Determination of the Material Intrinsic Length Scale of Gradient Plasticity...

International Journal for Multiscale Computational Engineering, 2(3)377-400(2004)

Determination of the MaterialIntrinsic Length Scale of Gradient

Plasticity TheoryRashid K. Abu Al-Rub & George Z. Voyiadjis*

Department of Civil and Environmental EngineeringLouisiana State University Baton Rouge, LA 70803

ABSTRACT

The enhanced gradient plasticity theories formulate a constitutive framework on thecontinuum level that is used to bridge the gap between the micromechanical plastic-ity and the classical continuum plasticity. The later cannot predict the size effectssince it does not posses an intrinsic length scale. To assess the size effects, it is indis-pensable to incorporate an intrinsic material length parameter `into the constitutiveequations. However, the full utility of gradient-type theories hinges on one’s abilityto determine the constitutive length-scale parameter ` that scales the gradient effects.Thus, the definition and magnitude of the intrinsic length scale are keys to the de-velopment of the new theory of plasticity that incorporates size effects. The classicalcontinuum plasticity is also unable to predict properly the evolution of the materialflow stress since the local deformation gradients at a given material point are not ac-counted for. The gradient-based flow stress is commonly assumed to rely on a mixedtype of dislocations: those that are initially randomly or statistically distributed,which are referred to as statistically stored dislocations (SSDs), and those formedto account for the additional strengthening mechanism associated with the deforma-tion gradients, which are referred to as geometrically necessary dislocations (GNDs).In this work two micromechanical models to assess the coupling between SSDs andGNDs are discussed. One in which the SSDs and GNDs are simply summed (model-I) and one in which, implicitly, their accompanying strength are added (model-II).These two dislocation interaction models, which are based on Taylor’s hardening law,are then used to identify the deformation-gradient-related intrinsic length-scale pa-rameter ` in terms of measurable microstructural physical parameters. The paperalso presents a method for identifying the material intrinsic length parameter ` frommicro hardness results obtained by conical or pyramidal indenters.

KEY WORDS

Gradient plasticity; size effects; intrinsic material length scale; statistically storeddislocations; geometrically necessary dislocations; microhardness

*Corresponding author: [email protected]

0731-8898/04/$20.00 © 2004 by Begell House, Inc. 377

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378 ABU AL-RUB & VOYIADJIS

1. INTRODUCTION

Experimental work on particle-reinforced com-posites has revealed that a substantial increasein the macroscopic flow stress can be achievedby decreasing the particle size while keepingthe volume fraction constant [1–7]. A similarstrengthening effect associated with decreasingthe diameter of thin wires in microtorsion testsand thickness of thin beams in microbendingtests has been reported by Fleck et al. [8] andStolken and Evans [9], respectively. Moreover,micro- and nanoindentation tests have shownthat the material hardness increases with de-creasing indentation size [10–14]. These experi-ments have shown an increase in yield strengthwith decreasing size at the micron and submi-cron scales. The mechanical properties, suchas flow stress or hardness, in metallic materi-als whether in simple tension, torsion, bending,or indentation testing, are size dependent. Theclassical continuum plasticity theory [15] can-not predict this size dependency. On the otherhand, it is still not possible to perform quantumand atomistic simulations on realistic time andstructures. Moreover, the emerging area of nan-otechnology exhibits important differences thatresult from continuous modification of charac-teristics with changing size. These differencescannot be explained by traditional models andtheories. Much is known about the physicalproperties and behavior of isolated moleculesand bulk materials; however, the propertiesof matter at the nanoscale cannot necessarilybe predicted from those observed at larger orsmaller scales. However, this problem can beresolved by regularizing or adding an intrinsicmaterial length scale to the continuum. A con-tinuum plasticity theory (but not classical plas-ticity), therefore, is needed to bridge the gap be-tween the classical continuum plasticity theoryand the classical dislocation mechanics.

In the last ten years, a number of authorshave physically argued that the size depen-

dence of the material mechanical properties re-sults from an increase in strain gradients inher-ent in small localized zones, leading to geomet-rically necessary dislocations that cause addi-tional hardening [8, 10-12, 16]. Material defor-mation in metals enhances the dislocation for-mation, motion, and storage. Dislocation stor-age causes material hardening. The stored dis-locations, generated by trapping each other ina random way, are referred to as statisticallystored dislocations (SSDs), while the stored dis-locations that relieve the plastic deformationincompatibilities within the polycrystal causedby nonuniform dislocation slip are called geo-metrically necessary dislocations (GNDs); theirpresence causes additional storage of defectsand increases the deformation resistance by act-ing as obstacles to the SSDs [17]. SSDs are be-lieved to be dependent on the equivalent plas-tic strain, while the density of GNDs is di-rectly proportional to the gradient of the equiv-alent plastic strain [16, 18, 19]. During the lastdecade, the so-called strain gradient plasticitytheories have been modified based on the con-cept of geometrically necessary dislocations inorder to characterize the size effects. A lengthscale enters into the theory, but the discrete dis-location origin is rarely clear and its value is afree parameter.

The enhanced gradient plasticity theorieswith length scales formulate a constitutiveframework on the continuum level that is usedto bridge the gap between different physicalscale levels while it is still not possible to per-form quantum and atomistic simulations onrealistic time and structures [20]. This willreduce the computational cost, the prototyp-ing costs, and time to the market. A varietyof different gradient-enhanced theories are for-mulated in a phenomenological manner to ad-dress the aforementioned size effects throughincorporation of intrinsic length-scale measuresin the constitutive equations, mostly based oncontinuum-mechanics concepts. Moreover, the

International Journal for Multiscale Computational Engineering


DETERMINATION OF THE MATERIAL INTRINSIC LENGTH SCALE 379

gradient theories are nonlocal in the sense thatthe interstate variable at any location within asolid does not depend only on the state at thatlocation, as in conventional plasticity theory,but also on the state at any other point withinthe solid. Gradient approaches typically retainterms in the constitutive equations of higher-order gradients with coefficients that representlength-scale measures of the deformation mi-crostructure associated with the nonlocal con-tinuum. Aifantis [21] was one of the first tostudy the gradient regularization in solid me-chanics. The gradient methods suggested byLasry and Belytschko [22] and Mühlhaus andAifantis [23] provide an alternative approachto the nonlocal integral equations [24–26]. Thegradient terms in several plasticity models areintroduced through the yield function [7, 8, 17,19, 21, 23, 27, 28]. The gradient concept hasbeen extended to the gradient damage theorythat has been developed for isotropic damage[29] and for anisotropic damage [30–34]. Thegradient plasticity theories have given reason-able agreements with the aforementioned sizedependence encountered in composite mate-rial experiments [19, 35–39], micro- and nano-indentation experiments [17, 40–45] as wellas with the microbend and microtwist experi-ments [17, 46]. There are also many gradient-enhanced models that were proposed as a local-ization limiter, i.e., in order to avoid a spurioussolution of the localization problems and exces-sive mesh dependence in conventional plastic-ity [21–23, 27, 31–34, 47, 48].

Although there has been a tremendous the-oretical work to understand the physical roleof the gradient theory, this research area is stillin a critical state with numerous controversies.Moreover, most of the computational applica-tions of gradient theories focus on the qual-itative behavior rather than both the qualita-tive and quantitative behaviors. This is due,to some extent, to the difficulty in calibrationof the different material properties associated

with the gradient-dependent models, which isimpossible for certain cases, but, more impor-tantly, have been due to the difficulty of car-rying out truly definitive experiments on crit-ical aspects of the evolution of the disloca-tion, crack, and void structure. Furthermore,it is believed that the calibration of the con-stitutive coefficients of a gradient-dependentmodel should not only be based on stress-strain behavior obtained from macroscopic me-chanical tests, but should also draw informa-tion from micromechanical gradient-dominanttests, such as micro- and nanoindentation tests,microbend tests, and/or microtorsion tests,accompained by metallographic studies andstereology-based quantification methods usingtomography images. However, indentation ex-periments are more attractive because they arethe most controllable method to examine theplastic behavior of metals under the inden-ter [49]. From dimensional consideration, ingradient-type plasticity theories, length scalesare introduced through the coefficients of spa-tial gradients of one or more internal variables.Hence the full utility of the gradient-basedmodels in bridging the gap between model-ing, simulation, and design of products hingeson one’s ability to determine the constitutivelength parameter that scales the gradient ef-fects. The work we report here aims at reme-dying this situation.

