Gradient plasticity theory with a variable length scale...

International Journal of Solids and Structures 42 (2005) 3998–4029

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Gradient plasticity theory with a variable lengthscale parameter

George Z. Voyiadjis *, Rashid K. Abu Al-Rub

Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, USA

Received 4 December 2004; received in revised form 6 December 2004Available online 21 January 2005

Abstract

The definition and magnitude of the intrinsic length scale are keys to the development of the gradient plasticity the-ory that incorporates size effects. However, a fixed value of the material length-scale is not always realistic and differentproblems could require different values. Moreover, a linear coupling between the local and nonlocal terms in the gra-dient plasticity theory is not always realistic and that different problems could require different couplings. This workaddresses the proper modifications required for the full utility of the current gradient plasticity theories in solvingthe size effect problem. It is shown that the current gradient plasticity theories do not give sound interpretations ofthe size effects in micro-bending and micro-torsion tests if a definite and fixed length scale parameter is used. A general-ized gradient plasticity model with a non-fixed length scale parameter is proposed based on dislocation mechanics. Thismodel assesses the sensitivity of predictions to the way in which the local and nonlocal parts are coupled (or to the wayin which the statically stored and geometrically necessary dislocations are coupled). In addition a physically-based rela-tion for the length scale parameter as a function of the course of deformation and the material microstructural featuresis proposed. The proposed model gives good predictions of the size effect in micro-bending tests of thin films and micro-torsion tests of thin wires.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Gradient plasticity; Material length scale; Size effects; Micro-bending; Micro-torsion

1. Introduction

Material length scales or size effects (i.e. the dependence of mechanical response on the structure size) areof great importance to many engineering applications. Moreover, the emerging area of nanotechnology

0020-7683/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijsolstr.2004.12.010

* Corresponding author. Tel.: +1 225 578 8668; fax: +1 225 578 9176.E-mail address: [email protected] (G.Z. Voyiadjis).

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G.Z. Voyiadjis, R.K. Abu Al-Rub / International Journal of Solids and Structures 42 (2005) 3998–4029 3999

exhibits important strength differences that result from continuous modification of the material microstruc-tural characteristics with changing size, whereby the smaller is the size the stronger is the response. Thereare many experimental observations which indicate that, under certain specific conditions, the specimen sizemay significantly affect deformation and failure of the engineering materials and a length scale is requiredfor their interpretation. An extensive review of the experimental observations of size effects has been sum-marized in Abu Al-Rub and Voyiadjis (2004,) and Voyiadjis and Abu Al-Rub (2004). This dependence ofmechanical response on size could not be explained by the classical continuum mechanics since no lengthscale enters the constitutive description. However, the gradient plasticity theory has been successful inaddressing the size effect problem. This success stems out from the incorporation of a microstructurallength-scale parameter in the governing equations of the deformation description.

Gradient approaches typically retain terms in the constitutive equations with specific order of spatialgradients and coefficients that represent length-scale measures of the deformed microstructure associatedwith the nonlocal continuum. Aifantis (1984) was one of the first to study the gradient regularizationin solid mechanics. An extensive review of the recent developments in gradient theories can be found inVoyiadjis et al. (2003, 2004) and Abu Al-Rub and Voyiadjis (2004,). A short review of the developmentsof the gradient-dependent theory is presented here. Other researchers have contributed substantially tothe gradient approach with emphasis on numerical aspects of the theory and its implementation in finiteelement codes: Lasry and Belytschko (1988), Zbib and Aifantis (1988), and de Borst and co-workers(e.g. de Borst and Muhlhaus, 1992; de Borst et al., 1993; Pamin, 1994; de Borst and Pamin, 1996). Gradientthermodynamic damage models were also introduced by Fremond and Nedjar (1996) and Voyiadjis et al.(2001, 2003, 2004).

In parallel, other approaches that have length-scale parameters in their constitutive structure (commonlyreferred to as nonlocal theories) have appeared as an outgrowth of earlier work by Eringen (e.g. Eringenand Edelen, 1972) and Bazant (e.g. Pijaudier-Cabot and Bazant, 1987; Bazant and Pijaudier-Cabot, 1988).Nonlocal models also abandon the assumption that the stress at a given point is uniquely determined by thehistory of strain and temperature at this point only. They take into account possible interactions with othermaterial points in the vicinity of that point. Theoretically, the stress at a point can depend on the strainhistory in the entire body, but the long-range interactions certainly diminish with increasing distance,and can be neglected when the distance exceeds the length of interaction. Early studies on nonlocal elastic-ity, motivated by homogenization of the atomic theory of Bravais lattices, aimed at a better description ofphenomena taking place in crystals on a scale comparable to the range of interatomic forces. They showedthat nonlocal continuum models can approximate the dispersion of short elastic waves and improve thedescription of interactions between crystal defects such as vacancies, interstitial atoms, and dislocations.Eringen (e.g. Eringen and Edelen, 1972) introduced the idea of nonlocal continuum in elasticity and phe-nomenological hardening plasticity; while, Bazant (e.g. Pijaudier-Cabot and Bazant, 1987; Bazant andPijaudier-Cabot, 1988) extended the nonlocal concept to strain softening materials and introduced thenonlocal damage theory.

Another class of gradient theories have advocated in the last decade that the stress tensor of the resultingthree-dimensional constitutive equations is an asymmetric stress tensor. These theories assume higher-ordergradients of the displacement field (e.g. Fleck et al., 1994; Fleck and Hutchinson, 1993, 1997, 2001; Nix andGao, 1998; Gao et al., 1999a,b; Huang et al., 2000a; Gao and Huang, 2001; Hwang et al., 2002). This groupof theories is in fact a particular case of generalized continua, such as micromorphic continua (Eringen,1968), or continua with microstructure (Mindlin, 1964), which were all inspired by the pioneering work ofthe Cosserat brothers (Cosserat and Cosserat, 1909). The Cosserat continuum (or micropolar continuum) en-hances the kinematic description of deformation by an additional field of local rotations, which can dependon the rotations corresponding to the displacement field, i.e. on the skew-symmetric part of the displace-ment gradient for the small displacement theory, or on the rotational part of the polar decomposition inthe large-displacement theory. In this connection, a similarly motivated strain gradient theory of plasticity

4000 G.Z. Voyiadjis, R.K. Abu Al-Rub / International Journal of Solids and Structures 42 (2005) 3998–4029

based on incompatible lattice deformations was recently advanced by Acharya and Bassani (2000), Bassani(2001), and Gurtin (2002, 2003).

All the theories mentioned so far include in their structure explicit material length scale measures. How-ever, incorporation of rate-dependent viscous terms introduces an implicit length scale measure and limitslocalization in dynamic or quasi-static problems (e.g. Perzyna, 1963; Needleman, 1988; Wang et al., 1998;Voyiadjis et al., 2003, 2004).

The gradient theory has been applied to interpret size-dependent phenomena including, shear banding,micro- and nano-indentation, twist of thin wires, bending of thin films, void growth, crack tip plasticity,fine-grained metals, strengthening in metal matrix composites, multilayers, etc. [see Qiu et al. (2003) fora detailed review]. However, the full utility of the gradient-type theories in bridging the gap between mod-eling, simulation, and design of modern technology hinges on one�s ability to determine accurate values forthe constitutive length-scale parameter that scales the effects of strain gradients. The study of Begley andHutchinson (1998) and Shu and Fleck (1998) indicated that indentation experiments might be the mosteffective test for measuring the length-scale parameter ‘. Nix and Gao (1998) estimated the material lengthscale parameter ‘ from the micro-indentation experiments of McElhaney et al. (1998) to be ‘ = 12 lm forannealed single crystal copper and ‘ = 5.84 lm for cold worked polycrystalline copper. Yuan and Chen(2001) proposed that the unique intrinsic material length parameter ‘ can be computationally determinedby fitting the Nix and Gao (1998) model from micro-indentation experiments and they have identified ‘ tobe 6 lm for polycrystal copper and 20 lm for single crystal copper. By fitting micro-indentation hardnessdata, Begley and Hutchinson (1998) have estimated that the material length-scale associated with thestretch gradients ranges from 1/4 to 1 lm, while the material lengths associated with rotation gradientsare on the order of 4 lm. Other tests also have been used to determine ‘. Based on Fleck et al. (1994) mi-cro-torsion tests of thin copper wires and Stolken and Evans (1998) micro-bend tests of thin nickel beams,the material length parameter is estimated to be ‘ = 4 lm for copper and ‘ = 5 lm for nickel. Recently,Abu Al-Rub and Voyiadjis (2004,) and Voyiadjis and Abu Al-Rub (2004) proposed a dislocation mechan-ics-based analytical model of a solid being indented with a spherical or pyramidal indenter to obtain valuesfor the length scale parameter. The values of ‘ inferred from micro and nano-hardness results for a numberof materials lies within the range of 1/4–5 lm, with the hardest materials having the smallest values of ‘.

