Strain-gradient modelling of grain size effects on fatigue of CoCr alloy

Strain-gradient modelling of grain size e?ects on fatigue of CoCr alloy

C.A.Sweeney a ,?,B.O’Brien b ,F.P.E.Dunne c ,P.E.McHugh b ,S.B.Leen a

a Mechanical Engineering,College of Engineering and Informatics,National University of Ireland,University Road,Galway,Ireland b

Biomedical Engineering,College of Engineering and Informatics,National University of Ireland,University Road,Galway,Ireland

c

Department of Materials,Imperial College,London,UK

Received 21February 2014;received in revised form 11June 2014;accepted 18June 2014

Available online 30July 2014

Abstract

A strain-gradient crystal plasticity framework based on physical dislocation mechanisms is developed for simulation of the experimen-tally observed grain size e?ect on the low cycle fatigue of a CoCr alloy.Finite-element models of the measured microstructure are pre-sented for both as-received and heat-treated CoCr material,with signi?cantly di?erent grain sizes.Candidate crystallographic slip-based parameters are implemented for prediction of fatigue crack initiation.The measured bene?cial e?ects of ?ne grain size on both cyclic stress–strain response and crack initiation life are predicted.The build-up of geometrically necessary dislocations as a result of strain-gradients,leading to grain-size-dependent material hardening,is shown to play a key role.ó2014Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.

Keywords:Size e?ect;Fatigue behaviour;Crystal plasticity;Geometrically necessary dislocations;Finite element

1.Introduction

It is well established that fatigue damage and fatigue crack initiation (FCI)in metals are sensitive to microstruc-ture [1,2].Experimental evidence indicates that grain size is a key microstructural feature a?ecting fatigue performance.Morrison and Moosbrugger [3],for example,presented an experimental study on the fatigue of coarse-and ?ne-grained polycrystalline nickel,reporting larger stress ampli-tudes and longer fatigue lives for the ?ne-grain material under strain-controlled plasticity conditions.While ?ne-grained materials are FCI-resistant,coarse grains have been shown to give an increased threshold stress intensity factor range and,thus,improved resistance to fatigue crack growth [4,5].Consequently,numerous industrial applica-tions take advantage of the grain size e?ect,with compo-nents designed to have ?ne-grained surfaces,where FCI

is more likely to occur [1],and coarse-grained interiors,for example,to resist crack growth.In the case of micro-scale components,for which microstructural features are comparable in size with component sectional dimensions,such as stents or micro-electromechanical devices,FCI typically dominates fatigue life and,therefore,a ?ne-grained structure is desirable throughout the load-carrying sections [6].

FCI is known to originate within persistent slip bands (PSBs)formed as a result of the build-up of mobile disloca-tions during irreversible cyclic slip [2,7].This paper demon-strates that the key physical basis of the grain size e?ect in fatigue lies in the development of immobile geometrically necessary dislocations (GNDs)to accommodate curvature of the crystal lattice in response to plastic strain gradients,which hinder the movement of mobile dislocations.Little-wood and Wilkinson [8]presented an electron backscatter di?raction (EBSD)study on cyclically deformed Ti–6Al–4V in which lattice rotation was measured and used to identify GND distributions via a cross-correlation tech-nique.The study provided experimental evidence of high

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1359-6454/ó2014Acta Materialia Inc.Published by Elsevier Ltd.All rights reserved.

?Corresponding author.Tel.:+353(0)91493020;fax:+353(0)91

563991.

E-mail address:c.sweeney4@nuigalway.ie (C.A.Sweeney).

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GND densities in regions of large strain gradients near grain boundaries and within grains.Finer-grained micro-structures produce higher gradients of plastic strain and, thus,associated regions of increased GND densities.As a result,these regions experience increased cyclic hardening and,therefore,exhibit reduced plastic slip,improving fatigue performance.

While this paper focuses on manifestation of the grain size e?ect via strain-gradient plasticity,other size e?ects can equally in?uence the fatigue behaviour of metals. Geers et al.[9]categorized size e?ects into(i)strain-gradi-ent e?ects,(ii)intrinsic size e?ects,(iii)statistical size e?ects and(iv)surface constraint size e?ects.As previously men-tioned,strain-gradient e?ects result from the storage of GNDs to allow lattice curvature.Intrinsic e?ects,also known as microstructural size e?ects,refers to the in?uence of intrinsic length scales in the microstructure,such as the Burgers vector magnitude,obstacle size and spacing,grain size and grain boundary width[10,11].For example, strengthening due to the Orowan mechanism is dependent on obstacle spacing and,thus,can be classi?ed as an intrin-sic size e?ect.Strengthening as a function of grain size as a result of the pile-up of dislocations at grain boundaries is another such e?ect(modelled empirically using the Hall–Petch relation).Statistical size e?ects arise when device or specimen dimensions are comparable with grain size.The mechanical performance of such devices is often dictated by the behaviour of relatively few grains.Hence,features such as crystallographic orientation can signi?cantly a?ect mechanical performance.Grogan et al.[12]investigated statistical size e?ects on the performance of stents,to develop a series of equations ensuring safe stent design, including a relationship between grain size,strut length and maximum strain.Finally,surface constraints become increasingly important in the presence of a high ratio of surface area to volume.The behaviour of material at an interface or along a free surface can signi?cantly a?ect the behaviour of a thin specimen,as illustrated by Geers et al.[9].

