Strain gradient effects on cyclic plasticity

ARTICLE IN PRESS Journal of the Mechanics and Physics of Solids 58 (2010) 542–557

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Strain gradient effects on cyclic plasticity Christian F. Niordson , Brian Nyvang Legarth Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

a r t i c l e i n f o

abstract

Article history: Received 5 October 2009 Received in revised form 8 January 2010 Accepted 23 January 2010

Size effects on the cyclic shear response are studied numerically using a recent higher order strain gradient visco-plasticity theory accounting for both dissipative and energetic gradient hardening. Numerical investigations of the response under cyclic pure shear and shear of a finite slab between rigid platens have been carried out, using the finite element method. It is shown for elastic–perfectly plastic solids how dissipative gradient effects lead to increased yield strength, whereas energetic gradient contributions lead to increased hardening as well as a Bauschinger effect. For linearly hardening materials it is quantified how dissipative and energetic gradient effects promote hardening above that of conventional predictions. Usually, increased hardening is attributed to energetic gradient effects, but here it is found that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening. Furthermore, it is shown that dissipative gradient effects can lead to both an increase and a decrease in the dissipation per load cycle depending on the magnitude of the dissipative length parameter, whereas energetic gradient effects lead to decreasing dissipation for increasing energetic length parameter. For dissipative gradient effects it is found that dissipation has a maximum value for some none zero value of the material length parameter, which depends on the magnitude of the deformation cycles. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Cyclic plasticity Strain gradient plasticity Size effects

1. Introduction In metals strain gradient effects lead to significant strengthening on the micron scale. Experimental investigations on the size-effects in metals have been carried out for different materials and under different loading conditions such as ¨ bending (Stolken and Evans, 1998; Haque and Saif, 2003; Lou et al., 2005), torsion (Fleck et al., 1994), indentation and contact compression (Ma and Clarke, 1995; Swadener et al., 2002; Wang et al., 2006). While some experiments suggest that the yield strength increases with decreasing size (e.g. Fleck et al., 1994; Swadener et al., 2002), other experiments show that the size-effect is mainly affecting the material hardening behavior (e.g. Xiang and Vlassak, 2006). Some experiments even show size-effects on both yield strength and hardening behavior, such as the bending experiments by Haque and Saif (2003). Also for polycrystals, size-effects are generally observed both for the yield strength (the Hall–Petch effect) and for the material hardening behavior (e.g. Tsuji et al., 2002; Yu et al., 2005). Much research has been devoted to modeling observed size-effects. This includes modeling of the above mentioned experiments (Fleck and Hutchinson, 1997, 2001; Huang et al., 2000; Qu et al., 2006) in addition to studies of size-effects in void growth (Liu et al., 2005; Wen et al., 2005; Niordson, 2007), fiber reinforced materials (Bittencourt et al., 2003; Niordson, 2003; Legarth and Niordson, 2009) and fracture problems (Wei and Hutchinson, 1997, 1999). Different

 Corresponding author.

E-mail addresses: [email protected] (C.F. Niordson), [email protected] (B.N. Legarth). 0022-5096/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2010.01.007

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approaches have been followed in modeling the observed size-effects. Discrete dislocation methods (see e.g. Deshpande et al., 2003) have been used to model a variety of the above mentioned problems. However, most attempts to model sizeeffects in metals have been based on higher order continuum modeling, and different theories, both phenomenological (Fleck and Hutchinson, 1997, 2001; Gudmundson, 2004; Gurtin and Anand, 2005; Lele and Anand, 2008) and microstructurally based (see e.g. Gao et al., 1999; Gurtin, 2002) have been developed. In this paper the details of a finite element formulation of the strain gradient visco-plastic theory by Gudmundson (2004) are laid out. Recently, the mathematical foundation of this theory has been explored in detail by Fleck and Willis (2009). The implemented framework is capable of accounting for both dissipative and energetic gradient effects, using the isotropic orthogonalization of Smyshlyaev and Fleck (1996). To this end three dissipative and three energetic length parameters are needed. Until now, modeling of size-effects has mainly focused on monotonic loading conditions. It is the aim of the present study to analyze size-effects under cyclic loading conditions, which enables the analyses of size-effects on both the effective yield strength, material hardening, Bauschinger effects as well as dissipation properties. The problem studied is that of a thin metallic layer between rigid platens subjected to cyclic shearing. Most of the results presented are for elastic– perfectly plastic materials, but also results for linearly hardening materials are discussed. As a linearly hardening material does not show saturation in hardening during cyclic deformation the limit response will be elastic. Although this is not realistic for cyclic behavior, no attempt is made in the present paper, to investigate later stages of cyclic hardening, as has been done by e.g. Chaboche (1986, 1989) and Ohno and Wang (1993a, b) for conventional cyclic material behavior. Here, focus is retained on size-effects in the earliest stages of cyclic deformation. Hence, results are presented for the initial single load cycle only. For the elastic–perfectly plastic studies presented, this first load cycle is identical to the any following load cycles, due to the absence of cyclic hardening. The results obtained are presented in terms of response curves, from which properties like the effective yield strength, hardening behavior, Bauschinger effects and dissipation are extracted. It will be shown that dissipative gradient effects lead to increased yield strength, whereas energetic gradient contributions lead to increased hardening as well as a Bauschinger effect. Usually, increased hardening is attributed to energetic gradient effects, but here it is found that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening, exemplified by a linearly hardening material. The dissipation per load cycle is quantified, and effects of both dissipative and energetic length parameters are studied. It is shown that dissipative gradient effects can lead to both an increase and a decrease in the dissipation per load cycle depending on the magnitude of the dissipative length parameter, whereas energetic gradient effects lead to decreasing dissipation for increasing energetic length parameter. 2. Material model The material model used is that proposed by Gudmundson (2004), for which the principle of virtual work in Cartesian components is expressed as Z Z ðsij deij þ ðqij sij Þdepij þ mijk drpijk Þ dV ¼ ðTi dui þ Mij depij Þ dS ð1Þ V