In spite of the fact regarding the crucial importance of the length-scale parameter in gradient theory,very limited work focused on the physical origin of this length-scale parameter. Therefore, the criticalimportance of the length scale parameter has not been properly dealt with in the literature until now. Whenconsidering the microstructure with zones of high inherent gradients, gradient-dependent behavior is ex-pected to play an important role once the length-scale associated with the local deformation gradients be-come sufficiently large when compared with the controlling microstructural feature (e.g. mean spacingbetween inclusions relative to the inclusion size when considering a microstructure with dispersed inclu-sions, size of the plastic process zone at the front of the crack tip, the mean spacing between dislocations,the grain size, etc.). All the gradient plasticity theories degenerate to classical plasticity when the controllingmicrostructural feature becomes much larger than the intrinsic material length-scale. Voyiadjis et al. (2003,2004) developed a general thermodynamic framework for the analysis of heterogeneous media that assessesa strong coupling between rate-dependent plasticity and anisotropic rate-dependent damage. They showedthat the variety of plasticity and damage phenomena at small-scale level dictates the necessity of more thanone length parameter in the gradient description. They expressed these material length measures in terms ofmacroscopic measurable material parameters. Similar phenomenological expressions have been assumed byAifantis and co-workers (Konstantinidis and Aifantis, 2002; Tsagrakis and Aifantis, 2002). Nevertheless,an initial attempt has been made recently to relate ‘ to the microstructure of the material. Based on theTaylor model in dislocation mechanics which incorporates a nonlinear coupling between the statisticallystored dislocations (SSDs) and the geometrically necessary dislocations (GNDs), Abu Al-Rub andVoyiadjis (2004) and Voyiadjis and Abu Al-Rub (2004) found ‘ to be proportional to the mean path of


the dislocation (LS). Abu Al-Rub and Voyiadjis (2004) derived an evolution equation for ‘ as a function oftemperature, strain, strain-rate, and a set of measurable microstructural physical parameters. Moreover,Nix and Gao (1998) identified ‘ as L2

Sy=b, where LSy is the average spacing between statistically storeddislocations (SSDs) at plastic yield, and b is the magnitude of the Burgers vector.

However, it is questionable whether a unique value of the internal length scale can describe the size effectfor different problems. There are indications that a fixed value of the material length scale is not alwaysrealistic and that different problems could require different values. (See for example the work of Aifantis(1999) and Tsagrakis and Aifantis (2002).) Moreover, the findings of Abu Al-Rub and Voyiadjis (2004),Voyiadjis and Abu Al-Rub (2004), and Nix and Gao (1998) that the material length scale is proportionalwith the mean free path suggests that the material length scale is not a fixed material parameter but changeswith the deformation of the microstructure because of the variation of the mean free path with dislocationevolution. The change in the length-scale magnitude is also physically sound since the continuous modifi-cation of material characteristics with deformation. Begley and Hutchinson (1998) showed that ‘ has dif-ferent values for different hardening exponents m. Stolken and Evans (1998) also showed that ‘ does notchange if m is constant. Abu Al-Rub and Voyiadjis (2004) showed a dependence of ‘ on the plastic strainlevel, as well as on the hardening level. Aifantis and co-workers (e.g. Tsagrakis and Aifantis, 2002; Zaiserand Aifantis, 2003; Zbib and Aifantis, 2003) have used different values of the length scale parameter forcopper material to fit the Fleck et al. (1994) micro-torsion test results and different values for nickel tofit Stolken and Evans (1998) micro-bending test results. Haque and Saif (2003) showed that the length scaleparameter is not fixed and depends on the grain size. Some authors also argued the necessity of a length-scale parameter in the gradient theories that change with plastic strain in order to achieve an efficient com-putational convergence while conducting multiscale simulations (e.g. Pamin, 1994; de Borst and Pamin,1996; Yuan and Chen, 2001).

In this paper, we review different gradient-dependent plasticity models and we introduce a dislocation-based gradient-dependent plasticity model for ductile materials with a variable (non-fixed) material lengthscale parameter. This model assesses the sensitivity of predictions to the way in which the local and non-local parts in the gradient plasticity theory are coupled. Moreover, a physically-based expression for thelength scale parameter as a function of plastic strain, grain size, and hardening exponent is proposed.We use the proposed model to investigate the micro-bending of thin beams and micro-torsion of thin wires.We also show that the proposed modified gradient plasticity model provides accurate predictions whencompared to the experimental results.

The layout of this paper is as follows: In Section 2 a generalized gradient plasticity model is presentedand its physical bases are discussed in Section 3 based on dislocation mechanics. In Sections 4–6, we use thecurrent gradient plasticity models of Aifanits (Aifantis, 1984), Fleck-Hutchinson (Fleck and Hutchinson,1993), and Gao (Gao et al., 1999a) using different values of the length scale for each model in order to pre-dict the experimentally obtained size effect in micro-bending of thin Annealed Nickel (Stolken and Evans,1998), LIGA Nickel (Shrotriya et al., 2003) and Aluminum (Haque and Saif, 2003) beams and micro-tor-sion of thin Copper wires (Fleck et al., 1994). Moreover, in Section 7 we propose a physically-based expres-sion for the length scale parameter as a function of plastic strain, grain size, and hardening exponent.We also show that the proposed gradient plasticity model with a non-fixed length scale provides accuratepredictions of the size effects in micro-bending of thin beams and micro-torsion of thin wires.

2. Gradient plasticity theories

Many researchers tend to write the weak (i.e. gradient) form of the nonlocal (i.e. integral) conventionaleffective plastic strain (bp), which is the conjugate variable of the plasticity isotropic hardening, in terms ofits local counterpart (p) and the corresponding high-order gradients (g). However, the coupling between p


and g was presented in many different mathematical forms, which are discussed later in this section. Basedon these forms, the following modular generalization of bp can be defined as follows:

bp ¼ f ðpÞc1 þ gð‘ngnÞc2½ �1=c3 ð1Þ

where ‘ is a length parameter that is required for dimensional consistency and whose role will be examinedin detail later in this paper. In Eq. (1), f is a function of the effective plastic strain, p, and g(‘ng) is the mea-sure of the effective plastic strain gradient of any order, g. The superimposed hat denotes the spatial weaknonlocal operator. The power n relates to the order of the gradient used to represent g (i.e. if the first ordergradient is used then n = 1). Eq. (1) should ensure that bp ! p whenever p � ‘g and that bp ! ‘g wheneverp � ‘g. Moreover, since the exact form of coupling between strain hardening and strain gradient hardeningis not known, c1, c2, and c3 are assumed as phenomenological material constants, termed here as interactioncoefficients. These coefficients are introduced in order to assess the sensitivity of the predictions to the wayin which p and g are coupled. Abu Al-Rub and Voyiadjis (2004,) and Voyiadjis and Abu Al-Rub (2004)showed that these interaction coefficients set the proper coupling between the statistically-stored dislocationdensity, which is proportional to the effective plastic strain p, and the geometrically-necessary dislocationdensity, which is proportional to the effective plastic strain gradient g. Moreover, they showed that byincorporating these interaction coefficients in the gradient plasticity theory a suitable remedy is given tothe Nix and Gao (1998) and Swadener et al. (2002) indentation size effect models in predicting the hardnessvalues from micro/nano-indentation tests.

The homogenous flow stress rf without the effect of deformation gradients can be identified, in general,as follows (e.g. Nix and Gao, 1998; Huang et al., 2000a; Yuan and Chen, 2001):

rf ¼ rref þ r0f ðpÞ ð2Þ

where rref is the initial yield stress in uniaxial tension, and r0 is a measure of the hardening modulus in uni-axial tension. For the majority of ductile materials, the function f can be written as a power-law relation(e.g. Fleck and Hutchinson, 1997; Kucharski and Mroz, 2001; Hwang et al., 2002), such that:

f ðpÞ ¼ p1=m ð3Þ

where m P 1 is the hardening exponent which can be determined from a simple uniaxial tension test orindentation test.

However, since uniaxial tension tests exhibit homogenous deformation with no strain gradients, Eq. (2)cannot be used to describe applications where the non-uniform (heterogeneous) plastic deformation playsan important role (twisting, bending, deformation of composites, micro- or nano-indentation, etc.). Eq. (2)cannot then predict the size dependence of material behavior after normalization, which involves no inter-nal material length scales. Consequently, Eq. (2) needs to be modified in order to be able to incorporate thesize effects. This can be effectively done by replacing the conventional effective plastic strain measure p by itscorresponding nonlocal measure bp defined in Eq. (1), such that we consider a more general hardeningrelation:

rf ¼ rref þ r0 f ðpÞc1 þ gð‘ngnÞc2½ �1=mc3 ð4Þ

where rf can also be set equal to the effective or equivalent stress reff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3r0

ijr0ij=2

qfor the case of the von-

Mises type plasticity or reff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3r0

ijr0ij=2

qþ arkk for the case of a Drucker–Prager plasticity (r0

ij denotes the

deviatoric component of the stress tensor rij).It is noteworthy that the unified expression for the flow stress that appears in Eq. (4) is not phenome-

nological but it is physically based and derived from a set of dislocation mechanics-based considerations.The derivation of Eq. (4) is outlined later in Section 3.


The work of Aifantis and his co-workers (see for example Aifantis, 1984, 1987; Zbib and Aifantis, 1988;Muhlhaus and Aifantis, 1991 and references quoted therein) falls within the definition of c1 = c2 = 1,c3 = 1/m, f(p) = p1/m, and g(‘ngn) = ‘g. Two distinct expressions or a combination were proposed byAifantis and his co-workers for the gradient term g : ‘g ¼ ‘k$pk ¼ ‘

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi$p � $p

pand ‘g = c$2p with

‘ ¼ ffiffiffic

p, where $ and $2 are, respectively, the forward gradient and Laplacian operators. The latter form

has been used by de Borst and his co-workers and by many others in solving the localization problem(see for example de Borst and Muhlhaus, 1992; de Borst et al., 1993; de Borst and Pamin, 1996; Claudiaand Perego, 1996; Chen et al., 2000; Zervos et al., 2001; Voyiadjis et al., 2001, 2003, 2004; Engelenet al., 2003; and references quoted therein).

