Fatigue and monotonic loading crack nucleation and propagation in bimodal grain size aluminum alloy

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Acta Materialia 59 (2011) 3550–3570

Fatigue and monotonic loading crack nucleation and propagationin bimodal grain size aluminum alloy

Steven Nelson a, Leila Ladani b,⇑, Troy Topping c, Enrique Lavernia c

a Mechanical and Aerospace Engineering, Utah State University, Logan, UT 84322, USAb Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA

c Chemical Engineering and Material Science, University of California, Davis, CA 95616, USA

Received 11 October 2010; received in revised form 16 February 2011; accepted 16 February 2011

Abstract

Bimodal and nanocrystalline (NC), or ultra-fine-grain (UFG) aluminum alloys are being investigated as stronger replacements forconventional polycrystalline aluminum alloys. Higher strengths are achieved by reducing the grain size of a metal; however, as the grainsize is reduced the ductility diminishes. One solution that limits this decrease in ductility is the addition of a percentage of microcrystal-line or coarse grains (CGs) into a nanocrystalline alloy, creating a bimodal microstructure which offers a better balance of strength andductility. Two- and three-dimensional microstructural finite element (FE) simulations of monotonic and fatigue failures in Al 5083 hav-ing bimodal grain structures are conducted. To reduce the computational time and facilitate the modeling of microstructural features, aglobal–local model is developed. Macroscopic linear-elastic and nonlinear plastic properties for each of the bimodal compositions arefirst used to simulate the tensile and fatigue tests in a global FE model. Subsequently, a local model that represents a single elementat the center of the global model is built with distinct CGs distributed throughout an UFG matrix. 10% of the elements in this modelare defined as CGs, after which NC and polycrystalline properties are assigned to the UFG and CG regions, respectively. Available fati-gue test data are utilized to generate a low cycle fatigue damage model for bimodal grains size Al 5083 and obtain the damage modelconstants for varied levels of coarse grains. This fatigue damage model is then used in conjunction with a finite element continuum dam-age modeling approach, namely, successive initiation, to predict the damage and crack initiation sites and propagation paths in bimodalalloys. The successive initiation method is used to continually accumulate damage in elements and initiate and propagate the crackthrough grains that reach the failure criteria defined for monotonic and cyclic loading. It is observed from the monotonic FE model thatusing ultimate stress as the failure criteria, cracks initiate on the boundaries between CGs and UFGs, and propagate through the UFGmatrix around the CG until they become large enough to extend all the way through the UFG region. In the cyclic FE models, the crackis observed to initiate in a CG and propagate along the CG and the surrounding UFG matrix until it is large enough to cause failure.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Finite element analysis; Simulation; Fatigue; Plastic deformation; Aluminum

1. Introduction

Aluminum alloys are known for being lightweight andductile, although, in general, not particularly strong. Recentadvancements in fabrication techniques that more preciselycontrol the microstructure have demonstrated markedincreases in tensile strengths when employed. Cryomillingaluminum alloy powders to generate nanoscale grains, con-

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.02.029

⇑ Corresponding author. Tel.: +1 205 348 2604.E-mail address: [email protected] (L. Ladani).

solidating the powders into a billet of nanocrystalline (NC)or ultra-fine-grain (UFG) material, and finally plasticallydeforming the consolidated billet is a common way to pro-duce microstructures with very fine grains. Other availabletechniques such as equal channel angular extrusion (ECAE)and equal channel angular pressing (ECAP) are also viablefabrication options [1–4]. The cryomilling fabricationprocess has some advantages over the alternative. One isthat Al 5083 is thermally stable at the cryogenic tempera-tures induced by the liquid nitrogen during cryomilling.Additionally, cryomilled powders are easily blended with

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Table 1Elemental composition of Al 5083 (wt.%).

Al Si Fe Cu Mn Mg Cr Zn Ti

92.55–94.25 0.4 0.4 0.1 0.4–1.0 4.0–4.9 0.05–0.25 0.25 0.15

S. Nelson et al. / Acta Materialia 59 (2011) 3550–3570 3551

unmilled powders to create bimodal microstructures. UFGmaterials processed in this manner have been reported tohave ultimate tensile strengths up to three times greaterthan polycrystalline materials having equivalent alloy com-positions. This increase in strength generally results in a cat-astrophic loss in ductility, and fracture of these materials isnotably brittle.

Variability in the fabrication process outlined above canproduce broad ranges of microstructures, which in turnaffect mechanical properties. First, the liquid nitrogenslurry required to maintain cryogenic temperatures doesnot entirely evaporate away after completion of cryomil-ling. The gaseous byproducts must be extracted before con-solidation in a hot vacuum degassing procedure consistingof holding the powders at a high temperature (400–500 �C)under vacuum for a prescribed amount of time, which var-ies according to powder bed mass and geometry, as well asthe vacuum system employed. High temperatures can facil-itate grain growth in the powders, and negate some of theeffects of cryomilling. Depending on the temperature andlength of time held at that temperature, it has been demon-strated that the grains can increase in size by 70–150% [5].This undesirable increase in grain size is offset by the obser-vation that the higher degassing temperatures also serve toproduce a denser billet once consolidated, indicating thatthere is a trade-off between the high strength achieved withsmaller grains and the improved ductility achieved whenthe microstructure is free of voids and interstitial elements[5].

Once degassed, the cryomilled powders are consoli-dated. Hot isostatic pressing (HIP) and cold isostatic press-ing (CIP) are the most prevalent methods of consolidation,though there are forging methods that serve to combine theconsolidation and plastic deformation steps into a singleprocess (quasi-isostatic forging or Ceracon forging [5–9],and vacuum hot pressing (VHP)), and other specializedmethods like spark plasma sintering [10] that are some-times employed. Both CIP and HIP use high pressures(7–400 MPa) to consolidate the powder into a billet of solidmaterial, and some grain growth is inevitable given thesetemperatures and pressures. As is evident from its name,HIP uses high temperatures (250–400 �C) in conjunctionwith the high pressure, contributing to even more graingrowth. Since HIP employs a combination of temperatureand pressure to consolidate the cryomilled powders, thepressures utilized are generally much lower than CIPpressures.

Finally, the consolidated billet must be plasticallydeformed to break up prior particle boundaries and elimi-nate excess porosity that exists after consolidation. If theseartifacts are not eliminated, the as-consolidated materialwill be very porous and its ductility is generally very low[11,12]. Extrusion, rolling, and forging are all viable plasticdeformation methods, though extrusion is by far the mostdocumented [14–22]. All three serve to add a level of anisot-ropy to the microstructure as the grains are elongated in thedirection of deformation. Since extruded and rolled materi-

als have elongated grains in only the direction of extrusionor rolling, they can be treated in a transversely orthotropicmanner. Forged materials may have the same transverseisotropy, but since the forging process does not necessarilyhave a principal deformation direction like extrusion orrolling, the levels of anisotropy of a material using forgingas the primary method of plastic deformation must bedetermined on a case-by-case basis. For UFG materials,the level of anisotropy induced by plastic deformation is vir-tually negligible. But when the material is modified to havea bimodal microstructure, the difference in longitudinal andtransverse properties is significant [13–23].