S

Here, sij is the stress tensor, eij is the strain tensor, qij is the micro-stress and sij ¼ sij  13 dij skk is the stress deviator. The higher order nature of the theory manifests itself through the term mijk, which is the higher order stress (moment stress) work conjugate to the plastic strain gradient, rijk ¼ epij;k , where ( ),k signifies the partial derivative with respect to the coordinate xk. The right-hand side of the principle of virtual work includes the conventional traction vector Ti ¼ sij nj , work conjugate to the displacement vector ui, and the higher order traction Mij =mijknk, which is work conjugate to the plastic strain tensor epij . The strong form of (1) is the two equilibrium (see Gudmundson, 2004)

sij;j ¼ 0

ð2Þ

mijk;k þ sij qij ¼ 0

ð3Þ

of which the first is recognized as the conventional equilibrium equation, in the absence of body forces. The second equation is the higher order equilibrium equation that shows that in the presence of higher order stresses, mijk, the microstress, qij, and the stress-deviator, sij, are in general different. An additive decomposition of the total strain, eij , into and elastic part, eeij , and a plastic part, epij , is used

eij ¼ eeij þ epij

ð4Þ

In incremental form the principle of virtual work has the following form, where Dt is the time increment and ð_Þ ¼ ðd=dtÞð Þ Z Z _ ij de_ p Þ dS _ ijk dr_ ijk Þ dV ¼ Dt ðT_ i du_ i þ M Dt ðs_ ij de_ ij þ ðq_ ij s_ ij Þde_ pij þ m ð5Þ ij V

S

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Accounting for both dissipative and energetic gradient effects, the higher order stress is decomposed into a dissipative part, ~ ijk m ijk and an energetic part, m ~ ijk mijk ¼ m ijk þ m

ð6Þ

whereas the micro-stress is assumed to have a dissipative part, q ij , only, so that qij ¼ q ij

ð7Þ

In the following the isotropic visco-plastic theory will be developed. 2.1. Dissipative contributions To account for the dissipative terms, a visco-plastic potential is defined as Z E_ p p p0 p0 FðE_ ; Ep Þ ¼ sc ðE_ ; Ep ÞdE_

ð8Þ

0 p

where the effective stress, sc ¼ sc ðE_ ; Ep Þ, depends on the rate as well as the accumulated gradient enhanced effective plastic strain, with Ep defined from the isotropic incremental relation 3 X 2 p2 ðIÞ ðIÞ ðLdðIÞ Þ2 r_ ijk r_ ijk E_ ¼ e_ pij e_ pij þ 3 I¼1

ð9Þ

p Here, L(I) d are three dissipative constitutive length parameters. The plastic strain gradient tensor, rijk ¼ eij;k , is decomposed ðIÞ such that into three orthogonal tensors rijk 3 X

rijk ¼

ðIÞ rijk

ð10Þ

I¼1 ðIÞ ðJÞ according to Smyshlyaev and Fleck (1996), so that rijk rijk ¼ 0 if IaJ. Taking the variation of the visco-plastic potential gives p

p

dF ¼ sc ½E_ ; Ep dE_ ¼

p p 3 3 X sc ½E_ ; Ep  2 _ p _ p sc ½E_ ; Ep  X ðIÞ ðIÞ ðIÞ ðIÞ e ij de ij þ ðLdðIÞ Þ2 r_ ijk dr_ ijk ¼ q ij de_ pij þ m ijk dr_ ijk p p

E_

3

E_

I¼1

ð11Þ

I¼1

where the dissipative micro-stresses and higher order stresses are given by q ij ¼

p 2 sc ½E_ ; Ep  p e_ ij p 3 E_

p

and

ðIÞ m ijk ¼

sc ½E_ ; Ep  p E_

ðLdðIÞ Þ2 r_ ðIÞ ijk

ð12Þ

Using these relations together with (9) gives the following relation for the effective stress (see Gudmundson, 2004): 3 2

s2c ¼ q ij q ij þ

3 X

ðLdðIÞ Þ2 m ðIÞ m ðIÞ ijk ijk

ð13Þ

I¼1

2.2. Energetic contributions We assume that free energy is stored due to elastic strain, eeij ¼ eij epij , and due to gradients of plastic strain, but not due to plastic strain itself. Hence, the free energy is given by the expression ðIÞ ðIÞ C ¼ Cðeeij ; rijk Þ ¼ Cðeij epij ; rijk Þ

ð14Þ

from which the conventional stresses can be derived according to

sij ¼

@C @eeij

ð15Þ

Similarly, the energetic higher order stresses are derived according to ðIÞ ~ ijk m ¼

@C ðIÞ @rijk

ð16Þ

2.3. Constitutive equations Usually, plastic deformation is mainly considered to be a dissipative process. Hence, no free energy associated with the plastic strain itself was introduced in (14), and the micro-stress was assumed only to have a dissipative part in (7). This ensures that, at large length scales (small strain gradients), all energy associated with plastic deformation is dissipated.