A different strain gradient plasticity theory with coupled stresses has been introduced by Fleck-Hutch-inson and co-workers (see for example Fleck et al., 1994; Fleck and Hutchinson, 1993, 1997, 2001; Begleyand Hutchinson, 1998; and references quoted therein). This type of strain-gradient theory falls within thedefinition of c1 = 2m, c2 = c3 = 2, f(p) = p1/m, and g(‘ngn) = ‘g, with ‘g expressed as ‘g = c1giikgjjk + c2gijk-gijk + c3gijkgkji, where cn (n = 1, . . . ,3) are material coefficients of length-square dimension. The third-ordertensor g is defined as the second gradient of displacement u, such that g = $$u, or alternatively defined asthe first gradient of the plastic strain ep, such that g = $ep. The mechanism-based strain-gradient (MSG)plasticity theory and the Taylor-based nonlocal theory (TNT) of plasticity proposed by Gao and co-work-ers, which are based on the work of Fleck and his co-workers, correspond to the case of c1 = 2, c2 = 1,

c3 = 2/m, f(p) = p1/m, and g(‘ngn) = ‘g with g expressed as g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigijkgijk=4

qand ‘ = 18a2(G/r0)

2b, where a

is an empirical material constant ranging from 0.2 to 0.5, G is the shear modulus, and b is the magnitudeof the Burgers vector (see for example Nix and Gao, 1998; Gao et al., 1999a,b; Huang et al., 2000a,b; Gaoand Huang, 2001; Guo et al., 2001; Hwang et al., 2002; Qiu et al., 2003; Hwang et al., 2003; Huang et al.,2004 and references quoted therein).

Those types of strain-gradient theories have been used by many authors to solve the problem of size ef-fects encountered in a variety of problems of mechanical behavior at small scales, including the strength-ening of twisted wires of reducing diameter, the strengthening of bended beams of reducing thickness,particle reinforced composites, microelectromechanical systems (MEMS), indentation, crack tips, and voidgrowth (e.g. Begley and Hutchinson, 1998; Stolken and Evans, 1998; Saha et al., 2001; Gao et al., 1999a,b;Huang et al., 2000b; Gao and Huang, 2001; Guo et al., 2001; Chen and Wang, 2002; Konstantinidis andAifantis, 2002; Xue et al., 2002a,b,c; Wang et al., 2003; Hwang et al., 2003).

However, the proposed model in Eq. (1) is a clear departure from all the current gradient theories thatare based on the ideal assumption of all the obstacles being equally strong and equally spaced along astraight or curved contacting line. However, the real situation in experiments suggests that the hardeninglaw cannot be taken as a simple sum of the densities of statistically stored dislocations (SSDs) and geomet-rically necessary dislocations (GNDs). This has been the basis in proposing the model in Eq. (1). Moreemphasis is placed on this point in Section 3.

It is imperative to mention that, generally, there exist two frameworks of gradient plasticity theories tomodel size effects at the micron and sub-micron scales. The first framework involves higher-order stressesand higher-order (or additional) governing equations and therefore requires extra boundary conditions,such as the theory of mechanism-based strain gradient (MSG) plasticity of Gao et al. (1999a) and Huanget al. (2000a) established from the Taylor dislocation model and the work of Fleck and Hutchinson (1993,2001). The second framework does not involve the higher-order stresses and the equilibrium equationremains the same as those in the classical theory (e.g. Aifantis, 1984, 1999; Acharya and Bassani, 2000;Bassani, 2001; Chen and Wang, 2002; Gao and Huang, 2001; Gao, 2004; Abu Al-Rub and Voyiadjis,2004). Huang et al. (2000a) showed that the higher-order stresses have a little or essentially no effect. Infact, MSG theory was recently modified by Huang et al. (2004) such that it does not involve the higher-order stresses and therefore falls into the second category of the strain gradient plasticity that preserves


the structure of conventional plasticity theories. Furthermore, Bazant and Guo (2002) argue that theasymptotic behavior at small sizes is unreasonably strong in the first category of strain gradient plasticitytheories because of the presence of third-order stresses, sikj, in these models.

The theory presented in this paper falls in the second category, where no higher-order stresses are in-volved. This feature would make the strain gradient plasticity theories very attractive in applications, sincehigher-order boundary conditions may not be uniquely defined and/or can be difficult to satisfy if one usesthe first category of strain gradient plasticity theories. Moreover, we introduce here higher-order gradientsof the plasticity hardening state variable, which is the effective plastic strain, p, into the constitutive equa-tion for the flow stress, while leaving all other features of classical plasticity unaltered. This is different thanthe MSG theory, which introduces higher-order gradients of the plastic strain tensor epij. Therefore, the twotheories are different.

It is also imperative to mention that in the last ten years a number of authors have argued that the sizedependence of the material mechanical properties results from an increase in strain gradients inherent insmall localized zones which lead to the formation of geometrically necessary dislocations (GNDs) thatcause additional hardening (e.g. Ashby, 1970; Stelmashenko et al., 1993; De Guzman et al., 1993; Flecket al., 1994; Ma and Clarke, 1995; Arsenlis and Parks, 1999; Busso et al., 2000; Tymiak et al., 2001; Swa-dener et al., 2002; Gao and Huang, 2003). The nonlocal effects associated with the presence of GNDs areincorporated into Eq. (4) through the local deformation gradients g at a given material point. Arsenlis andParks (1999) showed that the effective strain gradient g that appears in Eq. (4) can be defined by g ¼ qGb=�r,where qG is the density of GNDs, �r is the so-called Nye factor, and b is the magnitude of the Burgers vector.Thus, the introduction of higher-order gradient terms in the conventional continuum mechanics (i.e. Eq.(4)) has led the bridging of the gap between conventional continuum theories and micromechanical models.This bridge is thoroughly demonstrated later in Section 3. In Sections 4 and 5 we will discuss some typicalanalytical solutions for simple geometries. The corresponding solutions for the strain gradient plasticitytheories are reported by Fleck and Hutchinson (1997) and Huang et al. (2000a). Moreover, the followingexpression for the nonlocal term in Eq. (4) will be used in obtaining these solutions, such that:

gð‘ngnÞ ¼ ‘g1 þ ‘2g2 ð5Þ
where the first-order gradient g1 = k$pk and the second-order gradient g2 = $2p of the effective plasticstrain are considered. In Eq. (5), g1 takes into account the presence of the GNDs that maintain the defor-mation compatibility and describe the size effects that occur at the micron and/or sub-micron scales. g2 (theLaplacian) gives objective results for localization problems (i.e. strain softening media) and it is successfulin describing the thickness or size of shear bands at the meso-scale when dynamic loading conditions areconsidered. Third-order gradients and higher (i.e. higher than the Laplacian) are not considered in thisstudy. This is because of lacking of solid physical interpretations for the presence and significance of suchgradients. However, these gradients could result from an inherent small-scale heterogeneity and they mayplay a dominant role in materials subjected to dynamic loading conditions. This is not the subject of thiswork and more studies are needed to search the significance of gradients that are higher than the Laplacian.
In the next section we outline the physical bases of the proposed gradient theory and in particular thephysical basis for the unified expression of the flow stress in Eq. (4).

3. Physical bases

Generally, it is assumed that the total dislocation density, qT, represents the total coupling between twotypes of dislocations which play a significant role in the hardening mechanism. Material deformationin metals enhances the dislocation formation, the dislocation motion, and the dislocation storage. The dis-location storage causes material hardening. The stored dislocations generated by trapping each other in a


random way are referred to as statistically-stored dislocations (SSDs), while the stored dislocations that re-lieve the plastic deformation incompatibilities within the polycrystal caused by non-uniform dislocation slipare called geometrically-necessary dislocations (GNDs). Their presence causes additional storage of defectsand increases the deformation resistance by acting as obstacles to the SSDs (Gao et al., 1999b). Therefore,as far as the experimental findings up to day, one can not assume that GNDs are similar to the SSDs sinceboth are different in nature. The SSDs are believed to be dependent on the effective plastic strain, while thedensity of GNDs is directly proportional to the gradient of the effective plastic strain (Ashby, 1970; Fleckand Hutchinson, 1997; Arsenlis and Parks, 1999).

The critical shear stress that is required to untangle the interactive dislocations and to induce a signif-icant plastic deformation is defined as the Taylor flow stress, sf. The Taylor hardening law, which relatesthe shear strength to the dislocation density, has been the basis of the mechanism-based strain gradient(MSG) plasticity theory (e.g. Nix and Gao, 1998; Gao et al., 1999a; Huang et al., 2000a). It gives a simpledescription of the dislocation interaction processes at the microscale (i.e. over a scale which extends fromabout a fraction of a micron to tens of microns). One method to enhance the coupling between SSDs andGNDs is to assume that the overall shear flow stress, sf, has two components; one arising from SSDs, sS,and a component due to GNDs, sG. The following general functional form for sf has been proposed byColumbus and Grujicic (2002):

sf ¼ sref þ sbS þ sbG� �1=b ð6Þ

where b is considered as a material constant, termed here as the interaction coefficient, and used to assess thesensitivity of predictions to the way in which the coupling between the SSDs and GNDs is enhanced duringthe plastic deformation process. The stresses sS and sG are given, respectively, by Taylor�s hardening law asfollows:

sS ¼ aSGbSffiffiffiffiffiqS

p; sG ¼ aGGbG

ffiffiffiffiffiffiqG

p ð7Þ

where qS is the SSD density, qG is the GND density, bS and bG are the magnitudes of the Burgers vectorsassociated with SSDs and GNDs, respectively, G is the shear modulus, and aS and aG are statistical coef-ficients which account for the deviation from regular spatial arrangements of the SSD and GND popula-tions, respectively. For an impenetrable forest dislocations, it is reported that aS � 0.85 (Kocks, 1966) andaG � 2.15 (Busso et al., 2000).

