportional to the inclusion space L. Thus, when amaterial has just suffered a small number of load cycle under a given stress ampli-tude of τ appl, the intrinsic strength is linearly proportional to the inclusion size r asthe term with cubic of the inclusion radius in (22) is small. This provides a quan-titative explanation of the result of Ref. [5] in which the inverse size effect to yieldstrength is reported. When the load cycle number becomes huge, the last term in (22)will become to dominate. So the intrinsic strength becomes smaller when r is largerdue to the negative sign of �̄

pileupE . This explains the phenomenon that the larger par-

ticle has less fatigue life [3]. Plotted in Figure 10 are the theoretical prediction of thefatigue crack incubation life (load cycles) against inclusion size with varying appliedload using (10, 13–22). As can be anticipated, the detail comparison between this pre-diction and experimental results show that the theoretically estimated fatigue life isabout 30–90% higher than test. This is reasonable because the defects like pores arenot taken into account in the dislocation model. According to the analysis of thedislocation pileups in Figure 7b, we have also obtained an estimate of the materialintrinsic resistance against failure τ intrin

ul from (10).


Figure 10. The computed fatigue life – particle size diagram using (22) under various applied load.

Figure 11. The computed relationship between matrix flow strength and silicon particle size.

Plotted in Figure 11 is the relationship of the estimated τ intrinul against hard parti-

cle radius at low cycle fatigue case (N < 100). This diagram indicates that the τ intrinul

is approximately proportional to the silicon particle size when the amplitude of theglobal applied stress is low. This is because a larger particle has stronger hardeningeffect. Under this situation the failure of the alloy is controlled by fatigue cracking.Increasing applied stress reduces the strengthening effect of large silicon particle. Fig-ure 12 are the results of experimental research conducted in Ref. [1]. In conventionalanalysis the safety factor of fatigue analysis is usually higher than 2. The predictionin Figure 10 shows the same trend as that in Figure 12. By comparing the numberin details, one can find the differences is about from 30 to 80%.

It merits notice that though the above discussion related to Figures 10 and 11 ismostly related to the particle size effect. It can be used for the size effect of den-drite cells. In fact, a dendrite cell is surrounded by a hard interdendritic region whichtakes a role against applied stress. This interdendritic eutectic region filled with sili-


Figure 12. Effect of particle size on fatigue crack incubation life for applied stress amplitudes of 110,133 and 165 MPa (Rε =−1, Dmax/Dmin =1.65, L=10 µm, B/L=1, α=0.5 and C =0.0211). After Fanet al. [3].

con particles has the size of dendrite cell. This inclusion size is the important size fordislocation pileup and failure. Actually, using single-stage replication techniquesPlumtree and Schafer reported [41] that fatigue cracks initiate at the silicon particlesin the eutectic region, generally at triple points and then through the eutectic region.From the point of view at the scale of dendrite cell, this cracking process is throughthe boundary of the dendrite which may be considered as an interfacial failure pro-cess of the dendrite cell. From Figures 10–12 we can conclude that the inclusion sizeis crucial for the mechanical properties of the alloy. Therefore, these figures providea guideline to find an optimized inclusion size according to prescribed applied loadand load cycles.

5. Summary for design-centered approach and conclusions

A design-centered approach is proposed in developing Al-Si-based light-weight alloywith enhanced fatigue life and strength. This proposed approach is an initial stepin updating the traditional alloy developing approach which is a highly empirical,time-consuming and expensive process. Often the results of such activities are poorlyunderstood and suffer failure during scale-up or in the field. Although this pro-posed renovative approach is in its infancy, it is promising. The basic elements ofthis renovative approach and our experience in formulating the design frameworkfor enhanced fatigue life and high strength of A356 Al alloy are summarized asfollows:(1) In this work, the objective is to develop a design-centered approach for maxi-

mizing both fatigue life and strength for cast aluminum alloy. Related issues arethe contradictory observations of the size effects. On the one hand, experimen-tal results show that the smaller the dendrite size, the softer the cast alloy, anabnormal or inverse size effect on material strength. On the other hand, it isreported that the larger the dendrite cell, the lower the fatigue life; also calcu-lated results show that the larger the silicon particle the smaller the fatigue lifefor crack incubation.