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However, when large plastic strain gradients appear, geometrically necessary dislocations (GNDs) are stored (see Ashby, 1970), which gives rise to free energy associated with the local stress field of the GNDs (see e.g. Gurtin, 2002; Ohno and Okumara, 2007) as well as increased dissipation when the GNDs move in the lattice. In the following the specific constitutive assumptions are laid out. A power-law relation for the visco-plastic behavior is assumed ! p m E_ p sc ðE_ ; Ep Þ ¼ gðEp Þ _ ð17Þ e0 where e_ 0 is a reference strain rate, m is the visco-plastic exponent and g(Ep) defines the hardening behavior of the material. In the present paper we focus on elastic–perfectly plastic solids and linearly hardening solids. For these materials the hardening function is defined according to p

gðEp Þ ¼ sy þ hE

ð18Þ

where sy is the initial yield stress, and h is the hardening modulus, which is zero for elastic–perfectly plastic material behavior. The free energy is defined according to the isotropic expression

C¼

3 3 1 e 1 X 1 1 X ðIÞ ðIÞ ðIÞ ðIÞ e L ee þ G ðLðIÞ Þ2 rijk rijk ¼ ðeij epij ÞLijkl ðekl epkl Þ þ G ðLðIÞ Þ2 rijk rijk 2 ij ijkl kl 2 I ¼ 1 e 2 2 I¼1 e

ð19Þ

where Lijkl is the isotropic elastic stiffness tensor, G is the elastic shear modulus and L(I e ) are three energetic constitutive length parameters. With these constitutive assumptions, the increments of the conventional stresses are given by

s_ ij Dt ¼ Lijkl e_ ekl Dt ¼ Lijkl ðe_ kl e_ pkl ÞDt

ð20Þ

The micro-stress increments are found from (12) as   2 g 0 ½Ep  1 p q_ ij Dt ¼ q_ ij Dt ¼ E_ p ðm1Þq ij DE_ þ sc De_ pij þ q DEp 3 g½Ep  ij

ð21Þ

p

p p with DE_ ¼ Dt E€ , and De_ pij ¼ Dt e€ ij . The dissipative and the energetic higher order stress increments are given by

_ Dt ¼ Dt m ijk

3 X

_ ðIÞ m ijk

ð22Þ

ðIÞ ~_ ijk m

ð23Þ

I¼1

~_ ijk Dt ¼ Dt m

3 X I¼1

with 0 p _ ðIÞ Dt ¼ E_ 1 ððm1Þm ðIÞ DE_ p þ sc ðLðIÞ Þ2 Dr_ ðIÞ Þ þ g ½E  m ðIÞ DEp m ijk p ijk d ijk g½Ep  ijk

ð24Þ

ðIÞ ðIÞ ~_ ijk Dt ¼ GðLeðIÞ Þ2 Drijk m

ð25Þ

obtained from (12), (16) and (19). Inserting these incremental stress quantities in the incremental principle of virtual work, (5), leads to the two equations Z Z Z Lijkl Dekl de_ ij dV ¼ Lijkl Depkl de_ ij dV þ DTi du_ i dS ð26Þ V

V

S

from which the displacement increments can be determined, and ! !# Z " 3 3 X 2 p p X 1 ðIÞ 2 ðIÞ ðIÞ p ðIÞ 2 ðIÞ p _ _E 2 2 e_ p De_ p þ dV ðL Þ r_ ijk Dr_ ijk ðm1Þðq mn de_ mn þ m mnl de_ mn;l Þ þ E p sc De_ de_ þ ðL Þ Dr_ ijk dr_ ijk p 3 ij ij I ¼ 1 d 3 ij ij I ¼ 1 d V " # Z Z 3 X g 0 ½Ep  p ðIÞ ¼ Dsij de_ pij  DE ðq ij de_ pij þm ijk dr_ ijk Þ GðLeðIÞ Þ2 Drijk dr_ ijk dV þ DMij de_ pij dS ð27Þ p g½E  V S I¼1 from which the increments in plastic strain rate can be determined, having already solved for the displacement increments, and computed the increment in the stress deviator, Dsij . Eqs. (26) and (27) suggest that the displacement increments and increments of the plastic strain rate are decoupled. However, as noted above, this is not so as (27) depends on Dsij which is only known after solving for the displacement increments in (26). In the present study, a forward Euler solution procedure is used, where the plastic strain rate and its gradient are assumed to be known at any time based on the micro-stress, qij ¼ q ij , and the dissipative part of the higher

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order stress, m ijk . This enables the solution of the displacement increment from (26), and the increment in the stress deviator can thus be found by updating the stresses using (20) with the plastic strain rate calculated from (12a). This field is then inserted into (27) which is used to find the next increment of the plastic strain rate. A detailed discussion of the solutions procedure specific for the finite element method is included in the following section, also including the algorithm used for the forward Euler time integration. 3. Numerical method The material model has been implemented in a plane strain finite element program employing quadratic shape functions for the displacement increments and bilinear shape functions for increments of the components of the plastic strain rates. 3.1. Finite element discretization Eight node serendipity elements are used to interpolate displacement increments from nodal values, DuN , so that a total of 16 shape functions, NN i , are used for the displacements in two dimensions 16 X

Dui ¼

NiN DU N

ð28Þ

N¼1

Increments of the total strain are then interpolated according to 16 X

Dekl ¼

N EN kl DU ¼

N¼1

16 X 1 N N ÞDU N ðN þ Nl;k 2 k;l N¼1

ð29Þ

1 N N where EN kl ¼ 2 ðNk;l þ Nl;k Þ is the strain–displacement matrix. Finite element interpolation is also used to interpolate increments of the components of the plastic strain rate from nodal values. The components De_ p11 , De_ p22 and De_ p12 are used as free variables and De_ p33 ¼ ðDe_ p11 þ De_ p22 Þ are calculated based on plastic incompressibility. Hence, for a four node bilinear element a total of 12 shape functions are used 12 X

De_ pij ¼

PijN De_ pN

ð30Þ

N¼1

The increment in the plastic strain rate gradient can be expressed by 12 X

Dr_ ijk ¼ Dr_ ijk ¼

N Pij;k De_ pN

ð31Þ

N¼1

In order to calculate the three orthogonal parts of Dr_ ijk the shape functions A(I),N are used ijk 12 X