The general form in Eq. (6) ensures that s ! sS whenever sS � sG and that s ! sG whenever sS � sG.Two values of b are generally investigated in the literature:

(a) b = 1 which corresponds to a superposition of the contributions of SSDs and GNDs to the flow stress; i.e.sf ¼ sS þ sG ¼ aSGbS

ffiffiffiffiffiqS

p þ aGGbGffiffiffiffiffiffiqG

p, where if aS = aG = a and bS = bG = b, then sf ¼ aGb

ffiffiffiffiffiqS

p þ�ffiffiffiffiffiffi

qGp Þ for which the total dislocation density is expressed as the sum of the square root of SSD andGND densities, qT ¼ ffiffiffiffiffi

qSp þ ffiffiffiffiffiffi

qGp

. This expression for the total dislocation density has been tested byColumbus and Grujicic (2002).

(b) b = 2 which, since the flow stress scales with the square root of dislocation density (i.e. s / ffiffiffiq

p), cor-

responds to the superposition of the SSD and GND densities, i.e. sf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2S þ s2G

p¼

GffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaSbSð Þ2qS þ aGbGð Þ2qG

q. Also if aS = aG = a and bS = bG = b, then sf ¼ Gab

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqS þ qG

pfor which

the total dislocation density is expressed as a simple arithmetic sum of the SSD and GND densities,qT = qS + qG. This expression for the total dislocation density has been tested by many authors (seee.g. Busso et al., 2000).

The above two statements in (a) and (b) do not express the same thing. The first implies that the totalflow stress results from the sum of the two contributions from SSDs and GNDs (i.e. sS + sG), while the


second statement implies that the total flow stress results from the sum of SSD and GND densities (i.e.qS + qG). Moreover, Eq. (6) constitutes the non-local micromechanical plasticity constitutive model dueto the presence of GNDs. It is also imperative to emphasize that the validity of the Taylor relationship,Eq. (7), has been verified by numerous theoretical and experimental studies (see e.g. Hirsch, 1975). There-fore, one may indeed use it as a starting point.

Substituting Eq. (7) into Eq. (6) yields a general expression for the overall flow stress in terms of SSDand GND densities, such that:

sf ¼ sref þ G a2Sb2SqS

� �b=2 þ a2Gb2GqG

� �b=2h i1=bð8Þ

Alternatively, Eq. (8) can be redefined in terms of a set of interaction coefficients b1, b2, and b3 as follows:

sf ¼ sref þ G a2Sb2SqS

� �b1 þ a2Gb2GqG

� �b2h i1=b3ð9Þ

so that

b1 ¼ b=2; b2 ¼ b=2 and b3 ¼ b ð10Þ
Using the functional form of the Taylor relationship as shown by Eq. (9), one can devise different waysof coupling the SSD and GND densities; however, all are special cases of Eq. (9). For instance, manyauthors tend to write sf ¼ sref þ abG
ffiffiffiffiffiqT

p, where qT is a linear sum of qS and qG, such that qT = qS + qG

(e.g. Ashby, 1970; Stelmashenko et al., 1993; De Guzman et al., 1993; Fleck et al., 1994; Ma and Clarke,1995; Nix and Gao, 1998; etc.). Eq. (9) reduces to this case when aS = aG, bS = bG, and b1 = b2 = 1 andb3 = 2. Another possible coupling between the SSD and GND densities has been proposed by Fleckand Hutchinson (1997), where they expressed qT as the harmonic sum of qS and qG, such that qT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q2S þ q2

G

p. Eq. (9) reduces to this case when aS = aG, bS = bG, and b1 = b2 = 2 and b3 = 4. Moreover,

Kocks (1976) proposed a general form as qT ¼ qlS þ ql

G½ �1=l. This can be obtained from Eq. (9) bysettingaS = aG, bS = bG, and b1 = b2 = l and b3 = 2l. Thus, all the mentioned possibilities of couplingbetween SSDs and GNDs are special cases of the general form in Eq. (9).

Arsenlis and Parks (1999), Gao et al. (1999a,b), and Huang et al. (2000a) showed that gradients in theplastic strain field are accommodated by the GND density, qG, so that the effective strain gradient g thatappears in Eq. (4) can be related to qG by the following relation:

qG ¼ �rgbG

ð11Þ

They showed that this expression allows g to be interpreted as the deformation curvature in bending andthe twist per unit length in torsion. In Eq. (11), �r is the Nye factor introduced by Arsenlis and Parks (1999)to reflect the scalar measure of GND density resultant from macroscopic plastic strain gradients. For FCCpolycrystals, Arsenlis and Parks (1999) have reported that the Nye factor has a value of �r ¼ 1:85 in bendingand a value of �r ¼ 1:93 in torsion. The Nye factor is an important parameter in the predictions of the gra-dient plasticity theories as compared to the experimental results (Gao et al., 1999b).

During plastic deformation, the density of SSDs increases due to a wide range of processes that lead toproduction of new dislocations. Those new generated dislocations travel on a background of GNDs whichact as obstacles to the SSDs. If LS is the average distance traveled by a newly generated dislocation, then therate of accumulation of strain due to SSDs scales with _p / LSbS _qS. Bammann and Aifantis (1982) definedthe plastic shear strain, cp, as a function of the mobile dislocation density. However, a similar relation canthen be assumed for the evolution of cp in terms of qS for proportional loading (Abu Al-Rub and Voyiadjis,2004 and Voyiadjis and Abu Al-Rub, 2004), such that:

cp ¼ bSLSqS ð12Þ


where LS is the mean spacing between SSDs. Furthermore, Bammann and Aifantis (1982) generalized theplastic strain in the macroscopic plasticity theory, ep, in terms of the plastic shear strain, cp, and an orien-tation tensor, M, as follows:

epij ¼ cpMij ð13Þ

where M is the symmetric Schmidt�s orientation second-order tensor. In expressing the plastic strain tensorat the macro level to the plastic shear strain at the micro level, an average form of the Schmidt�s tensor isassumed since plasticity at the macroscale incorporates a number of differently oriented grains into eachcontinuum point (Bammann and Aifantis, 1982).

The flow stress rf in Eq. (4) is the conjugate of the effective plastic strain variable p in macro-plasticity.For proportional, monotonically increasing load in plasticity, p is defined as:

p ¼ffiffiffiffiffiffiffiffiffiffiffiffi2

3epije

pij

rð14Þ

Hence, utilizing Eqs. (12) and (13) into Eq. (14), one can write p as a function of SSDs as follows:

p ¼ MbSLSqS ð15Þ
which is referred to as the Orowan-like equation (Abu Al-Rub and Voyiadjis, 2004 and Voyiadjis and AbuAl-Rub, 2004), where M ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2M : M=3

pcan be interpreted as the Schmidt�s orientation factor. It is clear

from Eq. (15) that the Burgers vector and the dislocation spacing are two physical length measures whichcontrol plastic deformation.

The tensile flow stress rf in Eq. (4) is related to the shear flow strength sf in Eq. (9) through the Taylorfactor Z, such that:

rf ¼ Zsf ¼ rref þ ZG a2Sb2SqS

� �b1 þ a2Gb2GqG

� �b2h i1=b3ð16Þ

where rref = Zsref is the initial yield stress as it appears in (4). The Taylor factor Z acts as an isotropic inter-pretation of the crystalline anisotropy at the continuum level; Z ¼

ffiffiffi3

pfor an isotropic solid and Z = 3.08

for FCC polycrystalline metals (Taylor, 1938; Kocks, 1976).Substituting qG and qS from Eqs. (11) and (15), respectively, into Eq. (16), yields the following expres-

sion for the flow stress rf:

rf ¼ rref þ ZGaSbSLSM

� �b1=b3

pb1 þ a2GbG�r� �b2 LsM

a2SbS

� �b1

gb2

" #1=b3

ð17Þ

One can rewrite the above equation as follows:

rf ¼ rref þ r0 pb1 þ ‘gð Þb2h i1=b3

ð18Þ

where

r0 ¼ ZGa2SbSLSM

� �b1=b3

; ‘ ¼ a2GbG�r=a2SbSLsM

� �b1=b2

ð19Þ

Substituting the interaction coefficients in Eq. (10), one can then rewrite the expressions in Eq. (19) asfollows:

r0 ¼ ZGaS

ffiffiffiffiffiffiffiffiffiffibSLSM

s; ‘ ¼ ðaG=aSÞ2ðbG=bSÞM�rLS ð20Þ


Comparing Eq. (18) with Eq. (4), one concludes that:

f ðpÞ ¼ p; g ‘ngnð Þ ¼ ‘g; b1 ¼ c1; b2 ¼ c2; b3 ¼ mc3 ð21Þ

Therefore, the proposed unified expression of the flow stress in Eq. (4) is physically based with strong dis-location mechanics-based interpretations. Moreover, the phenomenological measure of the yield stress inuniaxial tension, r0, and the microstructure length-scale parameter, ‘, are now related to measurable phys-ical parameters; namely: the average distance between statistically stored dislocations LS (characterizes thecharacteristic length of plasticity phenomenon), the Nye factor �r (characterizes the microstructure dimen-sion, e.g. grain size, grain boundary width, obstacle spacing and radius), the Schmidt�s orientation factor M(characterizes the lattice rotation), the Burgers vector b (characterizes the displacement carried out by eachdislocation), and the empirical constant a (characterizes the deviation from regular spatial arrangement ofthe SSD or GND populations). Therefore, the form and magnitude of ‘ depend on the dominant mecha-nism of plastic flow at the scale under consideration. It appears from Eq. (20)2 that the size effect and itsimplications on the flow stress and work-hardening is fundamentally controlled by the dislocation glide LS.Moreover, Nix and Gao (1998) showed that the length scale is proportional to LS as appears in Eq. (20)2.Therefore, it can be concluded that the spacing between dislocations, LS, is the main physical measure thatcontrols the evolution of the length scale in gradient plasticity theories.