As mentioned before, UFG materials tend to be strongbut brittle. Researchers have found that mixing unmilledcoarse-grained (CG) powders with the cryomilled UFGor NC powders before consolidation produces a bimodalmicrostructure that results in a much more ductile materialwithout significantly decreasing the strength supplied bythe UFGs. Bimodal microstructures result with CG regionsin which grain diameters range from 1 to 5 lm and areevenly (but randomly) distributed throughout UFGmatrix, where the grain diameters range from 100 to500 nm [11,12]. These CGs are much more suited to allowdislocation motion since they are an order of magnitudelarger than the UFGs, where dislocation motion is some-what impeded. Local regions of highly ductile CGs distrib-uted throughout a less ductile UFG matrix allow bimodalmaterials to retain much of the strength of a purely UFGmaterial while becoming ductile enough to use in a muchbroader range of engineering applications.

Al 5083 is a predominately aluminum–magnesium alloycontaining 4.5% magnesium (see Table 1 for precise com-position information). UFG samples of this material havebeen shown to have an ultimate tensile strength (UTS) of740 MPa, which is over twice the UTS of polycrystallineAl 5083 (290 MPa) [24]. This alloy already is one of thestronger aluminum alloys, and the increase in tensilestrength by a factor of more than two is substantial. Butwhen compared to polycrystalline Al 5083 having a maxi-mum tensile strain of 20%, the 2–8% reported for NC Al5083 hardly seems worth the increase in strength. By creat-ing a bimodal microstructure, a better balance of strengthand ductility is achieved. With increasing levels of CG con-tent, the strength of the bimodal material decreases whileits ductility increases. For Al 5083, there is an observedlimit between 15% and 30% CGs after which there arediminishing returns as the CG content is increased further[23].

The constitutive properties of some specific bimodal Al5083 alloys are now described. They are obtained frommodels described in the literature and the experimental

3552 S. Nelson et al. / Acta Materialia 59 (2011) 3550–3570

data used to derive or verify those models. Ideally, themonotonic and fatigue properties would be known for aspecific fabrication process. Unfortunately, the most appli-cable model of tensile properties is derived using a CIPpedmaterial [24], whereas the only reported fatigue behavior ofAl 5083 was obtained using a HIPped material [25], so bothare described below.

2. Constitutive material properties

2.1. Tensile plasticity model for CIPped Al 5083

Tensile properties for cryomilled, CIPped, and extrudedare reported for 0%, 10%, 30% and 50% CG contents of Al5083 in both the longitudinal and transverse directions [13].They are used by Joshi et al. [23] to develop a plasticitymodel. The linear-elastic region is modeled for all CG con-tents in both directions using an elastic modulus of 70 GPaand a Poisson’s ratio of 0.3. Nonlinear inelastic behavior ismodeled using an exponential Voce hardening law, shownin Eq. (1), where rpl denotes the stress in the plastic region,rY is the yield stress, Ro;R1 and b are material constants,and epl is the inelastic strain [26]:

Table 2CIPped Al 5083 Voce constants used in Eqs. (3)–(5).

C1

(MPa)C2

(MPa)C3

(MPa)C4

(MPa)C5 C6

Longitudinaldirection

580 �218 710 �180 0.0084 �0.0086

Transversedirection

580 �356 710 �329 0.0084 0.065

Table 3CIPped Voce material parameters for a range of CG contents [23].

100% CG 30% CG 20% CG 10% CG UFG

rS (MPa) 425 656 674 692 710rY (MPa) 295 514.6 536.4 558.2 580eC 0.035 0.00582 0.00668 0.00754 0.0084

Fig. 1. Stress–strain curves for bimodal, UFG, and CG CIPped Al 5083: (a) dand 100% CG content and (b) model compared to experimental data for 0%

rpl ¼ rY þ R0epl þ R1ð1� e�beplÞ ð1ÞIt is modified to describe the hardening of bimodal Al

5083 in Eq. (2) by setting Ro ¼ 0; R1 ¼ rS � rY , andb ¼ 1=eC, where rS is the saturation stress denoting theUTS of the alloy, and ec is termed the characteristic strainand is essentially a proportionality constant [23].

rpl ¼ rs � ðrs � rY Þe�epl=ec ð2ÞThe Voce parameters, yield stress, saturation stress, and

characteristic strain, are defined as a function of CG con-tent (fcg) by Eqs. (3)–(5), respectively. The constantsrequired in these equations are shown in Table 2 [23].

rY ¼ C1 þ C2fcg ð3ÞrS ¼ C3 þ C4fcg ð4ÞeC ¼ C5 þ C6fcg ð5Þ

Applying the constants shown in Table 2 to a range ofCG contents yields the Voce material constants shown inTable 3. Note that this hardening law agrees with experi-mental results for CG contents up to 30%, so the propertiesof 100% CG material are derived independently by Joshi,but also reported in Table 3.

The stress–strain curves generated using this model forCIPped bimodal Al 5083 in the longitudinal orientationare compared to those of UFG material and 100% CGmaterial in Fig. 1.

2.2. Tensile plasticity model for HIPped Al 5083

Since the properties of bimodal Al 5083 vary so muchwith the fabrication process, and the only available fatiguedata presented uses HIP for the consolidation step [25], aconstitutive model for HIPped Al 5083 is also developed.The tensile properties for the HIPped Al 5083 used in thefatigue experiment are reported, and are shown in Table 4.Extrapolating these properties for additional CG contentsand fitting the Voce hardening law to each fraction ofCGs, results in constants fit for use in Eqs. (3)–(5) (Table 5).

eveloped using Joshi’s Voce hardening model [23] for 0%, 10%, 20%, 30%and 30% CG content [23].

Table 4Tensile properties of UFG, CG, and bimodal Al 5083 used in fatigue tests[25], and modified properties used in tensile tests.

100% CG 15% CG UFG

rS (MPa) 300 450.05 482rY (MPa) 286 380.1 441emax 0.108 0.107 0.11eC 0.01 0.007 0.0084

Table 5HIPped Al 5083 Voce constants used in Eqs. (3)–(5).

C1 (MPa) C2 (MPa) C3 (MPa) C4 (MPa) C5 C6

580 �218 710 �180 0.0084 �0.0086


Applying these constants to a range of CG contentsyields the Voce parameters shown in Table 4. And finally,the stress–strain curves comparing UFG, CG and bimodalcompositions of HIPped Al 5083 are shown in Fig. 2. Notethat this sample of HIPped Al 5083 exhibits much moreplasticity than that of the CIPped Al 5083 previously out-lined. This is likely due to the variability due to processingtechnique, where HIP or quasi-isostatic forging and rollingare used.

2.3. Fatigue damage model

A low cycle fatigue (LCF) model for UFG materials wasdeveloped by Ding et al. using experimental data for UFGcopper [27]. It is a crack propagation model based on themacroscopic properties of the UFG material. Since it onlydepends on the macroscopic properties, extending it for useon bimodal materials is reasonable, but experimental fati-gue data for the material is required.

The foundation of the fatigue model is a cyclic stress–strain relationship that essentially defines the material’sflow stress amplitude as a function of the plastic strainamplitude applied in a fatigue scenario. This relationship

Fig. 2. Tensile stress–strain curves for UFG, CG and bimodal HIPped Al5083 used for fatigue modeling [25].

is shown in Eq. (6), where K0 and n0 are material constantsderived by fitting a power law relationship to experimentaldata. By plotting the flow stress at fatigue failure for everyplastic strain amplitude tested, the curve fit, and thereforethe material constants, are obtained.