Micro- and nanoindentation are widely usedexperimental methods to probe mechanicalproperties of materials at micron or submicronscales. As opposed to the nonuniform defor-mation encountered by indentation tests, thematerial properties of gradient theories cannotbe effectively determined using a typical ten-sion test, where uniform deformation is en-countered. The study of Begley and Hutchin-son [42] and Shu and Fleck [41] indicated thatindentation experiments might be the most ef-fective test for measuring the length-scale pa-rameter `. However, it appears that few re-

Volume 2, Number 3, 2004



searchers have considered the identification ofthe material intrinsic length-scale ` of gradient-enhanced theories from measurements of in-dentation tests. Nix and Gao [40] estimatedthe material length-scale parameter ` from themicroindentation experiments of McElhaney etal. [13] to be ` = 12 µm for annealed sin-gle crystal copper and ` = 5.84 µm for cold-worked polycrystalline copper. Yuan and Chen[44] proposed that the unique intrinsic materiallength parameter ` can be computationally de-termined by fitting the Nix and Gao [40] modelfrom microindentation experiments and theyhave identified ` to be ` = 6 µm for poly-crystal copper and ` = 20 µm for single crys-tal copper. By fitting microindentation hard-ness data, Begley and Hutchinson [42] have es-timated that the material length scale associ-ated with the stretch gradients ranges from 0.25to 0.5 µm , while the material lengths associ-ated with rotation gradients are on the order of4.0 µm. Other tests also have been used to de-termine `. Based on Fleck et al. [8] microtor-sion test of thin copper wires and Stolken andEvans [9] microbend test of thin nickel beams,the material length parameter is estimated to be` = 4 µm for copper and ` = 5 µm for nickel.When considering the microstructure with lo-calization zones, gradient-dependent behavioris expected to play an important role once thelength scale associated with the local deforma-tion gradients become sufficiently large whencompared with the controlling microstructuralfeature (e.g., mean spacing between inclusionsrelative to the inclusion size when consideringa microstructure with dispersed inclusions, sizeof the plastic process zone at the front of thecrack tip, the mean spacing between disloca-tions, the grain size, etc.). All the aforemen-tioned gradient plasticity theories degenerateto classical plasticity when the controlling mi-crostructural feature becomes much larger thanthe intrinsic material length scale `. There-fore, the transition from the discrete dislocation

structure to a continuum description is possiblewhen the length scale is sufficiently large com-pared to the characteristic length of the disloca-tion distribution. However, there is still a fun-damental issue that needs to be resolved first:how to retain the length scale from the char-acteristic discrete dislocation scale in the con-tinuum description? In spite of the fact of thecrucial importance of the length-scale param-eter in gradient theory of plasticity, very lim-ited work focused on the physical origin of thislength scale. In this work we are aiming at rem-edying this situation by finding this parameterfrom a set of dislocation mechanics-based con-siderations. The definition and magnitude ofthe length scale are thus keys to the modeling,simulation, and design of emerging micro- andnanosystems [20].

Recently, Voyiadjis et al. [34] developeda general thermodynamic framework for theanalysis of heterogeneous media that assesses astrong coupling between rate-dependent plas-ticity and anisotropic rate-dependent damage.They showed that the variety of plasticity anddamage phenomena at small-scale level dictatethe necessity of more than one length parame-ter in the gradient description. They expressedthese material length-scale parameters in termsof macroscopic measurable material properties.However, this work concerns with the identifi-cation of the material intrinsic length-scale pa-rameter ` for gradient isotropic hardening plas-ticity. This can be effectively done through es-tablishing a bridge between the plasticity at themicromechanical scale and with the plasticity atthe macromechanical scale [50]. This bridge ischaracterized by the gradient theory of plastic-ity. A constitutive framework is formulated us-ing two micromechanical models that assess, ina different manner, the coupling between statis-tically stored dislocations (SSDs) and geomet-rically necessary dislocations (GNDs). One inwhich the SSDs and GNDs are simply summed(model-I) and one in which, implicitly, their




accompanying strength are added (model-II).These two micromechanical models, which arebased on Taylor’s hardening law, are usedto link the strain-gradient effect at the mi-croscopic scale with the stress-strain behav-ior of an equivalent continuum with homog-enized plastic deformation at the macroscopiclevel. This constitutive framework yields ex-pressions for the deformation-gradient-relatedintrinsic length-scale parameter ` in terms ofmeasurable microstructural physical parame-ters. Moreover, we present a method for iden-tifying the material intrinsic length parame-ter from micro- and nanoindentation tests us-ing conical or pyramidal (e.g., Berkovich andVickers) indenters. In addition, there are in-dications that a constant value of the materiallength scale is not always realistic and that dif-ferent problems could require different values[46]. The change in length-scale magnitude isphysically sound since the continuous modi-fication of material characteristics with time.Some authors argued the necessity of a length-scale parameter in the gradient theories thatchange with time in order to achieve an ef-ficient computational convergence while con-ducting multiscale simulations [51]. An evolu-tion equation has been derived for the length-scale parameter that shows that the characteris-tic material length scale decreases with increas-ing strain rates, but it increases with tempera-ture decrease.

The overall objective of this paper is to helpunderstand what the basic structure of gradientplasticity theories should be.

2. ISOTROPIC HARDENING

Many researchers tend to write the nonlocalweak form of the conventional effective plas-tic strain _

p in terms of its local counterpart pand high-order gradient terms. The followingmodular generalization of _

p can then be de-fined through the modification of that proposed

by Fleck and Hutchinson [19],

_p = [pγ + (`η)γ ]

1/γ , (1)

where ` is a length parameter that is requiredfor dimensional consistency and whose phys-ical interpretation will be discussed in detaillater in this paper. η is the measure of the ef-fective plastic strain gradient of any order. Thesuperimposed hat denotes the spatial nonlocaloperator. γ is a constant, which can be inter-preted as a material parameter. Equation (1) en-sures that _

p → p whenever p >> `η and that_p → `η whenever p << `η. Two values of γare generally investigated in the literature: (a)γ = 1 which corresponds to a superposition ofthe contributions of the local plastic strain andthe higher-order gradients of the plastic strainto the flow stress; and (2) γ = 2, which sincethe effective plastic strain scales with the normof the plastic strain tensor, corresponds to a su-perposition of the effective plastic strain of thetwo types of local and nonlocal parts.

Aifantis [21, 52] used the magnitude of thegradient of the conventional effective plastic

strain p =√

2εpijε

pij

/3 as a measure of the strain

gradient, with η expressed as follows:

η = ‖∇ip‖ =√∇kp∇kp, (2)

where∇k designates the gradient operator, andγ = 1 in this case.

In the work of de Borst and his cowork-ers [27, 47] the gradient-dependent plasticitymodel expresses the flow stress in terms of thesecond-order gradient of the effective plasticstrain such that η is expressed as follows:

`η =√

`∗∇2p, (3)

where `∗ is the square of a length parameter as-sociated with second-order gradient and ∇2 isthe Laplacian operator, and γ = 2 in this case.




In the mechanism-based strain gradient(MSG) plasticity theory [19, 28, 53], the effectivestrain-gradient is defined in terms of the gradi-ent of the plastic strain tensor εp

ij as follows:

`η =√

c1ηiikηjjk + c2ηijkηijk + c3ηijkηkji (4)

so that γ = 1 and ηijk = ηjik is given as

ηijk = εpki,j + εp

kj,i − εpij,k (5)

c1 , c2 and c3 scale the three quadratic invari-ants for the incompressible third-order tensorηijk, which are determined by matching a se-ries of distinct dislocation models consisting ofplane strain bending, pure torsion, and two-dimensional axisymmetric void growth [17,54], which respectively results in c1 = 0, c2 =1/4, and c3 = 0, such that Eq. (4) can be writtenas

η =√

14ηijkηijk . (6)

Smyshlyaev and Fleck [55] and Fleck andHutchinson [19, 28] have shown that the incom-pressible strain-gradient tensor ηijk can be de-composed as

ηijk =3∑

m=1η

(m)ijk (7)

with

η(1)ijk = ηS

ijk − 15(δijη

Skpp + δikη

Sjpp + δjkη

Sipp) (8)

η(2)ijk = 1

6×(eikpejlmηlpm+ejkpeilmηlpm+2ηijk−ηjki−ηkij) (9)

η(3)ijk = 1

6×(−eikpejlmηlpm−ejkpeilmηlpm+2ηijk−ηjki−ηkij)

+15(δijη

Skpp + δikη

Sjpp + δjkη

Sipp)

(10)

In the above equations, δij is the Kroneckerdelta, eijk is the permutation tensor, and ηS

ijk isthe fully symmetric part of ηijk,

ηSijk = 1

3 (ηijk + ηjki + ηkij) . (11)

Fleck and Hutchinson [19, 28] showed that theeffective strain gradient defined in Eq. (4) canbe rewritten by substituting Eqs. (7)–(11) as fol-lows:

η =

√√√√ 1`2

3∑

m=1

`2mη

(m)ijk η

(m)ijk , (12)

where `21

/`2 = c2 + c3, `2

2

/`2 = c2 − 1

2c3 and

`23

/`2 = 5

2c1 + c2 − 14c3. (13)

Substituting c1 = 0, c2 = 1/4, and c3 = 0, asreported by Gao et al. [17, 54], gives

`1 = `2 = `3 = 12` (14)

Fleck and Hutchinson [19] proposed also to de-termine these three length parameters from mi-croexperiments. Begley and Hutchinson [42]obtained from fitting their model with the ex-perimental data of bending of ultrathin beams,torsion of thin wires, and microindentation, thefollowing values of the length parameters suchthat,

`1 = 18`, `2 = 1

2`, and `3 =√

524`. (15)

Two models are proposed in the subsequentsections used to link the strain-gradient effectsat the microscopic scale with the stress-strainbehavior of an equivalent continuum with ho-mogenized plastic deformation at the macro-scopic level. This link is described by the afore-mentioned gradient theory of plasticity, whichis characterized by the general expression de-fined by Eq. (1).




