Equivalent plastic strain gradient enhancement of single...

Proc. R. Soc. A (2012) 468, 2682–2703doi:10.1098/rspa.2012.0073

Published online 19 April 2012

Equivalent plastic strain gradient enhancementof single crystal plasticity: theory and numerics

BY STEPHAN WULFINGHOFF* AND THOMAS BÖHLKE

Institute of Engineering Mechanics, Karlsruhe Institute of Technology,Karlsruhe, Germany

We propose a visco-plastic strain gradient plasticity theory for single crystals. Thegradient enhancement is based on an equivalent plastic strain measure. Two physicallyequivalent variational settings for the problem are discussed: a direct formulation and analternative version with an additional micromorphic-like field variable, which is coupledto the equivalent plastic strain by a Lagrange multiplier. The alternative formulationimplies a significant reduction of nodal degrees of freedom. The local algorithm andelement stiffness matrices of the finite-element discretization are discussed. Numericalexamples illustrate the advantages of the alternative formulation in three-dimensionalsimulations of oligo-crystals. By means of the suggested formulation, complex boundaryvalue problems of the proposed plastic strain gradient theory can be solved numericallyvery efficiently.

Keywords: strain gradient plasticity; crystal plasticity; finite elements

1. Introduction

The continuum mechanics of single crystal models (Hill 1966; Lee 1969;Rice 1971; Ortiz & Stainier 1999) are based on a well-established physicalbasis. The discovery of dislocations as the fundamental carriers of plasticityjustifies the mesoscopic kinematical assumptions concerning the plastic slip rates(Orowan 1934) and the slip system geometry associated with these models.Experiments (Schmid & Boas 1935) provide evidence of the correlation betweenthe projected shear stresses and the slip system activity. Also non-Schmid effectscan be taken into account within this setting (Kocks 1987; Yalcinkaya et al. 2008).

In contrast, single crystal hardening models and the prediction of the closelyrelated dislocation micro-structure are still subjects of current research. Besidespurely phenomenological hardening laws (Taylor 1938; Koiter 1953; Hill 1966)local dislocation density evolution or balance equations (Gillis & Gilman 1965;Mecking & Kocks 1981; Franciosi & Zaoui 1982; Estrin 1996) allow to estimatethe evolution of the total dislocation line length per unit volume and to model thehardening behaviour of crystals more physically. These models are successfullyapplied at scales and specimen dimensions, where no size effect is observed.

*Author for correspondence ([email protected]).

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Plastic strain gradient enhancement 2683

However, the inability to predict the size-dependent behaviour at smaller scalesobserved in experiments, e.g. the Hall–Petch effect (Hall 1951; Petch 1953) or thesize-dependent strength of micro-specimens (Fleck et al. 1994; Stölken & Evans1998; Xiang & Vlassak 2006; Dimiduk et al. 2007; Gruber et al. 2008) motivatedmany authors to enrich the classical local theory, for example, based on Nye’sdislocation density tensor (Nye 1953; Bilby et al. 1955; Kröner 1958) or relatedmeasures associated with the gradients of the plastic slips. Nye’s tensor can becomputed from the spatial gradients of the plastic variables and introduces aninternal length into the theory. It can be interpreted as the total Burgers vectorper unit area, i.e. it has a clear physical meaning and must be distinguished fromthe above-mentioned dislocation density (the total line length per unit volume).Nye’s tensor or familiar gradient-type quantities (Fleck & Hutchinson 1993;Liebe & Steinmann 2001; Liebe 2004; Gurtin et al. 2007) provide the physicalmotivation for size-dependent hardening models. However, a generally acceptedcorrelation between the gradients of the variables and the hardening behaviour ofthe crystal could not be established, yet. Several thermodynamic gradient theoriesfor polycrystals have been proposed associating the gradients of plastic quantitieswith focus on an increase of the free energy (Fleck et al. 1994; Steinmann 1996;Menzel & Steinmann 2000) or dissipation (Gurtin & Anand 2005; Fleck & Willis2009). In the case of single crystals, a refined kinematical theory allows to computegeometrically necessary screw and edge dislocation densities for the different slipsystems, closely related to Nye’s tensor (Gurtin 2002), which has been generalizedto the context of large deformations by Gurtin (2006). Associated thermodynamictheories have been proposed by e.g. Cermelli & Gurtin (2002), Berdichevsky(2006), Ohno & Okumura (2007) or Ekh et al. (2007). The dislocation fieldtheory of Evers et al. (2004), see also Geers et al. (2006) and its generalization(Bayley et al. 2006) have been compared with thermodynamical approaches byErtürk et al. (2009) and Bargmann et al. (2010). Similar non-local theories are,for example, based on incompatibility-dependent hardening moduli (Acharya &Bassani 2000) or other non-work-conjugate formulations (Kuroda & Tvergaard2008). In the micromorphic approach of Forest (2009), a local inelastic variableis energetically coupled to an additional degree of freedom. A strong energeticalcoupling can formally be interpreted as penalty approximation of, e.g. a straingradient plasticity theory. However, the coupling parameter allows for a specificadjustment of the scaling behaviour of the model, which was investigated byCordero et al. (2010) and Aslan et al. (2011) for single crystal laminates and byCordero et al. (2012) for periodic single crystals. Based on a closely related large-deformation theory for isotropic elastic–plastic materials, Anand et al. (2011)regularized strain-softening phenomena by means of the gradient of an additionalvariable related to the equivalent plastic strain. Hochrainer (2006) generalizedNye’s concept to a higher-dimensional continuum dislocation theory, which wasimplemented by Sandfeld et al. (2010).

