mechanism-based strain gradient drucker–prager elastoplasticity for pressure-sensitive materials

International Journal of Solids and Structures 47 (2010) 2693–2705

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International Journal of Solids and Structures

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Mechanism-based strain gradient Drucker–Prager elastoplasticityfor pressure-sensitive materials

Lu Feng a,*, Yilan Kang a, Guang Zhang b, Shibin Wang a

a Department of Mechanics, School of Mechanical Engineering, Tianjin University, Weijin Road, Nankai, Tianjin 300072, People’s Republic of Chinab Department of Mechanics, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China

a r t i c l e i n f o

Article history:Received 9 November 2009Received in revised form 10 May 2010Available online 31 May 2010

Keywords:Strain gradient plasticityDrucker–Prager yield functionPressure-sensitive materialDislocations

0020-7683/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2010.05.022

* Corresponding author.E-mail address: [email protected] (L. Fen

a b s t r a c t

Modeling strain gradient plasticity effects has achieved considerable success in recent years. However,incorporating the full mechanisms of the pressure-sensitive yielding and the size dependence of plasticdeformation still remains an open challenge. In this work, a mechanism-based stain gradient (MSG) plas-ticity theory for pressure sensitive materials with a variable material length-scale parameter is presented.The flow theory of MSG plasticity based on the Drucker–Prager yield function is established following thesame hierarchical framework of MSG plasticity proposed by Gao et al. (1999), Huang et al. (2000) and Qiuet al. (2003) in order to link the strain gradient plasticity theory on the mesoscale to the Taylor disloca-tion model on the micro-scale. The incremental constitutive relation based on the associated flow rule isderived for the Drucker–Prager yield function on the micro-scale, including the higher-order stress intro-duced as the thermodynamic conjugate of strain gradient at the mesoscale. The proposed theory success-fully predicts the experimental results. The numerical results show that when the pressure-sensitivityindex defined by the Drucker–Prager yield function takes different values, the material response curvesare different and the material strength increases with the increase of pressure-sensitivity index. It provesthat this procedure is able to represent the material behavior of pressure-sensitive materials such asgeomaterials, polymeric materials, metallic foams and metallic glass.

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1. Introduction

Starting from Tresca (1868), von Mises (1913) and Hill (1950),the development of material models that adequately representthe material behavior has been one of the fundamental tasks ofthe experimental and theoretical investigations. The improve-ments of the earlier material models and formulation of the newones last over decades and continue today. Classical plasticitymodels, formulated within the classical framework, are notequipped with any length scale that would reflect the typical sizeand spacing of characteristic microstructural features, thereforethey are inherently incapable of describing size effects. While re-cent experiments have demonstrated that metallic materials dis-play strong size effects at the micron and sub-micron scales andmaterial length scale effects are of great importance for sufficientsmall test specimens of various loading situations (Fleck et al.,1994; Nix and Gao, 1998; Stolken and Evans, 1998; Shrotriyaet al., 2003). Similarly, experiment on particle-reinforced compos-ites has revealed that a substantial increase in the macroscopicflow stress can be achieved by decreasing the particle size while

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g).

keeping the volume fraction constant (Zhu et al., 1997; Nan andClarke, 1996). Compared with metals, the gradient-dependentcharacteristics in geomaterials are also evident by substantiveexperimental data that, during a typical localized failure process,the underlying micro-scale mechanisms, such as crystal dislocationand development of defects and microvoids, may account for theirreversible deformation in the material (Zhao et al., 2005). Thegradient dependency in geomaterials is also experimentally evi-dent by the observation of localization phenomena in granularsoils indicating that the deformed material is usually characterizedby strong spatial density variations. In such circumstances, theinfluence of gradient terms of constitutive variables associatedwith characteristic length scale may be significant and can no long-er be neglected. Moreover, there are many other well-known prob-lems that show strong size effects. For example, micro- and nano-indentation tests have shown that the material hardness increaseswith decreasing the indentation size (e.g. Lim and Chaudhri, 1999;Elmustafa and Stone, 2002; Swadener et al., 2002). Indentation ofthin films shows an increase in the yield stress with decreasingthe film thickness (Huber et al., 2002).

In the last years abundant strain gradient plasticity theorieshave been proposed and various proposals are quite different withrespect to the structure of the equations. An extensive review of

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the recent developments in gradient-dependent theories can befound in Hutchinson (2000), Needleman (2000), Voyiadjis et al.(2003), Abu Al-Rub and Voyiadjis (2004a,b). In the literature, dif-ferent frameworks of gradient-dependent plasticity theories maybe recognized and they can be divided into two distinct groups.The first one based on Mindlin’s (1964, 1965) framework of high-er-order continuum theories is established by Fleck and Hutchin-son (1997, 2001). The basic feature of this kind of theories is thatit involves the higher-order stress as the work conjugate of straingradient. So the order of equilibrium equations is higher than thatin the conventional continuum theories with additional boundaryconditions. This group of theories includes Gao et al. (1999),Huanget al. (2000), and Qiu et al. (2003) with different mathematicalstructures in comparison with Fleck and Hutchinson (2001). Theydeveloped a macroscale theory of mechanism-based strain gradi-ent (MSG) plasticity from the micro-scale Taylor dislocation model.The other group of strain gradient plasticity theories includes mod-els with gradients of internal variables which are conjugate to cer-tain dissipative thermodynamic forces that can enter the evolutionequations but do not appear in the equilibrium equations. Thus thelatter group of theories modifies only the constitutive descriptionwhile the kinematic and equilibrium equations remain standard.This kind of theories includes Acharya and Bassani (2000), Bassani(2001), Huang et al. (2004). Of course, certain models may have amixed character. For instance, Zervos et al. (2001) proposed a mod-el that can be interpreted as a strain gradient theory with softeninglaw enriched by the second gradient of an internal variable. Gud-mundson (2004) proposed a theoretical framework that has poten-tial to cover a large range of strain gradient plasticity effects inisotropic materials.

Most of gradient-enhanced theories aforementioned are suit-able for metallic materials and do not consider volume changesin the plastic phase, therefore cannot be suitable for pressure-sen-sitive materials, such as geomaterials, polymeric materials, metal-lic glass and metallic foam. As it is well-known, during the yieldingof plastic materials, a phenomenon called plastic flow occurs withthe plastic strain evolution. Usually, the plastic flow direction isestablished by defining a plastic potential. If one adopts a plasticpotential similar to the von-Mises model, the volume change inthe plastic phase is null. But if the analyzed material shows pres-sure sensitivity during the plastic evolution, the adopted plasticpotential is required to simulate this change. For this kind of mate-rials, a peculiarity is that yielding cannot be simply described bythe usual von-Mises or Tresca criteria, as for crystalline metals,since normal-stress components acting on the shear plane alsoplay a role on the plastic deformation (Schuh and Nieh, 2004; Ogat-a et al., 2006; Fornell et al., 2009). To account for the normal-stressor pressure dependence of the shear stress, both the Mohr–Cou-lomb (Vaidyanathan et al., 2001; Lund and Schuh, 2004) and theDrucker–Prager (Zhang et al., 2006; Ai and Dai, 2007) yield criteriahave been used in the literature. Bardia and Narasimhan (2006)developed a method for analyzing the characterization of pres-sure-sensitive yielding in polymers. Guo et al. (2008) studied thepressure-sensitive plastic response of a material in terms of theintrinsic sensitivity of its yield stress to pressure and the presenceand growth of cavities. Patnaik et al. (2004) conducted finite ele-ment simulations of spherical indentation of a metallic glass byusing an extended Drucker–Prager model with different levels ofpressure sensitivity. Narasimhan (2004) investigated the influenceof pressure-sensitive yielding on the development of plastic flow ofindentation. Subramanya et al. (2007) studied 3D crack tip fields inpressure-sensitive plastic solids under model I, small scale yieldingconditions by finite element method.