2.1 Model-I

Taylor’s hardening law, which that relates theshear strength to the dislocation density, hasbeen the basis of the mechanism-based strain-gradient (MSG) plasticity theory [17, 40, 43,54]. It gives a simple description of the dis-location interaction processes at the microscale(i.e., over a scale that extends from about a frac-tion of a micron to tens of microns). The criti-cal shear stress that is required to untangle theinteractive dislocations and to induce a signifi-cant plastic deformation is defined as the Taylorflow stress (e.g., [12, 17, 49]):

τ = αGb√

ρT , (16)

where α is an empirical constant usually rang-ing from 0.1 to 0.5 [18], G is the shear modulus,b is the magnitude of the Burgers vector, and ρT

is the total dislocation density due to differenttypes of dislocations that cause material hard-ening. If the von Mises flow rule is assumed,the normal flow stress can then be written asfollows:

σ =√

3αGb√

ρT , (17)

where σ is the normal flow stress, which isequivalent to the effective stress. For crystallinematerials (e.g., face-centered-cubic solids), amore accurate relationship accounting for theso-called Taylor factor [56] between the tensileflow stress and critical resolved shear stress inslip systems is given by σ = 3 τ . The Taylorfactor acts as an isotropic interpretation of thecrystalline anisotropy at the continuum level.The use of the assumption of isotropy of solidsis imperative since we are seeking a continuumdescription of the discrete dislocations scale.The isotropic hardening law in Eq. (17) is acontinuum property, such as the forest harden-ing that is inherited from a discrete dislocationscale.

Generally, it is assumed that the total dis-location density ρT represents the total cou-pling between two types of dislocations, which

play a significant role in the hardening mech-anism. Material deformation enhances dis-location formation, motion, and storage. Asmentioned dislocation storage causes mate-rial hardening. Stored dislocations generatedby trapping each other in a random way arereferred to as statistically-stored dislocations(SSDs), while stored dislocations required forcompatible deformation within the polycrystalare called geometrically-necessary dislocations(GNDs). Their presence causes additional stor-age of defects and increases the deformation re-sistance by acting as obstacles to the SSDs [54].The SSDs are created by homogenous strainand are related to the plastic strain, while theGNDs are related to the curvature of the crystallattice or to the strain gradients [18, 57]. Plas-tic strain gradients appear either because of ge-ometry, loading, and/or because of inhomo-geneous deformation in the material. Hence,GNDs are required to account for the perma-nent shape change. The nonlocal effective plas-tic strain in Eq. (1) is intended to measure thetotal dislocation density that accounts for both:dislocations that are statistically stored and ge-ometrically necessary dislocations induced bythe strain gradients [17, 28].

Gao et al. [17, 54] and Fleck and Hutchin-son [19] expressed the total dislocation densityρT as the sum of the SSD density ρS , and GNDdensity ρG such that,

ρT = ρS + ρG. (18)

Other couplings between ρS and ρG are possi-ble. Fleck and Hutchinson [19] proposed thatρT can be expressed as the harmonic sum of ρS

and ρG such that,

ρT =√

ρ2S + ρ2

G. (19)

However, a more general coupling of ρS and ρG

can be proposed in the spirit of Eq. (1) as fol-lows [56]:

ρT =[ρµ

S + ρµG

]1/µ, (20)




where µ is a constant that can be interpretedas a material parameter similar to the γ con-stant in Eq. (1). It is obvious from Eqs. (1)and (20) that µ and γ are related to each other.Furthermore, the above expression ensures thatρT → ρS whenever ρG << ρS and that ρT → ρG

whenever ρS << ρG. Thus, Taylor’s hardeninglaw given by Eq. (17) can be rewritten as

σ =√

3αGb[ρµ

S + ρµG

]1/2µ. (21)

It is imperative to note that the nonlocal effectsassociated with the presence of local deforma-tion gradients at a given material point are in-corporated into Eq. (21) through GND density[37]. Thus, Eq. (21) constitutes the nonlocalmicromechanical plasticity constitutive modelsimilar to the nonlocal macromechanical plas-ticity expression in Eq. (1). Moreover, at the mi-croscale, where dislocation densities are usedas the appropriate variables to describe plas-tic flow, the introduction of high-order gradientterms in the conventional continuum mechan-ics has led to bridge the gap between the con-ventional continuum and the micromechanicalplasticity theories. Nix and Gao [40], Gao et al.[17, 54], Arsenlis and Parks [16], and Huang etal. [53] showed that the gradient in the plasticstrain field is accommodated by the geometri-cally necessary dislocations ρG so that the effec-tive strain gradient η, which appears in Eq. (1),can be defined as follows:

η =ρG b

r. (22)

They showed that this expression allows η tobe interpreted as the curvature of deformationin bending and twist per unit length in torsion[53]. Whereas, the statistically stored disloca-tions ρS are dependent on the effective plas-tic strain p. r is the Nye factor introducedby Arsenlis and Parks [16] to reflect the scalarmeasure of GND density resultant from macro-scopic plastic strain gradients. For FCC poly-crystals, Arsenlis and Parks [16] have reported

that the Nye factor has a value of r = 1.85 inbending and a value of r = 1.93 in torsion.The Nye factor emphasizes the increased con-centration of GNDs by affirming that the dis-location density accumulated in a grain at acertain strain is higher once the grain size de-creases, which is inherent to the increased in-homogeneous deformation (i.e., strain gradi-ents) within the grain and the accompanyingdecreasing mean-free path of the dislocations[16]. The Nye factor, therefore, is an impor-tant parameter in the predictions of the gradi-ent plasticity theories as compared to the exper-imental results [17].

The uniaxial flow stress-plastic strain hard-ening relation without the effects of strain gra-dients can be identified, in general, as follows(e.g., [40, 44, 54]):

σ = k f (p) , (23)

where k is a measure of the yield stress in uni-axial tension and f is a function of the effectiveplastic strain p. For the majority of the duc-tile materials, the function f can be written asa power-law relation (e.g., [58]), such that

f (p) = p1/m, (24)

where m ≥ 1 is a material parameter that can bedetermined from a simple uniaxial tension test.

However, since uniaxial tension tests ex-hibit homogenous deformation, Eq. (23) can-not be used to describe applications where thenonuniform plastic deformation plays an im-portant role (e.g. twisting, bending, deforma-tion of composites, micro- or nanoindentation,etc.). Equation (23) cannot then predict thesize dependence of material behavior after nor-malization, which involves no internal materiallength scale. Therefore, Eq. (23) should be mod-ified to be able to incorporate the size effects.This can be effectively done by replacing theconventional effective plastic strain measure pby its corresponding nonlocal measure _

p de-fined by Eq. (1), such that Eqs. (23) and (24)




can be rewritten as

σ = k [pγ + lγηγ ]1/mγ . (25)

This gradient law exactly matches the Fleck-Hutchinson [19] phenomenological law in thecase of a single-material length scale, where γ istaken to be 2, but can be generalized to any arbi-trary number larger than 1 without destroyingthe basic theoretical framework. However, intheir work η is expressed as an effective straingradient given by

η =√

23χijχij , (26)

where χij is the so-called curvature tensor [19],which is related to the incompressible third-order tensor ηijk [Eq. (5)] by

χij = 12eiqrηjqr. (27)

Moreover, the Fleck and Hutchinson theory[19, 28] considers the strain gradients as inter-nal degrees of freedom and requires thermo-dynamic work conjugate higher-order stresses,which need additional boundary conditions tobe imposed. However, the proposed model inEq. (25) gives an alternative framework thatcould model size-dependent plasticity withouthigher-order stresses or additional boundaryconditions, so as to preserve the essential struc-ture of classical plasticity. This approach hasthe advantages that it is simpler overall and canbe easily implemented into existing finite ele-ment codes.

It is worth mentioning that Eqs. (21) and (25)imply that plasticity is the macroscopic out-come from the combination of many dislocationelementary properties at the micro- and meso-scopic scale. These two equations, therefore,represent the link of microscale plasticity to themicroscopic plasticity. The effective use of thislink will be demonstrated in what follows.

Next we use Eq. (25), which expresses theflow stress in the macroscale, along with Tay-lor’s hardening law [Eq. (21)], in the microscale

to derive the material-intrinsic length scale forisotropic hardening gradient plasticity in termsof physical microstructural parameters. Bam-mann and Aifantis [59] defined the plastic shearstrain γp as a function of the SSD density ρS asfollows:

γp = bLSρS , (28)

where LS is the mean spacing between SSDs,which is usually in the order of submicron. Fur-thermore, Bammann and Aifantis [59] general-ized the Orowan’s equation (with an orienta-tion factor) of the plastic strain in the macro-scopic plasticity theory in terms of the plasticshear strain and an orientation tensor as fol-lows:

εpij = γpMij , (29)

where Mij is the symmetric Schmidt’s orien-tation tensor. In expressing the plastic straintensor at the macrolevel to the plastic shearstrain at the microlevel, an average form of theSchmidt’s tensor is assumed since plasticity atthe macroscale incorporates a number of differ-ently oriented grains into each continuum point[52, 60]. Moreover, average values are used forthe magnitude of the Burgers vector b and themean spacing between SSDs LS .