The fact that size-dependent theories usually require the computation ofadditional gradients makes the numerical treatment by the finite-element methodmore complex and computationally expensive (compared with local theories)since the variables have to be approximated by (at least piecewise) differentiableshape functions. One of the first works on this issue was published by deBorst & Mühlhaus (1992). Shu et al. (1999) presented and compared differentelement formulations for the Toupin–Mindlin framework of strain gradient theory.

Proc. R. Soc. A (2012)



2684 S. Wulfinghoff and T. Böhlke

They introduced the displacement gradients as additional nodal degrees offreedom (associated with C 0-continuous shape functions) and coupled them tothe true gradients by Lagrange multipliers. Liebe & Steinmann (2001) proposeda thermodynamic framework, where the yield condition is treated in a weaksense, leading to a distinction between plastically active and inactive nodes.A multi-field incremental variational framework for gradient-extended standarddissipative solids, including a generalization of this framework, can be found inMiehe (2011). Becker (2006) and Han et al. (2007) propose a projection of theplastic variables from the integration points to the nodes which subsequentlyallows the computation of plastic strain gradients and an implementation close tolocal theories. Forest (2009) proposes a generalized micromorphic approach witha local variable and a micromorphic variable. In addition, Wieners & Wohlmuth(2011) propose a primal–dual finite-element approximation. Both approachespreserve a local evaluation of the yield function and facilitate the applicationof the radial return algorithm.

One drawback of gradient plasticity theories is that a unique and generallyaccepted correlation between the gradients of the variables and the hardeningbehaviour is missing. Another disadvantage of implementations, including theplastic variables as additional nodal variables, is the significant increase inthe computational effort owing to the extended number of degrees of freedom,especially in single crystal computations. Consequently, the application of thetheory to three-dimensional problems is numerically extraordinarily expensive.

The present work provides a gradient theory for crystals which leads toefficient numerics. The formulation and model behaviour are close to existinggradient theories. We choose an equivalent plastic strain measure (which unitesthe plastic strain history of all slip systems in one scalar quantity) for thegradient enhancement aiming at a significant decrease of additional nodal degreesof freedom compared with existing theories. One central motivation for thisproceeding is that equivalent plastic strain measures have been used successfullyin the context of phenomenological local hardening models. Most of the above-mentioned works are based on geometrically necessary dislocations (GND),given by Nye’s tensor which has a clearer physical interpretation than thegradient of the equivalent plastic strain. However, the present theory allows thesimulation of three-dimensional systems (consisting of several grains) at reducedcomputational cost, owing to the significant reduction of degrees of freedom. Theclose relation to other gradient theories will be shown, and as long as the exactdependence of the hardening behaviour on geometrically necessary dislocationdensity measures (if there is any) remains unclear, even more complex gradienttheories remain imprecise.

The outline of the paper is as follows: first, we discuss the basic kinematicaland energetic assumptions. Subsequently, two different but physically identicalvariational settings are introduced. Details of the finite-element implementationare discussed before some numerical results illustrate the performance ofthe alternative formulation in three-dimensional applications. Finally, weanalyse the ability of the theory to reproduce the experimentally observedsize effects.

Notation. A direct tensor notation is preferred throughout the text. Vectorsand second-order tensors are denoted by bold letters, e.g. a or A. A linearmapping of second-order tensors by a fourth-order tensor is written as A = C[B].Proc. R. Soc. A (2012)




The scalar product and the dyadic product are denoted, e.g. by A · B and A ⊗ B,respectively. The composition of two second-order tensors is formulated by AB.Completely symmetric and traceless tensors are designated by a prime, e.g. A′.Matrices are denoted by a hat, e.g. 3. The transpose and the inverse of a matrixare indicated by DT and D−1, respectively.

2. Equivalent plastic strain gradient enhancement of single crystal plasticity

(a) Constitutive equations

(i) Kinematics

We describe the motion of a body B by the displacement field u(x , t), whichmaps the positions x ∈ B of the particles to their associated displacements u attime t. Here, B is open with boundary vB and closure B := B ∪ vB. We restrictourselves to the small deformation context, i.e. ‖H‖ � 1, where H := Vu =vui/vxjei ⊗ ej . Furthermore, we define the strain tensor by 3 := sym(H ) = VSuand assume the classical relation

H p :=N∑

a=1

lada ⊗ na (2.1)

for the plastic part of the displacement gradient, where the scalars la are slipparameters and N is the number of slip systems which are characterized by slipplane normals na and slip directions da (orthogonal to na) of unit length. Thesymmetric part of H p is the plastic strain

3p := sym(H p) =∑

a

laM Sa. (2.2)

The Schmid-tensor M Sa of slip system a is defined by M S

a := sym(da ⊗ na). Inorder to capture the actual single crystals kinematics, the slip systems are definedpairwise with common slip plane normal and opposite slip directions. This impliesfor octahedral slip systems, e.g. N = 24. The slip parameters are constrained toincrease monotonously, i.e. la ≥ 0 with initial values equal to zero. The effectiveplastic slips (in each slip system) are given by the difference between two pairwisedefined slip parameters. It should be noted that alternatively 12 slip systemswith possibly positive and negative slip increments for face-centred cubic (FCC)-crystals can be introduced. For most gradient theories, this leads to a reducednumber of degrees of freedom in the finite-element implementation. The elasticstrain is defined by 3e := 3 − 3p. We denote the plastic domain of a slip system a

by Bacta ⊆ B (where la > 0).