Dr2¼ K 0

Depl

2

� �n0

ð6Þ

A grain boundary strengthening and a grain boundaryconstraint factor are defined in Eqs. (7) and (8), respec-tively. The first is the ratio of the UFG material yieldstrength, rYufg, to the yield strength of the CG version ofthe same material, rYcg, and the latter is one half the ratioof the flow stress amplitude to the effective stress ampli-tude. In Eq. (8) Dr=2 denotes the average stress amplitudeof the bulk material and Dreff =2 is the effective stressamplitude that contributes to local deformation withinthe cyclic plastic zone

F ¼ rYufg

rYcgð7Þ

C ¼ 1

2

Dr2

Dreff

2

!ð8Þ

The first quantity of interest, shown in Eq. (9), is the sizeof the cyclic plastic zone (CPZ) around the crack tip. HereDKeff is the stress intensity factor range, and k is a cyclicplastic zone correction factor.

rCPZ ¼ kDKeff

2rYcg

� �2

ð9Þ

The stress intensity factor range is defined as a functionof crack length, a, and effective stress amplitude in Eq. (10).

DKeff ¼Dreff

2

ffiffiffiffiffiffipap

ð10Þ

By solving Eq. (8) for the effective stress amplitude andusing Eq. (10), the size of the CPZ is determined and shownin Eq. (11).

rCPZ ¼kpaF 2

16C2

K 0

rYufg

� �2 Depl

2

� �2n0

ð11Þ

The stress field within the CPZ is defined by Eq. (12).Again, the propagation of the crack is the primary concernhere, so this relation is used to define the size of the fatiguedamaged zone (FDZ). Material is damaged when its localstress reaches the UTS of the material, and so it is reason-able to state that rðrFDZÞ ¼ rUTS .

rðrÞ ¼ Dr2

rCPZ

r

� � n0n0þ1 ð12Þ

Using the UTS of the UFG material, Eq. (6) and (11) inEq. (12) and solving for the size of the FDZ yields:

RFDZ ¼kpaF 2

16C2

K 03n0þ1

n0

rn0þ1

n0UTS

0@

1A Depl

2

� �3n0þ1

ð13Þ

Table 6Fatigue material constants for UFG, CG, and bimodal Al 5083.

Fraction CG UFG 10% CG 20% CG 30% CG 100% CG

K 0 (MPa) 509 485.7 467.9 455.7 370.5n0 0.0997 0.101 0.102 0.100 0.0902rY ;ufg (MPa) 441.0 400.3 374.5 363.4 286.0F 1.54 1.40 1.30 1.27 1C 0.25 0.25 0.25 0.25 0.25ai (mm) 0.0003 0.0003 0.0003 0.0003 0.0003af (mm) 3.175 3.175 3.175 3.175 3.175k 0.00083 0.00083 0.00088 0.00101 0.00126


Now the plastic strain field within the CPZ is defined inEq. (14), and the accumulated plastic strain in the CPZ isdefined in Eq. (15):

eplðrÞ ¼Depl

2

rCPZ

r

� � 1n0þ1 ð14Þ

e�pl ¼1

rFDZ

Z rFDZ

0

eplðrÞdr ð15Þ

Using these two relations, Eq. (11) and (13), the accu-mulated plastic strain simplifies as shown in Eq. (16):

e�pl ¼K 0

rUTS

�1n0

!n0 þ 1

n0

� �ð16Þ

This quantity is important because it helps define theinteraction energy that drives crack growth, which isdefined in Eq. (17). The UTS of the UFG material is usedfor r�, and integrating Eq. (17) using cylindrical coordi-nates centered on the crack tip yields Eq. (18).

V int ¼ �Z

r�e�pldV ð17Þ

V int ¼ prUTSK 0

rUTS

� ��1n0 n0 þ 1

n0

� �rFDZ

2

� �2

ð18Þ

In LCF, the J integral is often used to correlate crackgrowth [28]. It is defined in Eq. (19). Differentiating thisexpression relative to the FDZ and substituting Eq. (13)results in Eq. (20).

DJ ¼ �@V int

@rFDZð19Þ

DJ ¼ kp2F 2a

32C2

n0 þ 1

n0

� �K 03

r2Yufg

!epl

2

� �3n0þ1

ð20Þ

Finally the crack growth rate is defined in Eq. (21) interms of the range of crack tip opening displacement(DCTOD), which is subsequently a function of the J

integral:

dadN¼ DCTOD

2¼ 1

2

2DJ3rYufg

� �ð21Þ

Fig. 3. Cyclic stress–strain curves for UFG, CG, and bimodal Al 5083 [25]: (a)content and (b) model curves compared to experimental data for 0%, 15% an

Substituting Eq. (20) into this expression, separating thevariables, and integrating both sides yields the relationshipshown in Eq. (22).

Nf ¼96C2

kp2F 2

n0

n0 þ 1

� �rYufg

K 0� �3

lnaf

ai

� �Depl

2

� ��ð3n0þ1Þ

ð22Þ

Using the experimental data for UFG, 15% CG and100% CG Al 5083 presented by Walley et al. [25], the cyclicstress–strain curves and all the constants required in thismodel are developed and shown in Fig. 3 and Table 6,respectively. The extension for other CG contents is fairlysimple as it has been documented that the flow stressdecreases linearly as a function of CG content [12,17,23].

Using these relations, the fatigue life of every CG con-tent is computed and shown in Fig. 4. These LCF livesagree with the experimental data [25]. It is interesting thatthe CG content does not affect the fatigue life in a signifi-cant way, though material comprising purely CGs doesoutperform the rest. This is not unexpected as the fatiguelife of material is very much dependent upon the ductilityof the material and a 100% CG sample exhibits a muchhigher level of ductility than any of the samples containingUFGs.

Since Eq. (22) is unwieldy to use in the form presented,the constants are all combined to fit a power law form, sim-ilar to original Coffin–Manson model, shown in Eq. (23),where K is the power law constant and n is the power

curves produced from fatigue model for 0%, 10%, 20%, 30% and 100% CGd 100% CG content.

Fig. 4. S–N plots of fatigue life for UFG, CG, and bimodal Al 5083: (a) model fit for 0%, 10%, 20%, 30% and 100% CG content and (b) comparison ofexperimental data to model for 0%, 15% and 100% CG content.


law exponent. The simplified constants for UFG, CG, anda range of bimodal Al 5083 are shown in Table 7. For thefinite element (FE) fatigue model, the UFG and 100% CGbehavior are of primary interest.

Nf ¼ KDepl

2

� ��n

ð23Þ

3. Finite element model

3.1. Global–local modeling

It is a relatively simple matter to build and solve a finiteelement model using constant, isotropic material proper-ties. Although bimodal alloys are not strictly isotropicand should not be modeled as such, the FE analyzes beingdone here are uniaxial with the load axis parallel to thedirection of grain elongation. Therefore the properties inthe orientation of interest can be modeled in an isotropicmanner. The macroscopic material properties of bimodalAl 5083 have been investigated at length, so modeling amacroscopic tensile or fatigue scenario using FE is not dif-ficult. But when the interactions between the CGs and theUFG matrix are desired, a macroscopic model using prop-erties averaged from both the coarse and fine grains is inad-equate. To more accurately predict the stress fields betweenthe CGs and UFG matrix, the CGs are treated as inhomo-genities that have different material properties, and difficul-ties arise in the size of the mesh required to model CGs.

These difficulties are resolved by using a local–globalmodeling technique. First, the average macroscopic prop-erties are used to build a global model that emulates a

Table 7Coffin–Manson power law parameters used to predict fatigue life.