The flow stress σ is the conjugate of theeffective plastic strain variable p in macro-plasticity. For proportional, monotonically in-creasing plasticity, p is defined as

p =√

23εp

ijεpij . (30)

Hence, utilizing Eqs. (28) and (29) in Eq. (30)one can write p as a function of SSDs as follows:

p = bLSρSM, (31)

which is referred to as Orowan’s equation,where M =

√2MijMij/3 can be interpreted

as Schmidt’s orientation factor, usually takenequal to 1/2. It is clear from Eq. (31) that theBurgers vector and the dislocation spacing are




two physical length scales that control plasticdeformation.

Substituting ρG and ρS from Eqs. (22) and(31), respectively, into Eq. (21) yields the follow-ing expression for the flow stress:

σ = αG

√3b

LSM

[pµ + LsM rηµ

]1/2µ. (32)

Comparing Eq. (32) with Eq. (25) yields the fol-lowing relations:

γ = µ , m = 2, k = αG

√3b

LSM,

` = LS M r. (33)

The rational for obtaining γ = µ is based onthe following argument. The flow strength de-pends on the total dislocation density ρT , asassumed by Eq. (17). One coupling form be-tween ρS and ρG is presented by Eq. (18) asρT = ρS +ρG with µ = 1 [17, 19, 54], and that ρG

and ρS are linear in the strain gradient [Eq. (22)]and the effective plastic strain [Eq. (31)], respec-tively. One concludes then that an appropriatescalar measure of hardening is given by Eq. (25)with γ = µ = 1. Moreover, another couplingbetween ρS and ρG is presented by Eq. (19) asρ2

T = ρ2S + ρ2

G with µ = 2 [19], where ρS andρG are obtained from Eqs. (31) and (22), respec-tively. Hence one concludes that the harden-ing law [Eq. (25)] is appropriately expressedwith γ = µ = 2. Thus, in general, an appro-priate scalar measure of hardening is given byEq. (25) with γ = µ. Furthermore, the conditionm = 2 is not unreasonable for some materials[40], particularly for some annealed crystallinesolids. The phenomenological measure of theyield stress in uniaxial tension k and the mi-crostructure length-scale parameter ` are nowrelated to measurable physical parameters LS ,b, r, M , α, and G. Moreover, by substitutingLS M from Eq. (33)3 into Eq. (33)4 one can ob-tain a similar relation for ` as proposed by Gao

et al. [17] such that:

` = 3α2br

(G

k

)2

. (34)

We note that the above equation implies thatthe length-scale parameter may vary with thestrain-rate and temperature for a given material

for the case k =_

k(p, T ), where p =√

2εpij ε

pij

/3.

However, for most metals, the flow stress in-creases with the strain rate and decreases withtemperature increase. Thus causing the intrin-sic material length scale to decrease with in-creasing strain rates and to increase with tem-perature decrease. However, opposite behavioris concluded for the gradient term η.

2.2 Model-II

Another approach to form the coupling be-tween SSDs and GNDs is to assume that theoverall flow stress τ has two components; onearising from SSDs τS and a component due toGNDs τG [61]. The following general functionfor τ is then chosen as follows:

τ =[τβS + τβ

G

]1/β(35)

with τS and τG are given by Taylor’s hardeninglaw as follows:

τS = αSGbS√

ρS , (36)

τG = αGGbG√

ρG, (37)

where bS and bG are the magnitudes of theBurgers vectors associated with SSDs andGNDs, respectively, and αS and αG are statisti-cal coefficients, which account for the deviationfrom regular spatial arrangements of the SSDand GND populations, respectively. For an im-penetrable forest, it is reported that αS ≈ 0.85[66] and αG ≈ 2.15 [37].

This general form ensures that τ → τS when-ever τS >> τG and that τ → τGwhenever τS <<




τG . Two values of β are generally investigatedin the literature: (a) β = 1, which correspondsto a superposition of the contributions of SSDsand GNDs to the flow stress (e.g., [61]); and(2) β = 2, which, since the flow stress scaleswith the square root of dislocation density, cor-responds to the superposition of the SSD andGND densities (e.g., [37]). Moreover, Eq. (35)constitutes the nonlocal micromechanical plas-ticity constitutive model due to the presence ofGNDs similar to the nonlocal macromechanicalplasticity constitutive model defined by Eq. (25)due to the presence of strain gradients η.

Expressing Eq. (35) in terms of Eqs. (36) and(37) yields a general expression for the overallflow stress in terms of SSD and GND densities,such that

τ = αSGbS

[ρ

β/2S +

(αGbG

αSbS

)β

ρβ/2G

]1/β

. (38)

Alternatively, Eq. (38) can be redefined in termsof an equivalent total dislocation density ρT asfollows:

ρT =

[ρ

β/2S +

(α2

Gb2G

α2Sb2

S

ρG

)β/2]2/β

(39)

so thatτ = αSGbS

√ρT . (40)

It is important to emphasize once again that thenonlocal effects associated with the presence oflocal deformation gradients at a given materialpoint are incorporated into Eq. (40) through theGNDs density.

Substituting ρG and ρS from Eqs. (22) and(31), respectively, into σ =

√3τ , where τ is

given by Eq. (40), yields the following expres-sion for the flow stress:

σ = αSG

√3bS

LSM×

[pβ/2 +

(α2

GbGLsM r

α2SbS

η

)β/2]1/β

. (41)

Note that we set b = bG in Eq. (22) and b = bS inEq. (31) since the plastic strain gradient and theplastic strain are related to the SSDs and GNDs,respectively. Comparing Eq. (41) with Eq. (25)yields the following relations:

γ = β/2,m = 2, k = αSG

√3bS

LSM,

` = (αG/αS)2 (bG/bS) LS M r. (42)

Moreover, by substituting LS M from Eq. (42)3into Eq. (42)4 one can obtain a relation for ` asfunction of the shear modulus G and the ther-mal stress k, such that

` = 3α2GbGr

(G

k

)2

. (43)

It is noteworthy to mention that by comparingthe results retained from model-I and model-II, Eqs. (42) and (44) with Eqs. (33) and (34),one can notice that model-II gives qualitativelymore general and rational expressions for k and`. k is essentially controlled by SSDs and `hasa more general form as obtained from model-II, Eq. (42)4, as compared to ` obtained frommodel-I, Eq. (33)4. Moreover, Eq. (44) showsthat the material length scale is directly relatedto the storage of geometrically necessary dis-locations. However, both models are identicalwhen αS = αG = α, bS = bG = b, and β = 2µ.Moreover, the condition m = 2 is not unreason-able for some materials as we mentioned ear-lier. However, the authors believe that the ori-gin of this condition, as it is concluded fromthe mathematical setup of model-I and model-II, comes out from the assumption that Taylor’sflow stress is directly proportional to the squareroot of the dislocation density (i.e., τ ∞

√b2ρT ,

which can be rewritten as ρ1/mT with m = 2.

For more generality, we can assume that τ ∝(b2ρT )1/m with m ≥ 1 . This is not the subject ofthis work, but further study needs to be carriedout to show the importance of this found.




3. IDENTIFICATION FROMMICRO-HARDNESS EXPERIMENTS

The conventional hardness test is one of theoldest and simplest methods of material qual-ity evaluation. Tabor [67] and Atkins and Ta-bor [68] showed in their experiments that theelasto-plastic material response in tensile test-ing could be correlated to the response in coni-cal/pyramidal (e.g., Berkovich and Vickers) orspherical (or Brinell) indentation. The funda-mental parameters for indentation tests by con-ical/pyramidal indenter are: the force appliedto the indenter P , the residual contact radius ofindentation a, the contact pressure (hardness)H = P

/πa2, the permanent penetration depth

h, the total penetration depth ht, and the inden-ter geometry (i.e., the angle between the surfaceof the conical indenter and the plane of the sur-face θ). The unloading process in the inden-tation experiment is the most important for aproper specification of these geometric parame-ters. Thus, the residual penetration depth h andcontact radius a should be used as measurabledata in the hardness H calculation.

It is well known by now that indentationtests at scales on the order of one micron haveshown that measured hardness increases sig-nificantly with decreasing indent size. Thishas been attributed to the evolution of geomet-rically necessary dislocations associated withgradients. Next we present a simple procedureto identify the intrinsic material-length param-eter that scales the effect of GNDs using conicalor pyramidal indenter.