Furthermore, by substituting LSM from Eq. (20)2 into Eq. (20)1 one obtains a relation for ‘ as a functionof G and r0 similar to that proposed by Nix and Gao (1998) and Gao et al. (1999a,b), such that:

‘ ¼ Z2a2GbG�rGr0

� �2

ð22Þ

One can note that Eq. (22) implies that the length-scale parameter may vary with the strain-rate and tem-

perature for a given material for the case r0 ¼ br0ð _p; T Þ, where _p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2_epij _e

pij=3

q. However, for most metals,

the yield stress increases with the strain rate and decreases with temperature increase. This causes ‘ to de-crease with increasing strain-rates, but to increase with temperature decrease (Abu Al-Rub and Voyiadjis,2004). This is not the subject of this paper but more work needs to be done in this direction.

4. Bending of thin beams

Stolken and Evans (1998); Shrotriya et al. (2003), and Haque and Saif (2003) performed bending tests ofultra-thin beams with different thicknesses and observed that the bending strength of beams significantlydecreased with the beam thickness increase. This size effect cannot be explained using the classical plasticitytheory which does not possess an intrinsic material length scale. In this section, we will use the generalizedgradient plasticity model presented in the last section to investigate the strength of thin beams in pure bend-ing. For simplicity, we assume that the beam is under plane-strain deformation and is made of an incom-pressible solid.

Let x1 be the neutral axis of the beam and the bending occurs in the x1–x2 plane. The curvature of thebeam is designated j, the thickness is h, and the width in the out-of-plane (x3) direction is b. From classicalstrength of materials, the displacement field of the beam under plane-strain bending (the out-of-plane widthin the x3 direction is much larger than the thickness in the x2 direction) can be defined as follows:

u1 ¼ �jx1x2; u2 ¼ �j x21 þ x22� �

=2; u3 ¼ 0 ð23Þ

The associated non-vanishing strain components are given by:

e11 ¼ �e22 ¼ jx2 ð24Þ


Due to the postulated proportional loading in the bending problem, we can use the deformation theory ofplasticity which coincides with the flow theory of plasticity. In the deformation theory of plasticity there isno formal distinction between elastic and plastic components of strain and the change in the plastic strain

can be formally integrated. We can then express the effective strain, p ¼ffiffiffiffiffiffiffiffiffiffiffi23eijeij

q, and the effective strain

gradients, g1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiriprip

pand g2 = $2p, using Eq. (24) as follows:

p ¼ 2ffiffiffi3

p jjx2j; g1 ¼2ffiffiffi3

p j; g2 ¼ 0 ð25Þ

By adapting the deformation theory of plasticity, which assumes the same structure of the classical plastic-ity theory, the deviatoric stress tensor, r 0, can then be defined as follows:

r0ij ¼

2rf

3peij ð26Þ

The constitutive equation, Eq. (26), then gives the non-vanishing deviatoric stresses as:

r011 ¼ �r0

22 ¼rfffiffiffi3

p x2jx2j

ð27Þ

The non-vanishing stresses, rij ¼ r0ij þ 1

3rkkdij, can be expressed as:

r11 ¼2rfffiffiffi3

p x2jx2j

; r33 ¼rfffiffiffi3

p x2jx2j

ð28Þ

where the flow stress in a power-law hardening material, can be expressed by substituting Eq. (25) intoEq. (4), as follows:

rf ¼ rref þ r0

2

3jjx2j

� �c1

þ 2‘ffiffiffi3

p j

� �c2� 1=mc3

ð29Þ

The pure bending moment M can be determined from the integration of the normal stress r11 over thecross-section of the beam as:

M ¼ 2bffiffiffi3

pZ h=2

�h=2rf jx2jdx2 ð30Þ

Substituting Eq. (29) into the above equation with the aid of variable substitution (i.e. y = x2/h), it follows:

M

bh2¼ rref

2ffiffiffi3

p þ 4r0ffiffiffi3

pZ 0:5

0

4ffiffiffi3

p esy� �c1

þ 4bffiffiffi3

p es

� �c2� 1=mc3

y dy ð31Þ

where es = jh/2 is the surface curvature and b = ‘/h. In the limit of h � ‘, M degenerates to that for clas-sical plasticity, such that:

M0

bh2¼ M0

bh2ð‘ ! 0Þ ¼ rref

2ffiffiffi3

p þ cðesÞ1=m with c ¼ 21=mmr0

3ðmþ1Þ=2mð2mþ 1Þð32Þ

with c1 = c2 = c3 = 1 (since no coupling is necessary when the strain gradients vanish).

5. Torsion of thin wires

A systematic experiment in reference to the size dependence of material behavior in micro-torsion ofhigh-purity thin copper wires has been reported by Fleck et al. (1994). In these experiments it is observed


that the scaled shear strength increases by a factor of 3 as the wire diameter decreases from 170 to 12 lm.However, Fleck et al. (1994) observed that in simple tension tests the corresponding increase in work-hard-ening with decrease of wire size is negligible. This size effect in torsion cannot be explained by the classicalcontinuum plasticity theory, which possesses no intrinsic material length scale. In this section we use thegeneralized gradient plasticity model proposed in Section 2 to investigate the strength of thin wires intorsion.

The Cartesian reference frame is set such that the x1 and x2 are in the plane of the cross-section of thewire, and the x3 axis coincides with the axis of the wire. The twist per unit length is designated j and theradius of the wire is a. The displacement field as in the classical torsion problem can be assumed as follows:

u1 ¼ �jx2x3; u2 ¼ �jx1x3; u3 ¼ 0 ð33Þ
The associated strain components are given by:
e13 ¼ e31 ¼ � 1

2jx2; e23 ¼ e32 ¼

1

2jx1; e11 ¼ e22 ¼ e33 ¼ 0 ð34Þ

where the strain field is obtained by adopting the assumption of incompressibility. Due to the postulatedproportional loading in the torsion problem, we can express the local effective strain, p, and the effectivestrain gradients, g1 and g2, using Eq. (34) as follows:

p ¼ 1ffiffiffi3

p jr; g1 ¼1ffiffiffi3

p j; g2 ¼1ffiffiffi3

p j1

rð35Þ

where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix21 þ x22

pis the radius in polar coordinates (r,h,z). The deformation theory of plasticity assumes

the same structure of the classical plasticity theory such that the deviatoric stress r 0 is defined by Eq. (26).The non-vanishing deviatoric stresses are then defined as follows:

r013 ¼ r0

31 ¼2rf

3pe13; r0

23 ¼ r032 ¼

2rf

3pe23 ð36Þ

where the flow stress in a power-law hardening material can be expressed by substituting Eq. (35) intoEq. (4) as follows:

rf ¼ rref þ r0

1ffiffiffi3

p jr� �c1

þ ‘ffiffiffi3

p j

� �c2

1þ ‘

r

� �c2� 1=mc3

ð37Þ

The torque can be obtained from the integration over the cross-section of the torques induced by the shearstresses r0

13 and r023 as

Q ¼ 2pffiffiffi3

pZ a

0

rf r2 dr ð38Þ

Substituting Eq. (37) into the above equation with the aid of variable substitution (i.e. y = r/a), it follows:

Qa3

¼ 2pffiffiffi3

p rrefffiffiffi3

p þ r0

Z 1

0

1ffiffiffi3

p esy� �c1

þ bffiffiffi3

p es

� �c2

1þ by

� �c2� 1=mc3( )

y2 dy ð39Þ

where es = ja is the surface angle of twist and b = ‘/a. In the limit of a � ‘, Q degenerates to that for clas-sical plasticity, such that:

Qa3

¼ Qa3

ð‘ ! 0Þ ¼ 2prref

3ffiffiffi3

p þ cðesÞ1=m with c ¼ 2pmr0

3ðmþ1Þ=2mð3mþ 1Þð40Þ

where c1 = c2 = c3 = 1 (since no coupling is necessary when the strain gradients vanish).


6. Comparing with experiments

In the present paper, the gradient plasticity models of Aifantis and co-workers, Fleck and co-workers,Gao and co-workers are referred to as AGP model, FGP model, and GGP model, respectively, whereas thepresent generalized gradient plasticity model is referred to as VGP model.