(2) The underlying scientific principle for this specific design of maximizingfatigue life and strength is that plastic deformation originates from disloca-tion movements and that fatigue crack nucleation from microplasticity causedby dislocations. Based on this principle, a dislocation pileup model is proposedby the present authors [21]. More specifically, as shown in Eqs. (22) and (23),the material resistance to interfacial debonding is proportional to the inclusionsize, r, at static loading. This resistance will be reduced when the cyclic number,N , increases due to the increase of stored energy of dislocation pileups which isproportional to the cubic of inclusion radius, i.e., r3. These two opposite effectsof inclusion size on material resistance explain the underlying mechanism of sizeeffects on strength which are opposite from size effects on fatigue life. As cyclicnumber increases the dislocation pileup energy increases. When this stored energyreaches the surface adhesion energy the interfacial crack advances. Otherwise,the crack may have arrested but continuous cycling at a certain location foradditional cycles. This cycling with fixed crack location will not stop until theinterface dislocation energy accumulated again to reach the critical energy sothat a further debonding occurs and crack advances for a further step. Thismulti-step debonding process may explain the extremely smooth and roundeddebonding surface as shown in Figure 8.

(3) Based on the basic scientific principle and theoretical development described in(2), a maximum critical stress criterion without microplasticity is proposed fortailoring microstructure to reach enhanced fatigue life. This criterion states thatamong several designed microstructures, the one with the largest critical stress,σCR, under given cyclic loading conditions is a good candidate for optimizationof fatigue performance. Because this critical stress, σCr, can be easily determinedto be the maximum stress amplitude for no microplasticity within the microstruc-ture, this criterion is effective in the initial stages of designing cast alloys forenhanced fatigue performance. Furthermore, using Eqs. (22), (23) and Figures 10and 11 a design with optimization for both enhanced fatigue life and strengthcan be realized. This is because these equations and figures quantitatively con-nect fatigue life of crack incubation, strength, particle and dendrite size and meanspacing of these inclusions. Therefore, it can be used to optimize the contradic-tory requirements in component design of Al-Si-based cast alloy.

(4) The microstructure-property relationships are emphasized in this design-centeredapproach. This is one of the keys to establish the optimized relationshipsamong the five fundamental elements which are alloy selection, process design,microstructure design, properties optimization, and performance. Due to thecomplexity of the geometry and morphology of microstructures in a cast alloy,it is difficult to analytically and experimentally determine these relationships. Thefruitful tool is to use microstructural finite element analysis as a tool for tailoringmicrostructure by the proposed maximum critical stress criterion.

(5) The design-centered approach can be realized through different way. As an exam-ple, a tentative FDS map is developed in this work. In this map there are criticalpore sizes, above which fatigue life is controlled by pores; otherwise controlled byother mechanisms such as silicon particles. In the latter case, the proposed criteriaare the foundations of a guideline in the design-centered approach to maximizeboth the fatigue life and the strength.


(6) The design-centered computer-aided renovative approach proposed in this workis still in its infancy. Here, the term “design-centered” is used to distinguish thetraditional trial-and-error approach by formulating a clear objective, offering ascientific foundation, and developing a computer-aided effective tool for the newalloy development. This ultimate goal is to combine numerical simulation, scien-tific principle, database or parametric study, and traditional design for reachinga clear objective of high performance of the light-weight alloy. The objective canbe different for different needs such as enhancing corrosion-resistant and creep-resistant capability for magnesium alloy, or enhancing fatigue life and strength ofAl-Si-Based alloy in this work. Based on the clearly described objective a relevantscientific principle should be searched to lay down a solid scientific foundationfor the criteria or guideline of alloy design. Furthermore, a simulation model andan accurate computation technique should be developed.