Additionally, the equivalent plastic strain measure is given by the expression

geq(l) :=∑

a

la, (2.3)

which maps the slip parameters l := (l1, l2, . . . , lN )T to the equivalent plasticstrain geq. Recall that (•) denotes the use of matrix notation. Note that, we choosedefinition (2.3) for simplicity and that other definitions of geq(l) are possible.





The theory can easily be generalized to more complex (e.g. tensorial) quantitieslike the plastic part of the displacement gradient H p(l). This generalizationallows the application of the theory presented in the following to more complexmodels, which assume the free energy to depend, for example, on Nye’s dislocationdensity tensor.

(ii) Free energy and dissipation

We assume the free energy per unit volume to be of the additive form (formallysimilar to, e.g. Steinmann 1996)

W (3, l, Vgeq) = We(3, l) + Wh(l) + Wg(Vgeq), (2.4)

with We(3, l) := 1/2(3 − 3p(l)) · C[3 − 3p(l)], where C is the elastic stiffnesstensor. The second part Wh(l) phenomenologically accounts for the isotopichardening behaviour of crystals. Physically, this observation is primarilyattributed to the multiplication of dislocations, which are mutually acting asobstacles leading to an increase of the macroscopically observed yield stress withproceeding plastic deformation. The third part Wg(Vgeq) introduces an internallength into the theory, i.e. it leads to a size-dependent mechanical responseof the model. In the case of metals, size effects are observed experimentallyfor, e.g. micro specimens, steels with varying grain sizes or nanoindentationtests. The size effect can partially be explained by geometrically necessarydislocation configurations represented by Nye’s tensor, which is closely relatedto the gradients of the plastic slips. The dissipation D (dissipated power per unitvolume) is assumed to be a superposition of the dissipation contributions of theindividual slip systems

D =∑

a

Da =∑

a

tdala, (2.5)

where tda are dissipative forces which extend power over the plastic slip rates

(cf. Cermelli & Gurtin 2002).

(b) Variational form and strong form of the field equations

The basis for the subsequent theoretical development and for the numericalimplementation is the principle of virtual power, which states that for a givenstate, described by the primary variables (u, l), the virtual power of the externalforces is equal to the virtual power of the internal forces. The formal treatmentis similar to that of Gurtin et al. (2007). The external forces are assumed tobe given by the traction field t on the associated Neumann boundary vBt ⊂ vB.The tractions t generate power over v := u. The micro-tractions X are definedanalogically, i.e. X generates power over geq = ∑

a la on the Neumann boundaryvBX ⊆ vB. The primary variables (u, l) are assumed to be known on the Dirichletboundary vBu := vB \ vBt and vBgeq := vB\vBX, respectively. Here, we restrictourselves to micro-hard Dirichlet boundary conditions l = 0 ∀x ∈ vBgeq .





We choose v∗ and ˙l∗ = (l∗

1, l∗2, . . . , l∗

N )T to be the independent virtualkinematical quantities and require both of them to respect the Dirichlet boundaryconditions on vBu and vBgeq . The dependent virtual rates can be expressed by

3∗ := VSv∗; 3e∗ := 3∗ −∑

a

l∗aM

Sa and g∗

eq :=∑

a

vgeq

vla

l∗a =

∑a

l∗a. (2.6)

The principle of virtual power states∫B(W ∗ + P∗

dis) dv =∫

vBt

t · v∗ da +∫

vBX

Xg∗eq da ∀ v∗, ˙

l∗, (2.7)

with

W ∗ = vWv3

· 3∗ + vW

vl· ˙

l∗ + vWv(Vgeq)

· Vg∗eq (2.8)

and the virtual power of the dissipative forces

P∗dis =

∑a

tdal∗

a. (2.9)

Expression (2.8) motivates the introduction of the micro-stress

x := vWg

v(Vgeq), (2.10)

which is energetically conjugate to Vgeq.Based on these results, equation (2.7) can be expressed in terms of the primary

variables (u, l)

∫B

{(VSv∗ −

∑a

l∗aM

Sa

)· C

[VSu −

∑a

laM Sa

]+

∑a

vWh

vla

l∗a +

∑a

tdal∗

a

}dv

+∫B

x ·(∑

a

Vl∗a

)dv =

∫vBt

t · v∗ da +∫

vBX

X

(∑a

Vl∗a

)da ∀v∗, ˙

l∗. (2.11)

Using Gauss’ theorem and the chain rule, we obtain

∫B

{−div(C[3e]) · v∗ −

∑a

(M S

a · C[3e] + div(x) −(

tda + vWh

vla

))l∗

a

}dv (2.12)

+∫

vBt

(C[3e]n − t) · v∗ da +∫

vBX

(x · n − X)∑

a

l∗a da = 0 ∀v∗, ˙

l∗. (2.13)

For vanishing ˙l∗, this expression yields

∫B

div(s) · v∗ dv −∫

vBt

(sn − t) · v∗ da = 0, (2.14)





where the Cauchy stress s = C[3e] has been introduced. Since v∗ is arbitrary theintegrands of both integrals must vanish independently and point-wise. The firstintegral in equation (2.14) implies

div(s) = 0 ∀x ∈ B. (2.15)

The second integral in equation (2.14) yields the Neumann boundary conditions

sn = t ∀x ∈ vBt . (2.16)

Choosing now v∗ = 0 and l∗b = 0 ∀ b ∈ {1, 2, . . . , N } \ {a}, we find

∫B

(M S

a · s + div(x) −(

tda + vWh

vla

))l∗

a dv +∫

vBX

(X − x · n)l∗a da = 0. (2.17)