Use is made of four sets of micro tests reported by Stolken and Evans (1998) for micro-bending of thin99.994% pure Annealed Nickel films, Shrotriya et al. (2003) for micro-bending of thin LIGA Nickel films,Haque and Saif (2003) for micro-bending of nano 99.999% pure Aluminum films, and Fleck et al. (1994) formicro-torsion of 99.99% pure Copper wires. Note that it was reported by the experimentalists that no dam-age occurred in the material such that the measured strength data provides a true measure of the plasticproperties of the specimen. Micro-bending and micro-torsion thus provide a convenient tool for the iden-tification of the plasticity intrinsic material length-scale, when damage is avoided.

Fig. 1 compares the predictions of AGP, GGP, and FGP with the micro-bending test of thin Ni films byStolken and Evans (1998), with foil width b = 2.5 mm, length L = 6 mm, and thicknesses h = 12.5, 25, and50 lm. The experimental results are fitted with r0 = 1167 MPa, m = 1, and rref = 103, 75, 56 MPa forh = 12.5, 25, and 50 lm, respectively. Bazant and Guo (2002) argue that the asymptotic behavior at smallsizes is unreasonably strong in GGP and FGP models because of the presence of third-order stresses inthese models. In what follows, the third-order stresses in GGP and FGP models are neglected as Bazantsuggested and discussed earlier in Section 2.

Moreover, if we use the AGP model where rf = r0(f(p) + g(‘ngn)) with g(‘ngn) = ‘k$pk, g(‘ngn) = ‘2$2p

or g(‘ngn) = ‘k$pk + ‘2$2p, we cannot capture the size effect in bending. Tsagrakis and Aifantis (2002)modified AGP model accordingly, such that:

rf ¼ rref þ r0 f ðpÞ þ c1krpkq þ c2r2p� �

ð41Þ

where c1 = ‘q and c2(p) = ‘2p(1�m)/m. This equation is a special case of the generalized gradient plasticitypresented in Eq. (4) such that:

gð‘ngnÞ ¼ ð‘krpkÞq þ ‘2r2p and c1 ¼ c2 ¼ 1; c3 ¼ 1=m ð42Þ
Fig. 1(a) shows good predictions with q = 0.64.
However, it is noteworthy that Fig. 1(a)–(c) show that different specimen sizes need different values ofthe material length scale to fit the corresponding experimental data. For example, Fig. 1(a) shows that threedifferent values of the material length scale are needed to simulate the experiments using the AGP theory:‘ = 6.8 lm is used for specimen size of h = 12.5 lm, ‘ = 8.1 lm for h = 25 lm, and ‘ = 10.5 lm forh = 50 lm. Stolken and Evans (1998) used an average value of the material length scale of ‘ = 5.2 lm tofit these results. However, their fit is not as good as appears in Fig. 1(a). The same can be said aboutthe simulations of GGP and FGP theories (Fig. 1(b) and (c)), where one value of the material length scaleis not sufficient to fit well the experimental results of Stolken and Evans (1998). In other words, ‘ is not afixed parameter but it depends on the microstructure of the material.

Fig. 2 compares the predictions of AGP, GGP, and FGP with the micro-bending test of thin LIGA Nifoils by Shrotriya et al. (2003), with foil width b = 0.2 mm, length L = 1.50 mm, and thickness h = 25, 50,100, and 200 lm. The experimental results are fitted with r0 = 1030 MPa, m = 1, q = 0.64, and rref = 400,305, 218, 191 MPa for h = 25, 50, and 100, 200 lm, respectively.

It is also noted from Fig. 2 that four different values of the material length scale are used to simulate theexperimental data, which correspond to the four beam thicknesses. For example, Fig. 2(a) shows that fourdifferent values of ‘ are needed to simulate the experiments using the AGP theory: ‘ = 10.8 lm is used forspecimen size of h = 25 lm, ‘ = 20 lm for h = 50 lm, ‘ = 28 lm for h = 100 lm, and ‘ = 60 lm forh = 200 lm. Shrotriya et al. (2003) used an average value of the material length scale of ‘ = 5.6 lm to fitthese results. However, their fit is very poor compared to the one that appears in Fig. 2(a). The same

Fig. 1. Comparison of the experiment (Stolken and Evans, 1998) and the predicted moment-curvature values by (a) AGP model, (b)GGP model, and (c) FGP model for different specimen sizes (h h = 12.5 lm, � h = 25 lm, n h = 50 lm). Solid lines are thepredictions from gradient plasticity theories for different values of the material length scale.


can be said about the simulations of GGP and FGP theories (Fig. 2(b) and (c)), where four values of ‘ areused to fit well the experimental results of Shrotriya et al. (2003). This confirms that ‘ is not a fixed para-meter but it depends on the microstructure of the material.

Fig. 3 compares the predictions of AGP, GGP, and FGP with the micro-bending test of thin 99.99%pure Aluminum films by Haque and Saif (2003), with film width b = 10 lm, length L = 275 lm, and thick-nesses h = 0.1, 0.2, and 0.485 lm. The experimental results are fitted with r0 = 5717 MPa, m = 2.22,q = 0.68, and rref = 0 MPa. It can be seen that the experimental normalized moment-curvature data com-pares very well with the predictions of AGP, GGP, and FGP models only after using three different values

Fig. 2. Comparison of the experiment (Shrotriya et al., 2003) and the predicted moment-curvature values by (a) AGP model, (b) GGPmodel, and (c) FGP model for different specimen sizes (n h = 25 lm, h h = 50 lm, � h = 100 lm, � h = 200 lm). Solid lines are thepredictions from gradient plasticity theories for different values of the material length scale.


of the length scale parameter ‘ for the three specimens. For example, Fig. 3(a) shows that three differentvalues of ‘ are needed to simulate the experiments using the AGP theory: ‘ = 23 lm for h = 0.1 lm,‘ = 8 lm for h = 0.15 lm, and ‘ = 0 lm for h = 0.485 lm. Also, Haque and Saif (2003) used different val-ues of ‘ to fit their experimental data. This, again, confirms our previous conclusion that ‘ is not a fixedparameter but it depends on the microstructure of the material.

Fig. 4 compares the predictions of AGP, GGP, and FGP with the micro-torsion test of thin Copperwires by Fleck et al. (1994), with wire diameters 2a = 12, 15, 20, 30, and 170 lm. The experimental results

Fig. 3. Comparison of the experiment (Haque and Saif, 2003) and the predicted moment-curvature values by (a) AGP model, (b) GGPmodel, and (c) FGP model for different specimen sizes (n h = 0.1 lm, h h = 0.15 lm, � h = 0.485 lm). Solid lines are the predictionsfrom gradient plasticity theories for different values of the material length scale.


are fitted with r0 = 226 MPa, m = 5, q = 1, and rref = 0 MPa. It can be seen that the experimental normal-ized torque-twist data compares very well with the predictions of FGP models and fairly well the predic-tions of AGP and GGP models only after using five different values of the length scale parameter ‘ forthe five specimens. For example, Fig. 4(a) shows the predictions of AGP model for different values of‘:‘ = 22.8 lm for 2a = 12 lm, ‘ = 16.5 lm for 2a = 15 lm, ‘ = 8 lm for 2a = 20 lm, ‘ = 7.5 lm for2a = 30 lm, and ‘ = 4.3 lm for 2a = 170 lm. Fleck et al. (1994) found a range of 2.6–5.1 lm for ‘ that

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4ε s =κ a

Q/a

3 (M

pa)

= 4.3 m

= 7.5 m

= 8 m

= 16.5 m

= 22.8 m

AGP predictions using first-order gradient

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4εs =κa

Q/a

3 (M

pa)

= 8.5 µµ

µ

m= 15 m

= 15 m

= 45 m

= 66 m

GGP predictions using first-order gradient

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4ε s =κa

Q/a

3 (M

pa)

= 40 µµ

µ

m= 42 m

= 43

µ

m

= 262.5

µ

m

= 732 m

FGP predictions using first-order gradient

µ

µ

µ

µ

µ

µ

µ

Fig. 4. Comparison of the experiment (Fleck et al., 1994) and the predicted torque-twist values for Copper by (a) AGP model, (b)GGP model, and (c) FGP model for different specimen sizes (* 2a = 12 lm, � 2a = 15 lm, n 2a = 20 lm, � 2a = 30 lm, h

h = 170 lm). Solid lines are the predictions from gradient plasticity theories using first-order gradient (g1 = k$pk) for different values ofthe material length scale.


fitted well these experimental data and assumed that a mean value of 3.7 lm should be satisfactory to refitthese data. However, better predictions are obtained when different values for ‘ are used as shown in Fig.4(c). Again, this confirms that ‘ is not a fixed parameter but it depends on the microstructure of thematerial.