Consider the indentation by a rigid cone, asshown schematically in Fig. 1. First, in thisanalysis we assume that the density of geo-metrically necessary dislocations is integratedby the geometry of the indenter and the in-dentation is accommodated by circular loopsof GNDs with Burgers’ vectors normal to theplane of the surface. Since model-II gives moregeneral expressions than model-I, we assume,

in what follows, that the densities of statisti-cally stored dislocations and geometrically nec-essary dislocations are coupled as proposed inmodel-II [Eq. (35)]. Based on the simple modelof GNDs developed first by Stelmashenko et al.[10] and DeGuzman et al. [11] and recently re-called by Nix and Gao [40], an analytical ex-pression for the density of GNDs (ρG) underthe conical indenter can be obtained in terms ofthe material properties and the conical indentergeometry. The development of ρG is presentedin detail here for the convenience of the reader.Following Begley and Hutchinson [42], the in-dentation profile in the unloaded configurationcan be described by

w (r) = r (tan θ)− h for 0 ≤ r ≤ a (44)

where h is the indentation depth and θ is theangle between the surface of the conical inden-ter and the plane of the surface. This angle isrelated to the indentation depth h and the ra-dius of the contact area of the indentation a bytan θ = h/a (see Fig. 1). Both h and a are mea-sured in the unloaded configuration and char-acterized as the residual values after unload-ing. If we assume that the individual dislo-cation loops of GNDs as being spaced equallyalong the surface of the indentation, then it iseasy to show that

Geometrically necessary

dislocations

a

h

q

P

Specimen

Indenter

Original free surface

position

r

w

FIGURE 1. Axisymmetric rigid conical indenter.Geometrically necessary dislocations created dur-ing the indentation process




∣∣∣∣dw

dr

∣∣∣∣ =bG

s= tan θ =

h

a, s =

bGa

h, (45)

where s is the mean spacing between individ-ual slip steps on the indentation surface. If Lis the total length of the injected loops, then be-tween r and r + dr we have

dL = 2πrdr

s= 2πr

h

bGadr. (46)

Integrating from 0 to a gives the total length ofdislocation loops as

L =

a∫

o

2πrh

bGadr =

πah

bG. (47)

Moreover, it is assumed that the dislocationevolution during indentation is primarily gov-erned by a large hemispherical volume V thatscales with the contact radius a around the in-dentation profile. One can then assume that allthe injected loops remain within the hemispher-ical volume V such that:

V =23πa3. (48)

Therefore, the density of geometrically neces-sary dislocations becomes

ρG =L

V=

32bGh

tan2 θ. (49)

In the early hardness experiments, it was con-cluded that the relation between the hardnessH and the permanent penetration depth h fol-lows a power law. Making use of this observa-tion, Tabor [67] specified the mapping from thehardness-indentation depth curve (H−h curve)to the tensile stress-plastic strain curve (σ − p),such that one can assume the following:

H = κσ, (50)

where κ is the Tabor’s factor of κ = 2.8 andc is a material constant with a value of c =

0.4 [67]; while corresponding numerical resultfrom Biwa and Storakers [69] is κ = 3.07. Thisrelation has been extensively verified and usedby many authors (e.g., [40, 49, 58, 69–75]).

Moreover, Based on the assumption of a self-similar deformation field [69, 76], it was shownby using the conical/pyramidal indenter thatthe displacement is proportional to the inden-tation depth h. Based on this observation, Xueet al. [77] showed from numerical experimentsthat the strain field should depend only onthe normalized indentation depth, h/a, and thenormalized position (x/a, y/a, z/a, where x, y,and z are the Cartesian coordinates such thatone can assume that the effective plastic strainp is defined by

p = c

(h

a

)= c tan θ (51)

where c is a material constant on the order of1.0 [77].

Considering the results derived from model-II, the substitution of Eq. (38) into Eq. (50) withσ =

√3τ and Eqs. (51) and (42)4 into Eq. (31),

yield the following expressions for hardness Hand statistically stored dislocations (ρS), respec-tively, as follows:

H =√

3καSGbS

[ρ

β/2S +

(αGbG

αSbS

)β

ρβ/2G

]1/β

,

(52)

ρS =crα2

GbG tan θ

`b2Sα2

S

. (53)

Note that we set b = bS in Eq. (31) since theplastic strain is related to the SSDs. Moreover,we can define the macrohardness Ho as thehardness that would arise from SSDs alone inthe absence of strain gradients, such that [40]

Ho =√

3κτS =√

3καSGbS√

ρS . (54)

With these relations we can now write the mi-crohardness from the conical indenter using




Eqs. (50)–(54) as

(H

Ho

)β

= 1 +(

h∗

h

)β/2

, (55)

where h∗ is a material specific parameter thatcharacterizes the depth dependence of thehardness and depends on the indenter geom-etry as well as on the plastic flow, such that it isderived as

h∗ = ζ` (56)

withζ =

32cr

tan θ. (57)

Equation (56) shows that h∗ is a linear functionof the length-scale parameter `. Thus, h∗ is acrucial parameter, which characterizes the in-dentation size effects, and its accurate exper-imental measure gives a reasonable value forthe length-scale parameter ` obtained by usingEq. (56). Moreover, substituting Eq. (53) intoEq. (54) along with Eq. (44), one can obtain asimple relation to predict the macrohardnessHo as

Ho = κk√

c tan θ, (58)

where k can be obtained by using Eq. (33)3.Equations (56) and (58) are similar to the phe-nomenological relations suggested by Yuanand Chen [44] based on finite element com-putations and the experimental results of Be-gley and Hutchinson [42]. Moreover, Eq. (58)can be used to determine the c parameter ifHo is provided from the indentation test. Ho

corresponds to the saturation value where thehardness H does not change as the indentationdepth h increases (or with further load increase)in the H versus h indentation test curve.

We note that if β = 2 in Eq. (55), one re-tains the relation originally proposed by Nixand Gao [40], where they found a linear de-pendence of the square of the microhardnessH2 to the inverse of the indentation depth 1/h.Nix and Gao [40] also suggested that h∗ andHo are dependent and related through h∗ =

(81/2)bα2 tan2 θ(G/Ho)2. Their relation thus

gives a similar argument to that of Eq. (56),which suggests that h∗ is dependent on theshape of the indenter as well as on the mate-rial property. Furthermore, Yuan and Chen [44]modified the Nix and Gao [40] relation by in-troducing another empirical term (ς∗/h)2 that issubtracted from the right-hand side of Eq. (55)with β = 2, where ς∗ is another material pa-rameter that depends on both material prop-erty and the indenter geometry. However, theyshowed numerically that the effect of this addi-tional parameter ς∗ is generally much smallerthan h∗, and it can be neglected in many cases.

The characteristic form for the depth depen-dence of the hardness presented by Eq. (55)gives a straight line when the data are plot-ted as (H/Ho)β versus h−β/2, the intercept ofwhich is 1 and the slope is h∗β/2. Nix andGao [40] showed for β = 2 that the plot of(H/Ho)2 versus 1/h compares well with the ex-perimental data from micro-indentation testsusing a conical indenter. The length-scale pa-rameter ` = h∗/ζ can then be calculated usingEq. (56), where ζ is determined in terms of theshape of the conical indenter (i.e., tan θ) andthe material properties (i.e., r and c ), whichare known. Therefore, by using Eq. (55) to fitthe hardness experimental data obtained fromindentation tests, one can simply compute theintrinsic length-scale parameter that character-izes the size effects using Eq. (56).

All the experiments used here in identifyingthe length-scale parameter ` were conducted atroom temperature and using a Berkovich trian-gular pyramidal indenter for which the nomi-nal or projected contact area varies as [78]

Ac = 24.5h2 = πa2. (59)

Using this relation together with tan θ = h/ayields

tan θ =√

π

24.5= 0.358. (60)




Note that it was reported by the experimental-ists that no damage occurred in the material be-neath the indenter such that the measured mi-crohardness data provides a true measure ofthe plastic properties of the specimen. Micro-hardness thus provides a convenient tool forthe identification of the plasticity intrinsic ma-terial length scale, where damage is avoided.

Following the method proposed by Nix andGao [40], the hardness results obtained frommicroindentation tests can be displayed as aplot of (H/Ho)β versus h−β/2, as shown forthe (111) single-crystal copper and cold-workedpolycrystalline copper in Fig. 2, for (100) and(110) single crystals of silver in Fig. 3, andannealed and work-hardened alpha brass inFig. 4.

Table 1 shows the calculated values of ` forthe six sets of indentation data described byEq. (55) in Fig. 2–4. The ζ parameter is calcu-lated by using Eq. (57), assuming that c = 1.0[77] and the Nye factor r = 2 [16]. The calcula-tion of `, however, depends strongly on the ma-terial parameters c and r. The resulting values

for the material intrinsic length scale outlinedin Table 1 are in the range of some microme-ters, which confirms well with the observationsof Nix and Gao [40], Begley and Hutchinson[42], Stolken and Evans [9], and Yuan and Chen[44]. Moreover, one can note that ` for the cold-worked sample is smaller than the value forthe annealed sample, indicating that spacingbetween statistically stored dislocations is re-duced in the hardened-worked material. Sim-ilar result has been reported by Nix and Gao[40], and Stolkend and Evans [9]. Apparently,numerical experiments of the indentation prob-lem using finite element method are requiredto verify those findings. However, this is notthe subject of this study and will be shown in aforthcoming paper [33, 34].