Acknowledgments

The first author would like to thank Professors D. L. McDowell, M. F. Horstemeyer,K. Gall for many constructive discussions during his work at Georgia Tech duringthe period of 1997–2000. Previous work was supported by Sandia National Labo-ratories under U.S. DOE contract DE-AC04-94AL85000 and recent work has beensupported by NSF of China through the project of No. 10372119 entitled “Mul-tiscale Modeling Scheme of Elastoplasticity” and by the Commission of Scienceand Technology of ChongQing for the project of “Fatigue and Creep Behavior ofLight-Weight Alloy”. The second author is grateful for the discussions with ProfessorJ. Weertman, Professor G. B. Olson, and Professor A. Freeman of Northwestern Uni-versity. The second author also thanks Dr. Singh of NRL for providing the DODPlane Wave Code and Dr. Mehl of NRL for providing the NRL Tight-BindingCode. The second author is grateful for the financial supports of the EMC2 sub-contract (project manager: Dr. Yong-Yi Wang) through the DOE project entitled“Advanced Integration of Multi-Scale Mechanics and Welding Process Simulation inWeld Integrity Assessment (DE-FC-04 GO14040, Project manager: Dr. Mahesh Jha).

Appendix A: The determination of critical pore size for strain amplitude of 0.2%(133 MPa)

The procedure was as follows. Six fatigue specimens of a pore-controlled castingwere microstructurally examined which were tested at Westmoreland Inc. at the strainamplitude of 0.2%. The fatigue lives of these six specimens were: 60547, 56340,104000, 123417, 150353, and 182474 cycles. A critical pore size is the maximum poresize below which the failure mechanism is controlled by other mechanisms such assilicon particles. Therefore, the maximum fatigue life which is controlled by mecha-nisms of silicon particles should be used for determination of this size. For the abovetesting data, the average life, 166414 cycles, of the last two larger fatigue life, i.e.,150353 and 182474 cycles were used to get the estimated critical pore size of 100 µm.In deed, a detailed microstructure examination for the specimen with life of 150353shows that the fatigue crack nucleation and failure were not controlled by pores butby trapped Al oxides, intermettalic and Si particles [17].


References

1. Horstemeyer, M.F., McDowell, D.L. and Fan, J., From atoms to autos – A new design paradigmusing microstructure-property modeling Part 2: Cyclic fatigue, Sandia Report, SAND2000-8661,Sandia National Laboratories (2001).

2. McDowell, D.L., Gall, K.A. Horstemeyer, M.F. and Fan, J., Eng. Frac. Mech., 70 (2003) 49–80.3. Fan, J., McDowell, D.L., Horstemeyer, M. and Gall, K., Eng. Frac. Mech., 70 (2003) 1281–1302.4. Fan, J., McDowell, D.L., Horstemeyer, M. and Gall, K., Eng. Frac. Mech., 68 (2001) 1687–1706.5. Benzerga, A.A., Hong, S.S., Kim, K.S., Needleman, A., and Van der Giessen, Acta Materialia,

45 (2001) 3071–3083.6. Mclellan, D.L., J. Testing Evaluation, 8(1980), 170–176.7. Olson, G.B., Science, 277 (1997) 1237–1242.8. Hao, S., Liu, W.K., Moran, B., Vernerey, F. and Olson, G.B., Compu. method appl. mech. eng.,

193 (2004) 1865–1908.9. Hao, S., Moran, B., Liu, W.K. and Olson, G.B., J. Compu.-Aided Mater. Design, 10(2), (2003)

99–142.10. Torquato, S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties

Springer-Verlag (2002).11. Gokhale, A.A., presentation at the quarterly review meeting of USCAR/SCMD project, March,

2004.12. Curtin, W.A., Kumar, H. and Briant, C.L., Collections of Abstracts of the Int. Conference on

Heterogeneous Mater. Mech., (2004) 113.13. Socie D.F., Critical Plane Approaches for Multiaxial Fatigue Damage Assessment, in Advances

in Multiaxial Fatigue, ASTM STP 1191, edited by D.L. McDowell, and R. Ellis, (1993) 7–36.14. Anderson, T.L., Fracture Mechanics: Fundamentals and Applications, Second Edition, CRC