Nye[13]and later Ashby[14]proposed the storage of GNDs as the mechanism by which strain gradients are accommodated in the crystal lattice.Fleck et al.[15]devel-oped a strain gradient plasticity theory based on disloca-tion mechanisms,describing the storage of GNDs as a result of strain-gradients,and applied it to torsion of circu-lar-section wire.Macroscopic continuum models have since been developed to deal with the presence of strain gradient e?ects.For example,one study homogenised the behaviour of a weld steel material in?nite element(FE) fatigue analyses by assigning strength to homogenisation units based on the empirical Hall–Petch relation and a sta-tistical grain size distribution[16],to circumvent the expli-cit modelling of strain gradients in di?erent-sized grains. Taking this a step further,a crystal plasticity(CP)formu-lation,describing plastic slip on individual slip systems in the crystal lattice,has also been developed to include a Hall–Petch term with a threshold stress for fretting fatigue of Ti–6Al–4V[17–19].However,more accurate and realistic predictive models require a more fundamental, physically based approach,based on GND evaluation,to capture length scale and,hence,grain(and other micro-structure)size e?ects.Various strain-gradient CP formula-tions,including the work of Fleck et al.[15],have been developed[20–25],via inclusion of hardening mechanisms based on the evolution of GND density.One such CP for-mulation has been used by Cheong et al.[20]to predict the e?ect of grain size on monotonic stress–strain response for copper polycrystals.Ma et al.[22]applied a similar approach to predicting the stress?eld during shearing of an aluminium single crystal,while Dunne and co-workers [25]successfully predicted the measured GND density?eld near a carbide particle in a nickel single crystal.However, evidence has yet to be presented on strain-gradient CP modelling of the experimentally observed grain size e?ect on fatigue behaviour.

A key objective of this study is the assessment of fatigue performance via prediction of fatigue life.Various macro-scale criteria have been developed for assessment of total life fatigue performance(e.g.[26–28]).However,these approaches have limited viability for micro-scale design. Fatigue life(N f)can be decomposed into number of cycles to crack initiation(N i),microstructurally small crack growth,physically small crack growth and long crack growth[2];the?rst two are microstructurally driven pro-cesses[1,2],while the latter can be largely predicted via fracture mechanics.Signi?cant scatter is observed in fati-gue due to microstructural inhomogeneity,which cannot be captured by macro-scale models.Microstructure-based approaches,e.g.CP modelling,which take into account these inhomogeneities,have the potential to capture scatter in fatigue performance(e.g.[29,30]).Microstructure-based fatigue criteria are attractive for the assessment and design of components on the micro-scale.Mura[31]proposed a model for FCI based on the stability of dislocation struc-tures in a PSB,evaluated via an energy balance in a math-ematical model for the evolution of dislocation structures. Sangid and co-workers[32–34]adopted a detailed physi-cally based energy balance approach for FCI in a slip band spanning a grain,including terms to account for applied stress?eld,work hardening,formation of dislocation dipole structures,dislocation nucleation at grain bound-aries,penetration of grain boundaries by dislocations, and interruption of lattice stacking sequence in the matrix and precipitates.Macro-scale criteria have been adapted for use at the micro-scale.For example,the Smith–Wat-son–Topper and Fatemi–Socie criteria have been used as fatigue indicator parameters(FIPs)in conjunction with a critical plane approach to predict fatigue life and number of cycles to FCI,respectively,in CP simulations of fretting fatigue[18,19,35,36].More recently,microstructure-sensitive FIPs have been developed speci?cally for CP fati-gue predictions.One example of this is the e?ective crystal-lographic slip parameter presented by Manonukul and Dunne[37],used to successfully predict numbers of cycles

342 C.A.Sweeney et al./Acta Materialia78(2014)341–353

to FCI across both LCF and HCF regimes,for various cyclic loading ratios and temperatures.The applicability of this FIP has been further investigated with strain-gradi-ent CP to predict experimentally observed FCI for several four-point bending,ferritic steel specimens,with a view to investigating the role of strain gradients,anisotropic elas-ticity and plastic slip in FCI[38].Strong correlation was demonstrated between distributions of the e?ective crystal-lographic slip FIP and the experimentally observed crack sites,and the study also revealed a strong correlation between computationally predicted peaks of GND density and experimental FCI sites,suggesting that GND density itself could be employed as an FIP.A crystallographic work parameter has also been developed for use with CP models[39],which addresses the need for stress–strain criteria.

In this study,a physically based strain-gradient CP methodology is developed and is shown to successfully predict the e?ect of grain size on both the cyclic stress–strain(hysteresis)and crack initiation fatigue behaviour of a biomedical grade CoCr alloy.LCF test data are presented for both?ne-grained(as-received)and coarse-grained(after heat treatment)test specimens of the material,to quantify the e?ect of grain size on cyclic stress–strain curves and initiation life.The strain-gradient constitutive formulation of Ref.[25]is modi?ed through the inclusion of statistically stored dislocation(accumula-tion and annihilation)evolution equations.The measured grain morphology statistics are implemented in unit cell FE models of the coarse-and?ne-grained materials and combined with the constitutive model for comparison with the measured hysteresis and FCI behaviour vis-a`-vis grain size and strain range e?ects.An additional novel aspect is the combined use of a damage evolution curve for FCI identi?cation(from experimental damage data) and microstructure-sensitive FIPs,based on crystallo-graphic slip and work,for prediction of crack initiation life.

2.Experiments

The material tested is the L605CoCr alloy,trade-named HAYNES25alloy[40],which has a face-centred cubic (fcc)crystal structure[41].The material was received as a hot-rolled bar,solution-annealed at1230°C,with chemical composition of20wt.%chromium,15wt.%tungsten, 10wt.%nickel,3wt.%iron, 1.5wt.%manganese, 0.4wt.%silicon,0.1wt.%carbon and the balance consist-ing of cobalt.The test programme includes(i)heat-treat-ment of the alloy to increase the grain size,(ii) microstructural characterisation via microscopy of the as-received and heat-treated material and(iii)mechanical test-ing of the two material conditions to measure the e?ect of microstructure on the fatigue behaviour.Some microscopy and mechanical testing of the as-received material were reported in a previous study[30],but are presented here also for comparison.2.1.Heat-treatment and microscopy

Heat-treatment of the L605material was employed to e?ect a signi?cant increase in grain size.The initial iterative heat-treatment tests were based on solution annealing tem-peratures suggested for the L605alloy[42].Based on the observed changes in grain size,a?nal heat-treatment pro-tocol was implemented;this corresponds to annealing at a1250°C for2.5h.