It can be seen from the results that as thedepth of indentation h becomes much largerthan the material-length scale parameter `, thegradient effects become smaller and the corre-sponding hardness does not exhibit any inden-tation size effect. Moreover, if the material isintrinsically hard (i.e., shallow indentation

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

h-b/2

(mm-b/2

)

(H/H

o)b

111 Single Crystal Cu

Cold-Worked Polycrystal Cu

Experimental Data (McElhaney et al. 1997)

(H/581)2=1+(1.538/h)

(H/834)2=1+(0.444/h)

FIGURE 2. Comparison of the experimental results and the prediction of Eq. (55) to determine theintrinsic material length scale ` for copper




0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 1 2 3 4 5 6h

-b/2(mm

-b/2)

(H/H

o)b

110 Single Crystal Ag

100 Single Crystal Ag

Experimenta Data (Ma and Clarke 1995)

(H/340)2.12

=1+(0.813/h)1.06

(H/361)1.9

=1+(0.398/h)0.95

FIGURE 3. Comparison of the experimental results and the prediction of Eq. (55) to determine theintrinsic material length scale ` for silver

1.00

1.10

1.20

1.30

1.40

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

h-b/2

(mm-b/2

)

(H/H

o)b

Work-Hardened Alpha Brass

Annealed Alpha Brass

Experimental Data (Elmustafa and Stone 2002)

(H/1060)2.1

=1+(0.604/h)1.05

(H/1325)2=1+(0.359/h)

FIGURE 4. Comparison of the experimental results and the prediction of Eq. (55) to determine theintrinsic material length scale ` for alpha brass




Table 1. Calculation of the length-scale parameter ` from the fitted microhardness data

Material β Ho

(MPa)h∗

(µm)ζ(Eq. (57))

` = h∗/ζ(µm)

(111) single crystal Cu (annealed) 2.00 581 1.538 0.268 5.74(12.00)1

(20.00)2

Polycrystalline Cu (cold-worked) 2.00 834 0.444 0.268 1.66(5.84)1

(6.00)2

(100) signle crystal Ag 2.12 340 0.813 0.268 3.03(0.34-0.39)3

(110) single crystal Ag 1.90 361 0.398 0.268 1.49(0.19-0.22)3

Alpha brass (annealed) 2.10 1060 0.604 0.268 2.25Alpha brass (cold-worked) 2.00 1325 0.359 0.268 1.34

1 Nix and Gao [40]2 Yuan and Chen [44]3 Begley and Hutchinson [42]

depths or nanoindentation), the microhardnessmodel given by Eq. (55) gives more accurateresults than Nix and Gao [40] model. Thisimprovement in the model predictions is at-tributed to β, which can be considered as amaterial parameter that depends on the crystalstructure. The role of the material parameterβ in the gradient plasticity models and in pre-dicting the micro- and nanohardness have beenstudied thoroughly by Abu Al-Rub and Voyi-adjis [45]. It is found that β constant, which as-sesses the coupling between statistically storeddislocations and geometrically necessary dislo-cations in Taylor’s hardening law, plays a cru-cial role in correlating the hardness data fromnano- and microindentations, where the Nixand Gao [40] model failed to do so. Apparently,further evelopments of the gradient plasticitytheory are required.

Moreover, it is obvious that numerical ex-periments of the indentation problem are re-quired to verify the above findings. Examples

of two-dimensional verses three-dimensionalindentation simulations can be found in a re-cent work by Muliana et al. [79]. In addition,numerical examples showing the effectivenessof the proposed equations in capturing the size-dependent behavior as compared to the experi-mental results are required. Examples for suchproblems are microbending of thin beams, mi-crotorsion of thin wires, growth of microvoids,shear bands, and rate-dependent micro- andnanoindentation of thin films. However, thisis not the subject of this study and will be ad-dressed in a forthcoming work by the authors.

4. ON THE EVOLUTION OF THEMATERIAL-INTRINSIC LENGTHSCALE

The aforementioned conclusion in Section 2,which states that Eq. (34) or (44) implies thatboth the strain-rate effect and temperature vari-ation are crucial to the reliability of the es-




timated length-scale parameter `, motivatesthe following argument. To the authors’ bestknowledge, neither a numerical investigationnor an experimental study has been carried outfor the influence of strain-rate effect and tem-perature variation on the strain-gradient plas-ticity, or more specifically, on the size effect ex-hibited by the presence of geometrically nec-essary dislocations. For example, the plasticzone ahead of the crack tip will decrease withincreasing yield stress, which for small-scaleyielding is of the order of microns. Therefore,the consideration of strain-rate effect and tem-perature variation on gradient plasticity, par-ticularly in dynamic problems, becomes morenecessary. Existing theories of gradient plastic-ity, however, have failed to explain such behav-ior. Equations (34) or (44) shows that ` is notconstant for a given material, but is requiredto change as the flow stress in the absence ofstrain-gradient changes. This is crucial for theclassical viscoplasticity theory since it cannotpredict size effect, too. Therefore, the consider-ation of the evolution of the material intrinsic-length parameter with time ` in dynamic prob-lems becomes more necessary. Moreover, thereare indications that a constant value of the ma-terial length scale is not always realistic, andthat different problems could require differentvalues (see for example the work of Aifantis[80] and Tsagrakis and Aifantis [46]). Someauthors argued the necessity of a length-scaleparameter in the gradient theories that changewith time in order to achieve an efficient com-putational convergence while conducting mul-tiscale simulations [51].

The rate and temperature dependence of `for metal crystals can be explained by differ-ent physical mechanisms of dislocation motion.Taking the time derivative of the general rela-tion for `, Eq. (42)4, yields

˙ = AυS , (61)

where υS = LS is the average dislocation speed

and A = (αG/αS)2 (bG/bS) M r. In obtainingEq. (61), it was assumed that the slip systemdoes not change with time such that the timederivative of the Burgers vector magnitude b,Schmidt’s orientation factor M , and the Nyefactor r is zero. One might think that this as-sumption is limiting in that plastic strains, evenin small deformation theory, will cause a ma-terial rotation and then strain-gradient evolu-tion, as stated by Eq. (26), which is taken fromthe work of Fleck and Hutchinson [19]. There-fore, such an assumption implies that there isno change in the geometrically necessary dis-locations; thus, the absence of strain-gradientchanges. However, this assumption is imper-ative since we need to study the effect of strainrates and temperature variations on ` such that` is not constant for a given material, but is re-quired to change as the flow stress in the ab-sence of strain gradient changes [see Eq. (25)].

Since plastic flow occurs by the motion ofdislocations, the rate at which it takes placedepends on how fast the dislocations move,how many dislocations are moving in a givenvolume, how much displacement is carried byeach dislocation, and how much rotation eachcrystal undergoes. The theory of crystal dis-locations shows that if a dislocation (in thiscase a statistically stored dislocation) is mov-ing through rows of barriers formed by obsta-cle dislocations (in this case geometrically nec-essary dislocations), then its velocity can be de-termined from the following expression:

υS =LS

tw + tt, (62)

where the total transient time of a dislocation isequal to the sum of the waiting time tw spent atthe obstacle and the travel time between obsta-cles tt. If the ratio tw/tt increases, then the straingradients become more important and the dis-location velocity υS , in Eq. (62), can be approx-imated by the expression

υS = νLS , (63)




where ν = 1/tw is the frequency of successfuljumps or the rate at which the SSDs overcomethe GNDs. This is defined from statistical con-siderations as follows (e.g., [59, 81]):

ν = νo exp(−U (σ∗)

kT

), (64)

where νo is the fundamental vibrational fre-quency of the dislocation (considerably lowerthan the vibrational frequency of the atom), k isthe Boltzmann’s constant, T is the absolute tem-perature, and U is the activation energy, whichmay depend not only on the applied thermalstress σ∗ but also on the temperature and the in-ternal structure. Expressions for the activationenergy for various types of obstacles are givenby Kocks et al. [56], where a generalized equa-tion for these shapes with two parameters, a1

and a2, has been proposed,

U = Uo

[1−

(σ∗

σ∗o

)a1]a2

. (65)

Substitution of Eqs. (63)–(65) into Eq. (61) alongwith Eq. (42)4 gives the following expressionfor the evolution of the length-scale parameter˙ as

˙ = `νo exp[−Uo

kT

{1−

(σ∗

σ∗o

)a1}a2

]. (66)

This equation shows that ` is temperature de-pendent. Moreover, it is well known that theflow thermal stress σ∗ of most metallic materi-als increases with strain rate, which also makesEq. (66) strain-rate dependent. There are dif-ferent empirical and physically based dynamicconstitutive models that have been proposed toestimate the dynamic thermal flow stress σ∗ formetallic materials at high-strain rates, which in-corporate strain, strain rate, temperature, dam-age history, plastic strain rate history, dislo-cation dynamics, grain size, and other intrin-sic variables (e.g., [81-84]). These models can

be combined with Eq. (66) to study the size-effect phenomena in dynamic problems. Fur-thermore, when plasticity occurs (i.e., σ∗ = σ∗o ,˙ becomes ˙ = `νo, where ` is given as the esti-mate identified from microhardness results.

5. CONCLUSIONS

This paper addresses the problem of determin-ing the intrinsic length-scale parameter of thegradient plasticity theory. The definition andmagnitude of the intrinsic length scale are keysto the development of the new theory of plastic-ity that incorporates size effects. It also sets upthe main assumptions to determine the lengthscales that will be essentially the basis for ac-curate qualitative and quantitative multiscalemodeling, simulation, and design of emergingmicro- and nanosystems. Two different mi-cromechanical flow stress models to assess thecoupling between statistically stored disloca-tions (SSDs) and geometrically necessary dislo-cations (GNDs) are discussed. One in which theSSDs and GNDs are simply summed (model-I)and one in which, implicitly, their accompany-ing strengths are added (model-II). These twodislocation interaction models, which are basedon Taylor’s hardening law, are used to iden-tify the deformation-gradient-related intrinsiclength-scale parameter ` in terms of measur-able microstructural physical parameters. Thisis done through using the gradient theory tobring the microstructural (described by Tay-lor’s hardening law) and continuum (describedby the strain-hardening power law) descrip-tions of plasticity closer together. As a result` is defined in terms of the average distance be-tween statistically stored dislocations LS (char-acterizes the characteristic length of plasticityphenomenon), the Nye factor r (characterizesthe microstructure dimension, e.g., grain size,grain boundary width, obstacle spacing, andradius), Schmidt’s orientation factor M (charac-terizes the lattice rotation), the Burgers vector b




(characterizes the displacement carried out byeach dislocation), and the empirical constant α(characterizes the deviation from regular spa-tial arrangement of the SSD or GND popula-tions).