A direct and thorough investigation of size effects for pressuresensitive materials by gradient theories, which may reveal thedistribution of all field variables, needs to be addressed. This is

the motivation of this paper. In this connection, Zhou et al.(2002, 2005) developed a strain-gradient-enhanced damage mod-el for rock-like geomaterials with application to shear band anal-ysis. Zhao et al. (2006) further constructed a framework of straingradient plasticity by an internal-variable approach with normal-ity structures to account for the influence of microstructures onthe overall macroscopic mechanical behavior of gradient-depen-dent solids. Fornell et al. (2009) proposed a new approach tomodel the indentation size effect of bulk metallic glass usingthe free volume concept. Han and Nikolov (2007) presented amicromechanically motivated model for the indentation sizeeffect in polymers. Bei et al. (2010) investigated specimen sizeeffects on mechanical behaviors of bulk metallic glasses by com-pression and nano-indentation tests. Similar research was doneby Li et al. (2008).

In this work, a mechanism-based gradient-dependent plasticitymodel for pressure-sensitive materials with a variable materiallength-scale parameter is derived. The flow theory of MSG plastic-ity based on Drucker–Prager yield function is established followingthe same multiscale, hierarchical framework of mechanism-basedgradient (MSG) plasticity proposed by Gao et al. (1999), Huanget al. (2000) and Qiu et al. (2003). A multiscale, hierarchical frame-work is adopted to link the strain gradient plasticity theory on themesoscale to the Taylor dislocation model on the micro-scale. Theincremental constitutive relation with the associated flow rule isderived for the Drucker–Prager yield function on the micro-scale.The micro-scale plastic deformation is assumed to obey the Taylorwork hardening relation. By the mesoscale analysis, the constitu-tive law including higher-order stresses introduced as thermody-namic conjugate to strain gradients at the mesoscale level isformulated. The interaction between the strain gradient effectand hydrostatic pressure is established. It is shown that the pro-posed gradient plasticity theory provides accurate predictionswhen compared to the experimental results of Stolken and Evans(1998) for micro-bending of annealed nickel thin films and Shro-triya et al. (2003) for micro-bending of LIGA nickel thin films whenthe pressure-sensitivity index takes zero. Extensive numericalstudies are made to investigate the influence of pressure-sensitiv-ity index on material strength when the length-scale effect existsat the same time. It proves that this procedure is able to representthe material behavior of pressure-sensitive materials such asgeomaterials, polymeric materials, metallic foams and metallicglass.

2. Basic model

In this section, the classic rate-independent plasticity and theassociative Drucker–Prager yield criterion are briefly reviewed.The local plasticity is considered under isothermal and small strainconditions. The total stress and strain, rij and eij, are decomposedinto deviatoric and volumetric components as follows:

rij ¼ r0ij þ rmdij with rm ¼13rkk ð1Þ

eij ¼ e0ij þ13evdij with ev ¼ ekk ð2Þ

where r0ij; e0ij;rm and ev are the deviatoric stress, deviatoric strain,hydrostatic stress and volumetric strain, respectively.

Some steels, aluminum alloys and polymers show pressure-sen-sitive yielding and exhibit different stress–strain curves under ten-sion and compression. Pressure-sensitive yield functions usuallyinvolve the hydrostatic tension and the second invariant of thedeviatoric stress (Spitzig and Richmond, 1979, 1980). In this study,we adopt the Drucker–Prager yield function to characterize the

-3 -2 -1 0 1 2

-3

-2

-1

0

1

2σσ 22

σσ11

Fig. 2. The plane stress yield contour based on the Drucker–Prager yield function.

L. Feng et al. / International Journal of Solids and Structures 47 (2010) 2693–2705 2695

plastic behavior of pressure sensitive materials and the differentstress–strain curves under tension and compression.

The original Drucker–Prager’s criterion, formulated in 1952, is asimple modification of the von-Mises’ criterion, where the influ-ence of the hydrostatic stress component on the failure is intro-duced by the inclusion of an additional term (Choi and Pan,2009). The Drucker–Prager yield function for the initial loadingprocess is expressed as

Fðrij;rÞ ¼ re þffiffiffi3p

arm � r ¼ 0 ð3Þ

where a is the pressure-sensitivity index, r is the flow stress and re

is the effective stress defined as

re ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32r0ijr0ij

rð4Þ

These relationships are available in many references, e.g. (Choi andPan, 2009). Drucker–Prager’s criterion defines a cone in the princi-pal stress space as presented in Fig. 1(a). As shown in Fig. 1(b), theradius of the yield surface in the p-plane is

ffiffiffiffiffiffiffi2rp

and the radius ofthe yield surface in any other deviatoric plane will be

ffiffiffi2pðr� apÞ,

where p is the hydrostatic stress rm. In this figure, r01, r02 and r03are the projections of the principal axes on the deviatoric plane.

Under plane stress conditions with r12 = 0, Eq. (3) can be ex-pressed as

Fðrij;rÞ ¼ ðr211 þ r2

22 � r11r22Þ1=2 þ affiffiffi3p ðr11 þ r22Þ � r ¼ 0 ð5Þ

The normalized yield contour based on Eq. (5) is an ellipse as shownin Fig. 2. The value of the pressure-sensitivity index a is arbitrarilychosen. In this figure, the major axis of the ellipse is oriented 45�from the r11 axis, which is the same as that of the Mises yield func-tion. It is noted that, for the ellipse based on Eq. (5), the geometriccenter is not at the origin in the r11 � r22 coordinate system, but isshifted in the negative directions of the r11 and r22 axes by thesame amount, due to the pressure-sensitivity index a. Therefore,the center of the yield contour is not the same as the geometric cen-ter of the yield contour as shown in Fig. 2.

In the flow theory of Drucker–Prager plasticity, the constitutiveequations are expressed in rate form. The total strain and volumet-ric strain increments are considered as the sum of elastic and plas-tic part

_eij ¼ _eeij þ _ep

ij ð6Þ_ekk ¼ _ee

kk þ _epkk ð7Þ

Note that, in contrast to the von-Mises criterion, the volumetriccomponent is not treated elastically. The total strain rate can begiven by the generalized Hook’s law

(a)

1σ−

2σ−

3σ−Hydrosta

axis

Fig. 1. Drucker–Prager’s yield surface in: (a) prin

_eij ¼1

2l_r0ij þ

19K

_rkkdij þ _epij ð8Þ

here K is the elastic bulk modulus K = E/3(1 � 2c), l is the shearmodulus l = E/2(1 + c), E is the Young’s modulus and c is the Pois-son’s ratio. The elastic volumetric strain rate is related to the hydro-static stress rate as

_eekk ¼

_rkk

3Kð9Þ

In an associative flow rule, the plastic strain increment is nor-mal to the yield surface and its length _k is defined as

_epij ¼ _k0

@F@rij¼ _k

a3

dij þffiffiffi3p

2re_r0ij

!ð10Þ

Considering _r0ii ¼ 0, one obtains

_epkk ¼ a _k ð11Þ

_epkk is the plastic volumetric strain rate and then _k is given by

_k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

92a2 þ 3

r_ep ¼ b _ep ð12Þ

where _ep ¼ffiffiffiffiffiffiffiffiffiffiffiffi23

_epij_ep

ij

qis the effective plastic strain rate, and the coef-

ficient is b ¼ffiffiffiffiffiffiffiffiffiffi

92a2þ3

q.