Figs. 5 and 6 show the effect of incorporating the second-order gradient term $2p on the predictions ofmicro-torsion size effect. The results in Figs. 5 and 6 suggest that material strengthening is associated with a


h = 170 lm). Solid lines are the predictions from gradient plasticity theories using second-order gradient (g2 = $2p) for different values

of the material length scale.


first-order gradient in strain. There appears to be little reason to incorporate higher order strain gradientswhen considering the existence of geometrically necessary dislocations in the material microstructure. Mostof the experimental observations have indicated that size effect is associated with the presence of geomet-rically necessary dislocations, which are proportional to the first order gradient of strain. Aifantis and hisco-workers assume strengthening is associated with terms of order $2p and $4p. In bending these higher


h = 170 lm). Solid lines are the predictions from gradient plasticity theories using first- and second-order gradients (g1 = k$pk andg2 = $

2p) for different values of the material length scale.


order gradients vanish and no size effects are predicted. Moreover, it can be seen from Figs. 5 and 6 that thelength scales oscillate in magnitude with monotonic changes in diameter size. In the numerical analysis ofmicro-indentation, Yuan and Chen (2001) found in the vicinity near the indenter tip, the Laplacian of theequivalent plastic strain strongly oscillates and is over hundreds times of the strain itself. Similar phenom-ena can be found in crack tip field analysis of ductile materials. This suggests that using a constant material


length scale makes numerical computations extremely difficult. In the analysis of strain-softening, Pamin(1994) suggested that the gradient coefficient associated with the Laplacian of the equivalent plastic strainshould be a function of the effective plastic stain in order to obtain robust convergence of the numericalsolution.

It can be seen from the predictions of all of the present gradient plasticity theories shown in Figs. 1–4that different values of the length scale parameter should be used in order to predict the size effects properly;i.e. the value of the length scale parameter changes as the specimen size change. Although the values are ofthe same order, no unique value could be used to simulate the size effect behavior for different specimensizes of the same material. Therefore, the results in Figs. 1–4 suggest that the length scale parameter ‘ ingradient-dependent plasticity is not a fixed material parameter but it depends on the material microstruc-ture and its evolution during deformation. We thus conclude that different values of the length scale param-eter are needed to predict the strength for different specimen sizes and using a fixed value of the materiallength-scale in the current gradient plasticity theories is not realistic.

Moreover, Figs. 1–4 show that a linear coupling between the local part (p) and the nonlocal part (g) inthe current gradient-dependent plasticity theories (i.e. setting c1 = c2 = c3 = 1) is not always realistic. Figs.1–3 show that the size effect in micro-bending of thin beams can be predicted best by setting c1 = c3 = 2 andc2 = 1. Whereas, Fig. 4 shows that the size effect in micro-torsion of thin wires can be predicted best bysetting c1 = c2 = c3 = 2. This suggests that the ideal assumption of the SSD and GND densities being cou-pled in a linear sense is a gross assumption and gives poor predictions of the size effects in some structuralproblems. Therefore, the real situation in experiments suggests that the hardening law should enhance anonlinear coupling between the local and nonlocal parts in the current gradient plasticity theories in orderto be able to predict the size effects reasonably well.

7. A non-fixed material length scale

It is concluded from the previous section that the current gradient plasticity theories do not give soundinterpretations of the size effects in micro-bending and micro-torsion tests if a definite and fixed length scaleparameter is used. There are several microstructural features that may affect the magnitude of the lengthscale parameter. We, therefore, put forward a hypothesis and substantiate it by experimental results thatthe length scale ‘, which characterizes the strain gradient effect on materials strength, is not a fixed param-eter but depends on the microstructure of the material.

Fig. 7 shows a very important observation about the variation of the material length scale parameterfrom the predictions of the current gradient plasticity theories. Fig. 7 illustrates the effect of the mean num-ber of grains of diameter d through the specimen macroscopic characteristic size D (i.e. the ratio D/d whereD = h for micro-bending specimens and D = 2a for micro-torsion specimens) versus the non-dimensional-ized material length-scale (‘/D) predicted by the gradient plasticity models. It can be concluded from thisfigure that for a constant characteristic size D the length scale parameter ‘ increases as D/d decreases, orequivalently, as the grain size d increases (i.e. ‘ / d). However, the grain size alone does not govern thestrengthening effect. The specimen size D also has an effect as shown in Fig. 8.

Fig. 8 shows the variation of the non-dimensional strength versus the non-dimensional length scale ‘/D.This figure indicates that the strength size effect increases with increasing of the length scale or decreasing ofthe macroscopic dimension D. Thus, we conclude that ‘/D decreases with increasing D. This conclusion canbe also inferred from Fig. 7 if the grain size is set constant such that ‘/D / 1/D. Moreover, Fig. 8 shows, forconstant D, as the grain size increases the strength increases as well as the length scale increases. This con-firms the previous conclusion that ‘ / d for a constant value of D.

Taking into consideration the two concluded features of the length scale parameter, which are inferredfrom Figs. 7 and 8, we can generally state, for the same size of the specimen, that the length scale parameter

Fig. 7. Effect of the mean number of grains through the specimen characteristic size upon the non-dimensionalized material length-scale predicted by gradient plasticity theories for (a) annealed nickel, (b) LIGA nickel, (c) aluminum, and (d) copper.


decreases with an increase in the mean number of grains (i.e. decreasing the grain size) through the spec-imen dimension; i.e. ‘ / d/D. This conclusion is schematically illustrated in Fig. 9. Weather we assume afixed D or a fixed d, it is shown that ‘ / d/D. Therefore, a general statement such as the strengtheningincreases as ‘ increases is inaccurate unless D is set constant. Moreover, a general statement such as ‘increases as D decreases or d increases is inaccurate unless d is fixed or D is fixed, respectively.

Furthermore, the above conclusion can be inferred if the non-dimensional strength is plotted against themean number of grains D/d (Fig. 10). It is shown from Fig. 8 that the size effect is larger with an increase in‘; whereas, Fig. 10 shows that the size effect is larger with a decrease in D/d. Therefore, from the results inFigs. 8 and 10, it can be also concluded that ‘ / d/D. and we need an evolution for the length scale para-meter in terms of d/D.

Moreover, by relating the length scale parameter to the grain size we still do not dismiss the dislocationdefect structure. However, we see that since the accommodation of the strain gradient in the deformationfield requires the presence of the geometrically necessary dislocations (GNDs), the strain gradient effect,quantified by the value of ‘, diminishes with decreasing grain size d and vanishes when GNDs cannot besustained within the grain. Smaller specimen dimensions increase the magnitude of the strain gradientsso that its effect is pronounced. However, the characteristic dimension of the specimen alone does not gov-ern the strengthening effect because the dislocation defect structure requires the grain size to be large en-ough to accommodate dislocations. Therefore, we expect very small gradients in a very small specimenwith very small grain size, which is the conclusion of Fig. 7. However, we can expect pronounced straingradient effect in very thin specimens with large grain sizes.

As shown in Fig. 11, for small strains the effect of the strain gradient term rapidly increases with thestrain field and at large strain levels the effect of the strain gradient term diminishes. Since the accommo-dation of the strain gradient in the deformation field requires the presence of GNDs, the strain gradients

Fig. 8. Effect of the non-dimensionalized material length-scale predicted by gradient plasticity theories upon the normalized strengthfor (a) annealed nickel, (b) LIGA nickel, (c) aluminum, and (d) copper.


effect on strengthening, quantified by the value of ‘, diminishes with increasing strain and vanishes whenGNDs cannot be sustained within the grain boundaries. This implies that at high strain values the micro-structural changes are retarded and lower size effects are encountered. As a consequence, the length scalechanges with the course of deformation and the size effect is significantly affected by the microstructuralchanges and features. Moreover, in torsion of a solid wire the length scale parameter ‘ must be plasticstrain-dependent for the stress to be finite at the wire center, Eq. (39). In most known applications theintrinsic material length parameters are assumed constant, which will lead the shear stress to be singularas r ! 0 in torsion. This result implies that the gradient plasticity theory should possess a length scaleparameter that is not fixed but varies with changes in the plastic strain.

According to this demonstration the length scale should vary with the deformation related microme-chanical features. The results in Figs. 7–11 showed that the length scale parameter of the gradient plasticitytheory should be a function of the strain (p) and mean number of grains through the macroscopic charac-teristic size of the structure (i.e. the ratio D/d). In the following we will consider a dislocation mechanics-based interpretation of the material length scale that changes with the course of the plastic deformation,average number of grains through the characteristic size, and the material hardening. The very existenceof such a length scale parameter can only be due to the presence of an internal structure which definesthe overall character of the nonhomogeneities of the plastic deformation and internal stresses.

Based on the Taylor model in dislocation mechanics, it is shown in Section 3 (Eq. (20)2) that the materiallength scale ‘ is identified to be in the order of the average distance between dislocations (i.e. that ‘ / LS)such that

‘ ¼ �hLS with �h ¼ ðaG=aSÞ2ðbG=bSÞM�r ð43Þ

Fig. 9. Schematic illustration of the length scale dependence on the D/d ratio. (a) If D is fixed, then ‘1 < ‘2 < ‘3 and ‘ / (d/D). (b) If d isfixed, then ‘1 < ‘2 < ‘3 and ‘ / (d/D).


where LS is the mean path of the dislocation, bS and bG are the magnitudes of the Burgers vectors associ-ated with the SSDs and GNDs, respectively, aS and aG are statistical coefficients which account for the devi-ation from the regular spatial arrangements of the SSD and GND populations, respectively, �r is the Nyefactor (�r � 2 Arsenlis and Parks, 1999), and M is the Schmidt�s orientation factor (usually taken equalto 1/2).

Begley and Hutchinson (1998) and Abu Al-Rub and Voyiadjis (2004) inferred from the values of ‘ thatwere obtained from micro- and nano-indentation tests for a number of materials that the hardest materialsare having the smallest values of ‘. This is consistent with the fact that the mean free path of dislocation(LS) decreases with the hardness increase, and that ‘ is related to the mean free path distance. If we considerthe case of bS = bG, aS = aG, �r ¼ 2, and M ¼ 1=2, then Eq. (43) reduces to ‘ = LS. This means that thematerial length scale is on the order of the mean free slip distance. Therefore, as a scale measure wemay use the mean distance between adjacent dislocations. Note that this distance has a lower bound, be-cause positive and negative dislocations will annihilate each other if they come too close to each other (e.g.a few nm).