ðIÞ Dr_ ijk ¼

ðIÞ;N Aijk De_ pN

ð32Þ

N¼1

where ð1Þ;N N N N N N N N N N N N N Aijk ¼ 13 ðPij;k þPjk;i þ Pki;j 15ðdij ðPkp;p þPpp;k þPpk;p Þ þ dik ðPjp;p þ Ppp;j þ Ppj;p Þ þ djk ðPip;p þPpp;i þ Ppi;p ÞÞÞ

ð33Þ

ð2Þ;N N N N N ¼ 16ðekip ðepqh Pjh;q þ ejqh Pph;q Þ þ ekjp ðepqh Pih;q þ eiqh Pph;q ÞÞ Aijk

ð34Þ

ð3Þ;N N N N N N ¼ 13 ð12 ekip ðepqh Pjh;q ejqh Pph;q Þ þ 12 ekjp ðepqh Pih;q eiqh Pph;q Þ þ 15ðdij ðPkp;p Aijk N N N N N N N N þ Ppp;k þ Ppk;p Þ þ dik ðPjp;p þPpp;j þ Ppj;p Þ þ djk ðPip;p þ Ppp;i þPpi;p ÞÞÞ

ð35Þ ðIÞ A133

Here, dij is Kronecker’s delta and eijk is the permutation tensor. Note that even for plane strain ðIÞ ðIÞ ¼ A323 a0. For further details see Smyshlyaev and Fleck (1996). A233 Using the above relations in (26) and (27) gives the following two systems of discretized equations

ðIÞ ¼ A313 a0,

and

N M KMN e DU ¼ DF1

ð36Þ

M _p KMN p De N ¼ DF2

ð37Þ

and

where ¼ KMN e

Z V

N Lijkl EM kl Eij dV

ð38Þ

ARTICLE IN PRESS C.F. Niordson, B.N. Legarth / J. Mech. Phys. Solids 58 (2010) 542–557

is the symmetric elastic stiffness matrix and Z Z p M DFM ¼ L D e E dV þ DTi NiM dS ijkl 1 kl ij V

547

ð39Þ

S

is the sum of the conventional visco-plastic loading vector and the external loading vector. The symmetric plastic system matrix is given by ! !# Z " 3 3 X X ðIÞ 2 ðIÞ ðIÞ;N ðIÞ 2 ðIÞ;N M M M _ 1 sc 2 P N P M þ _ 2 2 e_ p P N þ _ KMN ðm1Þðq dV ¼ ðL Þ r A P þm P Þ þ E ðL Þ A P E mnq mn mn mn;q p p p ij;k ijk ijk d ijk 3 ij ij I ¼ 1 d 3 ij ij V I¼1 ð40Þ and the higher order loading vector is given by # Z Z " 3 X g 0 ½Ep  p ðIÞ M M M M ðIÞ 2 DFM ¼ D s P  D E ðq P þm P Þ ðL Þ G D r P DMij PijM dS ij ij ijk ij;k ij ij e 2 ij;k dV þ ijk p g½E  V S I¼1

ð41Þ

The two systems of equations can be solved successively, such that (36) gives the solution for the displacement increments, which can then be used in relation to (37) when solving for the increments of the components of the plastic strain rate. 3.2. Numerical algorithm To start the numerical algorithm the plastic strain rate and all stresses are initialized to zero. An elastic increment is then carried out according to (36). Based on this solution the conventional stresses, sij , are obtained and the effective stress, sc , is calculated as the von Mises stress. The micro-stress, qij, is calculated as the stress deviator, sij, since the higher p order stress is taken to be zero initially. The effective plastic strain rate, E_ , is then computed from (17). This has initialized the solution procedure, and subsequent increments are carried out according to the following algorithm: ðIÞ 1. Based on the micro-stress, qij, and the orthogonalized dissipative part of the higher order stress, m ijk , the plastic strain p ðIÞ rate, e_ ij , and its orthogonalized gradient tensors, r_ ijk , are calculated using (12). This enables calculation of the increment in plastic strain and its gradient by multiplication by the time step. 2. The displacement increments are calculated according to (36). 3. Stress increments are calculated using (20). 4. Increments of the components of the plastic strain rate are calculated using (37). 5. The micro-stress and the higher order stress are updated using (21), (24) and (25).

Initializing the algorithm using an elastic step introduces an initial error in the micro-stress, qij, and in the higher order stress, mijk, and hence also in the effective stress, sc . This is due to the fact that the plastic strain distribution is calculated from (12a) which leads to a plastic strain rate distribution inconsistent with (12b). In the following time steps this inconsistency dies out and the error goes toward zero, as the micro-stress and the higher order stress converge toward their correct distributions consistent with the plastic strain rate field. A convergence study has revealed that it takes about 20 time steps before the relative error has died out to a level less than one percent. As this error only affect the first few time steps, and only the higher order stress quantities, the effect on the solutions can be eliminated completely by taking sufficiently small time steps initially, so that at least 20 are used in the elastic deformation regime. In the studies to be presented more than 500 increments are used in the elastic regime, which efficiently suppresses any error from the initialization procedure to a negligible level. Another option that could be used to initialize the solution procedure is to formulate a deformation theory for the first time step. However, this has not been pursued in the present paper. 4. Results and discussion A micron scale metallic layer between rigid platens is analyzed under plane strain deformation. The layer has the height 2H and width 2W, and it occupies the region: W rx1 rW and 0 r x2 r 2H (see Fig. 1). The top and bottom edges along x2 =0 and x2 =2H are bonded to rigid platens that are displaced horizontally according to u1 =u2 = 0 on x2 =0 and u1 =2U(t) on x2 =2H with u2 =0. The ends of the slab along x1 ¼ 7 W are traction free, both in terms of conventional tractions and higher order tractions. The material is modeled as elastic–perfectly plastic (unless otherwise stated), with a ratio of the yield stress to Young’s modulus of sy =E ¼ 0:004, Poisson’s ratio n ¼ 0:3, a visco-plastic exponent of m =0.05, and a reference strain rate of e_ 0 ¼ 0:005 s1 . A cyclic displacement is prescribed so that the amplitude of the overall shear strain is given by Umax =H ¼ 5s  e_ 0 with an overall shear strain rate of jg_ 12 j ¼ j2e_ 12 j ¼ e_ 0 . As discussed by Niordson and Hutchinson (2003), gradients in this problem arise from two sources: (i) the traction free ends; (ii) any constraint on plastic flow at the rigid platens.