The evolution law for the plastic strain is achieved by substitut-ing Eq. (12) into Eq. (10), as follows:

_epij ¼ b _ep a

3dij þ

ffiffiffi3p

r0ij2re

!ð13Þ

(b)

tic

1'σ

3'σ

2'σ

σ2

( )pασ −2

cipal stress space and (b) deviatoric planes.


Combining Eqs. (6)–(13), the following equation is obtained

_r0ij ¼ 2l _e0ij � b _ep

ffiffiffi3p

2rer0ij

!ð14aÞ

So we have

_rij ¼ _r0ij þ13

_rkkdij ¼ K _ekkdij þ 2l _e0ij �ffiffiffi3p

lbre

_epr0ij � Kab _epdij ð14bÞ

For proportional loading, the constitutive relation can be writ-ten as

e0ij ¼1

2lþ

ffiffiffi3p

bep

2re

!r0ij ð15Þ

3. Hardening rule based on the Taylor dislocation model

Different hardening rules were proposed in the past to specifythe evolution of the yield surface during plastic deformation. Thesehardening rules allow the yield surface to expand (isotropic hard-ening), contract (isotropic softening), translate (kinematic harden-ing) and change shape in the stress space.

In general, the uniaxial stress–strain relation can be written as

r ¼ rY gðeÞ ð16Þ

where rY is a measure of yield stress in uniaxial tension and g(e) is afunction of strain. For most ductile materials, the function g(e) canbe written as a power-law relation,

gðeÞ ¼ EerY

� �1=n

ð17Þ

here n is the plastic work hardening exponent (n P 1).Nix and Gao (1998) used the von-Mises rule to relate the tensile

flow stress to the shear flow stress as r ¼ffiffiffi3p

s, where the factorffiffiffi3p

results from isotropy of solids. The dislocation model of Taylor(1934) and Bailey and Hirsch (1960) gave the shear flow stress sin terms of the dislocation density q as

s ¼ a1lbffiffiffiffiqp ð18Þ

where b is the magnitude of the Burgers vector, and a1 is an empir-ical coefficient. It is assumed that total dislocation density q is di-vided into the density of statistically stored dislocations qS andthat of geometrically necessary dislocations qG, i.e.

q ¼ qS þ qG

The statistically stored dislocations are accumulate by trappingeach other in a random way (Ashby, 1970) and the geometricallynecessary dislocations are required for compatible deformation ofvarious parts of the non-uniformly deformed material (Ashby,1970; Gao and Huang, 2001).

For isotropic materials, a more accurate relationship between thetensile flow stress and critical resolved shear stress in slip systems is

r ¼ 3s ¼ 3a1lbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqS þ qG

pThe density of statistically stored dislocations qS can be determined(Nix and Gao, 1998) from the uniaxial stress–strain law in the ab-sence of strain gradient effects,

r ¼ 3a1lbffiffiffiffiffiqSp ¼ rref f ðeÞ ð19Þ

which can be typically represented by an elastic-power law harden-ing relation

r ¼ rref f ðeÞ ¼Ee e 6 rY

E

rref e1=n e P rYE

8><>: ð20Þ

the reference stress rref and f(e) can be written as

rref ¼ rYErY

� �1=n

; f ðeÞ ¼ e1=n ð21Þ

On the other hand, the density of geometrically necessary disloca-tions qG can be related to an effective strain gradient g and dependson non-uniform deformations.

The flow stress including strain gradient plasticity is given byHuang et al. (2000), Gao and Huang (2001)

r ¼ rref

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2ðeÞ þ lg

qð22Þ

where

l ¼ 18a21

lrref

� �2

b ð23Þ

is defined as the material length introduced by Fleck and Hutchin-son (1993, 1997). For metals, this length scale is in the order of mi-crons, but for concrete and other highly heterogeneous compositematerials, it is substantially larger.

There are two types of expression for l in most references, oneexpression for l is defined by Eq. (23), which indicates that thelength-scale parameter is fixed and does not change with the plas-tic deformation. Voyiadjis and Abu Al-Rub (2005, 2006) showedthat l depended on the grain size, the effective plastic strain andother factors, which were confirmed by Begley and Hutchinson(1998), Abu Al-Rub and Voyiadjis (2004a) and Haque and Saif(2003).

The unified expression of the flow stress in Eq. (22) is physicallybased with strong dislocation mechanics-based interpretations. Tobe noted, in most cases, the contribution of friction stress or inter-nal lattice resistance denoted as r0 to the tension flow stress exists,and the expression for the flow stress r can be written as (Qiuet al., 2001; Abu Al-Rub and Voyiadjis, 2005)

r ¼ r0 þ rref

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2ðeÞ þ lg

qð24Þ

Eq. (24) degenerates to Eq. (22) for a vanishing friction stress r0 = 0.For the flow theory of plasticity, it is useful to write the uniaxial

stress–strain relation in terms of the plastic strain ep as

r ¼ rref fpðepÞ ð25Þ

here fp is a function of plastic strain and related to the f(e) by

fpðepÞ ¼ f ðeÞ ¼ f ðee þ epÞ ¼ frEþ ep

� �¼ f

rref

EfpðepÞ þ ep

h ið26Þ

For a power-law hardening solid, fp(ep) takes the form (Huang et al.,2004)

fpðepÞ ¼ 1þ ErY

ep

� �1=n

ð27Þ

So the flow stress can be written as

r ¼ rref

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2p ðepÞ þ lðepÞgp

qð28Þ

where the effective strain gradient gp is given by (Huang et al.,2004)

gp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14gp

ijkgpijk

rð29Þ

with

gpijk ¼ ep

ik;j þ epjk;i � ep

ij;k ð30Þ

Plastic deformation in metallic materials reflects the collectivebehavior of a vast number of dislocations from a microscopic pointof view. But for a variety materials, such as metallic glasses,


metallic foam and some geomaterials, no dislocations are formedduring deformation, alternative approaches (like strain softeningassociated with a net production of free volume) need to be consid-ered to understand the size effect in these materials. Although dis-location networks cannot be created in these materials toaccommodate plastic strains, plastic flow in these material at roomtemperature and moderate strain rate is typically inhomogeneousand proceeds via formation and propagation of shear bands (Anandand Su, 2005; Jiang et al., 2008; Schuh et al., 2007). Consequently,the hardening rule based on the Taylor dislocation model is suit-able for a variety of materials besides metals.

4. Constitutive equations of MSG theory based on Drucker–Prager plasticity

Gao et al. (1999) and Huang et al. (2000) proposed a mecha-nism-based theory of strain gradient plasticity (MSG) based on amultiscale framework linking the micro-scale notion of statisticallystored and geometrically necessary dislocations to the mesoscalenotion of plastic strain and strain gradient. This theory is then gen-eralized by Qiu et al. (2001) to account for the effect of intrinsic lat-tice resistance in the Taylor dislocation model. The flow theory ofMSG plasticity was established by Qiu et al. (2003) and the plasticshear localization was studied via this flow theory (Shi et al., 2009).In this section, a gradient-dependent Drucker–Prager elastoplasticmodel based on MSG theory is presented. The important effects ofthe pressure-sensitive yielding or the unsymmetrical behavior intension and compression can be shown unambiguously.