While such an increasing interest in gradient-enhanced theories can be understood in viewof the aforementioned remedies they provided,actual experiments for the direct measurementof the material-intrinsic length parameter intro-duced by the phenomenological gradient coef-ficients are lacking. When considering the mi-crostructure with localization zones, gradient-dependent behavior is expected to become im-portant once the length scale associated withthe local deformation gradients becomes suffi-ciently large when compared with the charac-teristic dimension of the system. It follows thatsuch experiments could be difficult to designand interpret. However, microhardness testsprovide a method to determine the material-intrinsic length-scale parameter ` in the gradi-ent plasticity models.

This paper, therefore, provides an initial ef-fort in this direction, where we discuss the is-sue of size effect and the calibration of gra-dient theory in terms of micro- and nanoin-dentation experiments to determine the valueof the material-intrinsic length parameter thatscales the gradient effects. We show, in par-ticular, that gradient plasticity models can beused for the interpretation of size effect experi-ments in micro- and nanoindentation. In fact,the aforementioned gradient models are cali-brated by fitting the corresponding length-scaleparameter to the experimental data. Micro-and nanoindentation tests can thus be used forthe experimental determination of the gradientlength-scale parameter. Moreover, an evolutionlaw has been derived for the intrinsic-materiallength scale ` in gradient plasticity in termsof the flow stress and temperature. This lawsuggests that the material-intrinsic length scaledecreases with increasing strain rates, and in-

creases with temperature. However, additionalexperimental measurements and numerical in-vestigations are needed to fully verify the de-rived micromechanical equations for the mate-rial intrinsic length parameter `.

The gradient law proposed in Eq. (25) isbased on dislocation-mechanics considerations,which is a departure from all the current gradi-ent theories that are based on the ideal assump-tion of all the obstacles being equally strongand equally spaced along a straight or curvedcontacting line (i.e., ρT = ρS + ρG). The real sit-uation in experiments suggests that the hard-ening law cannot be taken as a simple sumof the densities of SSDs and GNDs such thatρT =

[ρβ

S + ρβG

]1/β.This general form of the

gradient plasticity theory encompasses most ofthe gradient plasticity theories found in theopen literature and forms a promising basis forfuture multiscale computations. In addition,this model gives an alternative framework thatcould model size-dependent plasticity withouthigher-order stresses or additional boundaryconditions, so as to preserve the essential struc-ture of classical plasticity. This approach hasthe advantages that it is simpler overall and canbe easily implemented into existing finite ele-ment codes.

Finally, no one up until now can claim thathe obtained the actual material-intrinsic lengthscale because of the lack of experimental pro-cedures to do so and the lack of solid physicalinterpretations of the material-intrinsic lengthscale. Therefore, there still an open question forresearchers: What is the physical interpretationof the constitutive length parameter `?

ACKNOWLEDGMENT

The authors gratefully acknowledge the finan-cial support by the Air Force Institute of Tech-nology, at Wright Patterson Air Force Base,Ohio.




REFERENCES

1. Lloyd, D.J. Particle reinforced aluminum andmagnesium matrix composites. Int. Mater. Rev.39:1–23, 1994.

2. Rhee, M., Hirth, J.P., and Zbib, H.M. A su-perdislocation model for the strengthening ofmetal matrix composites and the initiation andpropagation of shear bands. Acta Metall. Mater.42:2645–2655, 1994.

3. Rhee, M., Hirth, J.P., and Zbib, H.M. On thebowed out tilt wall model of flow stress andsize effects in metal matrix composites. ScriptaMetall. Mater. 31:1321–1324, 1994.

4. Zhu, H.T., and Zbib, H.M. Flow strength andsize effect of an Al–Si–Mg composite modelsystem under multiaxial loading. Scripta Met-all. Mater. 32:1895–1902, 1995.

5. Nan, C.-W., and Clarke, D.R. The influence ofparticle size and particle fracture on the elas-tic/plastic deformation of metal matrix com-posites. Acta Mater. 44:3801–3811, 1996.

6. Kiser, M.T., Zok, F.W., and Wilkinson, D.S. Plas-tic flow and fracture of a particulate metal ma-trix composite. Acta Mater. 44:3465–3476, 1996.

7. Zhu, H.T., Zbib, H.M., and Aifantis, E.C. Straingradients and continuum modeling of size ef-fect in metal matrix composites. Acta Mechan.121:165–176, 1997.

8. Fleck, N.A., Muller, G.M., Ashby, M.F., andHutchinson, J.W. Strain gradient plasticity: the-ory and experiment. Acta. Metallurgica et Mate-rialia 42:475–487, 1994.

9. Stolken, J.S., and Evans, A.G. A microbendtest method for measuring the plasticity length-scale. Acta Mater. 46:5109–5115, 1998.

10. Stelmashenko, N.A., Walls, M.G., Brown, L.M.,and Milman, Y.V. Microindentation on W andMo oriented single crystals: an STM study. ActaMetall. Mater. 41:2855–2865, 1993.

11. De Guzman, M.S., Neubauer, G., Flinn, P., andNix, W.D. The role of indentation depth onthe measured hardness of materials. Mater. Res.Symp. Proc. 308:613–618, 1993.

12. Ma, Q., and Clarke, D.R. Size dependent hard-ness in silver single crystals. J. Mater. Res.10:853–863, 1995.

13. McElhaney, K.W., Valssak, J.J., and Nix, W.D.Determination of indenter tip geometry and in-dentation contact area for depth sensing inden-tation experiments. J. Mater. Res. 13:1300–1306,1998.

14. Elmustafa, A.A., and Stone, D.S. Indentationsize effect in polycrystalline F.C.C metals. ActaMater. 50:3641–3650, 2002.

15. Hill, R. Mathematical Theory of Plasticity, OxfordUniversity Press, Oxford, 1950.

16. Arsenlis, A., and Parks, D.M. Crystallo-graphic aspects of geometrically-necessary andstatistically-stored dislocation density. ActaMater. 47:1597–1611, 1999.

17. Gao, H., Huang, Y., and Nix, W.D. Modelingplasticity at the micrometer scale. Naturwis-senschaften 86:507–515, 1999.

18. Ashby, M.F. The deformation of plasticallynon-homogenous alloys. Phil. Mag. 21:399–424,1970.

19. Fleck, N.A., and Hutchinson, J.W. Strain gradi-ent plasticity. Adv. Appl. Mech. 33:295–361, 1997.

20. Voyiadjis, G. Z., Aifantis, E. C., and Weber, G.Constitutive modeling of plasticity in nanos-tructured materials. In Nano-Scale Mechanics ofNanostructured Materials for the LaRC Nanotech-nology Program - NASA, Kluwer Academic Pub-lishers, Dordrecht, 2003.

21. Aifantis, E.C. On the microstructural origin ofcertain inelastic models. J. Eng. Mater. Technol.106:326–330, 1984.

22. Lasry, D., and Belytschko, T. Localization lim-iters in transient problems. Int. J. Solids Struct.24:581–597, 1988.

23. Mühlhaus, H.B., and Aifantis, E.C. A varia-tional principle for gradient plasticity. Int. J.Solids Struct. 28:845–857, 1991.

24. Kroner, E. Elasticity theory of materials withlong range cohesive forces. Int. J. Solids Struct.24:581–597, 1967.

25. Eringen, A.C., and Edelen, D.G.B. On non-localelasticity. Int. J. Eng. Sci. 10:233–248, 1972.

26. Pijaudier-Cabot, T.G.P., and Bazant, Z.P. Non-local damage theory. ASCE J. of Engng. Mech.113:1512–1533, 1987.

27. de Borst, R., and Mühlhaus, H.-B. Gradient-




dependent plasticity formulation and algorith-mic aspects. Int. J. Numer. Met. Engng. 35:521–539, 1992.

28. Fleck, N.A. and Hutchinson, J.W. A reformula-tion of strain gradient plasticity. J. Mech. Phys.Solids 49:2245–2271, 2001.

29. Peerlings, R.H.J., de Borst, R., Brekelmans,W.A.M., and de Vree, J.H.P. Gradient enhanceddamage for quasi-brittle materials. Int. J. Nu-mer. Met. Engng. 39:3391–3403, 1996.

30. Kuhl, E., Ramm, E., and de Borst, R. Ananisotropic gradient damage model for quasi-brittle materials. Comp. Met. Appl. Mech. Eng.183:87–103, 2000.

31. Voyiadjis, G.Z., Deliktas, B., and Aifantis, E.C.Multiscale analysis of multiple damage me-chanics coupled with inelastic behavior of com-posite materials. ASCE J. Eng. Mech. 127(7):636–645, 2001.