Introducing the resolved shear stress ta := s · M Sa, equation (2.17) yields the

field equations

tda = ta − vWh

vla

+ div(x) ∀ x ∈ B. (2.18)

Note that the contribution div(x) is equal for all slip systems due to the specialchoice of the scalar equivalent plastic strain geq. For the Neumann boundaryconditions, we obtain

x · n = X ∀ x ∈ vBX. (2.19)

Intending to use a visco-plastic overstress constitutive formulation, we define theyield criteria (cf. Miehe 2011) by

fa := tda − tc

0, (2.20)

with the initial yield stress tc0. From equation (2.18), it follows that

fa = ta + div(x) −(

tc0 + vWh

vla

)∀ x ∈ B. (2.21)

Rate-independent single crystal plasticity theories suffer from the problem ofpossibly occurring linearly dependent constraints (Kocks 1970). In order tocircumvent this problem, we base our formulation on a visco-plastic constitutiveflow rule of the Perzyna-type (Perzyna 1971)

la = g(fa)h

, (2.22)

where g(x) is monotone with x ≤ 0 ⇔ g(x) = 0 and h is a viscosity parameter(cf. Simo & Hughes (1998) or Miehe (2011) in the context of gradient theories).We define the viscous stress by tvis(la) := g−1(hla) ∀x ∈ Bact

a , i.e. tvis(la) = fa

in the case of plastic yielding, and find from equation (2.21) the shear stress





Box 1. Equivalent plastic strain gradient theory.

balance lawsbalance of linear momentum 0 = div(s) ∀x ∈ B

dissipative stress tda = ta − vWh/vla + div(x) ∀x ∈ B

Neumann boundary conditions sn = t on vBt , x · n = X on vBX

constitutive equationsfree energy W = We(3, l) + Wh(l) + Wg(Vgeq)

Cauchy stress s = vWe/v3 = C[3e]non-local stress x = vWg/v(Vgeq)yield criterion fa = td

a − tc0

flow rule la = g(fa)/h

equilibrium in the plastic zones Bacta

ta + div(x) −(

tc0 + vWh

vla

+ tvis(la))

= 0 ∀ x ∈ Bacta . (2.23)

Note that the negative of div(x) can formally be interpreted as a back-stress (cf.Gurtin et al. 2007), which is slip system independent, since Wg depends only onthe gradient of the equivalent plastic strain.

From the equations (2.5), (2.18) and (2.23), the dissipation can be deduced

D =∑

a

(tc0 + tvis(la))la. (2.24)

This result formally corresponds to the result of the local theory (i.e. Wg = 0).Hence, compared with a purely local theory, the dissipative mechanisms due toplastic slip rates are not altered in the theory at hand. In the case of plasticyielding, the dissipative stress td

a, as obtained in equation (2.18), is

tda = tc

0 + tvis(la) ∀x ∈ Bacta . (2.25)

The introduction of a dissipation potential (here discussed only for linearviscosity), e.g. by

f( ˙l) =∑

a

supta

(tala − 〈f (ta)〉2

2h

), (2.26)

allows to construct a global incremental potential of the model (cf. Miehe 2011).However, this procedure is not part of the present work. The model equationsintroduced in this section are summarized in box 1.

(c) Alternative formulation

In the following, we define a model in terms of four primary variables (u, l, z, p)which is (concerning its mechanical behaviour) physically equivalent to the modelintroduced in §2b. We introduce an additional field variable z and weakly enforceits equivalence to geq by a Lagrange parameter p. This apparently complex





approach leads to a massive reduction of the computational effort when solvingthe problem by use of the finite-element method. This idea is inspired by the workof Simo et al. (1985) (also by Forest 2009), who introduced the weak enforcementof the equality of two (a priori different) field variables by a force-like field whichis treated like an additional field variable (see also Shu et al. 1999).

The free energy is assumed to take the form

W (3e, l, z, Vz, p) = We(3e) + Wh(l) + Wg(Vz) + p(geq(l) − z). (2.27)

We define x := vWg/v(Vz) for the subsequent developments and start from theweak form

∫B

{(VSv∗ −

∑a

l∗aM

Sa

)· s(VSu, l) +

∑a

vWh

vla

l∗a + x · Vz∗ +

∑a

tdala

∗}

dv

+∫B

˙p∗(geq(l) − z) dv +∫B

p(g∗eq − z∗) dv =

∫vBt

t · v∗ da +∫

vBX

X z∗ da.

(2.28)

Application of Gauss’ theorem and the chain rule yields

∫B

{−div(s) · v∗ −

∑a

((M S

a · s − p) −(

tda + vWh

vla

))l∗

a

}dv

+∫B

˙p∗(geq(l) − z) dv −∫B

z∗(p + div(x)) dv

+∫

vBt

(sn − t) · v∗ da +∫

vBX

(x · n − X)z∗ da = 0 ∀ v∗, ˙l∗, ˙p∗, z∗. (2.29)

The field equations and Neumann boundary conditions associated with (2.29) aresummarized in box 2. Additionally, the yield criteria and flow rule are listed. Itshould be noted that the slip parameters can alternatively be handled as internalvariables. The associated evolution rules can then be derived based on the reduceddissipation inequality (for further details, see Forest (2009)).

Similar to the case before (equation (2.23)), we obtain the relation

(ta − p) −(

tc0 + vWh

vla

+ tvis(la))

= 0 ∀x ∈ Bacta (2.30)

for the plastic zones.Note that formally, p = −div(x) can be interpreted as a back stress again and

vWh/vla is a local hardening stress. The back stress p is equivalent for all slipsystems due to the reduction to only one scalar variable z and the special choiceof geq(l).