Optical microscopy employed for both as-received and heat-treated material,using an etching solution of100ml HCl and5ml30%H2O2,to discern information on grain size and grain area distributions.Optical micrographs of the as-received and heat-treated material can be seen in Fig.1a;the grain sizes were found to be32and243l m, respectively.Twin boundaries are visible in both microstructures,formed during the annealing process. Microscopy images of both transverse and longitudinal cross-sections of the as-received and heat-treated bar por-tions revealed no discernible di?erence in grain structure and,thus,an equiaxed microstructure was inferred for the two conditions.Scanning electron microscopy(SEM) was used to investigate particles in the L605matrix, including energy dispersive X-ray(EDX)spectroscopy for identi?cation of particle composition,and crystallographic texture,via EBSD,for the two material conditions.As reported previously[30],the as-received material contained precipitates with high tungsten and carbon content (relative to the L605matrix)with a total area fraction of 0.52%.Microscopy of the heat-treated material yielded no signs of precipitates.This is in agreement with a previ-ous study on L605precipitates[43],which reported that almost no precipitates were visible after heat-treating at temperatures>1200°C.EBSD orientation maps provided in Fig.1b for both the as-received and heat-treated material conditions for views parallel and perpendicular to the central axis of the cylindrical CoCr bar indicate the absence of crystallographic texture(i.e.random crystal-lographic orientations)for the two material conditions. 2.2.Mechanical testing

The fatigue test programme consisted of strain-controlled LCF tests on the as-received and heat-treated materials at strain ranges±0.5%and±1.0%.The specimen design followed ASTM standards[44]for strain-controlled fatigue;the test rig employed was an Instron8500servo hydraulic system with hydraulic grips and V-shaped jaws at a strain rate of1.0%sà1.The test was stopped in each case when a30%drop in maximum load was observed following stabilization of initial cyclic hardening.

https://www.sodocs.net/doc/1412602103.html,putational framework

This section describes the development of the FE model, including both strain-gradient CP constitutive formulation and microstructure geometry,used to simulate the LCF

C.A.Sweeney et al./Acta Materialia78(2014)341–353343

tests in Section 2for the as-received and heat-treated materials.

3.1.CP constitutive model

The strain-gradient CP constitutive formulation of Dunne et al.[25]is adopted for this study.The equations used in the formulation describe the movement of mobile dislocations,or dislocation glide,through the crystal lat-tice,with the application of stress.Hardening occurs owing to the presence of immobile dislocations,including both statistically stored dislocations (SSDs)and GNDs,which behave as obstacles in the path of mobile dislocations.The presence of GNDs introduces the strain-gradient e?ect into the model,as shown below.The schematic in Fig.2shows the mechanism proposed here to explain the role of the development of GNDs in causing a grain-size e?ect on fatigue behaviour.This schematic is necessarily a simplistic two-dimensional (2-D)representation of a com-plex three-dimensional (3-D)crystallographic phenomena.An arbitrary grain in a polycrystal is hypothesised to experience a spatial gradient in plastic deformation,which is due,for example,to the orientation and geometry of adjacent grains,as illustrated in Fig 2a.In this case,the plastic deformation in the x -direction (signi?ed by the xx component of the plastic deformation gradient F p )at A is greater than at B,i.e.a gradient in F p xx forms in the y -direction.A simpli?ed view of the hypothesised deformed crystal lattice within this grain is provided in Fig.2b.Dis-locations of a particular sign must be stored in order to accommodate the curvature in the lattice,consistent with the gradient in plastic deformation.Therefore,these dislo-cations are geometrically necessary (i.e.GNDs).While the lattice also experiences stretch and additional rotation under applied and/or residual stress,these forms of defor-mation are elastic.From Fig.2b,the density of GNDs is dependent on the length of the lattice L over which the gra-dient in plastic deformation exists,as the same number of GNDs is required for a given change in plastic deforma-tion,regardless of the distance over which the change takes place.Two grains under the same applied loading ?eld,in an identical scaled microstructure morphology,

will

micrographs of the as-received material and heat-treated material conditions,and (b)EBSD orientation maps views both parallel and perpendicular to the central axis of the CoCr bar.

develop GND densities dependent on their respective grain sizes,d 1and d 2,as shown in Fig.2c.The same number of GNDs must form along the d 2dimension of the ?ner grain as along the d 1dimension of the coarser grain.Hence,GND densities are higher in ?ner-grained materials,result-ing in increased hardening and,thus,reduced cyclic plasticity.