The MSG plasticity (Gao et al., 1999) provided a method of link-ing Taylor’s model to continuum theories, which is based on a mul-tiscale, hierarchical framework as shown in Fig. 3. A unite cell,named the mesoscale cell, is attached to each material point x onthe mesoscale, where x is the global coordinate of such point. A lo-cal coordinate system ~x centered at x characterizes the micro-scalevariables within the cell. Each point within the cell is considered asa micro-scale sub-cell, within which dislocation interaction is sup-posed to obey the Taylor relation that the strain gradient law of Eq.(22) applies. At this level of analysis, the stress and strain tensorsare defined in the classical sense and are defined as ~rðxþ ~xÞ and~eðxþ ~xÞ, respectively. Higher order stresses and strain gradientsassociated with strain gradient plasticity are introduced at themesoscale analysis. In such a way, mesoscale variables includethe stress r, strain e, higher-order stress s and strain gradient gwhich are independent of the local coordinate x.

Because the mesoscale cell size le (Fig. 3) is much smaller thanthe intrinsic material length l in Eq. (22), the displacement fieldwithin this cell is assumed to be

Fig. 3. The multiscale framework for MSG plasticity; the micro-scale strains ~e andstresses ~r are linked to dislocation interactions via the Taylor relation; straingradients g and higher-order stresses s are introduced as local variables on themesoscale level.

~uk ¼ eikxi þ12gijkxixj þ oðx3Þ ð31Þ

where xi denotes the local coordinate with origin at the center of thecell and gijk is the second gradient of the displacement field. Whenthe cell is sufficient small, considering the Taylor expansion of astrain component eij in the neighborhood of point x, we have

~eijðxþ ~xÞ ¼ eijðxÞ þ eij;m~xm ¼ eijðxÞ þ12ðgkijþgkji

Þ~xk ð32Þ

The micro-scale strain ~eij is thus related to the mesoscale strain eij

and strain gradient gkij.From energetic considerations, the mesoscale stress r(x) and

higher-order stress s(x) are the work conjugate of e(x) and g(x).They are linked to the micro-scale stress ~rðxþ ~xÞ by the workequality

ðrijdeij þ sijkdgijkÞVcell ¼Z

Vcell

~rijd~eij dV ð33Þ

where d represents the virtual variation, and Vcell is the volume ofthe mesoscale cell. The mesoscale stress rij and higher-order stresssijk in terms of the micro-scale stress ~rij in the cell can be obtainedby substituting Eq. (32) into Eq. (33) as follows:

rijðxÞ ¼1

Vcell

ZVcell

~rijðxþ nÞdV ð34Þ

sijkðxÞ ¼1

2Vcell

ZVcell

~rikðxþ nÞ~xj þ ~rjkðxþ nÞ~xi� �

dV ð35Þ

The constitutive law of strain gradient plasticity can be determinedby the micro-scale stress–strain relation (~r versus ~eÞ, since themesoscale stress r and higher-order stress s can be obtained fromthe mesoscale strain e and strain gradient g.

In the case of a cubic representative cell centered at x, with theedge length le, there exist relations

1Vcell

ZVcell

dV ¼ 1;1

Vcell

ZVcell

xkdV ¼ 0;1

Vcell

ZVcell

xkxmdV ¼ 112

l2e dkm

ð36Þ

4.1. Microscale analysis

In the flow theory of present gradient plasticity, the constitutiveequations are expressed in rate form. The total strain is composedof the elastic part _~ee

ij and plastic part _~epij,

_~eij ¼ _~eeij þ _~ep

ij ¼1

2l_~r0ij þ

19K

_~rkkdij þ _~epij ð37Þ

On the micro-scale, we consider the Drucker–Prager yield criterion

f ¼ffiffiffi3p

a~rm þ ~re � ~r ¼ 0 ð38Þ

where ~rm ¼ ~rkk=3 is the micro-scale hydrostatic stress, ~re ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

~r0ij ~r0ijq

is the effective stress, ~r0ij is the micro-scale deviatoric stres-

ses, ~r is the micro-scale flow stress derived from the Taylor modelin dislocation mechanics to account for the effect of geometricallynecessary dislocations and is given by

~r ¼ rref

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2p ðepÞ þ lg

qð39Þ

The rate form of above equation during plastic loading gives

~r _~r ¼ r2ref fpf 0p

_~ep þ 12

lg� �

ð40Þ

here f 0p ¼ @fp=@~ep.According to Eq. (13), the associated flow rule based on the

Drucker–Prager yield criterion, the micro-scale plastic strain ratecan be expressed as


_~epij ¼ b _~ep a

3dij þ

ffiffiffi3p

2~re~r0ij

!ð41Þ

Substituting Eq. (41) into Eq. (37), one gets

_~eij ¼1

2l_~r0ij þ

19K

_~rppdij þ b _~ep a3

dij þffiffiffi3p

2~re~r0ij

!ð42Þ

Multiplying ~r0ij of both sides of above equation, we obtain

~r0ij _~eij ¼1

2l_~r0ij ~r

0ij þ

ffiffiffi3p

3b _~ep ~re ð43Þ

Eq. (38) can be written as

~r ¼ ~re þffiffiffi3p

a~rm ð44Þ

The rate form of Eq. (44) is

_~r ¼ _~re þffiffiffi3p

a~rm ð45Þ

Multiplying the above two equations yields

~r _~r ¼ _~re ~re þ 3a2 _~rm ~rm þffiffiffi3p

a _~rm ~re þffiffiffi3p

a~rm_~re ð46Þ

Combining Eqs. (40) and (46) leads to

_~re ~re þ 3a2 _~rm ~rm þffiffiffi3p

a _~rm ~re þffiffiffi3p

a~rm_~re

¼ r2ref fpf 0p

_~ep þ 12

l _g� �

ð47Þ

The micro-scale effective plastic strain rate _~ep can be obtained fromEqs. (43), (47) in terms of strain rate and strain gradient rate

_~ep ¼ 1aþ b0

1~re

~r0ij _~eij þ ~M � 16l

r2ref l _g

� �ð48Þ

Here we have used the relationship _~re ~re ¼ 32

~r0ij _~r0ij, and

a ¼r2

ref fpð~epÞf 0pð~epÞ3l~re

; ~M ¼ a2

l_~rm ~rm þ

ffiffiffi3p

a3l

_~rm ~re þffiffiffi3p

a3l

~rm_~re;

b0 ¼ b=ffiffiffi3p

The micro-scale stress is then given by

_~rij ¼ K _~ekkdij þ 2l _~e0ij � Kbadij_~ep �

ffiffiffi3p

lbre

~r0ij _~ep ð49Þ

4.2. Mesoscale analysis: zeroth-order average

Let ~p be the micro-scale variable in the mesoscale cell and theaverage of ~p is defined by

pð0Þ ¼ 1Vcell

ZVcell

~pdV

which is also called the zeroth-order average of ~p.According to this definition, the zeroth-order average of micro-

scale strain ~e and stress ~r are the mesoscale strain e and stress r,respectively. It is straightforward to show that the micro-scalerelations given in Section 4.1 keep unchanged when experiencingthe zeroth-order average. For example, the zeroth-order averageof Eq. (37) gives the mesoscale strain rate

_eij ¼ _eeij þ _ep

ij ¼1

2l_r0ij þ

19K

_rkkdij þ _epij ð50Þ

The mesoscale plastic strain rate _epij can be obtained from Eq. (41)

_epij ¼ b _ep a

3dij þ

ffiffiffi3p

2rer0ij

!ð51Þ

We can see the form of mesoscale strain rate Eq. (50) and plasticstrain rate Eq. (51) is the same as that of Eqs. (8) and (13) described

in Section 2. But the mesoscale effective strain _ep is different fromthat defined in classic plasticity that the length effect has not beenconsidered and then it can be obtained from Eq. (48) by the zeroth-order average

ep ¼ 1aþ b0

1re

r0ij _eij þM � 16l

r2ref l _g

� �ð52Þ

with M ¼ a2

l_rmrm þ

ffiffi3p

a3l

_rmre þffiffi3p

a3l rm _re. The mesoscale stress rate

is then given by

_rij ¼ K _ekkdij þ 2l _e0ij � Kbadij þffiffiffi3p

lbre

r0ij

!_ep ð53Þ

4.3. Mesoscale analysis: first-order average

The constitutive relation between strain gradient and higher-order stress can be deduced from the first-order average of mi-cro-scale variable ~p within the mesoscale cell by

pð1Þk ¼1

Vcell

ZVcell

~p~xk dV

where ~xk denotes the local Cartesian coordinate which origins at themesoscale cell center.

