Comput Mech (2015) 55:755–769

DOI 10.1007/s00466-015-1134-5

ORIGINAL PAPER

A computational investigation of a model of single-crystal gradient

thermoplasticity that accounts for the stored energy of cold work

and thermal annealing

A. McBride · S. Bargmann · B. D. Reddy

Received: 2 September 2014 / Accepted: 16 February 2015 / Published online: 3 March 2015

© Springer-Verlag Berlin Heidelberg 2015

Abstract A theory of single-crystal gradient thermoplas-

ticity that accounts for the stored energy of cold work and

thermal annealing has recently been proposed by Anand et al.

(Int J Plasticity 64:1–25, 2015). Aspects of the numerical

implementation of the aforementioned theory using the finite

element method are detailed in this presentation. To facilitate

the implementation, a viscoplastic regularization of the plas-

tic evolution equations is performed. The weak form of the

governing equations and their time-discrete counterparts are

derived. The theory is then elucidated via a series of three-

dimensional numerical examples where particular emphasis

is placed on the role of the defect-flow relations. These rela-

tions govern the evolution of a measure of the glide and geo-

metrically necessary dislocation densities which is associated

with the stored energy of cold work.

Keywords Gradient single crystal plasticity · Thermo-

plasticity · Finite element method · Cold work · Annealing

1 Introduction

Purely-mechanical models of single-crystal gradient plas-

ticity have received considerable attention during the last

A. McBride (B) · B. D. Reddy

Centre for Research in Computational and Applied Mechanics,

University of Cape Town, 5th Floor, Menzies Building,

Private Bag X3, Rondebosch 7701, South Africa

e-mail: andrew.mcbride@uct.ac.za

B. D. Reddy

e-mail: daya.reddy@uct.ac.za

S. Bargmann

Institute of Continuum Mechanics and Materials Mechanics,

Helmholtz-Zentrum Geesthacht, Hamburg University of Technology

& Institute of Materials Research, Geesthacht, Germany

e-mail: swantje.bargmann@tu-harburg.de

two decades. A widely-adopted and notable contribution is

the class of thermodynamically consistent gradient theories

of Gurtin and co-workers, and related works (see e.g. [22–

24,27]). A variational formulation of the Gurtin [22] the-

ory has been developed in [34,35]. Ertürk et al. [17] show

how the theory of Gurtin et al. can be related to the more

physically motivated theories due to Evers et al. [18–20] and

Bayley et al. [7]. The finite element method has been used

extensively to provide additional insight into both the theory

and the complex response of metals at small length scales

(see e.g. [10]). Algorithms for the efficient solution of large-

scale three-dimensional problems in gradient plasticity have

recently been presented by Wulfinghoff and Böhlke [44],

Reddy et al. [36] and Miehe et al. [32].

By contrast, coupled thermomechanical models for single-

crystal gradient plasticity have received relatively little atten-

tion to date. An exception is the recent work of Bargmann

and Ekh [4] which extends the model of gradient crystal plas-

ticity due to Ekh et al. [15] to account for thermal and cou-

pling effects. The evolution of the temperature profile due

to thermomechanical coupling effects at high strain rates

was demonstrated using numerical examples solved using

the finite element method. Both material and geometric non-

linearities were considered.

An extension of the Gurtin [22] theory of single-crystal

gradient plasticity to account for the stored energy of cold

work, thermal annealing and other thermodynamic effects

has recently been developed by Anand et al. [1]. When

a metal is cold-worked, most of the plastic work is con-

verted into heat. The plastic work not converted to heat is

the stored energy of cold work. The stored energy of cold

work plays an important role in the long-term evolution of

the defect structure of ductile metals. Given the importance

of the stored energy of cold work, numerous experimental

investigations have been performed in an attempt to quan-

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756 Comput Mech (2015) 55:755–769

tify it, see, e.g. the review by Bever et al. [9] and the ref-

erences therein. Accounting for stored energy of cold work

is important when modelling a range of processes involving

thermoplastic phenomena. Rosakis et al. [37] presented a

one-dimensional model for the classical (non-gradient) case.