UFG 10% CG 20% CG 30% CG 100% CG

K (MPa) 290.2 232.31 211.98 212.33 228.08n 1.444 1.544 1.587 1.589 1.599

dog-bone test specimen (Fig. 5), the likes of which arefrequently used in tensile and fatigue testing. This modelis solved either monotonically in tension, or cyclically tosimulate a fatigue scenario. As shown in Fig. 5, symmetryis used to reduce the model to an eighth its total size. Themesh is biased toward the gauge section since that is theprimary area of interest. Multipoint constraint elementsare utilized to best simulate force applied to the pinholeby a rigid pin. This method of force application was chosenexperimentally. At this scale it is very difficult to apply theforce in any other way without causing large amounts ofmathematically unreconcilable deformation. Stress concen-trations and subsequent deformations at the pinholeproved to be the simplest to accurately account for duringtensile testing. Both two-dimensional (2-D) and three-dimensional (3-D) models are developed and solved usingANSYS. For the 3-D model SOLID186 elements withthree translational degrees of freedom and 20 nodes areused. The 2-D model uses PLANE193 elements with twotranslational degrees of freedom and eight nodes. Plasticityis modeled in both the 2-D and 3-D models using a multi-linear approach, where a tabulated version of the stress–strain data for each bimodal composition is input.

For monotonic loads, the applied load is incrementedfrom the macroscopic bimodal yield strength to the bimo-dal macroscopic UTS. Starting the initial load at, orbeneath, the macroscopic yield strength is importantbecause in the local model the CG and UFG properties dif-fer from the macroscopic bimodal properties, and stressesmay be much higher locally. A crack could start in oneregion or the other when, or immediately after, the macro-scopic yield strength is reached, though failure will notoccur until well after this yield strength. Similarly, for cyc-lic loads, a load that produces a constant plastic strainamplitude is chosen, and then applied in a cyclic manner.When solutions for each load step are complete, the defor-mation from the element at the centermost portion of themodel is applied to a local model that is the same size asthe single element in the global model.

Fig. 5. Global model of dog-bone tensile specimen with loads and constraints described.

Fig. 6. Local 2-D model with randomly sized CGs in random locations.Fig. 7. Local 3-D model with two large CG bands: one at the symmetryintersection of the model and one at the center of the model.


In the local 2-D models, CGs of random sizes are ran-domly interspersed throughout the UFG region such thatthe appropriate CG ratio is attained, as shown in Fig. 6.Conical coarse grains with diameters between 0.4 and1.2 lm and lengths between 2.0 and 4.0 lm are modeledin this random manner. Care is taken to ensure that theCGs do not overlap nonsymmetric boundaries, as themodel is unable to properly account for the interaction ofthe CGs and UFGs at the free edges.

When the random CG generator is used with the 3-Dmodels, the resulting menagerie of CGs proves to be excep-tionally difficult to post-process in a way that shows thebehavior of each microstructural region and their interac-tions with each other. So, two large CG bands are definedinstead. One long and narrow conical CG band having adiameter of 6 lm and a length of 28 lm is defined at theintersecting of all the symmetry axes, and a short andobtuse conical CG band having a diameter of 5.4 lm and

a length of 12 lm is defined nearly at the center of themodel, as shown in Fig. 7. In this figure, only the UFG ele-ments that make up the model’s outer edges are shown sothat the locations of the CGs are easily seen relative to theUFG framework.

Instead of using macroscopic properties defined forbimodal microstructures, macroscopic properties forUFG and CG Al 5083 are applied to appropriate regionsof the model shown in Figs. 6 and 7. As previously alludedto, symmetry is used on all three axes. Local deformationhistory obtained from the global model is applied to thelocal model, allowing solutions to be obtained. Stress, strainand displacement fields in both the UFG and CG regionsare analyzed, and elements are eliminated from the modelsbased on either monotonic or fatigue failure criteria.

One final note about local–global modeling: in 2-D, thesize of the center element is small enough (0.01 mm � 0.01mm � 0.01 mm) to apply the displacements directly from

Fig. 8. Crack initiation in tension corresponding to a gauge section stress of 560 MPa (load step 1) in the global model: (a) eliminated elements in themicrostructure, (b) stress fields in the UFG matrix and CG regions and (c) and (d) enlarged views of the areas indicated in (a).


the global model to the local model. But for the 3-D model,the size of the center element (0.1 mm � 0.1 mm � 0.1 mm)is still a thousand times larger than the local model canrealistically be. So an intermediate model that is local com-pared to the global model but still global relative to thelocal model must be used. This intermediate model hasthe same properties as the global model, and displacementsare applied to the intermediate model from the globalmodel and from the intermediate model to the local modelin the same way described above.

3.2. Successive initiation

Successive initiation is a damage initiation–propagationmodeling technique that has been successfully used in pre-dicting cracks in several applications [29–33]. It involvesfinite element modeling in conjunction with a damagemodel or a failure criterion. The damage or failure criterion

is applied element-wise, and failed elements are eliminatedfrom the structure by reducing their stiffness to close to airstiffness. In monotonic modeling, depending on the physicsof the application, the failure criteria could be yield stress,ultimate strength, yield strain, ultimate strain or totalstrain energy. In fatigue modeling, typically a continuumfatigue damage model is used to evaluate the accumulateddamage in elements.

3.2.1. Successive initiation in monotonic loadingThe CIPped and HIPped material properties described

in Sections 2.1 and 2.2, respectively, are used in the FEmodel to simulate monotonic loading in tension. First,the 10% CG properties are applied to the global modeland it is loaded until the stress across the model’s gaugesection is equivalent to the model’s respective yieldstrength. Displacement values for all the nodes comprisingthe element in the center of the gauge length are stored.

Fig. 9. Crack propagation in tension corresponding to a gauge section stress of (a) 583 MPa (load step 6), (b) 595 MPa (load step 9), (c) 618 MPa (loadstep 14) and (d) 630 MPa (load step 17) in the global model.


Then the load is increased for 30 steps in a way such thatthe stress across the model’s gauge length reaches the mod-el’s respective ultimate tensile strength, and the subsequentnodal displacements of the central element are also stored.All the stored displacements are in turn applied to the localmodel, which comprises two distinct regions: the UFGmatrix and the CGs. After each set of displacements isapplied and a solution reached, the equivalent stress ineach UFG element and each CG element in the model iscompared to the appropriate UTS. If the stresses in the ele-ment exceed the respective UTS, the element is eliminatedfrom the model. Each successive load step increases thenumber of eliminated elements, thereby propagating thepath of tensile fracture. It should be noted that the applica-tion of loads and displacements is unidirectional. The dis-placements recorded on the center element of the globalmodel are applied to the local model, but the resulting

microstructural displacements and stresses are not usedto update the global model in any way.

3.2.2. Successive initiation in cyclic loading

Successive initiation in fatigue analyzes requires severalsteps. The damage initiation site is first identified with thehelp of a Coffin–Manson damage model, which predictsthe cycles to failure as a function of plastic strain ampli-tude. After a single cycle of the model, the plastic strainamplitude in every element is known, and the number ofcycles to failure throughout the model is determined forall the elements. All elements that have fatigue lives lessthan a threshold value are eliminated and identified asthe damage initiation zone. Damage is accumulated inevery other element throughout the model according toEq. (24), where D* indicates the accumulated damage afterthe current load step, D0 represents damage accumulated in

Fig. 10. Crack initiation in tension corresponding to a gauge section stress of 595 MPa (load step 9) in the global model: (a) eliminated elements in themicrostructure and (b) stress fields in a UFG matrix cross-section taken at the x-location of the initial crack.


previous load steps, Nf is the number of cycles to failurepredicted in the current load step, and Ncyc denotes thenumber of cycles that the current load step simulates.The accumulated damage is stored and added with eachsuccessive simulated FE cycle. Whenever the accumulateddamage reaches unity in a certain element, the element isassumed to have failed and eliminated.