As the calculation of GND densities is dependent on spatial gradients of deformation,the CP equations are implemented in a user-element subroutine (UEL)in Aba-qus to provide access to the necessary element information.The UEL implementation of Dunne et al.[25]is used here.Deformation of the crystal lattice is described in the formu-lation using a deformation gradient F ,with elastic and plastic components:F ?F e áF p

e1T

The second Piola–Kirchho?stress S is used in the imple-mentation to facilitate large deformation computations.The stress–strain relation is de?ned as S e ?D R E e

e2T

where D R is the fourth-order elasticity tensor referred to the reference frame,E is the Green strain tensor,and superscript e denotes variables evaluated at an intermediate frame.Cubic elastic constants are used to construct D R ,owing to the fcc structure of the L605lattice.The second Piola–Kirchho?stress is calculated from the Cauchy stress by

S e ?det eF e TF e à1r F e àT

e3T

The shear stress on a slip system a is de?ned as a func-tion of S :

s a ?s a 0 n a

0àá:S

e

e4Twhere s a 0and n a 0denote the slip system direction and nor-mal,respectively,evaluated in the undeformed reference frame.The fcc L605alloy has twelve {111}h 110i slip sys-tems.The plastic velocity gradient can be expressed as

L p ?_F p áF p à1?X

a

_c a s a 0 n a 0

e5Twhere _c

a is the slip rate on a slip system a .The ?ow rule relating slip rate to shear stress s a on a slip system is de?ned by

_c

a

?q SSD ;m b 2m exp àD H kT

sinh es a às c Tc 0D V kT e6Twhere q SSD,m is the mobile SSD density on a slip system,b

is Burgers vector magnitude,m is the frequency of attempts by mobile dislocations to overcome energy barriers,D H is the Helmholtz free energy,k is the Boltzmann constant,T is the temperature,s c is the critical resolved shear stress,c 0is a reference strain,and D V is the activation volume.The ?ow rule of Eq.(6)is developed from the Gibbs creep rate equation [45],as described by Manonukul et al.[46],in which an exponential function developed by

Granato

development of spatial gradient of plastic strain in a grain owing to the orientation of surrounding grains,(c)storage of di?erent GND densities in grains of di?erent size.

et al.[47]de?nes the probability of an energy ?uctuation at a given temperature that could allow a dislocation to over-come an energy barrier.The Helmholtz free energy is asso-ciated with the activation of pinned dislocation jumps,successful or otherwise,used by Gibbs to establish average dislocation glide speed.Forward and backward activation events are taken into account by the hyperbolic sine term of Eq.(6).The activation volume,on which the stress ?eld operates,can be de?ned by D V ?lb 2

e7T

where the pinning distance between dislocations l is described as a function of both immobile SSD density q SSD,i and GND density q GND

l ?1X a

q a SSD ;i tq a

GND

1=2,

e8T

Thus,the ?nal ?ow rule is de?ned by

_c a

?q SSD ;m b 2m exp àD H kT

sinh es a às c Tc 0b 2kT ????????????????????????????????????????X a

q a SSD ;i

tq

a

GND

r e9T

The slip system GND density is de?ned here as a func-tion of the elastic deformation gradient

X a

b a q a GND

àá?curl F e à1àáT

e10Twhere b a is Burgers vector,and q a GND is composed of both

edge and screw dislocation density components.Eq.(10)is solved via a least squares minimisation scheme,described fully in previous work [25],whereby the sum of the squares of dislocation densities is minimised.An evolution equa-tion for the immobile SSD density,similar to that pre-sented by Evers et al.[48],has been implemented here in the UEL,to account for both accumulation and annihila-tion of dislocations:

_q a SSD ;i ?j _c a j b X a

H ab q a SSD ;i tH ab q a GND à2cy c q a

SSD ;i

"#

e11T

where H ab are interaction coe?cients describing the mutual

immobilisation between dislocations of di?erent slip sys-tems,c is a constant,and y c is the critical annihilation dis-tance.The rationale of Ohashi et al.[49]is adopted for assigning H ab ;dislocations on the same slip system and on coplanar slip systems are assumed not to contribute to the mean free path,and thus a coe?cient of 0is assigned,while a coe?cient of 1is assigned for all other combinations.

The temperature T is set at room temperature (293K)for the CP simulations,while the frequency m is set at 1.0?1011s à1,chosen to be two orders of magnitude smaller than the Debye frequency [50].The mobile SSD density q SSD,m and the initial immobile SSD density

q a SSD ;i 0

are both taken to be 5?1010m à2,based on data presented in Ref.[15].The critical resolved shear stress s c ,calculated by dividing the yield stress of the material (reported in previous work [30])by the Taylor factor for an fcc crystal,is taken as 173.5MPa.The reference strain is taken as a constant at 0.001.The constant c used in Eq.(11)is set at 0.01,within the range suggested by Ohashi et al.[51],while the critical annihilation dis-tance y c is taken to be 2?10à9m,similar to that reported by Essmann and Mughrabi [52].The Helmholtz free energy D H is identi?ed via comparison of the cyclic micromechanical FE analyses with experiments.For sim-ulation of tests at room temperature,as in the present study,thermal activation is unlikely to be important,so that in fact the rate sensitivity (relating to D H )in the model is negligibly small.

A key aspect of the present work is the prediction of numbers of cycles to crack initiation N i and the e?ect of grain size on this.Microstructure-sensitive FIPs are imple-mented for this purpose in the UEL,based on variables in the CP equations above.The importance of scale compat-ibility between constitutive model and FIP-based predictive methods has been discussed previously for CP modelling [29,30].The e?ective crystallographic slip parameter of Manonukul and Dunne [37],de?ned by

_p ?23L p

:L p 1=2;p ?Z t 0_pdt e12T

has been related to the formation of PSBs in fatigue [2].Although the method has been used to predict locations of FCI [38],the present work is the ?rst attempt,to the authors’knowledge,to correlate this parameter with crack initiation life within the context of a physically based length-scale-dependent CP constitutive framework.The crystallographic work parameter W [39]