From the set of equations in box 2, the equations in box 1 can be derived. Inthis sense, the models (2.11) and (2.28) can be considered to be equivalent.





Box 2. Equivalent plastic strain gradient theory, alternative formulation.

balance lawsbalance of linear momentum 0 = div(s) ∀x ∈ B

dissipative stress tda = ta − vWh/vla − p ∀x ∈ B

back stress p = −div(x) ∀x ∈ Bequivalent plastic strain z = geq(l) ∀x ∈ B

Neumann boundary conditions sn = t on vBt ,x · n = X on vBX

constitutive equationsfree energy W = We(3, l) + Wh(l) + Wg(Vz)

+ p(geq(l) − z)Cauchy stress s = vWe/v3 = C[3e]

non-local stress x = vWg/v(Vz)yield criterion fa = td

a − tc0

flow rule la = g(fa)/h

The original system of equations (box 1) includes three scalar partialdifferential equations for the linear momentum balance and N (e.g. N = 24/12in the case of FCC-crystals) partial differential equations for the micro forcebalances, whereas the total number of scalar partial differential equations of thenew set of equations (box 2) is only four. It is emphasized that this massivelyreduced degree of non-locality makes the new set of equations (box 2) interestingfor a numerical implementation via finite elements. In contrast, the discretizationof the first model leads to elaborate element stiffness matrices and activeset search algorithms on the nodal level which have barely been studied forcrystals, yet. Therefore, only the second model is implemented. The numericalimplementation is presented in the following.

3. Numerical implementation

In this section, the local (integration point) algorithm, the algorithmic tangentmoduli and the element stiffness matrices for the alternative formulation arederived. It is shown that the number of degrees of freedom per node increasesonly by one compared with a local theory. For the numerical implementation,explicit expressions for the free energy are required. We assume quadratic formsfor simplicity

We = 12

3e · C[3e], Wh(l) =∑a,b

12

lahablb and Wg(Vz) = 12KgVz · Vz, (3.1)

with constant hardening moduli hab and Kg.The starting point for the discretization via finite elements is the time-

discretized problem (2.28), evaluated at time step tn+1 (for convenience, the index





‘n + 1’ is dropped in the following), given by the residuals associated with thevariations of the displacements

Gu :=∫B

VSv∗ · s dv −∫

vBt

t · v∗ da != 0, (3.2)

of the variable z

Gz :=∫B(Vz∗ · x − pz∗) dv −

∫vBX

Xz∗ da != 0, (3.3)

of the back stress p

rp := z − geq(l) != 0 ∀x ∈ B (3.4)

and of the slip parameters la

rla := ta −⎛⎝tc

0 +∑

b

hablb + tvis

(Dla

Dt

)+ p

⎞⎠ != 0 ∀x ∈ Bact

a . (3.5)

Here,Dla

Dt= la − la,n

Dt(3.6)

is the slip rate which is assumed constant during the time interval [tn , t]. Forsimplicity, we prescribe a linear-viscous behaviour

la = 〈fa〉h

⇒ tvis(la) = hla. (3.7)

(a) Local algorithm

The structure of the set of residuals allows for an algorithmically decoupleddetermination of the quantities (u, z) and the local quantities (l, p), whichis close to standard iterative procedures in computational inelasticity, solvingthe discretized momentum equations and updates of the local variablesseparately (e.g. Simo & Hughes (1998); Miehe & Schröder (2001), among manyothers). For the computation of the local quantities, we assume the solution (u, z)to be given. The local problem, given by equations (3.4) and (3.5), is constrainedby the requirement Dla ≥ 0, which will be exploited to identify the local set ofactive slip systems J (containing the indices of the active systems). For themoment, assume an active set J �= ∅ to be given. The aim here is the developmentof an algorithm which allows the computation of

xS := (l(1), l(2), . . . , l(nact), p)T = xSn + DxS, (3.8)

where nact is the number of active slip systems. Recall that (•) denotes the use ofmatrix notation. The subscripts in the parenthesis and the superscript ‘S’ accountfor the active slip parameters.





To identify the solution, we compute the residuals (3.4) and (3.5) and thereforeuse the following matrix notation

RS := (rl(1) , rl(2) , . . . , rl(nact) , rp)T Def.= ((rl)T, rp)T, (3.9)

based on the state (u, z, ln , pn). The solution xS can then be obtained as the

solution of the linear system of equations ASDxS = RS with the symmetric tangent

AS:= −vRS

vxS= −

⎛⎜⎜⎜⎝

vrl

vlS

vrl

vp(vrp

vlS

)T

0

⎞⎟⎟⎟⎠ . (3.10)

The explicit elements of AS

are given by

−vrl(a)

vl(b)= M S

(a) · C[M S(b)] + h(a)(b) + d(a)(b) h

Dtand

−vrl(a)

vp= vgeq,n+1

vl(a)= − vrp

vl(a)= 1. (3.11)

For convenience, the complete local algorithm is summarized in box 3 containinga standard active set search procedure (see Ortiz & Stainier (1999); Miehe &Schröder (2001), and references therein).