Gracio (1994) speculated that LS decreases when the plastic strain increases and being equal to the grainsize at the beginning of the deformation, and saturating towards values on the order of micrometer at largestrains. Thus, we concluded from Eq. (43) that the length scale parameter is not fixed but depends on theaccumulation of the plastic strain. However, if a solid is not a single crystal, but a polycrystalline aggregate,

Fig. 10. Effect of the mean number of grains through the specimen characteristic size upon the normalized strength for (a) annealednickel, (b) LIGA nickel, (c) aluminum, and (d) copper.

Fig. 11. Experimental variation of the normalized strength with deformation for (a) annealed nickel, (b) LIGA nickel, (c) aluminum,(d) copper.



then we have the additional scale of grains; i.e. the relevant length is the mean grain diameter d. Moreover,Begley and Hutchinson (1998) showed from comparisons with micro-indentation results that ‘ increases asthe hardening exponent m decreases. Stolken and Evans (1998) also showed that ‘ does not change if m isconstant. Abu Al-Rub and Voyiadjis (2004) showed that ‘ decreases with increase in the plastic strain level(p) and the hardening exponent (m). Haque and Saif (2003) showed that ‘ is not a fixed parameter and de-pends on the mean grain size d. This is in line with the dependence of the mean free path distance LS on theplastic strain p, the hardening exponent m, and the grain size d. We, therefore, require an evolution equa-tion for the material length scale that is consistent with the experimental trends, such that the followingexpression for the length scale parameter ‘ can be adopted:

Fig. 12throug

‘ ¼ �hDdDþ d p1=m

ð44Þ

This equation is plotted in Fig. 12 for different values of p, d, D/d, and m with �h ¼ 1. Eq. (44) shows that thelength scale parameter decreases with strain, increases with grain size, increases with D, and decreases withthe hardening exponent. It also gives the important finding that ‘/D decreases if the ratio D/d increases. Eq.(44) is a simple relation that gives feasible interpretations of the length scale in terms of the strain, grainsize, mean number of grains through size D, and hardening as compared to the above experimental obser-vations. Moreover, Eq. (44) shows that the intrinsic material length-scale decreases from an initial value‘ = ⁄d at yield to a final value of ‘ ! 0 at very high values of D or p (corresponds to the classical, localplasticity limit).

Moreover, it can be shown that for plastic flow localization phenomena (e.g. shear bands) this lengthscale in Eq. (44) could be the space period (wavelength) of deformation localization. Some authors tend toexpress the wavelength of deformation localization as a function of strain and grain size (Zuev et al., 2003).

. The material length scale parameter dependence on (a) the plastic strain p, (b) the grain size d, (c) the mean number of grainsh size D, and (d) the hardening exponent m.


Therefore, the variation in the length scale parameter of the plastically deforming medium may be relatedto one of the most complicated problems concerned with elucidation of the specific features of deformationlocalization.

For micro-bending and micro-torsion tests, the dependence of the material length scale on the course ofdeformation, grain size, and hardening exponent turns out to be universal in nature. Figs. 13 and 14 indi-cate that the proposed formula given by Eq. (44) gives reasonable remedy to the current gradient-dependent

Fig. 13. Comparison of the micro-bending experiment and the predicted moment–curvature values by the VGP model for differentspecimen sizes ‘‘h’’. Solid lines are the predictions from gradient plasticity theories for different grain size values ‘‘d ’’. The dashed linedare the predictions if strain gradients effects were neglected. Micro-bending tests by (a) Stolken and Evans (1998), (b) Shrotriya et al.(2003), and (c) Haque and Saif (2003).

Fig. 14. Comparison of the micro-torsion experiment (Fleck et al., 1994) and the predicted torque-twist values for Copper by the VGPmodel for different specimen sizes ‘‘2a’’. Solid lines are the predictions from gradient plasticity theories for different grain size values‘‘d ’’.


plasticity theories. The predictions are obtained by using the VGP model with a non-fixed length scale givenby Eq. (44).

Moreover, Figs. 13 and 14 show that the coupling between the local part (p) and the nonlocal part (g) ingradient-dependent plasticity theories varies from one problem to another. There is a dependency on thetype of strain gradients produced in the given problem ranging from shear strains to normal strains. Themicro-bending size effect results in Fig. 13 are predicted by setting the interaction coefficients c1 = c3 = 2and c2 = 1, while the micro-torsion size effect results in Fig. 14 are obtained by setting c1 = c2 = c3 = 2.

8. Conclusions

The problem in developing a macroscopical model embedded with a micromechanical-based theory ofelastoinelasticity which could be used as an engineering theory for both the analysis and in computer-aideddesign of materials, is a topical and still unsolved material science problem. Attempts to construct such atheory are faced with the difficulties in describing the microscopic structure of materials in terms of mac-roscopic mechanics. When load is applied, the elastic and inelastic deformation of materials that occur inmost cases is not homogeneous, but reveal fluctuations on various space scales. This heterogeneity plays akey role in determining the macroscopical properties of materials. Gradient-dependent theory is developedas a remedy for this situation. However, the full utility of the gradient theory hinges on our ability to deter-mine the physical nature of the intrinsic material length scale and the proper coupling between the local andnonlocal terms.

This article addresses the proper modification required for the full utility of the current gradient plastic-ity theories in solving the size effect problem. A generalized gradient plasticity model with a non-fixedlength scale parameter is proposed. This model assesses the sensitivity of predictions to the way in whichthe local and nonlocal parts are coupled. The model also proposes a length scale parameter that changesduring the course of deformation and is associated with the corresponding material microstructural fea-tures. This model is successful in predicting the size effect encountered in micro-torsion tests of thin wiresand micro-bending tests of thin films.

It is inferred from the size effects observed in experimental trends that for small strains the effect of thestrain gradient term rapidly increases with plastic strains and at large strain stages the gradient does notbecome as prevailing as the strain. Since the accommodation of the strain gradient in the deformation fieldrequires the presence of GNDs, the strain gradients effect on strengthening, quantified by the value of ‘,


diminishes with increasing strain and vanishes when GNDs cannot be sustained within the grain bound-aries. This implies that at high strain values the microstructural changes are retarded and lower size effectsencountered. Moreover, it is concluded that the size effect increases with the material length scale (‘),decreases with the strain history (p), decreases with specimen size (D), decreases with the grain or particlesize (d), and increases with the mean number of grains through D (i.e. D/d ratio).

A fixed value of the material length-scale is not always realistic and that different problems could requiredifferent values. The current gradient plasticity theories do not give sound interpretations of the size effectsin micro-bending and micro-torsion tests if a definite and fixed length scale parameter is used. The exper-imental results from size effect tests (micro/nano indentation, micro-torsion of thin wires, micro-bending ofthin beams) showed that the intrinsic material length-scale ‘ is not fixed but evolves with the course ofdeformation. A semi-empirical evolution law for ‘ in terms of the grain size (d), the specimen size (D),the equivalent strain, and the hardening level has been proposed in this paper. Moreover, a linear couplingbetween the local and nonlocal terms in gradient plasticity theory is not always realistic and that differentproblems could require different couplings.

It is concluded that both the Laplacian and the first-order strain gradient terms have the ability ofstrengthening the material. However, it appears inappropriate to incorporate the Laplacian when consid-ering the existence of geometrically necessary dislocations in the material microstructure or consideringsolving size effect problems in hardening materials. In bending the Laplacian vanishes and no size effectsare predicted. Moreover, it causes the material length scale to oscillate in magnitude with monotonicchanges in the specimen size.

As the strain gradient plasticity theories involve the gradient of the plastic strain in the constitutive equa-tions, the order of the governing equations becomes higher such that additional boundary conditions mustbe imposed. The proposed model in this paper gives an alternative framework which could model size-dependent plasticity without higher-order stresses and strains, so as to preserve the essential structure ofclassical plasticity. This approach has the advantages that it is simpler overall and can be easily imple-mented into existing finite element codes.

Although the focus here has been on the role of gradient plasticity theory in predicting the size effectaccommodated by the presence of deformation gradients, it should be emphasized that size effect associatedwith macroscopically homogeneous deformation (i.e. no gradients), as in simple tension, cannot be ex-plained by the current gradient theories. Nonlocal theories that account for such size effect have not beendeveloped yet. Moreover, the various theories of gradient plasticity differ significantly in their mathematicalstructure. The actual forms of terms and the values of the parameters depend on the problem investigated.It can also be emphasized that incorporating the size effect into a phenomenological theory of plastic flow isnot necessarily a matter of adding one or more additional parameters to an existing theory. It may requirethe reformulation of the entire theoretical framework of the gradient-dependent media.

Acknowledgement

The authors gratefully acknowledge the financial support by the Air Force Institute of Technology,WPAFB, Ohio. The authors also acknowledge the financial support under grant number M67854-03-M-6040 provided by the Marine Corps Systems Command, AFSS PGD, Quantico, Virginia. They thankfullyacknowledge their appreciation to Howard ‘‘Skip’’ Bayes, Project Director.

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Gradient plasticity theory with a variable length scale...

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