Press (1995).15. Wulf, J., Steinkopff, Th. and Fischmeister, H.F., Acta Mater. 44 (2000) 1765–1779.16. Couper, M.J., Neeson, A.E. and Griffiths, J.R., Fatigue. Frac. Eng. Mater. Struct., 13 (1990)

213–227.17. Gall, K., Yang, N., Horstemeyer, M., McDowell, D.L. and Fan, J., Fatigue. Fract. Eng. Mater.

Struct., 23 (2000), 159–172.18. Wang, Q.G., Apelian, D. and Griffiths, Microstructure Effects on the Fatigue Properties of Alu-

minum Castings, in Advances in Aluminum Casting Technology, ASM International, ConferenceProceedings from Materials Solutions, edited by Murat Tiryakioglu and John Campbell, (1998)217–223.

19. Doglione, R., Douziech, J.L., Berdin, C. and Francois D., Mater. Sci. Forum, 217 (1996) 1435–1440.

20. Polmear, I.J., Light Alloys, Halsted Press, New York 1996.21. Hao, S., Olson, G.B. and Fan, J. Two competing mechanisms in alloys with particles strength-

ening, Technical Report, NWU-ME-MTL-03–2004, Northwestern University, Evanston, 2004.22. Hirth, J.P. and Lothe J., Dislocation Theory, Wiley, New York, 1986.23. Weertman J., Dislocation Based Fracture Mechanics, World Scientific, 1996.24. Cottrell, A.H., Dislocations and Plastic Flow in Crystals, Clarendon Press, Oxford, 1961.25. Ham, R.K., Trans. Japan Inst. Met., Supplement of v. 9, (1968) 52–59.26. Pettifor, D.G., Cottrell, A.H. (edt.), Electronic theory in alloy design, the Institute of Material,

1992.27. Bernstein, N., Mehl, M.J., Phys. Rev. B, 62 (2000) 4477–4487.28. Martsinovich, N., Rosa, A.L., Heggie, M.I., Ewels, C.P., Briddon, P.R., Physica B, 340–342

(2003) 654–658.29. NRL, DoD Plane Wave: A General Scalable Density Functional Code. 2002.30. Krakauer, H. and Freeman, A.J., Phys. Rev. B, 19, (1979) 1706–1719.31. Fleck, N.A. and Hutchinson, J.W., Adv. Appl. Mech., 33 (1997) 295–361.32. Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W., J. Mech. Phys. Solids., 47 (1999) 1239–

1263.33. Eshelby, J.D., Frank, F.C. and Nabarr, F.R.N., Phil. Mag., 42 (1951), 351–364.


34. Rice, J.R., J. Mech. Phys. Solids, 40 (1992) 239–271.35. Adklman, N.T., Dudurs, J. (advisor), The instability of dislocation pile-ups and its application

to yielding and fracture. Ph. D. Thesis, Northwestern University: Evanston, 1971.36. Leibfried, G., Z. Physik, 135 (1953) 23–43.37. Leibfried, G., Z. Angw. Physik, 6 (1954) 251–254.38. Jing-Gwo Kuang and Mura, T. (advisor), Dislocation Pile-up in two-phase materials. Ph. D. The-

sis, Northwestern University, Evanston, 1967.39. Mott, N.F. and Nabarro, F.R.N., Proc. Phys. Soc., 52 (1940) 86–89.40. Mura, T., Micromechanics of defects in solids, Nijhoff, Boston, 1987.41. Plumtree, A. and Schafer, S., Initiation and Short Crack Behavior in Aluminum Alloy Castings,

In. K.J. Miller, R. De Los Rios (Eds.) The Behavior of Short Fatigue Cracks, EGF Pub. 1.Mechanical Engineering Publications, London (1986) 215–227.

a design-centered approach in developing al-si-based light

Documents