(b) Algorithmic tangent moduli

The local update procedure is coupled to the global algorithm via a generalizedform of the classical algorithmic tangent moduli

C s3 = vs

v3, C sz = vs

vz, C p3 = vp

v3and C pz = vp

vz. (3.12)

In order to compute the algorithmic tangent, consider the definition y :=(3T, z)T. The basis for the computation of the algorithmic tangent moduli is therequirement, that the residuals RS remain zero, if the quantities (u, z) change

dRS = vRS

vy︸︷︷︸=:B

dy + vRS

vxS︸︷︷︸−AS

dxS != 0. (3.13)

Accordingly, the differential of the local variables dxS can be reconstructedexplicitly as a function of the differential of the quantities dy

dxS = AS−1Bdy = Ddy. (3.14)





Box 3. Local algorithm.

Local algorithm

(i) Get the quantities {3n , zn , ln , Jn , pn} from the previous time step as well as theincrements {D3, Dz, Dt}, compute 3 = 3n + D3 and z = zn + Dz. Set C s3 = C .

(ii) Compute the plastic strain from the previous time step 3pn = ∑

a la,nM Sa.

(iii) Calculate the trial stress str = C [3 − 3pn ] and trial back stress ptr = C pz

el (z − geq,n).

(iv) Check the yield conditions based on the trial state

f tra = (str · M S

a − ptr) −⎛⎝tc

0 +∑

b

hablb,n

⎞⎠ , a = 1, . . . , N .

(v) If f tra < tol ∀a ∈ {1, 2, . . . , N } (elastic update):

s = str; p = ptr; l = ln ; J = ∅; C p3 = 0; C pz = C pzel ;

fa = f tra and EXIT.

(vi) Else (plastic update) compute the residuals

Ra = M Sa · str −

⎛⎝tc

0 +∑

b

hablb,n + pn

⎞⎠ , a = 1, . . . , N ;

RN+1 = z − geq(ln),

and the tangent components

Aab = (M Sa)TC M S

b + hab + dab h

Dt; a, b = 1, . . . , N ,

as well asA(N+1)a = Aa(N+1) = 1, a = 1, . . . , N and A(N+1)(N+1) = 0.

(vii) Define k : Rnact → R

N , which maps the active slip indices {1, . . . , nact} to the global slipsystem indices {1, . . . , N }. Activate the maximum loaded system with f = fmax.

(viii) While active set search not converged:

(a) Build submatrix ASab = Ak(a)k(b) and subresidual RS

a = Rk(a).

(b) Compute solution DxS = AS−1RS.

(c) Identify minimum slip increment aMin = arg(

mina∈J

Dla

).

If mina∈J Dla < tol : J ← J \ {amin}. Update k(a). CONTINUE.(d) Set preliminary solution

l = ln + Dl, p = pn + Dp, 3p =∑

a

laM Sa, s = C [3 − 3p].

(e) Identify maximum loaded system

∀a /∈ J : fa = M Sa · s −

⎛⎝tc

0 +∑

b

hablb + p

⎞⎠ , amax = arg(maxa/∈J fa),

i.e. compute fa only, if system a and its pairwise defined counterpart areinactive.

(f) Eventually, update active set and repeat computationfamax > tol : J ← J ∪ {amax}, update k(a), CONTINUE.





Hence, the tangent operator D, containing the partial derivatives

D =

⎛⎜⎜⎝

vlS

v3

vlS

vz(vpv3

)Tvpvz

⎞⎟⎟⎠ =

⎛⎝vlS

v3

vlS

vz

C p3 C pz

⎞⎠ , (3.15)

can be computed as a matrix product and allows to identify two of the requiredalgorithmic tangent moduli. The matrix B is given by the expression

B =

⎛⎜⎜⎝

vrl

v3

vrl

vz(vrp

v3

)Tvrp

vz

⎞⎟⎟⎠ =

(vrl

v30

0T 1

),

vrl(a)

v3= C M S

(a). (3.16)

The identification of the remaining algorithmic moduli C s3 and C sz can beeffectuated based on the upper entries of D, which allow the direct computationof the increment of the plastic parameters

dlS = vlS

vzdz + vlS

v3d3. (3.17)

Finally, the computation of the stress-increment

ds = C d3 − C∑

a

dlaM Sa

=(

C −∑

a

C M Sa

(vla

v3

)T)

d3 +(

−∑

a

C M Sa

vla

vz

)dz

= C s3 d3 + C sz dz (3.18)

leads to the identification of the algorithmic tangents C s3 and C sz. It should benoted that due to the symmetry of the problem, the equivalence C p3 = −C sz canbe shown.

In the case of an elastic update, i.e. fa < 0 ∀ a ∈ {1, 2, . . . , N }, we set

C s3 = C , C sz = 0, C p3 = 0 and C pz = C pz

el . (3.19)

Here, C pz

el is a large (negative) number which penalizes a deviation of z from geq.Alternatively, the equation z = geq(l), evaluated at the elastic integration point,can be interpreted as kinematical constraint on the nodal values associated withz of the corresponding element.

(c) Element stiffness matrices

With the algorithmic moduli at hand, the element stiffness matrices associatedwith the linearized global residuals Gu and Gz can be computed. The linearization





of Gu with respect to u yields

keuu =

∑p

W ep (Bep

3 )TC s3Bep

3 . (3.20)

Here, W ep is the weight due to the numerical quadrature scheme associated

with integration point p of element e. The matrix Bep3 is the standard

interpolation operator, projecting the nodal displacements to the strain at theintegration points.