D� ¼ D0 þ N cyc1

Nfð24Þ

Section 2.3 defines the HIPped fatigue properties of Al5083, and Section 2.2 defines the corresponding constitu-tive properties used in the fatigue experiments. The globaland local models are set up in an identical manner as themodels for the tensile. First the global model is subjectedto a tension–compression load cycle in such a way as toproduce a plastic strain amplitude of 0.1%. Nodal displace-ments in tension and compression of the centermost globalelement are stored for application to the local model. Aftereach local cycle in tension and compression, the plasticstrain and plastic strain amplitudes experienced by everyelement are stored. Using Eq. (23) with the appropriateUFG or CG properties to predict the fatigue lives of everyelement after each load step allows the successive initiationcriteria described in Eq. (24) to be used to determine thestate of damage in every element. The crack is initiated inthe first step by eliminating the elements with exceptionallyshort fatigue lives. Damage in the rest of the elements isaccumulated based on the fatigue lives of the eliminatedelements (Ncyc) and the fatigue lives of each remaining ele-ment (Nf). As the model is cycled further, elements areeliminated when their value of accumulated damagereaches or exceeds unity. Again, it should be noted thatthe local results are not used to update the global model.The global model is solved once, and the center element

displacements recorded and applied to every cycle in thelocal model. Locally, the model is updated as elements fail,causing its deformation to change despite the constantapplied loads.

4. Finite element results

This chapter contains many microstructural FE figures,all of which are color-coded in the same manner. UFGregions are shown in blue, and CG regions in purple.Red failure paths indicate crack in UFG materials, and yel-low-green failure paths denote crack in CG elements. Thiscolor scheme applies to both the 2-D and 3-D models.Also, it is difficult to illustrate the path of crack propaga-tion in the 3-D models using 2-D figures. So, for the sakeof visual clarity, the failed elements and the CG elementsare shown framed by only the outer edge of the model,which comprises almost entirely UFG elements. Thoughonly the outer edge UFG elements are shown, all the emptyspace in the figures is occupied by live UFG elements.

Furthermore, all the results presented here are the high-est resolution possible given the computational power ofthe machines being used. For the 2-D models, a grid sizeof 200 � 200 is used, making the size of each element inthe models 50 � 50 nm2. The 3-D models have grid sizesof 20 � 20 � 20, or individual element sizes of500 � 500 � 500 nm3. When compared to results using 2-D resolutions of 100 � 100 and 3-D resolutions of15 � 15 � 15, the site of crack initiation and the path ofpropagation do not change. Cracks formed a load step ear-lier and propagated a load step faster in the high resolutionmodels, particularly when loaded cyclically. This is attrib-uted to the smaller elements on the grain boundaries beingsubjected to a larger portion of the stress concentrationslocated there. But because the crack nucleation site and

Fig. 11. Crack propagation in tension corresponding to gauge section stresses of (a) 612 MPa (load step 13), (b) 616 MPa (load step 14), (c) 625 MPa (loadstep 16) and (d) 630 MPa (load step 17) in the global model.


propagation path are of primary interest in this study, it isreasonable to conclude that no more grid refinement isrequired.

4.1. Tensile failure of CIPped Al 5083

4.1.1. 2-D model

To model the tensile failure in two dimensions loads areapplied to the global model starting at the macroscopicyield strength (560 MPa) and ending at the macroscopicultimate strength (690 MPa) of bimodal Al 5083 compris-ing 10% CGs. Thirty steps are solved, each accountingfor 4.3 MPa of sequential loading.

Cracks initiate after one load step when the centralgauge stress in the global model is 560 MPa. They areshown in Fig. 8a as single failed UFG elements on the

boundaries of CGs. Higher magnitude views of the failedelements are shown in Fig. 8c and Fig. 8d and corre-spond to the boxed regions in Fig. 8a. Stresses through-out the local model are shown in Fig. 8b. It isimmediately apparent that the high stress regions arelocalized to the UFG matrix near the CG boundaries.Many of these regions are very near the UTS(690 MPa) of the UFG elements, indicating that failurecould be imminent in all of these high stress areas. Alsonote that the stresses in the CGs are well below the CGultimate tensile strength of 390 MPa. The lower stress inthe CGs is directly attributed to their higher level of duc-tility. In other words, given equivalent strains in theUFG and CG elements, the corresponding stress levelin the UFGs will be much closer to their ultimatestrength than the stress level in the CGs.

Fig. 12. Crack initiation in tension corresponding to a gauge section stress of 387–394 MPa (load steps 3 and 5) in the global model: (a) eliminatedelements in the microstructure, (b) stress fields in the UFG matrix and CG regions, both after the fifth load step and (c) and (d) enlarged views of the areasindicated in (a) after the third and fifth load steps, respectively.


Five load steps later, the cracks grow in the UFGmatrix, extending toward the nearest neighboring CGs,as shown in Fig. 9a. They also breach the boundaries intothe CGs they started next to. Also, another crack forms onthe opposite side of the most heavily affected CG. After theninth load step the growing crack is shown completing itspassage through the CG and extending into the UFGmatrix in Fig. 9b. Also note that the second crack has com-pleted it progression between its two nearest CGs. Fivesteps later the crack extension is almost complete, as shownin Fig. 9c. The two cracks have moved toward each other,through every CG they were in contact with, as well as onethat was previously unaffected, and extend through theUFG matrix to the boundaries of the model. The modelfails completely three steps later in Fig. 9d when the stressin the global model’s gauge section is 630 MPa.

In summary, the crack starts at a boundary between theUFG matrix and a CG. This occurs because the brittle

UFGs cannot withstand deformation as well as the ductilecoarse grains. Once the crack is started, the surroundingUFGs fail very soon, and the stress concentrations at thecrack tip that is pressed up against a CG cause said CGto fail. This process repeats itself as the crack propagatesthrough the UFG matrix toward surrounding CGs, swal-lowing them up as soon as contact is made.

4.1.2. 3-D model

The 3-D successive initiation solution is determined byincreasing the load for 30 steps from the global yieldstrength (560 MPa) to the global ultimate strength(690 MPa) of bimodal Al 5083 comprising 10% CGs. Thisresults in a step size of 4.3 MPa applied sequentially.

Similarly to the 2-D model, the crack initiates in theUFG matrix at the boundary of a CG, as shown inFig. 10a. This figure and all following figures in the 3-Danalyzes show the failed elements and the CGs framed by

Fig. 13. Crack propagation in tension corresponding to a gauge section stress of (a) 401 MPa (load step 7), (b) 408 MPa (load step 9), (c) 415 MPa (loadstep 11) and (d) 422 MPa (load step 13) in the global model.


the edges of the UFG matrix. A cross-sectional slice of themodel showing the stress distribution in the UFG matrixsurrounding the initial crack is shown in Fig. 10b. CGregions are omitted to provide a clear picture of the stressin the UFG matrix. The stresses in the CGs are far lowerthan their ultimate strength and do not largely impactthe stresses at the location of crack initiation, thereforethey are not shown here. Stress levels in the UFGs increasewith their proximity to the CG, peaking at the boundary,particularly, where the failure starts. As stated previouslyin the 2-D model, UFG stresses at the interface are signif-icantly higher because the interactions at the boundarycause the UFG region to displace more than it would inthe absence of the CG. This increase in strain correspondsto a higher level of stress, and ultimately causes the UFGmaterial to fail at the boundary.