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Fig. 1. (a) A slab of material between rigid platens is analyzed under shear deformation. The elastic–plastic solid occupies the region W r x1 r W and 0 r x2 r 2H, and the platens are displaced the distance 2U(t) relative to each other in the direction of the x1-axis. (b) A quarter of the slab is analyzed (marked by the dashed rectangle). To the right appropriate higher order boundary conditions for the quarter problem are shown.

Fig. 2. Cyclic pure shear response curves for an elastic–perfectly plastic solid for (a) dissipative gradient effects and (b) energetic gradient effects.

Consider first the infinitely long slab (W-1) of an elastic–perfectly plastic material, which has no gradients of strain in the x1-direction. This case is modeled using a single column of 80 elements through half the thickness of the layer (H) and _ 12 ¼ e_ p ¼ e_ p ¼ 0). Here, any gradient effects are appropriate periodic boundary conditions at the sides of the column (M 11 22 triggered by constraints on plastic flow at the platens. Fig. 2a shows the cyclic response in terms of the shear stress as a (I) function of the overall shear strain, considering dissipative gradient effects with pffiffiffi the three length parameters equal, Ld = Ld, I= 1,2,3. The stress is normalized by the yield stress in pure shear ty ¼ sy = 3, and the strain is normalized by the yield strain in pure shear, gy ¼ ty =G, with the shear modulus, G ¼ E=2ð1 þ nÞ. It is seen that increasing Ld leads to an increased effective yield strength, teff , as was also found by Frederiksson and Gudmundson (2007). A parametric study reveals that the effective yield strength increases slowly with Ld for small values of the material length parameter, but for larger values an almost linear relationship is predicted. For energetic gradient effects with L(I) e = Le, I= 1,2,3, Fig. 2b shows cyclic response curves for different values of Le. It is seen that increasing Le leads to increased material hardening, with no effects on the yield strength. For the dissipative case (Fig. 2a) no Bauschinger effect is observed, as the yield strength in reverse loading is the same as during initial loading. However, for the energetic case (Fig. 2b) a significant Bauschinger effect is observed with a yield strength during reverse loading is significantly different from that at the onset of unloading. This kinematic hardening effect is a natural ingredient in the present model due to the influence of higher order stresses on the effective stress (see (13)). In some other studies, where energetic higher order effects are incorporated through gradients of the effective plastic strain (e.g. Lele and Anand, 2008) the Bauschinger effect qffiffiffiffiffiffiffiffiffiffiffiffiffi is not captured, since the effective plastic strain quantities are defined by incremental positive relations like e_ p ¼ 23 e_ pij e_ pij . It should be noted that the solution procedure presented in the present paper rests on visco-plastic gradient effects. Hence, for the solutions involving energetic gradient effects, a small amount of visco-plastic gradient effects have been included, by specifying a value of Ld/H on the order of a few percent. This leads to a negligible effect on the effective yield strength of the material. In general, the three length parameters for the dissipative and energetic cases can be different (see Hutchinson, 2000). For the pure shear problem the material length parameters combine into two effective material length parameters that

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govern the problem. These are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 3 ð3Þ 2 ðLð1Þ Þ2 þ 16 ðLð2Þ Þ2 þ 10 ðLd Þ Ld ¼ 15 d d Le ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ 2 ð2Þ 2 8 1 3 ð3Þ 2 15 ðLe Þ þ 6 ðLe Þ þ 10ðLe Þ