W ?X a

Z t

s a _c a dt e13T

is also investigated here.This energy-based parameter takes into account di?erent aspects of work carried out at a crystallographic level,de?ned by the terms of Eq.(9).This includes plastic work done by the applied stress ?eld,due to s a ,work hardening with increase in density of GNDs and immobile SSDs,and activation energy for unpinning of mobile dislocations.Both components of the crystallographic work parameter are a?ected by GND density;slip rate is directly in?uenced in accordance with Eq.(9),decreasing with increasing GND density,owing to increased microstructural barriers to slip,and slip sys-tem shear stress increases with GND density,as increased energy is required to overcome barriers.In the present work,crack initiation is deemed to correspond to critical values of the FIPs,p crit and W crit ,being reached.Cyclic val-ues of the FIP (i.e.accumulated over a single stabilized cycle)are used to assess when these critical values are

reached,i.e.N i ?W crit

W cyc

(similar for p ).346 C.A.Sweeney et al./Acta Materialia 78(2014)341–353

3.2.FE microstructure geometry

Microscopy data acquired in Section2were used for the generation of realistic polycrystal models for the microme-chanical FE analyses.Conversion of the grain area distri-butions extracted from optical microscopy in Section2.1 to grain volume distributions was required for generation geometries.A polycrystal model was then identi?ed with grain volume distribution that correlated closely with the measured microstructure distribution for the as-received. This model was then scaled to obtain the heat-treated microstructure geometry according to average grain vol-umes of the two material conditions,i.e.eV pTcoarse

eV pTfine

?eV gTcoarse

eV gTfine

,

representations of grain area and volume for conversion of distributions,comparison of experimental and FE microstructures of(b)the as-received material and(c)the heat-treated material,(d)FE microstructure geometry,

simulations of as-received material,and(e)inverse pole?gures for each realisation of the polycrystal model.

C.A.Sweeney et al./Acta Materialia78(2014)341–353347

in Fig.3d,along with the seven-element

into the space of a regular element in the

was inserted randomly into the mesh,at a

of locations to maintain measured

tion.The precipitates were assumed to

bides,based on the EDX

accordingly assigned a high Young’s

[54].

Each FE polycrystal comprised28

grains.The mesh for the heat-treated

of3375(153)20-noded brick elements

dimensions of(866l m)3.Replacement of

with the seven-element unit for the

resulted in a mesh composed of

Fig.3d),with polycrystal dimensions of

re?nement was carried out based on a

mesh sensitivity study by Cheong et al.[20].The?nal mesh

used here employs an average of120elements per grain; results from Cheong et al.[20]indicated that macroscopic polycrystal response showed little sensitivity to mesh re?ne-ment above8elements per grain for grain sizes greater than 30l m.Periodic displacement boundary conditions were applied to the polycrystal to simulate macroscopic behav-iour.Owing to the prohibitively large computational expense associated with cyclic3-D strain-gradient CP FE analyses,a de?nitive representative volume element (RVE)has not been used here.However,it has been shown [55]that periodic displacement boundary conditions applied to a volume element smaller than an RVE provide a response between the upper and lower limits of the mac-roscopic response of that material.

The polycrystal FE analyses used the strain-gradient CP constitutive model to simulate LCF loading of the as-received and heat-treated materials at the two experimental strain ranges of±0.5%and±1.0%,until stabilization of the cyclic stress–strain response.In order to predict micro-structure-induced results,three realisations of random crystallographic orientation distributions,as represented by the inverse pole?gures in Fig.3e,are presented for each strain range and material condition.Assignment of ran-dom orientations corresponds to a lack of crystallographic texture,corroborated by the EBSD maps in Fig.1b.

4.Results

4.1.LCF testing

The?ne-grain(as-received)material was found to exhi-bit superior fatigue behaviour for both applied LCF strain ranges.The stress range history of the two materials can be seen in Fig.4,where the cyclic stress amplitudes(D r/2)of the?ne-grain material are$100MPa greater than that of the coarse-grain(heat-treated)material across both applied strain ranges.

The measured total and plastic strain-life data is shown in Fig.5a and b,respectively,including additional results from previous work by the present authors[30]for the as-received material at applied strain ranges of±0.8% and±1.2%.A Co?n–Manson?t for the as-received mate-rial is also provided in Fig.5b.The total strain data in Fig.5a shows that the coarse-grain(heat-treated)material exhibits lower fatigue life.However,the?ne-and coarse-grain material data points appear to fall on the same Cof-?n–Manson curve.This demonstrates increased hardness and reduced cyclic plasticity of the?ne-grain material con-dition,and suggests that the material exhibits the same relationship between plastic strain and fatigue life for the two grain sizes.

4.2.Micromechanical simulations

The polycrystal FE simulations are used to identify the Helmholtz free energy D H for the strain-gradient CP con-stitutive model via comparison of the predicted and mea-sured responses,for the as-received material,using a least squares objective function.All other material parameters are based on physical considerations,as described in Sec-tion3.The complete set of constitutive constants used in the strain-gradient CP formulation is provided in Table1.

GND density distributions from the CP LCF simula-tions are presented for a sample microstructure realisation in Fig.6for both grain sizes for the two strain ranges. Clearly,signi?cantly higher q GND values are predicted for the?ne-grain cases.The average q GND distribution is an order of magnitude higher for the?ne-grain material.A comparison of the predicted and experimental stabilized stress–strain hysteresis loops for one microstructure reali-sation(one random crystallographic orientation set)is shown in Fig.7;excellent agreement is achieved generally. Cyclic stress range data for experiments and simulations are compared in Fig.8,including results for all three microstructure realisations.Signi?cantly lower stress ranges are clearly predicted for the coarse-grain material.

The measured evolution of cyclic maximum stress response is used here to compute the evolution of fatigue damage and,hence,identify the experimental number of cycles to FCI,N i.N i is considered to occur at a critical Fig.4.Experimental stress range histories for LCF tests.