Benzerga et al. [8] present a numerical investigation into the

stored energy of cold work using discrete dislocation plas-

ticity.

The primary contribution to the stored energy is the energy

associated with the evolving dislocation density and sub-

structure of the material. The measure of the net dislocation

density proposed by Anand et al. [1] is a continuum descrip-

tion of both glide and geometrically-necessary dislocations.

The evolution of the net dislocation density is governed by

the defect-flow relation which accounts for accumulation due

to plastic deformation and recovery due to thermal annealing.

The objectives of the current presentation are twofold: to

develop a numerical model of the Anand et al. theory using

the finite element method—this has not been done before;

and second to elucidate aspects of the physical behaviour

based on the Anand et al. theory and via a series of numerical

examples.

The plastic flow relations proposed by Anand et al. [1] are

rate-independent. A viscoplastic reformulation of the rate-

independent theory, in the spirit of the original formulation

of Gurtin and co-workers (see e.g. [26]), is developed here

to circumvent algorithmic problems that arise for realistic

crystal structures (see e.g. [39,40] for further details). For

the particular viscoplastic model used it is well known that

the rate-independent limit could be reached in the limit [14,

34]. The resulting regularized dissipation function acts as a

potential for both the energetic and dissipative generalized

stress measures.

The vast majority of the numerical solutions for prob-

lems in gradient single-crystal plasticity do not consider both

energetic and dissipative scalar and vectorial micro-forces as

proposed in the original theory of Gurtin[22]. Notable excep-

tions include [5,6,31,33]. The structure of the theory pre-

sented by Anand et al. [1] requires both scalar and vectorial

energetic non-recoverable stresses. Given that the general-

ized dissipative stress has the same structure as its energetic

non-recoverable counterpart, we solve for both its scalar and

vectorial parts.

The governing system of equations is highly nonlinear

and coupled. They are solved approximately using the finite

element method in conjunction with a Newton scheme and

time-step control. Automatic differentiation [30] is used to

compute parts of the resulting residual equations and the

complete tangent.

Key features of the theory are elucidated via a series of

numerical examples. The good performance of the finite ele-

ment algorithm for realistic crystal structures within rela-

tively complex domains is illustrated.

The structure of the presentation is as follows. In Sect. 2,

the kinematics of gradient crystal plasticity are recalled. The

balance relations are then summarised. The kinetics are pre-

sented in Sect. 4. A particular emphasis is placed on the

defect-flow relation. A viscoplastic reformulation of the rate-

independent plastic flow relations proposed by Anand et al.

[1] is developed in Sect. 5. The weak formulation of the prob-

lem and the associated incremental formulation are devel-

oped in Sect. 6. This is followed by details of the numeri-

cal implementation within the finite element framework. The

finite element model is then used to simulate a series of repre-

sentative numerical examples in Sect. 9. Finally, conclusions

are made and various extensions proposed in Sect. 10.

1.1 Notation and basic relations

Direct notation is adopted throughout. Occasional use is

made of index notation, the summation convention for

repeated indices being implied. When the repeated indices

are lower-case italic letters, the summation is over the range

{1, 2, 3}. The scalar product of two vectors a and b is denoted

a · b = [a]i [b]i . The scalar product of two second-order

tensors A and B is denoted A : B = [A]i j [B]i j . The

composition of two second-order tensors A and B, denoted

AB, is a second-order tensor with components [AB]i j =

[A]im[B]mj . The tensor product of two vectors a and b is

a second-order tensor D = a ⊗ b with [D]i j = [a]i [b] j .

The action of a second-order tensor A on a vector b is a

vector with components [a]i = [A]im[b]m . Any array asso-

ciated with the set of N slip systems is denoted by γ := {γ 1,

γ 2 . . . , γ N } . Summation over the slip systems will be abbre-

viated by∑

α . Index notation is not employed for summation

over slips systems. The unit basis vectors in the Cartesian

(standard-orthonormal) basis are {e1, e2, e3}. The general-

ized inner product between a pair A := (a, a) and a pair

B := (b, b) is defined by A • B := ab + a · b. The sym-

metric part of a tensor A is defined by Asym = 12[A + AT].