549

ð42Þ

ð43Þ

for the dissipative and energetic cases, respectively. For the dissipative case, any combination of the three material length parameters, L(I) d , will yield the same pure shear response as for the case where all three length parameters are chosen equal to Ld (L(I) d = Ld for I= 1,2,3). Similarly, it holds for the energetic case that any combination of the material length parameters, L(I) e , will yield the same pure shear response as for the case where all three energetic length parameters are chosen equal to Le. It is obvious from (42) and (43) that, L(1) is the most important length parameter for the pure shear problem, whereas L(2) is the least important parameter, for both the dissipative and the energetic cases. The effective yield strength is shown in Fig. 3a as a function of the normalized dissipative length scale. The markings on the curve show the computed points, and a piecewise linear curve is plotted as well. It is seen that for small values of Ld/H a slow increase in the yield strength is predicted, but for larger material length scales an almost linear relationship is predicted. The normalized effective hardening modulus, heff, is shown in Fig. 3b as a function of the energetic length parameter. A quadratic relationship is predicted, and it can be shown analytically for corresponding time-independent material behavior that the normalized effective hardening modulus, heff, is given by    1 heff G 9 Le 2 ¼3 1 ¼ ð44Þ G Gt 2 H where Gt is the tangent modulus of the response curves. Also in Fig. 3b, the markings indicate the computed points, but the curve is plotted according to (44), showing that viscous effects are negligible. The increase in yield strength for dissipative gradient materials is due to higher order stresses. Plastic deformation is governed by the effective stress, sc , which is influenced by higher order stresses, mijk, both directly and through the microstress, qij. This has the effect of lowering the effective stress on small scales that are comparable to or smaller than Ld. Fig. 4 shows the distribution of different normalized stress quantities through the half thickness of the layer for a dissipative gradient material with Ld = H at different deformation levels. Both the effective stress, sc , the micro-stress, q12 = q21, and the higher order stress, m122 =m212 are shown. Fig. 4a shows results at an overall deformation level equal to that at the onset of plasticity for a corresponding conventional material (point I in Fig. 2a), while Fig. 4b shows results at the maximum deformation level in Fig. 2a (point II). For comparison the effective stress level is shown for a corresponding conventional material. The results show that the effective stress, sc , is suppressed initially (Fig. 4a), when compared to conventional predictions, but it is enhanced at larger deformation levels (Fig. 4b). Furthermore, it is found that for the problem of pure shear studied here, all stress quantities scale proportionally, as the deformation increases. For the dissipative gradient dependent materials the results also show that a small increase in the effective stress close to the platens exists. Perhaps counter-intuitively, this means that plastic deformation (here measured by the gradient-enriched effective plastic strain, Ep) initiates at the platens where plastic deformation is imposed to vanish from the higher order boundary condition, epij ¼ 0. This result is an artifact of the visco-plastic material model, and decreases with the visco-plastic exponent, m, and it does not transfer to time-independent material behavior for the present problem.

Fig. 3. (a) Effective yield strength as a function of the dissipative length parameter, and (b) effective hardening modulus as a function of the energetic length parameter for an elastic–perfectly plastic solid under pure shear.

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Fig. 4. Distribution of different stress quantities through the half thickness of the layer for an elastic–perfectly plastic dissipative gradient material with Ld = H at different deformation levels. (a) Shows stress distribution at the overall deformation level at the onset of plasticity for a corresponding conventional material (point I in Fig. 2 a), and (b) shows the results at the maximum deformation studied (point II in Fig. 2 a). For comparison, the effective stress is shown for a corresponding conventional material.

Fig. 5. Distribution of plastic shear strain, gp ¼ 2ep12 through the half thickness of the layer, at the maximum deformation level studied in Fig. 2. Results are shown for a dissipative gradient material with Ld = H, an energetic gradient material with Le = H and a conventional material.

Fig. 5 shows the distribution of plastic shear strain, gp ¼ 2ep12 through the half thickness of the layer, at the maximum deformation level studied in Fig. 2 for both dissipative gradient behavior with Ld = H, energetic gradient behavior with Le = H and conventional material behavior. For conventional materials the plastic strain distribution is constant, while for energetic gradient material behavior the plastic strain distribution is parabolic. For the dissipative gradient material, a more distinct boundary layer is observed near the rigid platen, where the plastic flow constraint is imposed. In Fig. 6a results are presented for combined energetic and dissipative gradient material behavior. The energetic and the dissipative length scales are chosen equal, and results are presented for two different values of Ld =Le = H/2 and Ld = Le = H in addition to the conventional cyclic response curve. It is observed that combining the two different kinds of gradient behavior leads to additional hardening as well as an increase in yield strength. The distribution of stress quantities through the half thickness of the layer is shown in Fig. 6b, for Ld = Le = H at the maximum deformation level (point IV in Fig. 6a). Even though the dissipative higher order stress, m 122 , and the micro-stress, q12, are rather non-homogeneous, the resulting ~ 122 , is scaled down by a factor of 103 effective stress, sc , is rather uniformly distributed. The energetic higher order stress, m to show on the same figure, as it scales with the shear modulus according to (25), and not the yield stress as the other stress quantities. It is observed that the higher order stress quantities vanish at the symmetry boundary and have their

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Fig. 6. (a) Cyclic response curves for combined dissipative and energetic gradient effects for an elastic–perfectly plastic solid. (b) Distribution of different stress quantities through the half thickness of the layer.

Fig. 7. Normalized dissipation per load cycle as a function of material length parameters for an elastic–perfectly plastic solid. Results are shown for both pure dissipative gradient effects, pure energetic gradient effects and combined dissipative and energetic gradient effects with Ld = Le.

maximum value at the rigid platen, whereas the micro-stress has its maximum value at the symmetry boundary and vanishes at the platens. The predicted increase in yield strength and material hardening with diminishing size have a rather complicated combined effect on the dissipation during cyclic loading. Fig. 7 shows the dissipation per load cycle, D, as a function of the material length parameters for both dissipative and energetic gradient effects. The dissipation is normalized by that for a conventional material, D0. It is observed that for this specific magnitude of the deformation cycles considered, dissipation has a maximum value for Ld =H  1, whereas for the energetic gradient effects dissipation decreases with increasing Le/H. As energetic gradient effects lead to increased hardening and a large Bauschinger effect, they tend to decrease dissipation by decreasing the size of the hysteresis loop. On the other hand, for dissipative gradient effects, the yield strength is increased with no Bauschinger effect, which results in increasing dissipation for moderate material length scales, but decreasing dissipation for larger dissipative length scales as the response becomes more and more elastic. Combining dissipative and energetic gradient effects leads to increased dissipation for small length scales, but decreasing dissipation for larger length scales. For all the three cases studied here (pure dissipative, pure energetic and combined dissipative and energetic with equal length parameters) it is seen that dissipation goes toward zero for large material length parameters, which is a result that holds true in general.