Inserting Eq. (48) into Eq. (49) leads to

_~rij ¼ K _~ekkdij þ 2l _~e0ij �ffiffiffi3p

lb~r2

e

1aþ b0

~r0ij ~r0mn

_~emn

�ffiffiffi3p

lb~r2

e

~Maþ b0

~r0ij þffiffiffi3p

b6~r2

e

r2ref

aþ b0~r0ijl _g� Kbadij

_~ep ð54Þ

This equation can be written as

_~rij ¼ _~rð1Þij þ _~rð2Þij þ _~rð3Þij þ _~rð4Þij þ _~rð5Þij ð55Þ

where

_~rð1Þij ¼ K _~ekkdij þ 2l _~e0ij ð56Þ

_~rð2Þij ¼ �ffiffiffi3p

lb~r2

e

1aþ b0

~r0ij ~r0mn

_~emn ð57Þ

_~rð3Þij ¼ �ffiffiffi3p

lb~r2

e

~Maþ b0

~r0ij ð58Þ

_~rð4Þij ¼ffiffiffi3p

b6~r2

e

r2ref

aþ b0~r0ijl _g ð59Þ

_~rð5Þij ¼ �Kbadij_~ep ð60Þ

According to these micro-scale stresses, by using the first-orderaverage of Eqs. (35) and (54), we can obtain the higher-order stressat the mesoscale

_sijk ¼1

2Vcell

ZVcell

_~rkixj þ _~rkjxi

� �dV

¼ 12Vcell

ZVcell

_~rð1Þki xj þ _~rð1Þkj xi

� �þ _~rð2Þki xj þ _~rð2Þkj xi

� �hþ _~rð3Þki xj þ _~rð3Þkj xi



� �idV

¼ _sð1Þijk þ _sð2Þijk þ _sð3Þijk þ _sð4Þijk þ _sð5Þijk ð61Þ

By carefully derivation, we can get the concrete form of higher-or-der stress rate

_sijk ¼ c _gijk þ12ð _gkij þ _gkjiÞ þ

2Kl� 4

3

� �_gH

ijk

�ffiffiffi3p

lb2r2

e

1aþ b0

r0ki s0jmn þ s0jnm � s0mnj

� �h

þr0kj s0imn þ s0inm � s0mni

� �i_emn �

ffiffiffi3p

b4r2

e

caþ b0

r0kið _gjmn þ _gjnmÞ�


þr0kjð _gimn þ _ginmÞir0mn þ

ffiffiffi3p

br2

e

Mlaþ b0

s0ijk þffiffiffi3p

b6r2

e

r2ref

aþ b0l _gs0ijk

� Kbaaþ b0

12re

dki s0jmn þ s0jnm � s0mnj

� �hn

þdkj s0imn þ s0inm � s0mni

� ��_emn þ

l2e

24dki _gjmn þ _gjnm� ��

þdkj _gimn þ _ginmð Þ�r0mn

ð62Þ

Here, the derivation of above equation is detailed in Appendix.Finally, the constitutive equations for the flow theory based on theDrucker–Prager plasticity are assembled as follows.

If re < r, or _re < 0

_rij ¼ K _ekkdij þ 2l _e0ij

_sijk ¼ c _gijk þ12

_gkij þ _gkji

� �þ 2K

l� 4

3

� �_gH

ijk

If re = r, or _re P 0

_rij ¼ K _ekkdij þ 2l _e0ij � 2lbadij þffiffiffi3p

lbre

r0ij

!_ep

_sijk ¼ c _gijk þ12

_gkij þ _gkji

� �þ 2K

l� 4

3

� �_gH

ijk

�ffiffiffi3p

lb2r2

e

1aþ b0


� �h

þr0kj s0imn þ s0inm � s0mni

� �i_emn �

ffiffiffi3p

b4r2

e

caþ b0

"r0ki _gjmn þ _gjnm� �

þr0kj _gimn þ _ginmð Þ#r0mn þ

ffiffiffi3p

br2

e

Mlaþ b0

s0ijk þffiffiffi3p

b6r2

e

r2ref

aþ b0l _gs0ijk

� Kbaaþ b0

12re


� �hnþdkj s0imn þ s0inm � s0mni

� ��_emn

þ l2e

24dki _gjmn þ _gjnm� �

þ dkj _gimn þ _ginmð Þ� �

r0mn

)

g

σ

01 =x

Lx =1

Fig. 4. A schematic diagram of a bar subjected to a constant force g and uniformstress �r at the free end along the direction of the bar.

4.4. Equilibrium equations and boundary conditions

The rate form of equilibrium equations is

_r0ik;i � _s0ijk;ij þ _H;k þ _f k ¼ 0 ð63Þ

where fk is the body force and H is the combined measure of thehydrostatic stress and hydrostatic high-order stress according to

H ¼ 13rkk �

12sjkk;j ð64Þ

The increments of stress traction and higher-order stress traction onthe surface of the body are

_tk ¼ _Hnk þ ni _r0ik � _sijk;j

� �þ Dk ninjnp _s0ijp

� �� Dj ni _s0ijk

� �þ ninj _s0ijk � nkninjnp _s0ijp� �

Dqnq� �

ð65Þ

_rk ¼ ninj _s0ijk � nkninjnp _s0ijp ð66Þ

where ni is the unit normal to the surface, and Dj ¼ djk � njnk� �

@@xk

isthe surface gradient operator.

5. Applications

In this section, we employ the proposed gradient plasticity the-ory to investigate the size-dependent behavior and illustrate some

features of this theory for pressure-sensitive and pressure-insensi-tive materials. Especially, we investigate the effect of the pressure-sensitivity index on the material response via two examples of uni-axial tension and pure bending.

5.1. Uniaxial tension subject to a constant body force

We consider a simple bar subjected to a constant body force gand a uniform stress �r at the free end along the direction of thebar as shown in Fig. 4. This is the simplest, one-dimensional exam-ple displaying the strain gradient effect. The only non-vanishingstress, satisfying the equilibrium Eq. (63) and boundary conditionr11jx1¼0 ¼ �r, is

r11 ¼ gx1 þ �r ð67Þ

The non-vanishing deviatoric stresses are

r011 ¼23r11; r022 ¼ r033 ¼ �

13r11 ð68Þ

The effective stress is re = r11 and the hydrostatic stress is rm = rkk/3 = r11/3. The Drucker–Prager yield function under this stress con-dition is expressed asffiffiffi

3p

ar11=3þ r11 ¼ r ð69Þ

Inserting Eq. (28) into Eq. (69), we obtain

ffiffiffi3p

a=3þ 1� �

r11 ¼ rY

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2p ðepÞ þ lgp

qð70Þ

The function fp(ep) in Eq. (70) is given by Eq. (27).The non-vanishing plastic strain rates follow directly from Eq.