Components of vectors and tensors are expressed, where nec-

essary, relative to a fixed orthonormal basis and a Cartesian

coordinate system.

2 Kinematics

The theory presented is restricted to the geometrically lin-

ear problem. Consider a continuum body whose placement

is denoted by V ⊂ R3 at time t = 0. A typical material

point is identified by the position vector x ∈ V . The absolute

temperature at a material point is denoted by ϑ > 0. The

displacement of a material point is denoted by u(x, t). The

displacement gradient H := ∇u is decomposed (locally)

into elastic and plastic parts He and Hp according to

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Comput Mech (2015) 55:755–769 757

H = He + Hp.

The elastic displacement gradient He accounts for recov-

erable elastic lattice stretching, while Hp quantifies the plas-

tic distortion due to slip on the predefined slip planes. The

elastic strain Ee is given by the symmetric part of the elastic

displacement gradient as

Ee :=1

2

[

He + HeT]

.

The flow of dislocations through the crystal lattice is

described kinematically via the assumption that the plas-

tic distortion tensor Hp can be expressed in terms of the

slip strain γ α on the individual prescribed slip systems

α = 1, 2, . . . , N as

Hp =∑

α

γ αsα ⊗ mα =∑

α

γ αS

α.

The slip direction and slip plane normal of slip system α

are denoted by sα and mα , respectively, where sα · mα = 0

and |sα| = |mα| = 1. The slip plane associated with slips

system α is denoted by �α . The Schmid tensor is defined

by Sα = sα ⊗ mα . The plastic strain Ep is defined by the

symmetric part of the plastic distortion; that is

Ep :=1

2

[

Hp + HpT]

=1

2

∑

α

γ α[sα ⊗ mα + mα ⊗ sα]

=∑

α

γ αS

αsym.

The vector lα is defined by lα := mα × sα . Hence

{mα, sα, lα} constitute a local orthonormal basis.

Following Gurtin [24], the constitutive theory at the

microscopic scale accounts for a continuous distribution of

geometrically-necessary dislocations (GNDs). The disloca-

tions are either of edge or screw type and are characterised

in terms of their Burgers and line directions as follows:

• edge dislocation Burgers direction sα and line direction

lα;

• screw dislocations Burgers direction sα and line direction

sα .

The density of the edge and screw dislocations per unit

length (a measure widely adopted in the continuum mechan-

ics literature), denoted by ρα⊢ and ρα

⊙, respectively, can be

related to the slip gradient (see [3]) as follows:

ρα⊢ = −∇γ α · sα and ρα

⊙ = ∇γ α · lα,

and hence

|∇γ α| =[

|ρα⊢|2 + |ρα

⊙|2] 1

2.

Thus the rate of change of the GND density on �α is given

by the magnitude of the gradient of the corresponding slip

rate.

The generalized accumulated slip rate Ŵαacc is a measure

of the rate of slip of both glide dislocations and GNDs, and

is defined by

Ŵαacc :=

[

|γ α|2 + l2|∇γ α|2] 1

2, (1)

where l > 0 is a length scale. Central to the theory of Anand

et al. [1] is an evolution equation for the net dislocation den-

sity ρα ≥ 0 that accounts for dislocation accumulation (both

glide and GND) due to plastic flow and recovery due to ther-

mal annealing. The net dislocation densities are defined per

unit area as is common in the materials science literature (see

[25] for a detailed discussion on commonly used definitions

of dislocation density).

3 Balance relations

The symmetric stress tensor in the bulk is denoted by T .

Scalar and vector microscopic forces, denoted by πα and ξα

respectively, are postulated as conjugates to the slip rates and

their spatial gradients [21,22]. The resolved shear stress on

�α is denoted by τα := T : Sα . The power conjugate kinetic

and kinematic pairings are as follows:

T ↔ Ee (macroscopic stress),

πα ↔ γ α (scalar microscopic force),

ξα ↔ ∇γ α (vector microscopic force).