The linearization with respect to z is incorporated by

keuz =

∑p

W ep (Bep

3 )TC sz(N ep)

T = (kezu)

T, (3.21)

where N ep interpolates the nodal values of z to the integration points. Thesymmetry of the problem was exploited again in order to compute ke

zu . Finally,the linearization of Gz with respect to z leads to

kezz =

∑p

W ep (Kg(Bep)

TBep − C pzN ep(N ep)

T), (3.22)

where Bep interpolates the nodal values of z to its gradient in the integrationpoints. In the finite-element procedure, the matrices ke

uu , keuz, ke

zu and kezz represent

parts of the (symmetric) element stiffness matrix.

4. Numerical examples

We demonstrate the performance of the implementation of the alternativeformulation by the simulation results of micro-mechanical tension and torsiontests of micro-components which have been carried out with the in-housefinite-element program of the Institute of Engineering Mechanics (ContinuumMechanics).

In both cases, we use an oligo-crystal consisting of eight grains with asimplified grain geometry as shown in figure 1 and random crystal orientations.The hardening parameters are assumed to be given by hab = H [q + (1 − q)dab](Hutchinson 1970) with q ∈ [1; 1.4] as suggested by Kocks (1970). We use thefollowing material parameters

C11 C12 C12 Kg H q tc0 h

168 GPa 121 GPa 75 GPa 3 × 10−2 N 200 MPa 1.1 35 MPa 0.001 MPa s

The boundary conditions for the tension tests allow for free lateral contraction.We choose micro-free boundary conditions for z, i.e. x · n = 0 ∀x ∈ vBX. Allsimulations were performed with standard linear hexahedrons and adaptive timesteps. The total simulation time is 1 s. Figure 2 shows the results for the tensileand torsion test simulations for varying cube sizes. The model response for torsionis normalized in order to allow a comparison of the responses of the cubes with





0z

0.1 –35p (MPa)

137

Figure 1. Oligo-crystal with simplified grain shape and edge length L = 10 mm. Undeformed state,tensile and torsion test simulation. The tensile test simulation was performed with micro-hardgrain-boundary conditions, in case of the torsion test simulation micro-free boundary conditionswere applied. (Online version in colour.)

120 30

15

0

60

norm

al. t

orqu

e M

/L3

(MPa

)

0 0.01 0.02 0.03rotation angle

0.005 0.010aver

age

tens

ile s

tres

s s

= F

/A (

MPa

)

average tensile strain e = DL/L

(a)(b)

Figure 2. Mechanical responses in the case of tensile and torsion test simulations with micro-freeboundary conditions and for varying cube edge lengths L (blue filled squares, L = 10 mm; redopen circles, L = 15 mm; green filled circles, L = 30 mm; pink triangles, L = 100 mm; purple invertedtriangles, L = 300 mm). (Online version in colour.)

different dimensions. The results (figure 2) show that the size-dependence of themechanical response is stronger in the case of torsion (compared with tension).This difference, which is also observed in experiments (Fleck et al. 1994), occurs(here) due to increased strain gradients in the case of torsion, which are penalizedby Wg(Vz). On the integration point level, this effect is represented by the backstress p (cf. equation (2.30)). Figure 1(right) illustrates that p can be consideredas a hindrance for further plastic deformation in strongly deformed regions whileit augments plastic processes in little distorted parts of the body and therebyreduces gradients of z. For both load cases, the simulations show a strong sizeeffect for L � 100 mm. The negligible size effect in the case of larger bodies isdue to the fact that the strain gradients decrease with increasing dimensions, i.e.the theory at hand tends to a classical local theory for large bodies with smallgradients. An idealized (e.g. misorientation-induced) boundary slip resistancecan be introduced into the model by micro-hard boundary conditions. For the





700

350

0 0.005 0.010

aver

age

tens

ile s

tres

s s

= F

/A (

MPa

)

average tensile strain e = DL/L

Figure 3. Tensile test simulations with micro-hard boundary conditions. For the second simulationwith L = 30 mm∗, a larger initial time step was used, i.e. the elasto-plastic transition was not resolved(blue filled squares, L = 10 mm; red open circles, L = 15 mm; green filled circles, L = 30 mm; yellowfilled triangles, L = 30 mm∗; pink triangles, L = 100 mm; purple inverted triangles, L = 300 mm).(Online version in colour.)

130(a) (b)

450

400

350

3000

no. degrees of freedom70 000 140 000 210 000 280 000

120

110

100

mic

ro-f

ree

s =

F/A

(M

Pa)

mic

ro-h

ard

s =

F/A

(M

Pa)

0.028

0.016

z

Figure 4. Dependence of the global mechanical response on the number of nodal degrees of freedomof the model (a) and simulation of 512 grains (b) for 3max = 0.01. The diminution of d.f. due tothe Dirichlet boundary conditions is neglected (micro-free: blue filled squares, eight grains; opencircles, 512 grains; micro-hard: red filled circles, eight grains). (Online version in colour.)

simulation of the micro-component (cf. figures 1 and 3), we model the internalgrain boundaries (represented by G) as micro-hard, i.e. z = 0 ∀x ∈ G. Formally,this case is covered by the presented theory by considering the oligo-crystal as aunion of eight bodies. The implementation is effectuated by setting the prescribednodal values of z to zero and eliminating the associated lines and columns fromthe system matrix and the residual vector. The non-local hardening modulusKg has not been changed to allow a comparison with the micro-free boundaryconditions (figure 2). However, for more realistic applications, Kg should bereduced in the case of micro-hard grain boundaries, as the response shownin figure 3 is much too hard compared with experimental results. The mesh





dependence of the results is depicted in figure 4. The study was performed with acube size of L = 30 mm. The choice of different cube sizes resulted in comparableresults. The convergence study suggests that, in the case of the oligo-crystalwith simplified grain geometry, the mesh with approx. 37 000 d.f. (20 × 20 × 20elements), which has been used in figures 2 and 3, yields reasonable global resultsclose to the converged solution (especially for micro-free boundary conditions).A more realistic (and therefore more complex) grain geometry is expected torequire an increased number of degrees of freedom per grain. Additionally, amesh dependence study with 512 crystals (8 × 8 × 8 crystals) was carried out formicro-free boundary conditions and L = 30 mm (figure 4) to get a first impressionof the mechanical response of a polycrystal.