(51) as

_ep11 ¼ b _ep a

3þ

ffiffiffi3p

3

!ð71aÞ

_ep22 ¼ _ep

33 ¼ b _ep a3�

ffiffiffi3p

6

!ð71bÞ

When the applied stress �r at the free end of the bar (x1 = 0) and thebody force g are imposed proportionally, the plastic strain reducesto

ep11 ¼ bep a

3þ

ffiffiffi3p

3

!ð72Þ

ep22 ¼ ep

33 ¼ bep a3�

ffiffiffi3p

6

!ð73Þ

The non-vanishing components of the strain gradient tensor can beobtained from Eq. (30),

0.0 0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x1/L

α=0.0 l/L=0.1α=0.1 l/L=0.1α=0.2 l/L=0.1α=0.0 l/L=0.3α=0.1 l/L=0.3α=0.2 l/L=0.3

εp

Fig. 6. The plastic strain distribution in the bar when different values of pressure-sensitivity index are taken, with l/L = 0.1 and l/L = 0.3, respectively.


gp111 ¼ ep

11;1 ¼ ba3þ

ffiffiffi3p

3

!dep

dxð74aÞ

gp221 ¼ gp

331 ¼ �ep22;1 ¼ �ep

33;1 ¼ �ba3�

ffiffiffi3p

6

!dep

dxð74bÞ

gp122 ¼ gp

133 ¼ gp212 ¼ gp

313 ¼ ep22;1 ¼ b

a3�

ffiffiffi3p

6

!dep

dxð74cÞ

The effective plastic strain gradient gp in Eq. (29) becomes

gp ¼ b2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi79a2 þ 5

6� 4

ffiffiffi3p

9a

sdep

dxð75Þ

From Eqs. (67), (70), (27), (75), one obtains

ffiffiffi3p

3aþ 1

!ðx1=Lþ 1Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ E

rYep

� �1=n

þ lgp

sð76Þ

Eq. (76) can be solved numerically by Newton method, the centralfinite difference scheme is used to calculate dep/dx. Fig. 5 showsthe plastic strain distribution in the bar (ep versus x1/L) for severalratios of intrinsic material length l to the bar length L. The limitl/L = 0 corresponds to conventional plasticity. The material parame-ters are plastic work hardening exponent n = 5, Young’s modulusE = 500rY, pressure sensitivity index a = 0. All curves in Fig. 5 givea vanishing plastic strain at x1 = 0, but strong size effect is clearlyobserved as x1 increases such that it develops more plastic strainfor small value of l/L than that for large value of l/L.

It is shown that, when the pressure-sensitivity index a takeszero, the Drucker–Prager yield function equals to the von-Misesyield function. Under this condition, the strain gradient plasticitytheory developed here is the same as the well-known MSG plas-ticity developed by Gao et al. (1999), Huang et al. (2000) and Qiuet al. (2003). In order to investigate the effect of the pressure-sen-sitivity index a on strain gradient, we take three different valuesof a, a = 0, 0.1 and 0.2, respectively. As a comparison study, twovalues of l/L are considered in Fig. 6, each with three values ofa. Fig. 6 shows the larger the value of a takes; the more plasticdeformation develops along the bar and the plastic strain atx1 = 0 increases with the increase of a. This result is the same asthe research of Subramanya et al. (2007) that the pressure sensi-tivity enhances the plastic strain. These numerical results showthat our model can predict the interplay between the size effectand pressure sensitivity.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.01

0.02

0.03

0.04

0.05

0.06

εp

x1/L

l/L=0.0 l/L=0.2 l/L=0.4

Fig. 5. The plastic strain distribution in the bar; l is the instrinsic material length inthe strain gradient plasticity and L the bar length. The pressure-sensitivity index atakes zero.

Shi et al. (2001) studied the same problem using the MSG plas-ticity that involves the higher-order stress. Huang et al. (2004) alsostudied this problem using the CMSG plasticity theory withouthigher-order stress. Their studies showed that these two kinds oftheories agreed very well with each other except near the bound-ary. According to their research, we do not consider the influenceof the higher-order stress in the numerical examples in presentstudy.

5.2. Bending of thin beams

Stolken and Evans (1998), Shrotriya et al. (2003) and Haque andSaif (2003) observed strong size effect on plastic work hardening inbending of thin beams with different beam thicknesses. They drewa conclusion that the bending strength of beams significantly de-creased with the beam thickness increase. This size effect cannotbe explained using the classical plasticity theory which does notpossess an intrinsic material length scale. In this section, thestrength of thin beams in pure bending is considered by employingthe gradient plasticity model developed in Section 4. Although thisproblem has got extensive researches, it does not consider thepressure-sensitive yielding and is only suitable for pressure-insen-sitive materials which the von Mises yield criterion dominates.Here, we give more emphases on the effect of the pressure-sensi-tivity index on the strength of thin beams in the presence of sizeeffects.

Consider bending of ultra-thin beams under plane-strain condi-tions. Let x1 be the neutral axis of the beam and bending occurs inthe x1–x2 plane. The beam has a thickness h, the width in the out-of-plane (x3) direction is b and is subject to a bending curvature kas shown in Fig. 7(a).

From classical strength of materials, the displacement fieldof the beam under plane-strain bending can be defined asfollows:

u1 ¼ kx1x2; u2 ¼ �kðx21 þ x2

2Þ; u3 ¼ 0 ð77Þ

The associated non-vanishing strains are

e11 ¼ kx2; e22 ¼ �kx2; e12 ¼ 0 ð78Þ

Then we obtain

ekk ¼ 0; eij ¼ e0ij ð79Þ

The effective strain, ee ¼ffiffiffiffiffiffiffiffiffiffiffiffi23 eijeij

q, and the effective strain gradient,

g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffireereep

, can be expressed by using Eq. (79), as follows:

ee ¼2ffiffiffi3p kjx2j; g ¼ 2ffiffiffi

3p k ð80Þ

x3

x2

x1

11σ

11σM

x2

x3

x1

MM b

h

L

(a) (b)

Fig. 7. Schematic diagram of a beam subjected to plane-strain bending: (a) geometrical configuration and (b) the stress distribution across the section.

0.00 0.02 0.04 0.06 0.08 0.10 0.120

40

80

120

160

h=25μm

h=50μmM/b

h2

εs=κh/2

h=12.5μm

Fig. 8. Comparison of the present gradient theory with the micro-bendingexperiment of annealed nickel thin beams by Stolken and Evans (1998). The dotsare experimental data and the lines are the predictions.


The rate of stresses can be obtained from Eq. (53)

_r0ij ¼ 2l _e0ij �ffiffiffi3p

2reb _epr0ij

!ð81Þ

Under the condition of proportional loading, Eq. (81) can be writtenas

r0ij ¼ 2l e0ij �ffiffiffi3p

2rebepr0ij

!

This equation can be written as

r0ij ¼ Ne0ij ð82Þ

with N ¼ 2l1þlbep

ffiffi3p

re

.