3.1 Balance of forces

A balance of macroscopic (in the absence of inertial and body

forces) and microscopic forces yields

divT = 0 in V and t⋆(n) = T n on ∂Vt , (2)

divξα+τα−πα =0 in V and �α⋆(n) =ξα · n on ∂V�.

(3)

Equation (2) is the standard equilibrium equation, and t⋆

is the prescribed Cauchy traction on the Neumann part of

the boundary ∂Vt . Dirichlet boundary conditions on the dis-

placement u are prescribed on ∂Vu, where ∂V = ∂Vu ∪ ∂Vt

and ∂Vu ∩ ∂Vt = ∅. Furthermore, the boundary ∂V is subdi-

vided into complementary parts ∂V� and ∂Vγ . The standard

boundary condition on the micro-free part of the boundary

∂V� is that the scalar microscopic traction �α⋆ ≡ 0 while

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758 Comput Mech (2015) 55:755–769

on the micro-hard part of the boundary ∂Vγ homogeneous

conditions on the slip are prescribed. For additional details

on the microscopic boundary conditions see Ekh et al. [16],

Gurtin and Needleman [27]. The macroscopic and micro-

scopic force balances are coupled via the dependence of the

resolved shear stress τα on the macroscopic stress T .

3.2 Balance of energy

We assume from the onset that the entropy of dislocations is

quite small, and that, at ordinary and low temperatures, it is

negligible relative to the energetic contributions (see [1] and

the references therein). The heat source per unit volume of

V is denoted by q and the heat flux vector by q. Following

Anand et al. [1], the balance of energy gives the temperature

evolution equation, the flux boundary condition and the initial

condition as

cϑ = −divq + q +∑

α

τα γ α + M : Ee−

∑

α

Fαcwρα

+∑

α

div(γ αξα) in V, (4)

q⋆(n) = q · n on ∂Vq , (5)

ϑ(x, t = 0) = ϑ0(x) in V, (6)

where c is the specific heat per unit volume and Fαcw is a

thermodynamic force associated with dislocation evolution

on slip system α. The stress-temperature modulus is defined

by

M := ϑ∂T

∂ϑ.

Dirichlet boundary conditions on the temperature ϑ are

prescribed on ∂Vϑ , where ∂V = ∂Vϑ ∪ ∂Vq and ∂Vϑ ∩

∂Vq = ∅. The heat flux q ·n is prescribed on ∂Vq . In addition,

ϑ0 denotes the initial temperature distribution. The first four

terms on the right-hand side of Eq. (4) are present in classical

thermoplasticity. The final two terms account for temperature

changes due to thermal annealing and higher-order plasticity.

Consider a completely insulated domain with zero heat

sources and no thermoelastic coupling. In the spirit of Taylor

and Quinney [41,42], a traditional point-wise measure of the

amount of plastic energy (ignoring gradient effects) that goes

into heating is given by

βϑ :=cϑ

cϑ +∑

α Fαcwρα

.

Anand et al. [1] have extended this point-wise measure

to account for the additional terms present in their model.

It should be noted that the sign of the second term in the

denominator can be positive or negative.

4 Kinetics

The free energy � is composed of thermoelastic �e and

thermoplastic �p parts as follows

� = �e(Ee, ϑ) + �p(ρ, ϑ).

The rate of change of the free energy is thus

� =∂�e

∂ Ee : Ee +∂�

∂ϑϑ +

∑

α

[∂�p

∂ρα

]

︸ ︷︷ ︸

Fαcw

ρα,(7)

where Fαcw := ∂�p/∂ρα > 0 (see Eq. (4) where Fα

cw was

introduced).

A central relation in the theory proposed by Anand et al.

[1] is an evolution equation for the net dislocation density

given by

ρα = Aα(ρα)Ŵαacc − Rα(ρα, ϑ) with ρα

0 := ρα(t = 0),

(8)

where Aα(ρα) ≥ 0 is the dislocation-accumulation modulus

and Rα(ρα, ϑ) ≥ 0 is the recovery rate. Eq. (8) is termed the

defect-flow relation. Thus, Eq. (7) becomes

� =∂�e

∂ Ee : Ee+∂�

∂ϑϑ +

∑

α

Fαcw AαŴα

acc −∑

α

Fαcw Rα.