348 C.A.

damage D c .A high probability exists of FCI occurring at the free surface for the LCF regime [1,56].Therefore,D c is selected based on the ratio of cross-sectional area of

the specimen compromised by initiation depth equal to the standard element polycrystal models)in each repeated unit of the fatigue specimen.This leads 0.0125.Experimental damage curves for each test,as shown in Fig.9D ?1àr

r max ,based on the e?ective stress [57],where r is the cyclic maximum the maximum value of r for each test.of the FIPs,W crit and p crit ,are calibrated the ?ne-grain material at the ±1.0%W crit =W cyc N i (similar for p crit ).Maximum strain–life data,and (b)plastic strain–life data for the LCF tests,with ?tted Co?n–Manson curve.Plastic strain range D e ,stress range D r and Young’s modulus E :D e p =D e àD r /E .

Table 1

Strain-gradient CP constants.k 1.381?10à23J K à1T 293K

2.56?10à10m 1.0?1011s à15?1010m à25?1010m à217

3.5MPa 1.0?10à32?10à9m 0.01

2.85?10à20J

of GND density q GND in a cut-away view of the model for one microstructure realisation,with crystallographic the as-received model.

of the microstructure-sensitive FIP,W cyc and p cyc,are extracted from the micromechanical LCF simulations. The resulting identi?ed values for W crit and p crit are 9.19?104MJ mà3and164.3,respectively.Predictions of N i for the other grain size–strain range combinations using these critical values,along with the calibrated case,are shown in Fig.10.W and p both clearly capture the decrease in N i for the coarse-grain material(red symbols)for a given applied total strain range.W predicts signi?cantly more accurately than p for all validation cases.5.Discussion

The as-received,?ne-grained material

performance superior to the heat-treated

coarse-grained,heat-treated material

ranges and shorter lives for a given

as seen in Figs.4and5a,respectively.

points for the coarse-and?ne-grained

lie on the same Co?n–Manson line,as

can be postulated that this is due to the

ing and thus reduced cyclic plasticity

?ne-grain material,owing to the presence

contrast,Morrison and Moosbrugger

Co?n–Manson curves for?ne-and

(with similar grain sizes to those used

points provided in the LCF region investigated

little di?erence in the Co?n–Manson

Morrison and Moosbrugger suggests that,with further testing towards the HCF regime,the Co?n–Manson curves of the?ne-and coarse-grain materials will begin to diverge.

The cyclic hysteresis loops in Fig.7clearly show the ability of the strain-gradient micromechanical framework to capture the grain size e?ect in fatigue.A decrease in stress range is seen with increase in grain size in Figs.7 and8.The increased hardening of the?ne-grain material can be explained by the increased build-up of GNDs. The di?erences in GND densities of Fig.6,as a result of

Comparison of experimental and simulated stabilized hysteresis loops for one simulated microstructure https://www.sodocs.net/doc/1412602103.html,parison of experimental cyclic stress–strain curves with FE

simulations for three microstructure realisations.Filled symbols indicate

experimental data,while open symbols mark micromechanical

simulations.

grain size,corroborate the mechanism of GND formation due to spatial strain gradients proposed in Fig.2.The authors are not aware of any published measurements of GND density for the present material,particularly after cyclic loading;however,Littlewood and Wilkinson[8]mea-sured GND densities in Ti–6Al–4V following cyclic load-ing,and showed that the range was between1010and 1016mà2.The predicted values in Fig6lie within this range.The in?uence of hardening due to GNDs on fatigue behaviour is dependent on the relative densities of GNDs and immobile SSDs,as given by Eq.(9).For the coarse-grain simulations,the ratio of GND to immobile SSD average densities ranges from0.35to1;for the?ne-grain simulations,the range is from2.5to4.1.This indicates that,while a further decrease in grain size will result in a greater in?uence of GNDs,as expected,an increase in grain size will result in negligible further deterioration in fatigue performance.This is due to GND densities drop-ping further below immobile SSD densities and,thus, becoming negligible.This is consistent with the?ndings of Fleck et al.[15],who showed that strain-gradient e?ects are most in?uential in polycrystals for

20l m.

Although both microstructure-sensitive

fully predict the trend of fatigue performance

material conditions(Fig.10),i.e.higher

grain material,the crystallographic work

signi?cantly more accurate.This is

approach of Sangid and co-workers[32–34]

crack initiation in a PSB spanning a grain

sation of a summation of energy terms,

carried out in the formation of the slip

at the intersection with the grain boundary.

eter used in this work does not operate

of a PSB spanning a grain,but predicts

maximum crystallographic work.However,

slip rule in Eq.(9),it incorporates components

similar to terms used in the approach

workers,such as the applied stress?eld,work hardening due to immobile dislocation build-up and unpinning of immobile dislocations for glide.Other energetic mecha-nisms are omitted by the FIP,including dislocation activity explicitly at the grain boundary(i.e.dislocation nucleation and formation of extrusions)and the interruption of lattice stacking sequence by mobile dislocations during slip.The Sangid model is a more detailed one,including atomistic simulations for quantifying grain boundary energy terms, based on the analysis of a single slip band in each grain. It is expected that some of the additional terms may have less in?uence in the CP approach used here,where the highly organised dislocation dipole structures of a PSB and the PSB–grain boundary interface are not explicitly modelled.

While it may be postulated that the data in Fig.5b indi-cate that the Co?n–Manson relation can predict total fati-gue life for a material across multiple grain sizes,it is important to note that a Co?n–Manson life prediction (for non-laboratory conditions)is dependent on the pre-dicted cyclic plastic strain range and,thus,the constitutive model.The strain-gradient model presented here can

Fig.9.Experimental damage accumulation curves for the LCF tests. https://www.sodocs.net/doc/1412602103.html,parison of experimental number of cycles to FCI(taken at a

critical damage,D=D c)with FE predictions using the microstructure

sensitive FIPs,W and p.

clearly be used e?ectively for multiple grain sizes in con-junction with the Co?n–Manson relation.However,use of this approach with a de?nitive RVE polycrystal would not produce scatter in life predictions,as this is a bulk material prediction.In contrast,microstructure-sensitive FIPs o?er the bene?t of capturing localised e?ects,where FCI is predicted to occur at peaks in FIP distributions, present because of microstructural inhomogeneity.Hence, the combination of microstructure-sensitive constitutive model and microstructure-sensitive FIPs facilitates capture of scatter in life predictions and provides a statistical basis for fatigue design.