The non-vanishing deviatoric stresses can be obtained from Eq.(82) as

r011 ¼ Nkx2; r022 ¼ �Nkx2 ð83Þ

Then the effective stress and the mean stress can be expressed as

re ¼ffiffiffi3p

Nkx2 rm ¼13rkk ¼ r011 ¼ Nkx2 ð84Þ

According to the Drucker–Prager yield criterion, under uniaxial ten-sile loading conditions, the flow stress r has the form

r ¼ re þffiffiffi3p

arm ¼ffiffiffi3p

Nkx2ð1þ aÞ ð85aÞ

Under uniaxial compressive loading conditions, the flow stress rhas the form

r ¼ re �ffiffiffi3p

arm ¼ffiffiffi3p

Nkjx2jð1� aÞ ð85bÞ

The stresses, rij ¼ r0ij þ 13 rkkdij, with rkk ¼ 3r011, can be expressed as

r11 ¼ r011 þ rm ¼ 2Nkx2 ¼2ffiffiffi

3pð1þ aÞ

r for r11 > 0 ð86aÞ

r11 ¼ r011 þ rm ¼ �2Nkjx2j ¼ �2ffiffiffi

3pð1� aÞ

r for r11 < 0 ð86bÞ

The stress distribution across the section is shown in Fig. 7(b). To benoted, the Drucker–Prager yield function gives different stress–strain curves under tension and compression. The reason for thisdifference is owing to the different yield stresses and the differentvalues of the slopes of the yield surface under tension and compres-sion as shown in Fig. 2.

The flow stress in a power-law hardening material, can be ex-pressed by substituting Eq. (80) into Eq. (24) as follows:

r ¼ r0 þ rref

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffi3p kjx2j� �2=n

þ 2ffiffiffi3p lk

sð87Þ

here we have considered the influence of friction stress r0.The pure bending moment M can be obtained from the integra-

tion of the normal stress r11 in Eq. (86) over the cross-section ofthe beam as

M ¼ bZ h=2

�h=2r11x2dx2

Owing to half of the cross-section under uniaxial tensile loadingconditions, and the other under uniaxial compressive loading condi-tions, as shown in Fig. 7(b), the expression for M is

M ¼ bZ h=2

0r11x2dx2 þ

Z 0

�h=2r11x2dx2

" #

Here we assume the part with positive axis x2 is under uniaxial ten-sion and the part with negative axis x2 is under uniaxial compres-sion. So we obtain

M=bh2 ¼ r0

4ffiffiffi3pð1þ aÞ

þ r0

4ffiffiffi3pð1� aÞ

þ 2rrefffiffiffi3pð1þ aÞ

þ 2rrefffiffiffi3p

1� að Þ

" #

�Z 0:5

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ffiffiffi3p esy� �2=n

þ 2esdp

s� ydy ð88Þ

where es = kh/2 is the surface curvature and dp = l/h.In order to check the predictions of present gradient plasticity

with experimental results, comparisons the theory predictionswith the micro-bending test of thin Ni films by Stolken and Evans(1998) are made, with foil width b = 2.5 mm, length L = 6 mm, andthickness h = 12.5, 25 and 50 lm, respectively. The experimentalresults are fitted with rref = 1167 MPa, n = 1.54, r0 = 38, 10,6.2 MPa, l = 7.9, 8.0, 9.8lm for h = 12.5, 25, 50 lm, respectively. Be-cause Ni is pressure insensitivity material, the pressure-sensitivityindex a takes zero here. Fig. 8 gives the comparison of the experi-ment (Stolken and Evans, 1998) with the predictions from thepresent gradient theory.

0.00 0.02 0.04 0.06 0.08 0.10 0.120

40

80

120

160

εs=κh/2

M/b

h2

α=0.0 h=12.5μmα=0.1 h=12.5μmα=0.2 h=12.5μmα=0.3 h=12.5μmα=0.0 h=50μmα=0.1 h=50μmα=0.2 h=50μmα=0.3 h=50μm

Fig. 9. The normalized bending moment, M/bh2, versus the normalized curvature,es = jh/2, for several pressure sensitivity index a of annealed nickel thin beams.

0.00 0.02 0.04 0.06 0.08 0.10 0.120

50

100

150

200

250

εs=κh/2

M/b

h2

α=0.0 h=25μmα=0.1 h=25μmα=0.2 h=25μmα=0.3 h=25μmα=0.0 h=200μmα=0.1 h=200μmα=0.2 h=200μmα=0.3 h=200μm

Fig. 11. The normalized bending moment, M/bh2, versus the normalized curvature,es = jh/2, for several pressure-sensitive factor a of LIGA nickel thin beams.


We also take four distinct values of the pressure-sensitivity in-dex, a = 0, 0.1, 0.2, and 0.3 to evaluate the effect of pressure sensi-tivity index on material strength. Here a = 0 corresponds to theMSG plasticity theory as shown in Fig. 9 when other materialparameters are the same as those in Fig. 8. As a comparison study,two thicknesses are considered: h = 12.5 lm and h = 50 lm. It isobserved that the material response curves are different with dif-ferent values of a, and the material strength increases with the in-crease of pressure-sensitivity index. This result is the same as thework of Narasimhan (2004) that material strength increases whenthe material exhibits a pressure dependent plastic behavior. Con-sidering the interplay between the size effect and pressure sensi-tivity, we find that the size effect mainly dominates the level ofmaterial strength. Different pressure-sensitivity index can causethe difference of material strength, but this difference is less thanthat of size effect.

Fig. 10 compares the predictions from the present gradient plas-ticity with the micro-bending test of thin LIGA Ni foils by Shrotriyaet al. (2003), with foil width b = 0.2 mm, length L = 1.5 mm, andthickness h = 25, 50, 100, 200 lm. Because LIGA Ni is pressureinsensitive, the pressure-sensitivity index a takes zero. The exper-imental results are fitted with rref = 1030 MPa, n = 1, and r0 = 290,270, 230, 225 MPa for h = 25, 50, 100, 200lm, respectively. Thematerial scale length l takes 23, 21, 19, 21 lm for h = 25, 50, 100,200 lm, respectively. Fig. 11 shows the normalized bendingmoment, M/bh2, versus the normalized curvature, es = jh/2, for

0.00 0.02 0.04 0.06 0.08 0.10 0.120

40

80

120

160

200

240

h=200μm

h=25μm

h=50μm

M/b

h2

εs=κh/2

h=100μm

Fig. 10. Comparison of the present gradient theory with the micro-bendingexperiment of LIGA nickel thin beams by Shrotriya et al. (2003). The dots areexperimental data and the lines are the predictions.

four distinct values of pressure-sensitivity index with two differentthicknesses: h = 25 lm and h = 200 lm, respectively. All materialproperties are the same as those in Fig. 10. Here a = 0 correspondsto MSG plasticity. Similar to Fig. 9, it is observed that the materialstrength increases with the increase of a.

6. Conclusions

The enhanced gradient material theories formulate a constitu-tive framework on the continuum level that is used to bridge thegap between the micro- and macro-mechanical description. Inthe literature, different frameworks of gradient-dependent plastic-ity theories are presented and most of them are only suitable forpressure-insensitive materials. But many materials such as rock,concrete and newly developed metallic glasses exhibit some pecu-liarity that yielding cannot be simply described by the usual vonMises or Treasca criteria, as for crystalline metals, since normal-stress component acting on the shear plane or hydrostatic pressurealso plays a role on the plastic deformation. Alternatively, othercriteria e.g. the Mohr–Coulomb or Drucker–Prager, which take nor-mal stress and pressure effect into account, are invoked to properlydescribe the mechanical behavior of such materials. While they arenot equipped with any length scale that would reflect the typicalsize and spacing of characteristic microstructural features; there-fore, they are inherently incapable of describing size effects thatare experimentally observed.