Four load steps later, more UFG elements surround-ing the site of initiation have failed, as have elements

surrounding the other CG at the symmetry corner ofthe model (Fig. 11a). UFG elements between the two ini-tiation sites are also failing, indicating the initial path ofpropagation. This is further illustrated an additional steplater by Fig. 11b, where the connection between the CGsis almost complete. The crack has penetrated the CG onthe symmetry boundaries, but unseen in these figures isthe failure of elements in the center CG, which is alsooccurring. Two steps later, the crack spans the lengthof the model as shown in Fig. 11c, and the afflictionto the CGs is finally visible, particularly in the CG onthe symmetry boundaries. At this point, the model isas good as failed, as shown one step later by Fig. 11d.

The 2-D and 3-D models agree very well in that crackinitiation occurs in the UFG matrix on the boundary ofa CG. Furthermore, there are multiple points of nucleationafter initiation. In both cases the cracks initially movethrough the UFG matrix before afflicting the CG regions,

Fig. 14. Crack initiation in tension corresponding to a gauge section stress of 380 MPa (load step 1) in the global model: (a) eliminated elements in themicrostructure and (b) stress fields in the UFG matrix relative to CG locations.


and once the CGs are affected, both models show that theyfail very quickly thereafter. Failure is attributed to the brit-tleness of the UFGs. Given the same displacements, theywill reach their ultimate strength much sooner and fail.The crack that results from their failure creates zones ofconcentrated stress that cause a cascade effect on boththe surrounding UFGs and CGs.

4.2. Tensile failure of HIPped Al 5083

The tensile properties described for HIPped UFG andbimodal Al 5083 are primarily included as they pertain tothe fatigue model. They are used to predict the crack nucle-ation sites and propagation path in the same manner usedfor the CIPped material. In both the 2-D and 3-D models,the loads are applied to the global model starting at themacroscopic yield strength (380 MPa) and ending at themacroscopic ultimate strength (450 MPa) of bimodal Al5083 comprising 10% CGs. Since the difference betweenthe UTS and the yield strength of the HIPped material isonly 70 MPa (compared to the 130 MPa difference exhib-ited by CIPped Al 5083), only 20 load steps are solved,each accounting for 3.5 MPa of sequential loading. Thismaintains a similar step size in both the CIPped andHIPped solutions.

4.2.1. 2-D model

A crack nucleates when the stress in the global model is387 MPa (load step 3), which is uncharacteristically closeto the macroscopic yield stress, as shown in Fig. 12d. It ini-tiates in the CG and quickly spreads to the UFG matrixtwo steps later (394 MPa) shown in Fig. 12c. Both theseillustrations are magnified views of the full model shown

in Fig. 12a. Stresses in the model are illustrated after thethird step in Fig. 12b, where it is apparent why failureoccurs in the region it does. High stresses in the UFGregions between CGs are also visible, and additional nucle-ation sites are likely.

As expected, two load steps later (401 MPa) additionalnucleation sites are visible (Fig. 13a). Also, the initial crackis propagating through the UFG matrix toward the nearestCG. Fig. 13b–d respectively shows the next three loads stepof interest (9, 11 and 13, which correspond to global stres-ses of 408, 415 and 422 MPa respectively). At 408 MPa ofglobal load both the primary and secondary initiation sitesare expanding quickly. They coalesce at 415 MPa of globalload immediately before the model fails at 422 MPa.

When compared to the CIPped FE results, these HIPpedFE results are notably different. First, the CIPped modelfails at strengths much closer to the global ultimate tensilestrength. And second, the crack nucleates within the UFGmatrix on the boundary of a CG in the CIPped model, butthe first elements to fail in the HIPped model are CG ele-ments. This difference is attributed to ductile behavior ofHIPed material. Because the HIPped material describedhas exceptionally high ductility for bimodal Al 5083, theapplied deformations affects both the UFG and CGregions in almost the same way. The high elongations atfailure (approximately 10% for CG, UFG, and bimodalsamples) explain the difference in crack nucleation sites aswell.

Since both the UFG and CG regions fail after experienc-ing about the same level of deformation, it is impossible topredict which one will fail first. In this 2-D case, it is theCGs, where in both the CIPped cases it is the UFGs. The3-D HIPped case is now described, and the different site

Fig. 15. Crack propagation in tension corresponding to gauge section stresses of (a) 387 MPa (load step 3), (b) 394 MPa (load step 5) and (c–d) 401 MPa(load step 7): (c) interiors of CGs and (d) full model after failure.


of crack initiation verifies this unpredictability by agreeingwith both the 2-D and 3-D CIPped models.

4.2.2. 3-D model

In this model, the crack initiates in the UFG matrix asexpected (Fig. 14a) during the first load step (380 MPa ofglobal stress). Nucleation sites appear on both CG bound-aries and correspond to the stress distribution shown inFig. 14b. This result, surprisingly, agrees more with the3-D CIPped initiation location and less with the 2-DHIPped initiation location just presented.

Two load steps later (387 MPa of global stress) showssubstantial crack propagation through the UFG matrix(Fig. 15a). Fig. 15b shows the fifth load step (394 MPa),where the crack has propagated through the CG on thesymmetry axes, but has not yet affected the central CG.It has surrounded said CG, but it takes two more loadsteps (401 MPa) for it to fail as shown in Fig. 15Xc, where

the live CG elements have been omitted so the extent offailure in the CGs can be more readily seen. Fig. 15d illus-trates the path of failure in the full model after the samefinal load step.

As alluded to in the 2-D HIPped results, these 3-DHIPped results are more in line with the 3-D CIPpedresults. The crack nucleates in the UFG matrix andpropagates between the CGs before penetrating them.Ultimately, the only real agreement this 3-D model haswith the preceding 2-D model is the accelerate rate offailure. Both models fail much sooner than the CIPpedmodels do, relative to the global ultimate tensilestrengths. The disagreement between the 2-D and 3-DHIPped models is likely due entirely to the additionaldegree of freedom in the 3-D model. Though the crackcan be visualized much better in 2-D, the simplificationmay not be appropriate for this problem. The 3-D mod-els allow stress concentrations to be distributed across a

Fig. 16. Crack initiation and propagation in cyclic loading after (a) 100 simulated cycles and (b) 125 simulated cycles. Enlarged views, (c) and (d) of areasindicated in (a).


surface rather than at a point allowing them to developthrough the material in a more realistic manner. In thiscase, a 2-D stress concentration may have caused theCG element to fail prematurely, where the same stressconcentration in the 3-D model distributed itself acrossthe CG–UFG boundary, allowing the CG elements tosurvive just long enough to not be crack nucleationsite.

4.3. Cyclic failure

The successive initiation solution for a cyclic loadingscenario is determined using tensile and compressive dis-placements obtained from the global model given a plasticstrain amplitude of 0.01 and applying them to the localmodel. Each FE iteration simulates 25 cycles (2-D model)or 50 cycles (3-D model) of fatigue loading with damage

accumulated in an element by element basis. A larger cycleincrement is chosen because it takes far longer for damageto accumulate in the 3-D elements. Also, the 3-D modelrequires far more computational resources than the 2-Dmodel, and the largest step size possible that did not causea notable change in simulated behavior is desirable. All ele-ments start with an initial damage of zero, and are consid-ered to have failed when their damage value equals orexceeds unity.