Table 1 shows the Euclidean norm of the residual for a tensile test simulationwith micro-hard boundary conditions, a cube size of L = 30 mm and a 8 × 8 × 8-mesh. For time steps 3 and 4 (elasto-plastic transition with 16/13 iterations)not all values are represented. The total CPU time for the solution was 25.89 s(table 1) on a Pentium Dual Core PC with 3.0 GHz and 6 GB RAM (which limitsthe maximum number of d.f., in the case of a direct solver, to approx. 300 000).

5. Summary

A geometrically linear single crystal model extended by an energetic hardeningterm based on the gradient of the equivalent plastic strain geq hasbeen introduced. The direct finite-element implementation of the theory iscomputationally expensive due to the need of piecewise differentiable shapefunctions for the plastic slip parameters la. The introduction of an alternative(but physically equivalent) formulation based on a Lagrange multiplier p and amicromorphic-like variable z drastically reduces the number of partial differentialequations and nodal degrees of freedom of the finite-element implementation. Thelocal algorithm and its linearization, represented by the extended algorithmictangent moduli, have been discussed in detail. Numerical simulations ofthree-dimensional tension and torsion tests of oligo-crystals with a simplifiedgeometry illustrate the performance of the implementation. The overall modelresponse qualitatively mimics the size-dependent material behaviour observed inexperiments (Fleck et al. 1994; Stölken & Evans 1998; Xiang & Vlassak 2006;Gruber et al. 2008). The experimentally found size-dependence of the yield stressis not captured, a well-known consequence of the simple quadratic ansatz for thestrain gradient-dependent energy contribution, which can partially be remediedby more elaborate approaches (Ohno & Okumura 2007).

Overall, the proposed geometrically linear model allows computationallyrelatively cheap three-dimensional size-dependent simulations of oligo-crystal models.

The theory is not restricted to the special scalar variable geq. Instead, thepresented concepts can be generalized to more complex (e.g. tensorial) quantitieslike the full plastic part of the displacement gradient and thereby cover theoriestaking, e.g. Nye’s tensor as argument of the free energy.

The boundary conditions for the field z remain an issue for further research.The physical relevance of micro-free and micro-hard boundary conditions atthe grain boundaries is questionable. Both conditions seem to be applicable in





Tab

le1.

Exa

mpl

efo

rth

eE

uclid

ean

norm

ofth

ere

sidu

al(l

eft)

and

CP

U-t

ime

(rig

ht)

for

L=

30mm

.

Dt:

0.01

s0.

02s

0.04

s0.

08s

0.16

s0.

32s

0.33

3s

0.03

7s

elem

ents

CP

U(s

)

1.51

×10

43.

63×

104

7.26

×10

41.

58×

105

3.17

×10

56.

22×

105

6.17

×10

56.

55×

104

8×

8×

825

.89

1.31

×10

33.

01×

10−1

1—

—2.

38×

103

3.33

×10

31.

53×

103

2.97

×10

18

×8

×8a

10.5

14.

71×

10−1

2—

—3.

96×

102

4.80

×10

21.

73×

102

2.06

12×

12×

1297

.51

1.78

×10

25.

794.

81×

101

7.32

×10

11.

40×

101

2.32

×10

−112

×12

×12

a26

.73

3.63

×10

−83.

33×

10−7

4.29

9.47

2.99

×10

−12.

62×

10−7

20×

20×

2072

0.87

8.76

×10

−73.

27×

10−1

2.29

×10

−620

×20

×20

a19

1.06

2.02

×10

−632

×32

×32

a16

58.2

2a C

PU

-tim

ew

itho

utre

solu

tion

ofth

eel

asto

-pla

stic

tran

siti

on(c

ompa

refig

ure

3).





order to reproduce the global size-dependent behaviour. However, their validityis restricted to certain ranges of the deformation and size. Therefore, moreelaborated grain boundary models should be examined.

The authors acknowledge the support rendered by the German Research Foundation (DFG) undergrant BO 1466/5-1. The funded project ‘Dislocation-based gradient plasticity theory’ is part of theDFG Research Group 1650 ‘Dislocation-based plasticity’.

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http://dx.doi.org/doi:10.1002/1097-0207(20010120)50:2$<$273::AID-NME17$>$3.0.CO;2-Q

http://dx.doi.org/doi:10.1002/1097-0207(20010120)50:2$<$273::AID-NME17$>$3.0.CO;2-Q




http://dx.doi.org/doi:10.1007/BF01341480



http://dx.doi.org/doi:10.1016/0022-5096(71)90010-X

http://dx.doi.org/doi:10.1016/0022-5096(71)90010-X


http://dx.doi.org/doi:10.1002/(SICI)1097-0207(19990130)44:3$<$373::AID-NME508$>$3.0.CO;2-7

http://dx.doi.org/doi:10.1002/(SICI)1097-0207(19990130)44:3$<$373::AID-NME508$>$3.0.CO;2-7








Equivalent plastic strain gradient enhancement of single...

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