32. Voyiadjis, G.Z., and Dorgan, R.J. Gradient for-mulation in coupled damage-plasticity. Arch.Mech. 53(4-5):565–597, 2001.

33. Voyiadjis, G.Z., Abu Al-Rub, R.K., and Pala-zotto, A.N. Non-local coupling of viscoplas-ticity and anisotropic viscodamage for impactproblems using the gradient theory. Arch. Mech.55(1):39–89, 2003.

34. Voyiadjis, G.Z., Abu Al-Rub, R.K., and Pala-zotto, A.N. Thermodynamic formulationsfor non-local coupling of viscoplasticity andanisotropic viscodamage for dynamic localiza-tion problems using gradient theory. Int. J. Plas-ticity (in press).

35. Shu, J.Y., and Fleck, N.A. Strain gradient crys-tal plasticity: Size dependent deformation inbicrystals. J. Mech. Phys. Solids 47:297–324, 1999.

36. Shu, J.Y., and Barrlow, C.Y. Strain gradient ef-fects on microscopic strain field in a metal ma-trix composite. Int. J. Plasticity 16:563–591, 2000.

37. Busso, E.P., Meissonnier, F.T., and O’Dowd,N.P. Gradient-dependent deformation of two-phase single crystals. J. Mech. Phys. Solids48:2333–2361, 2000.

38. Bassani, J.L. Incompatibility and a simple gra-dient theory of plasticity. J. Mech. Phys. Solids49:1983–1996, 2001.

39. Xue, Z., Huang, Y., and Li, M. Particle size ef-

fect in metallic materials: a study by the theoryof mechanism-based strain gradient plasticity.Acta Mater. 50:149–160, 2002.

40. Nix, W.D., and Gao, H. Indentation size effectsin crystalline materials: a law for strain gradi-ent plasticity. J. Mech. Phys. Solids 46(3):411–425,1998.

41. Shu, J.Y., and Fleck, N.A. The prediction of asize effect in micro-indentation. Int. J. SolidsStruct. 35:1363–1383, 1998.

42. Begley, M.R., and Hutchinson, J.W. The me-chanics of size-dependent indentation. J. Mech.Phys. Solids 46:2049–2068, 1998.

43. Huang, Y., Xue, Z., Gao, H., and Xia, Z.C. Astudy of micro-indentation hardness tests bymechanism-based strain gradient plasticity. J.Mat. Res. 15:1786–1796, 2000.

44. Yuan, H., and Chen, J. Identification of theintrinsic material length in gradient plasticitytheory from micro-indentation tests. Int. J.Solids Struct. 38:8171–8187, 2001.

45. Abu Al-Rub, R.K., and Voyiadjis, G.Z. Ana-lytical and experimental determination of thematerial intrinsic length scale of strain gradi-ent plasticity theory from micro- and nano-indentation experiments. Int. J. Plasticity (inpress).

46. Tsagrakis, I., and Aifantis, E.C. Recent develop-ments in gradient plasticity-Part I: formulationand size effects. ASME Trans., J. Eng. Mat. Tech.124:352–357, 2002.

47. de Borst, R., Sluys, L.J., Mühlhaus, H.-B., andPamin, J. Fundamental issues in finite elementanalysis of localization of deformation. Eng.Comp. 10:99–121, 1993.

48. Bammann, D.J., Mosher, D., Hughes, D.A.,Moody, N.R., and Dawson, P.R. Using Spa-tial Gradients to Model Localization Phenomena,Sandia National Laboratories Report, SAND99-8588, Albuquerque, NM, and Livermore, CA,1999.

49. Poole, W.J., Ashby, M.F., and Fleck, N.A. Micro-hardness of annealed and work-hardened cop-per polycrystals. Scripta Mater. 34(4):559–564,1996.

50. Voyiadjis, G. Z., and Abu Al-Rub R. K. Lengthscales in gradient plasticity. In Multiscale Model-




ing and Characterization of Elastic-Inelastic Behav-ior of Engineering Materials, Morocco, October,Ahzi, S., and Cherkaoui, M., Eds. Kluwer Aca-demic Publishers, Dordrecht, 8 pages, 2002.

51. Pamin, J. Gradient-Dependent Plasticity in Nu-merical Simulation of Localization Phenomena,Ph.D. dissertation, Delft University Press, TheNetherlands, 1994.

52. Aifantis, E.C. The physics of plastic deforma-tion. Int. J. Plasticity 3:211–247, 1987.

53. Huang, Y., Gao, H., Nix, W.D., and Hutchinson,J.W. Mechanism-based strain gradient plastic-ity - II Analysis. J. Mech. Phys. Solids 48:99–128,2000a.

54. Gao, H., Huang, Y., Nix, W.D., and Hutchinson,J.W. Mechanism-based strain gradient plastic-ity - I. Theory. J. Mech. Phys. Solids 47(6):1239–1263, 1999a.

55. Smyshlyaev, V.P., and Fleck, N.A. The role ofstrain gradients in the grain size effect for poly-crystals. J. Mech. Phys. Solids 44:465–495, 1996.

56. Kocks, U.F., Argon, A.S., and Ashby, M.F. Ther-modynamics and Kinetics of Slip, Oxford, 1975.

57. Kroner, E. Dislocations and continuum me-chanics. Appl. Mech. Rev. 15:599–606, 1962.

58. Kucharski, S., and Mroz, Z. Identification ofplastic hardening parameters of metals fromspherical indentation tests. Mater. Sci. Eng.A318:65–76, 2001.

59. Bammann, D.J., and Aifantis, E.C. On a pro-posal for a continuum with microstructure.Acta Mech. 45:91–121, 1982.

60. Bammann, D.J., and Aifantis, E.C. A model forfinite-deformation plasticity. Acta Mech. 69:97–117, 1987.

61. Columbus, D., and Grujicic, M. A A com-parative discrete-dislocation/nonlocal crystal-plasticity analysis of plane-strain mode I frac-ture. Mater. Sci. Eng. A323:386–402, 2002.

62. Kocks, U.F. A statistical theory of flow stressand work hardening. Phil. Mag. 13:541, 1966.

63. Tabor, D. The Hardness of Metals, ClarendonPress, Oxford, 1951.

64. Atkins, A.G., and Tabor, D.T. Plastic indenta-tion in metals with cones. J. Mech. Phys. Solids13:149–164, 1965.

65. Biwa, S., and Storakers, B. An analysis of fullyplastic Brinell indentation. J. Mech. Phys. Solids43(8):1303–1333, 1995

66. Robinson, W.H., and Truman, S.D. Stress-straincurve for aluminum from a continuous inden-tation test. J. Mater. Sci. 12(1):1961–1965, 1977.

67. Tangena, A.G., and Hurkx, G.A.M. Determina-tion of stress-strain curves of thin layers usingindentation tests. ASME J. Eng. Mater. Technol.108(3):230–232, 1986.

68. Huber, N., and Tsakmakis, Ch. Determinationof constitutive properties from spherical inden-tation data using neural networks. Part II: plas-ticity with nonlinear isotropic and kinematichardening. J. Mech. Phys. Solids 47:1589–1607,1999.

69. Caceres, C.H., and Poole, W.J. Hardness andflow strength in particulate metal matrix com-posites. Mater. Sci. Eng. A332:311–317, 2002.

70. Swadener, J.G., George, E.P., and Pharr, G.M.The correlation of the indentation size effectmeasured with indenters of various shapes. J.Mech. Phys. Solids 50:681–694, 2002.

71. Oliver, W.C., Pharr, G.M. An improved tech-nique for determining hardness and elasticmodulus using load and displacement sensingindentation experiments. J. Mater. Res. 7:1564–1583, 1992.

72. Hill, R., Storakers, B., and Zdunek, A.B. A the-oretical study of the Brinell hardness test. Proc.Roy. Soc. (London) A423(1865):301–330, 1989.

73. Xue, Z., Huang, Y., Hwang, K.C., and Li,M. The influence of indenter tip radius onthe micro-indentation hardness. ASME J. Eng.Mater. Technol. 124:371–379, 2002.

74. McClintock, F.A., and Argon, A.S. MechanicalBehavior of Materials, Addison-Wesley Publish-ing Company, Reading, MA, 1966

75. Muliana, A., Steward, R., Haj-Ali, R., Saxena,A. Artificial neural network and finite elementmodeling of nanoindentation tests. Metallur-gical and Materials Transactions A33:1939–1947,2002.

76. Aifantis, E.C. Strain gradient interpretation ofsize effects. Int. J. Fract. 95:299–314, 1999.

77. Mayers, M.A., Benson, D.J., Vohringer, O., Kad,B.K., Xue, Q., and Fu, H.-H. Constitutive de-




scription of dynamic deformation: physically-based mechanisms. Mater. Sci. Eng. A322:194–216, 2002.

78. Johnson, G.R., and Cook H. W. Fracture char-acteristics of three metals subjected to variousstrains, strain rates, temperature and pressures.Engng. Fract. Mech. 21(1):31–48, 1985.

79. Zerilli, F.J., and Armstrong, R.W. Dislocation-mechanics-based constitutive relations for ma-terial dynamics calculations. J. App. Phys.61(5):445–459, 1987.

80. Steinberg, D.J., and Lund, C.M. A constitutivemodel for strain rates from 10−4 to 106. J. App.Phys. 65(4):1528–1533, 1989.



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