4.3.1. 2-D model

Initially, fatigue microcracks nucleate within the CGs,where they intersect with the UFG matrix after 100 sim-ulated cycles, as shown in Fig. 16a. Higher magnitudeviews of the failed regions are shown in Fig. 16c and dand correspond to the boxed areas of Fig. 16a. Mostof the model remains undamaged through the first

Fig. 17. Crack propagation in cyclic loading after (a) 150 simulated cycles, (b) 175 simulated cycles, (c) 200 simulated cycles and (d) 325 simulated cycles.


iteration, but isolated areas corresponding with the CGboundaries are seriously afflicted by damage. Thisobservation is in direct contradiction to the monotonicresults previously presented, but is not unexpected. Whensubjected to equivalent global plastic strain amplitudes,the CG regions experience higher levels of plastic defor-mation than the UFG regions. Since the UFG and CGdamage models do not differ in a large way, the higherplastic strain amplitude the CGs endure reduce theirfatigue lives significantly, therefore increasing the damagethey accumulate to the point of failure.

Note that in Fig. 16c and d, there are a few UFG ele-ments that are damaged enough to fail. These elementshave accumulated far lower amounts of damage when com-pared to the failed CG elements. In other words, the UFGelements do not fail at the same time as the CG elements,they fail as a result of the CG elements around them failing.The subsequent finite element iteration after 125 cycleshave been simulated is shown in Fig. 16b. This figure shows

how the crack initially propagates through the CGs with-out affecting the UFG regions at all.

After 25 additional simulated cycles (150 total), the mul-tiple nucleated microcracks begin to coalesce into fracturesystems (Fig. 17a). Coarse grains subjected to high levels ofdamage have cracked through. As the cycling continues,cracks form between affected CGs in close proximity toeach other, quickly spanning the UFG matrix in between(Fig. 17b). When the simulated cycles reach 200, the crackbegins to span the larger gaps between damaged CGs, asshown in Fig. 17c. Failure occurs after 325 simulated cycleswhen the crack spans the entire model (Fig. 17d). The crackpropagates through the remaining UFG matrix, taking thepath of least resistance and connecting through the CGsclosest in proximity to the crack tips.

In summary, the higher levels of plastic strain in bothtension and compression that the CGs endure cause themto accumulate damage, and ultimately fail, first. Localizedregions of UFGs around the failed CG elements then

Fig. 18. Crack initiation in cyclic loading after (a) 500 simulated cycles and (b) 550 simulated cycles.


undergo higher levels of plastic deformation themselvesand begin to accumulate damage at higher rates. Coarsegrains in close proximity to each other fail first, followedby the UFG matrix between them. Once the initial crackshave coalesced, they work their way through the remainingUFG matrix until fatigue failure occurs.

4.3.2. 3-D model

Similarly to the 2-D model, cracks nucleate on the edgesof CGs, as shown in Fig. 18a, where 500 cycles are simu-lated. 50 cycles later (Fig. 18b), it is clear that the CGsare failing first, which substantiates the 2-D results. Again,it is the higher levels of plastic strain suffered by the CGsthat initiate the crack.

Three FE iterations (150 simulated cycles) later, thecrack spans through both CGs, as shown in Fig. 19a,and breaches the UFG matrix 200 cycles later, as shownin Fig. 19b. The UFG matrix continues to fail, as shownin Fig. 19c, where 1050 simulated cycles are complete. Overthe next four FE iterations, it grows into the large crackseen in Fig. 19d that is about to fail the model. The 3-Dcrack grows outward from the CG in a way impossibleto visualize using the 2-D model. Unfortunately, the levelof detail possible in the 3-D model limits the number ofCGs that the crack can interact with, so essentially onlythe progression of the crack between two CGs is modeledin three dimensions.

The 2-D and 3-D fatigue models agree with and supporteach other in the same way the 2-D and 3-D tensile modelsdo. In both, the crack nucleates within a CG at the CG–UFG interface. It progresses through first the CG regions,where it nucleated and then through the surrounding UFGregions. Ultimately, cracks that started in different CGsjoin together and course through the rest of the materialuntil the model fails.

Notable simulation disagreements include the totalnumber of simulated cycles before failure, 325 in 2-D and1250 in 3-D, as well as these values being far smaller thanthe global fatigue lives of around 8000–9000 cyclesdescribed by the fatigue model. The first can be partlyexplained by the additional degree of freedom. Deforma-tions large enough to cause damage accumulation arehighly localized in the 2-D model. CG elements on theboundaries of UFGs exhibit far more deformation, strainand damage than the CG elements in the “far field”. Thesehighly localized deformations cause the boundary elementsto fail before adjacent elements are subjected to significantlevels of damage. In the 3-D model, the damage is of courseconcentrated at the CG–UFG interface, but the “far field”

levels of damage are not insignificant. This further demon-strates that 3-D modeling is much more realistic whenplane stress assumptions are not completely justified.

As for the disagreement with the global fatigue lives, thepoint must be taken into account that these are local mod-els at the nanometer-to-micrometer scale with loads anddisplacements applied from the location of highest stressand deformation in the millimeter scale global model. Ifthe differential pieces of material between this central highstress model and the free edge of the global model weresolved sequentially from center to surface, there wouldlikely be a nonlinear increasing gradient in local fatigue life.Elements on the free edge of the global model could poten-tially have fatigue lives much higher than the global values.But the locality of the model may not account for the entirelocal–global fatigue life discrepancy. A large part of it maybe a symptom of the process of element elimination. Therate at which elements fail increases with the number offailed elements, and there is no way to account for anycrack closures in this FE model. Once failure occurs, thesubsequent rate at which elements fail increases with each

Fig. 19. Crack propagation in cyclic loading after (a) 700 simulated cycles, (b) 900 simulated cycles, (c) 1050 simulated cycles and (d) 1250 simulatedcycles.


step. It is very difficult to precisely control the step sizetaken with each FE iteration in ANSYS, and even if thatelement of step size control was enabled, the shear compu-tational resources required to solve that many additionalFE iterations is beyond the scope of these simulations.Since the successive initiation approximations used wereemployed specifically to simulate crack nucleation sitesand propagation paths through the microstructure, modifi-cations to better match the local and global fatigue liveswere deemed unnecessary.

5. Summary and conclusions

Bimodal Al 5083 fails in very different ways when loadedmonotonically and cyclically. Nevertheless, the difference

in failure modes stem from the same microstructural char-acteristics. When subjected to a tensile load both the 2-Dand 3-D FE models agree that the crack nucleates inultra-fine-grain regions that are adjacent to coarse grains.In both models the crack propagates away from or aroundthe CG before directly afflicting it. Crack propagation thenaccelerates as elements in the model fail until the crackspans the model, running through both microstructuralregions. The 2-D HIPped tensile model does not agree fullywith these conclusions, but the differences are attributed tothe high levels of ductility reported for the HIPped materialand do not invalidate the findings reported in the CIPpedtensile models.

Conversely, when subjected to cyclic loads, the cracknucleates at the outer edges of the CGs in both the 2-D


and 3-D models. In the 2-D model it traverses the affectedCGs before expanding outward into the UFG matrix,whereas in the 3-D model only one CG is damaged initiallyand the crack propagates into the UFGs from there beforethe other CG is affected. Model size and resolution limitthe number of CGs in the 3-D model, limiting the correla-tion between the two models, but it is reasonable to expectthe same behavior in the 3-D model if it were possible tomodel more CGs without exceeding memory limitations.Once the crack connects the severely damaged CGs, thecrack expands through the UFG matrix until the modelfails entirely.