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In Fig. 8 the first load cycle is shown for linearly hardening materials with h/G= 0.50, and for different combinations of dissipative and energetic gradient effects. The material length parameters are chosen to be equal to H/3, since this value of the energetic length parameter would give the same effective hardening behavior (in the time independent limit) for a non-hardening material (h= 0) as that of the conventionally hardening material with h/G= 0.50, according to (44). This fact is confirmed in Fig. 8, where the conventional material with h/G= 0.50 shows the same hardening behavior as that of the non-hardening (h= 0) energetic strain gradient material with Le = H/3. However, during reverse loading no Bauschinger effect is seen for the conventional material that shows pure isotropic hardening, whereas the energetic gradient dependent material with h= 0 shows pure kinematic hardening. For hardening materials it is seen that the load cycle is not a closed loop as was seen in the results presented until now. This is due to the increase in yield stress upon plastic deformation. Furthermore, a difference in effective hardening is observed, with both energetic and dissipative gradient effects promoting the effective hardening of the material. In Table 1 the effective hardening modulus at the maximum load, evaluated according to (44), is shown for the different materials. It is observed that energetic gradient effects give rise to hardening even for a conventionally non-hardening material (h= 0), and that energetic and dissipative gradient effects promote hardening over the conventional hardening level. It is noted that the extracted hardening of the conventional material is slightly below the expected hardening level (heff =G  0:49 o 0:50), due to visco-plastic effects. Usually, promoted hardening is attributed to energetic gradient effects (e.g. Lele and Anand, 2008), but here it is seen that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening. This can be explained by the fact that the effective plastic strain, Ep, which defines the hardening behavior through g(Ep) in (18), is enhanced by dissipative gradient effects through the dissipative length parameters (according to (9)). While conventional hardening is usually included in a theory to model material hardening due to entanglement and trapping of dislocations, the extra hardening predicted here for Ld 4 0 can be thought to model the increased dissipation due to the motion of geometrically necessary dislocations introduced by plastic strain gradients. For a non-hardening (h= 0) finite slab of aspect ratio W/H=1, cyclic response curves are shown for dissipative gradient materials in Fig. 9a, and for energetic gradient materials in Fig. 9b. For the finite slab gradient effects are triggered due to the non-homogeneous stress state (due to the traction free ends) as well as due to the full constraint on plastic flow at the rigid platens. This leads to plastic strain gradients in both directions throughout the deformation history. As for the pure shear case (Fig. 2), dissipative gradient effects lead to increased yield strength, whereas energetic gradient effects lead to increased hardening and a significant Bauschinger effect. Corresponding response curves for combined dissipative and

Fig. 8. First load cycle for hardening materials with h/G= 0.50. Results are shown for a conventional material and for different gradient dependent materials with dissipative and energetic length parameters equal to H/3. For comparison, results for a non-hardening energetic gradient material with Le =H/3 are included. According to (44) the corresponding time-independent material has an effective hardening modulus of heff/G= 0.50.

Table 1 Effective hardening modulus for materials with two different hardening moduli, and for different combinations of the material length parameters. heff/G

Conventional

Ld = H/3

Le = H/3

Ld =Le =H/3

h/G= 0.00 h/G= 0.50

0.00 0.49

0.00 0.82

0.50 1.08

0.50 1.32

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Fig. 9. Cyclic shear response curves for an elastic–perfectly plastic finite slab of aspect ratio W/H= 1. Both results for (a) dissipative gradient effects and (b) energetic gradient effects are shown.

Fig. 10. Cyclic response curves for an elastic–perfectly plastic finite slab (W/H= 1) for combined dissipative and energetic gradient effects.

gradient effects are shown in Fig. 10 with Ld =Le, where gradient effects lead to both increased yield strength and hardening as well as to a Bauschinger effect. As for the pure shear case significant effects on the stress distribution are found due to gradient effects. In Fig. 11 contour of the effective stress is presented for (a) a conventional material (point V in Fig. 9a), (b) a dissipative gradient material with Ld/H= 0.5 (point VI in Fig. 9a), (c) an energetic gradient material with Le/H= 0.5 (point VII in Fig. 9b) and (d) a material with combined energetic and dissipative gradient effects with Ld/H=Le/H=0.5 (point VIII in Fig. 10). For the conventional case, the effective stress scales with the stress deviator. Hence, at the point (x1/W,x2/H)= (1,1) the effective stress must vanish, as the point is stress free due to the symmetries of the problem together with the traction free condition at x1/W=1. This results in the large variation in effective stress seen in Fig. 11a, where the numerical solution shows that the normalized effective stress, sc =sy , is in the interval [0.074;1.072]. However, for the gradient dependent results this interval is much smaller. For the dissipative gradient material the interval is [0.966;1.095] (Fig. 11b), with a maximum value at the platens, while the interval is [0.808;0.973] (Fig. 11c) for the energetic gradient material. For the combined dissipative and energetic materials the normalized effective stress is located in the very limited interval [0.956;0.985] (Fig. 11d). For the case with equal length parameters, an effective higher order stress can be defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð45Þ me ¼ mijk mijk

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Fig. 11. Contours of normalized effective stress, sc =sy , for an elastic–perfectly plastic slab (W/H= 1). Results are shown for (a) a conventional material, (b) a dissipative gradient hardening material with Ld/H= 0.5, (c) an energetic gradient hardening material with Le/H= 0.5, and (d) a material with both dissipative and energetic hardening with Ld/H =Le/H= 0.5.