It can be seen from the as-received models in Fig.6that peaks in q GND distributions often occur near precipitates. To investigate the e?ect of precipitates on fatigue perfor-mance,a polycrystal model for one microstructure realisa-tion of the as-received material without precipitates was analysed for the two strain ranges.The results indicate that precipitates have a negligible e?ect on the macroscopic hysteresis behaviour of the as-received material.Changes in predictions of N i upon removal of precipitates for the two FIPs and the two strain ranges are provided in Table2. For p cyc,the predicted e?ect of precipitate removal on N i is small,<3%,and does not show a consistent increase or decrease.N i is predicted to increase for both strain ranges using W cyc,with a maximum increase of nearly13%pre-dicted.However,the higher N i values still lie within the range of predictions for the three microstructural realisa-tions of the as-received model with precipitates.Nonethe-less,as W cyc predictions were found to give better correlation with the measured data in Fig.10,results from these additional simulations predict a bigger di?erence between the fatigue performance of the as-received and heat-treated conditions without the presence of precipitates and,thus,an even larger e?ect of grain size on FCI life.

This work does not address all types of size e?ects in fatigue, e.g.some mechanisms relating to intrinsic size e?ects.For example,a size e?ect due to build up of dislo-cations at grain boundaries is not captured.For the bulk material behaviour presented here,statistical size e?ects and surface constraint e?ects are not considered to be important.However,such e?ects may become important for micro-scale applications.Twin boundaries observed in both the as-received and heat-treated microstructures have not been modelled in this study.Simulations of polycrys-tals with a high frequency of twin boundaries by Sangid and co-workers[32,33],using the energy balance approach for FCI in PSBs,provided strong evidence of FCI occurring in grains with twin boundaries.More recent work presented CP FE simulations investigating the e?ect of twin boundaries on predicted FIP values[58].While this work showed that inclusion of twin boundaries in polycrys-tal simulations results in an increase in frequency of high FIP values in the vicinity of twin boundaries,the range of maximum FIP values observed appeared largely unaf-fected.Therefore,while it is anticipated that the inclusion of twin boundaries in the analyses presented here would generally act to reduce predicted crack initiation lives,it is expected that changes in N i predictions would be small, and predictions for both grain sizes would be equally a?ected without a change in trend of fatigue behaviour. Alternative candidate constitutive models exist in the liter-ature,which could improve hysteresis loop shapes,such as physically based kinematic hardening formulations(e.g. [23,48,59]).Also,while the mesh used in the present work is su?ciently re?ned to capture the macroscopic behaviour of a polycrystal[20],further mesh re?nement would undoubtedly a?ect local FCI predictions.However,the purpose of this study was to demonstrate the key role of strain gradients and GNDs in explaining the grain size e?ect on fatigue.

The micromechanical framework presented here is directly relevant to analysis of micro-scale devices,where fatigue performance is highly microstructure dependent, for the assessment and design of device geometry and microstructure.Micromechanics has been used to generate a set of design curves for deployment of cardiovascular stents[12],for example.The present framework can facili-tate development of a similar set of design rules for the fati-gue of stents as a function of grain size,crystallographic texture,precipitates,and geometry.

6.Conclusions

A strain-gradient CP methodology was used to predict the experimentally observed e?ect of grain size on cyclic hysteresis response and FCI life for a CoCr alloy,thus demonstrating the key role played by GNDs in the grain size e?ect on https://www.sodocs.net/doc/1412602103.html,putational simulations of exper-imental LCF tests indicate that,for the CoCr alloy tested, precipitates have a negligible e?ect on hysteresis behaviour and,while they appear to have an e?ect on fatigue damage accumulation,leading to crack initiation,it is relatively minor compared with the in?uence of grain size.A work-based crystallographic parameter was found to be most e?ective in predicting FCI;this is attributed to the depen-dence of PS

B formation on the dissipation of energy via dislocation activity.The framework presented is applicable to microstructure-sensitive fatigue design of devices and materials.

Acknowledgements

The authors acknowledge funding from the Irish Research Council under the EMBARK Initiative Scheme and the Irish Centre for High-End Computing

Table2

Percentage change in N i predictions for one microstructure realisation of

the?ne-grain model upon removal of precipitates.

N i(using W cyc)N i(using p cyc)

D e=±0.5%D e=±1.0%D e=±0.5%D e=±1.0%

+3.56%+12.89%à2.05%+1.26%

352 C.A.Sweeney et al./Acta Materialia78(2014)341–353

(ICHEC)for the provision of computational facilities and support.The EBSD work in this study was carried out at the Materials and Surface Science Institute at the Univer-sity of Limerick.

References

[1]Chan KS.Int J Fatigue2010;32:1428–47.

[2]McDowell DL,Dunne FPE.Int J Fatigue2010;32:1521–42.

[3]Morrison DJ,Moosbrugger JC.Int J Fatigue1997;19:51–9.

[4]Hertzberg RW.Deformation and fracture mechanics of engineering

materials.4th ed.Chichester:John Wiley;1996.

[5]Miller KJ.Mater Sci Technol1993;9:453–62.

[6]Connelley T,McHugh PE,Bruzzi M.Fatigue Fract Eng Mater Struct

2005;28:1119–52.

[7]Sangid MD.Int J Fatigue2013;57:58–72.