In this work, the flow theory of MSG plasticity based on theDrucker–Prager yield function for pressure-sensitive isotropicmaterials has been established via a multiscale, hierarchical frame-work. A systematic method of formulating the mesoscale law ofstrain gradient plasticity from the micro-scale Taylor model hasbeen proposed. We follow the gradient-dependent theory by Gaoet al. (1999) and Huang et al. (2000). The incremental constitutiverelation based on the associated flow rule is adopted for a generalyield function for pressure-sensitive materials at micro-scale.Detailed incremental constitutive relations based on the Druc-ker–Prager yield function are derived at mesoscale. Two scalevariables are linked by a virtual equating work statement. Our the-ory belongs to higher order continuum theories of strain gradientplasticity. The focus here has been on the role of gradient plasticitytheory in predicting the size effect accommodated by the presenceof deformation gradient for pressure-sensitive materials.

Then we employ the strain gradient plasticity developed here toinvestigate two sample applications, including uniaxial tensionsubject to a constant body force and bending of thin beams. Theproposed theory provides accurate predictions when compared


to the experimental results. It is shown that the present strain gra-dient plasticity theory can capture the size effect observed at themicron and sub-micron scales in experiments. The numerical re-sults illustrate that when the pressure sensitive index takes differ-ent values, the material response curves are different and thematerial strength increases with the increase of pressure-sensitiveindex. It is also found that the material strength increases whenmaterial exhibits a pressure dependent plastic behavior and thesize effect mainly dominates the level of material strength consid-ering the interplay between the size effect and pressure sensitivity.These computational simulations shed light on the interplay be-tween the two mechanisms and its enhanced effect on yieldstrength and plastic flow. It proves that this procedure is able torepresent the material behavior of pressure-sensitive isotropicmaterials such as geomaterials, metallic foams and metallic glass.The proposed technique is easy and versatile. It is recommendedthat it can be developed for further applications.

There are still many challenges for improving the present work.For instance, there are many new phenomena due to the introduc-tion of pressure sensitivity, such as instabilities, shear bands for-mation, indentation size effects and crack tip fields in pressure-sensitive plastic solids. The model should be implemented in astandard finite element programme and then we can address theabove issues.

Acknowledgements

This work was supported by the National Science Foundation ofChina (Grant Nos. 10672120/A020316 and 10732080/A020316).

Appendix A

This appendix contains the detailed derivation of Eq. (62). UsingEqs. (35) and (56), it obtains

_sð1Þijk ¼1

2Vcell

ZVcell


� �dV

¼ 12Vcell

ZVcell

K _~eppdki þ 2l~e0ki

� �xj þ K _~eppdkj þ 2l~e0kj

� �xi

h idV

¼ c _gijk þ12

_gkij þ _gkji

� �þ 2K

l� 4

3

� �_gH

ijk

That is

_sð1Þijk ¼ c _gijk þ12

_gkij þ _gkji

� �þ 2K

l� 4

3

� �_gH

ijk

ðA:1Þ

where c ¼ ll2e=12, here we have used the relations

~eij ¼ eij þ12ðgkij þ gkjiÞxk

~emm ¼ emm þ12ðgkij þ gkijÞ þ xk

~e0ij ¼ e0ij þ12

gkij þ gkji �23

dijgkmm

� �xk

By using Eqs. (35) and (57), we have

_sð2Þijk ¼1

2Vcell

ZVcell


� �dV

¼ 12Vcell

ZVcell

�ffiffiffi3p

lb~r2

e

1aþ b0

~r0ki~r0mn

_~emnxj

"

�ffiffiffi3p

lb~r2

e

1aþ b0

~r0kj~r0mn

_~emnxi

#dV

¼ 12Vcell

ZVcell

�ffiffiffi3p

lb~r2

e

1aþ b0

~r0ki~r0mn

_emnxj

"

�ffiffiffi3p

lb~r2

e

1aþ b0

~r0kj ~r0mn

_emnxi

#dV

þ 12Vcell

ZVcell

�ffiffiffi3p

lbr2

e

1aþ b0

~r0ki~r0mn

12ð _gpmn þ _gpnmÞxpxj

"

�ffiffiffi3p

lbr2

e

1aþ b0

~r0kj~r0mn

12

_gpmn þ _gpnm� �

xpxi

#dV

¼ �ffiffiffi3p

lb2r2

e

1aþ b0


� �hþr0kj s0imn þ s0inm � s0mni

� �i_emn

�ffiffiffi3p

b4r2

e

caþ b0

r0ki _gjmn þ _gjnm� �

þ r0kj _gimn þ _ginmð Þh i

r0mn

Thus we get

_sð2Þijk ¼ �ffiffiffi3p

lb2r2

e

1aþ b0



� �i_emn

�ffiffiffi3p

b4r2

e

caþ b0



r0mn ðA:2Þ

Similarly we have

_s 3ð Þijk ¼

12Vcell

ZVcell


� �dV

¼ 12Vcell

ZVcell

ffiffiffi3p

br2

e

Mlaþ b0

~r0kixj þ ~r0kjxi

� �dV ¼

ffiffiffi3p

br2

e

Mlaþ b0

s0ijk

That is

_sð3Þijk ¼ffiffiffi3p

br2

e

Mlaþ b0

s0ijk ðA:3Þ

Next

_sð4Þijk ¼1

2Vcell

ZVcell


� �dV

¼ 12Vcell

ZVcell

ffiffiffi3p

b6r2

e

r2ref

aþ b0l _g ~r0kixj þ ~r0kjxi

� �dV ¼

ffiffiffi3p

b6r2

e

r2ref

aþ b0l _gs0ijk

So we have

_sð4Þijk ¼ffiffiffi3p

b6r2

e

r2ref

aþ b0l _gs0ijk ðA:4Þ

Substituting Eq. (48) into Eq. (60) obtains

_~rð5Þij ¼ �Kbadij1

aþ b01re

~r0mn_~emn þ ~M�

r2ref

6ll _g

!ðA:5Þ

By conducing the first-order average of Eq. (35), we obtain

_sð5Þijk ¼1

2Vcell

ZVcell


� �dV

¼ 12Vcell

ZVcell� Kba

aþ b01re

~r0mn_~emn dkixj þ dkjxi� �

dV

¼ � Kbaaþ b

1re

12Vcell

ZVcell

~r0mn_emn dkixj þ dkjxi� �

dV�

þ 12Vcell

ZVcell

~r0mn12ð _gpmn þ _gpnmÞðdkixj þ dkjxiÞxp

dV�

Then we have

_sð5Þijk ¼ �Kba

aþ b01

2redki s0jmn þ s0jnm � s0mnj

� �þ dkj s0imn þ s0inm � s0mni

� �h i_emn

n

þ l2e

24dkið _gjmn þ _gjnmÞ þ dkjð _gimn þ _ginmÞ� �

r0mn

)ðA:6Þ


Combining Eqs. (A.1)–(A.6), it can be obtained that

_sijk ¼ _sð1Þijk þ _sð2Þijk þ _sð3Þijk þ _sð4Þijk þ _sð5Þijk

¼ c _gijk þ12

_gkij þ _gkji

� �þ 2K

l� 4

3

� �_gH

ijk

�ffiffiffi3p

lb2r2

e

1aþ b0



� �i_emn

�ffiffiffi3p

b4r2

e

caþ b0



r0mn

þffiffiffi3p

br2

e

Mlaþ b0

s0ijk þffiffiffi3p

b6r2

e

r2ref

aþ b0l _gs0ijk

� Kbaaþ b0

12re


� �"(

þdkj s0imn þ s0inm � s0mni

� �#_emn

þ l2e

24dki _gjmn þ _gjnm� �

þ dkj _gimn þ _ginmð Þ� �

r0mn

)ðA:7Þ

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mechanism-based strain gradient drucker–prager elastoplasticity for pressure-sensitive materials

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