For the purely dissipative gradient material contours are presented for this effective higher order stress in Fig. 12a in terms of the dimensionless parameter me =ðLd sy Þ. The figure shows that the normalized effective higher order stress varies from 0 to about unity, with the minimum value at the center (due to the symmetries of the problem) and the maximum value at the platens. For the case of combined dissipative and energetic gradient effects, corresponding results are shown in Fig. 12b. Here, much larger normalized higher order stresses exist (up to a level of around 2200), since the higher order stress has an energetic part that scales with the elastic shear modulus of the material, rather than with the yield stress which is the case for the dissipative part. However, the distribution is very similar to the dissipative case (Fig. 12a). Response curves for the finite square slab between rigid platens are shown in Fig. 13a for dissipative gradient effects and in Fig. 13b for energetic gradient effects. As could be shown analytically for the pure shear case in (42) and (43), L(1) has the largest effect on the response curves, while L(2) has the smallest effect for both dissipative and energetic gradient dependent materials. Response curves for the corresponding conventional material and materials with equal length parameters are included in the figure for comparison. 5. Discussion A numerical method has been developed to generate numerical finite element solutions for the most general version of the isotropic visco-plastic strain gradient plasticity theory proposed by Gudmundson (2004). The method enables

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Fig. 12. Contours of normalized effective higher order stress, me =ðLd sy Þ, for an elastic–perfectly plastic slab (W/H =1). Results are shown for (a) a dissipative gradient hardening material with Ld/H= 0.5, (b) a material with both dissipative and energetic hardening with Ld/H= Le/H= 0.5.

Fig. 13. Cyclic shear response curves for an elastic–perfectly plastic slab (W/H= 1) for each of the different length parameters activated one by one. Both results for (a) dissipative gradient effects and (b) energetic gradient effects are shown. For comparison, response curves are included for a corresponding conventional material as well as materials with all three length parameters set equal in the dissipative and energetic cases, respectively.

modeling of both dissipative and energetic gradient effects, with three length parameters characterizing each of the two kinds of gradient strengthening. The constitutive assumptions are based on an isotropic quadratic gradient contribution to the free energy as well as isotropic dissipative gradient contributions. The numerical model is implemented in a 2D plane strain setting and used to study size-effects during cyclic shearing of both an infinite layer of thickness 2H between rigid platens and a finite slab of material of aspect ratio W/H=1. In accordance with findings by Lele and Anand (2008) for an elastic–perfectly plastic solid, dissipative gradient effects lead to an increase in the effective yield strength of the material (see Fig. 3a), resulting from gradient contributions to the effective stress, whereas energetic gradient effects lead to an increase in material hardening (see Fig. 3b), with a quadratic dependence on the energetic length parameter, Le, for the pure shear case. Combining dissipative and energetic gradient effects leads to both an increase in the effective yield strength and material hardening. For the case of pure energetic hardening a Bauschinger effect is predicted which is governed by the energetic length parameters. This is different from earlier isotropic studies involving reversed loading as that by Lele and Anand (2008), where a Bauschinger effect resulting from energetic strain gradient effects is not predicted. This is due to their way of introducing gradients based on

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non-decreasing strain measures as the conventional effective plastic strain defined incrementally by the relation qffiffiffiffiffiffiffiffiffiffiffiffiffi e_ p ¼ 23 e_ pij e_ pij . On the other hand, in the present study, energetic gradient contributions originate from the full plastic strain tensor, epij . Hence, the free energy originating from gradients of plastic strain is regained from the system when the plastic strain gradients are removed. The increase in yield strength (over that of the conventional predictions) for dissipative gradient effects is found to vary almost linearly with Ld/H, for values of Ld/H above around one-half. Assuming a material length parameter on the micron scale this suggests that the increase in yield strength is inversely proportional to H. This is in good accordance with many tensile experiments for polycrystalline materials with grain sizes on the micron to sub-micron scale (e.g. Tsuji et al., 2002; Yu et al., 2005), where the increase in yield strength has been found to vary inversely proportional to the grain size, as noted by Ohno and Okumara (2007). On the other hand, the increase in hardening modulus with diminishing size for energetic gradient effects does not correspond well to these experimental studies, as they showed that the material hardening decreases with diminishing size. This suggests that great care should be taken when comparing idealized shear studies, as those carried out in the present paper, with the macroscopic tensile behavior of polycrystals. A central issue, which should be dealt with in greater detail, is the interplay between conventional hardening and energetic strain gradient hardening. In contrast, good qualitative agreement is found between the increase in hardening with diminishing size predicted in the present analyses and the experimental results by Xiang and Vlassak (2006) for the plane strain bulge test, where it was shown that a constraint on plastic flow by using passivation layers leads to increased material hardening with diminishing film thickness. Distributions of strain are presented and it is shown how dissipative higher order stresses scale with the yield stress, sy , while energetic higher order stresses scale with the shear modulus, G. For the pure shear case a maximum of the effective stress, sc , is predicted at the platens, where a full constraint on plastic flow is imposed. This seems to be an artifact of the visco-plastic formulation. Analyses not presented here indicate that, for a decreasing visco-plastic exponent, m, the effective stress goes towards a more constant level throughout the layer thickness. For the pure shear case, the dissipation per load cycle has been shown to decrease with increasing energetic gradient effects. On the other hand, for dissipative gradient effects, dissipation increases with Ld initially, and then decreases for larger values of Ld. The dissipative length scale at which there is maximum dissipation depends on the various parameters of the problem, including the magnitude of the load cycles as well as the value of the energetic length parameter, Le. For both pure shear and shear of a finite slab of aspect ratio W/H= 1, it was found that the three length parameters in the isotropic orthogonalization by Smyshlyaev and Fleck (1996) have different influence on the problem. For both energetic and dissipative gradient effects, it was found that L(1) was the most important parameter, while L(2) was the least important parameter. For linearly hardening materials it was found that both dissipative and energetic gradient effects promote hardening over that of conventional predictions. Usually, promoted hardening is attributed to energetic gradient effects, but here it was found that also dissipative gradient effects lead to additional hardening in the presence of conventional material hardening.

Acknowledgments This work is supported by the Danish Research Council for Technology and Production Sciences in a project entitled Plasticity Across the Scales.

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