[8]Littlewood PD,Wilkinson AJ.Acta Mater2012;60:5516–25.

[9]Geers MGD,Brekelmans WAM,Janssen PJM.Int J Solids Struct

2006;43:7304–21.

[10]Arzt E.Acta Mater1998;46:5611–26.

[11]Gil Sevillano J,Ocan?a Arizcorreta I,Kubin LP.Mater Sci Eng,A

2001;309–310:393–405.

[12]Grogan JA,Leen SB,McHugh PE.J Mech Behav Biomed Mater

2013;20:61–76.

[13]Nye JF.Acta Metall1953;1:153–62.

[14]Ashby MF.Phil Mag1970;21:399–424.

[15]Fleck NA,Muller GM,Ashby MF,Hutchinson JW.Acta Metall

Mater1994;42:475–87.

[16]Remes H,Varsta P,Romano?J.Int J Fatigue2012;40:16–26.

[17]Zhang M,Zhang J,McDowell DL.Int J Plast2007;23:1328–48.

[18]Zhang M,McDowell DL,Neu RW.Tribol Int2009;42:1286–96.

[19]Zhang M,Neu RW,McDowell DL.Int J Fatigue2009;31:1397–406.

[20]Cheong KS,Busso EP,Arsenlis A.Int J Plast2005;21:1797–814.

[21]Han C-S,Gao H,Huang Y,Nix WD.J Mech Phys Solids

2005;53:1188–203.

[22]Ma A,Roters F,Raabe D.Acta Mater2006;54:2169–79.

[23]Kuroda M,Tvergaard V.J Mech Phys Solids2008;56:1591–608.

[24]Counts WA,Braginsky MV,Battaile CC,Holm EA.Int J Plast

2008;24:1243–63.

[25]Dunne FPE,Kiwanuka R,Wilkinson AJ.Proc R Soc A

2012;468:2509–31.

[26]Smith KN,Watson P,Topper TH.J Mater1970;5:767–78.

[27]Brown MW,Miller KJ.Proc Inst Mech Eng1973;187:745–55.

[28]Fatemi A,Socie DF.Fatigue Fract Eng Mater Struct1988;11:149–65.

[29]Sweeney CA,McHugh PE,McGarry JP,Leen SB.Int J Fatigue

2012;44:202–16.

[30]Sweeney CA,O’Brien B,McHugh PE,Leen SB.Biomaterials

2014;35:36–48.[31]Mura T.Mater Sci Eng,A1994;176:61–70.

[32]Sangid MD,Maier HJ,Sehitoglu H.Int J Plast2011;27:801–21.

[33]Sangid MD,Maier HJ,Sehitoglu H.J Mech Phys Solids

2011;59:595–609.

[34]Sangid MD,Maier HJ,Sehitoglu H.Acta Mater2011;59:328–41.

[35]McCarthy OJ,McGarry JP,Leen https://www.sodocs.net/doc/1412602103.html,put Mater Sci

2011;50:2439–58.

[36]McCarthy OJ,McGarry JP,Leen SB.Tribol Int2014;76:100–15.

[37]Manonukul A,Dunne FPE.Proc R Soc London,A:Math Phys Sci

2004;460:1881–903.

[38]Sweeney CA,Vorster W,Leen SB,Sakurada E,McHugh PE,Dunne

FPE.J Mech Phys Solids2013;61:1224–40.

[39]Korsunsky AM,Dini D,Dunne FPE,Walsh MJ.Int J Fatigue

2007;29:1990–5.

[40]HAYNES International.HAYNES25Alloy;2004.

[41]Teague J,Cerreta E,Stout M.Metall Mater Trans A

2004;35:2767–81.

[42]Davis JR.Heat-resistant materials.Metals Park,OH:ASM Interna-

tional;1997.

[43]Poncin P,Gruez B,Missillier P,Comte-Gaz P,Proft JL.In:

Proceedings of the M aterials and P rocesses for M edical D evices

C onference.Metals Park,OH:ASM International;2005.

[44]ASTM E606-04.Annual book of ASTM standards.West Cons-

hohocken,PA:ASTM International;2004.

[45]Gibbs GB.Mater Sci Eng1969;4:313–28.

[46]Manonukul A,Dunne FPE,Knowles D.Acta Mater

2002;50:2917–31.

[47]Granato AV,Lu¨cke K,Schlipf J,Teutonico LJ.J Appl Phys

1964;35:2732–45.

[48]Evers L,Brekelmans WA,Geers MG.Int J Solids Struct

2004;41:5209–30.

[49]Ohashi T,Kawamukai M,Zbib H.Int J Plast2007;23:897–914.

[50]Cottrell AH.An introduction to metallurgy.London:Edward

Arnold;1979.

[51]Ohashi T,Barabash RI,Pang JWL,Ice GE,Barabash OM.Int J

Plast2009;25:920–41.

[52]Essmann U,Mughrabi H.Philos Mag A1979;40:731–56.

[53]MATLAB Release2010b.Natick,MA:The Math Works,Inc.;2010.

[54]Connolly P,McHugh PE.Fatigue Fract Eng Mater Struct

1999;22:77–86.

[55]Jiang M,Alzebdeh K,Jasiuk I,Ostoja-Starzewski M.Acta Mech

2001;148:63–78.

[56]Przybyla CP,McDowell DL.Acta Mater2012;60:293–305.

[57]Lema??tre J.A course on damage mechanics.2nd ed.Berlin:Springer

Verlag;1996.

[58]Castelluccio GM,McDowell DL.J Mater Sci2013;48:2376–87.

[59]Geers MGD,Brekelmans WAM,Bayley CJ.Model Simul Mater Sci

Eng2007;15:S133–45.

C.A.Sweeney et al./Acta Materialia78(2014)341–353353