These different locations of crack nucleation and modesof propagation, though both due to the CG–UFG matrixinterface, are symptoms of load. In tension case, CG withlower yield strength than the UFG yields first at muchlower stresses. Therefore, the UFG has to carry additionalload around weak regions of CG. At a given value of stress,the CG elements deform more, so the UFG elements adja-cent to them will be stressed even more, leading to theirearlier failure.

On the other hand, when subjected to cyclic loads, theplastic strain amplitude is held constant. Macroscopically,the amount of plastic deformation is more dependent uponthe UFGs than the CGs. Since the yield strength of theUFG region is higher, they will continue to deform elasti-cally as the CGs reach their yield strength and start todeform plastically. For the same macroscopic strain ampli-tude, the CG elements experience significantly more plasticdeformation in both the tensile and compressive cycles.Also, since the damage models are nearly equivalent inboth regions, the fatigue lives of the CGs are severelyreduced. Consequentially, the CG elements accumulatemore damage sooner and faster than the UFG elements.

These findings stress the importance of determining theproper CG content when using a bimodal material. Ofcourse the macroscopic properties of bimodal materialswill primarily affect the choice of CG content as therequired strengths and ductility will drive the choice. Intensile loading, it is safe to assume that the UFGs willalways fail first, regardless of the CG content, so for mono-tonic loading the strength required drives choice of bimo-dal material. But in cyclic loading, the CGs themselvesare the catalyst for crack nucleation. Macroscopically,the fatigue behavior of all levels of CG content are aboutthe same (as also shown by Walley [25]), but since the plas-tic deformation occurring locally severely limits the fatiguelives of CGs when interspersed between UFGs, there is areasonable concern that bimodal Al 5083 will not performas well as UFG or CG Al 5083 alone. Here, only a very lowplastic strain amplitude (0.1%) is tested, and at plasticstrain amplitudes this low the difference between the plasticstrain amplitudes felt by the CGs and UFGs is high. Athigher plastic strain amplitudes, this difference will not beso pronounced and these results may not be applicable.But at the upper limit of the low cycle fatigue regime,the addition of CGs can cause the inevitable fatigue

microcracks to nucleate far earlier than they would other-wise. Even if the material does not fail any sooner macro-scopically, the internal microstructural damage is notdesirable in any loading scenario.

In conclusion, the path of propagation in monotonicloading scenarios, though interesting to know, does not sig-nificantly affect the choice of CG content in bimodal Al5083. But in cyclic loading scenarios they are the mediumin which fatigue microcracks nucleate, so their addition willcause undesirable, even premature, levels of damage toaccumulate in the microstructure. Though the macroscopicfatigue lives of UFG, CG and bimodal Al 5083 do not varya lot, any amount of damage very early in the fatigue livesof materials will make them very susceptible to any loadingvariations and cause untimely failure. This is of particularconcern at low plastic strain amplitudes in the low cyclefatigue regime.

Acknowledgements

The authors would like to acknowledge the NationalScience Foundation for support of this research. Thismaterial is based upon work supported by the NationalScience Foundation under Grant No. 1053434.

References

[1] Iwahashi Y, Horita Z, Nemoto M, Langdon TG. Metall Mater TransA 1998;29A:2503–10.

[2] Hasegawa H, Komura S, Utsunomiya A, Horita Z, Furukawa M,Nemoto M, et al. Mater Sci Eng 1999;A265:188–96.

[3] Valiev RZ, Langdon TG. Prog Mater Sci 2006;51:881–981.[4] Valiev RZ, Alexandrov IV. Ann Chim Sci Mater 2002;27:3–14.[5] Ahn B, Newbery AP, Lavernia EJ, Nutt SR. Mater Sci Eng

2007;463:61–6.[6] Ahn B, Lavernia EJ, Nutt SR. J Mater Sci 2008;43:7403–8.[7] Ahn B, Nutt SR. Exp Mech 2009;50(1):117–23.[8] Newbery AP, Ahn B, Topping TD, Pao PS, Nutt SR, Lavernia EJ. J

Mater Process Technol 2008;203:37–45.[9] Newbery AP, Ahn B, Hayes RW, Pao PS, Nutt SR, Lavernia EJ.

Metall Mater Trans A 2008;39A:2193–205.[10] Ye J, Ajdelsztajn L, Schoenung JM. Metall Mater Trans A

2006;37A:2569–79.[11] Lavernia EJ, Han BQ, Schoenung JM. Mater Sci Eng

2008;493:207–14.[12] Han BQ, Ye J, Tang F, Schoenung J, Lavernia EJ. J Mater Sci

2007;42:1660–72.[13] Witkin DB, Lavernia EJ. Progress Mater Sci 2006;51:1–60.[14] Fan GJ, Wang GY, Choo H, Liaw PK, Park YS, Han BQ, Lavernia

EJ. Scripta Mater 2005;52:929–33.[15] Fan GJ, Choo H, Liaw PK, Lavernia EJ. Acta Mater

2006;54:1759–66.[16] Han BQ, Lee Z, Nutt SR, Lavernia EJ, Mohamed FA. Metall Mater

Trans A 2003;34A:603–13.[17] Han BQ, Lee Z, Witkin DB, Nutt SR, Lavernia EJ. Metall Mater

Trans A 2005;36A:957–65.[18] Han BQ, Mohamed FA, Bampton CC, Lavernia EJ. Metall Mater

Trans A 2005;36A:2081–91.[19] Hayes RW, Witkin D, Zhou F, Lavernia EJ. Acta Mater

2004;52:4259–71.[20] Lee Z, Witkin DB, Radmilovic V, Lavernia EJ, Nutt SR. Mater Sci

Eng A 2005;410:462–7.


[21] Witkin D, Lee Z, Rodriguez R, Nutt S, Lavernia EJ. Scripta Mater2003;49:297–302.

[22] Witkin D, Han BQ, Lavernia EJ. Metall Mater Trans A2006;37A:185–94.

[23] Joshi SP, Ramesh KT, Han BO, Lavernia EJ. Metall Mater Trans A2006;37A:2397–404.

[24] Youssef KM, Scattergood RO, Murty EJ, Koch CC. Scripta Mater2006;54:251–6.

[25] Walley JL, Lavernia EJ, Gibeling JC. Metall Mater Trans A2009;40A:2622–30.

[26] Voce E. J Inst Met 1948;74:537–62.

[27] Ding HZ, Mughrabi H, Hoppel HW. Fatigue Fract Eng Mater Struct2002;25:975–84.

[28] Rice JR. J Appl Mech 1968;35:379–86.[29] Ladani L, Dasgupta A. ASME J Electron Pack 2008;130(1):011008-

1–011008-11.[30] Ladani L. IEEE Trans Dev Mater Reliab (T-DMR)

2008;8(2):375–86.[31] Ladani L. ASME J Electron Pack 2010;132(December):041011-1–7.[32] Gyllenskog J, Ladani LJ. AIAA J Aircr, in press.[33] Gyllenskog J. Mechanical and aerospace engineering department.

Master’s thesis, Utah State University; March 2010.

Fatigue and monotonic loading crack nucleation and propagation in bimodal grain size aluminum alloy

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