(9)

The generalized stress �α and slip rate Ŵα

are defined by

�α := {πα, l−1ξα} and Ŵα

:= {γ α, l∇γ α}, (10)

such that

� • Ŵ =∑

α

[πα γ α + ξα · ∇γ α].

The magnitude of the generalized slip rate Ŵα

is the gen-

eralized accumulated slip rate defined in Eq. (1), that is

Ŵαacc = |Ŵ

α| =

Ŵα

|Ŵα|• Ŵ

α, for |Ŵ

α| �= 0. (11)

Combining Eqs. (11) and (9) gives the rate of change of

free energy as

� =∂�e

∂ Ee : Ee +∂�

∂ϑϑ +

∑

α

[

Fαcw Aα Ŵ

α

|Ŵα|

]

︸ ︷︷ ︸

�αnr

•Ŵα

−∑

α

Fαcw Rα.

(12)

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Comput Mech (2015) 55:755–769 759

The energetic non-recoverable part of the generalized

stress �α is defined by �αnr := Fα

cw AαŴα/|Ŵ

α|. The dis-

sipative part of the generalized stress is defined by �αdis :=

�α −�αnr. Substituting Eq. (11) into the defect-flow relation

(8) yields

ρα = Aα Ŵα

|Ŵα|• Ŵ

α− Rα. (13)

Combining the balance of energy and the second law of

thermodynamics gives the free-energy imbalance

�+ηϑ−T : Ee−∑

α

[

πα γ α+ξα · ∇γ α]

︸ ︷︷ ︸

�α•Ŵα

+1

ϑq · ∇ϑ ≤ 0,

(14)

where η is the entropy density.

Substituting Eq. (12) into the free-energy imbalance (14)

and following a procedure due to Coleman and Noll [12]

gives the stress T as the conjugate kinetic quantity to the

elastic strain Ee, and the entropy η as the conjugate quantity

to the temperature ϑ , that is

T =∂�e

∂ Ee and η = −∂�

∂ϑ.

Taking into account the constitutive relations, the free-

energy imbalance (14) reduces to the following reduced dis-

sipation inequality

∑

α

[

�αdis • Ŵ

α]

︸ ︷︷ ︸

Dα�

+∑

α

[

Fαcw Rα

]

︸ ︷︷ ︸

DαF

−1

ϑq · ∇ϑ ≥ 0. (15)

The final term in the reduced dissipation inequality is ren-

dered non-negative by assuming Fourier’s law of (isotropic)

heat conduction

q = −k∇ϑ, (16)

where k is the thermal conductivity. The second term in the

reduced dissipation inequality DαF ≥ 0 due to the assump-

tions that Fαcw > 0 and Rα ≥ 0. A thermodynamically admis-

sible form for �αdis is developed in the next section on the

plastic flow relations to ensure that Dα� ≥ 0.

Remark From the defect-flow relation (13), the rate of

change of the free energy (7) can be expressed as

� =∂�e

∂ E: Ee +

∂�

∂ϑϑ +

∑

α

�αnr • Ŵ

α−

∑

α

∂�p

∂ραRα.

(17)

The structure of a potential for the energetic non-

recoverable part of the generalized stress �αnr is discussed

in the next section.

5 Plastic flow relations

In order to complete the theory, the reduced dissipation

inequality (15) needs to be satisfied in a thermodynamically

consistent manner; that is, we require a (flow) relation for

�αdis such that Dα

� ≥ 0.

Consider first the rate-independent case. The yield func-

tion f (�αdis) defines the region of admissible dissipative

stresses on �α . The yield function and the flow law for the

generalized plastic slip rates are defined by

f α = |�αdis| − Y α, (18)

Ŵα

= λα ∂ f (�αdis)

∂�αdis

, (19)

where Y α > 0 is the slip resistance and λα ≥ 0 is a scalar

multiplier, together with the Kuhn–Tucker complementarity

conditions

f (�αdis) ≤ 0, λα ≥ 0 , λα f (�α

dis) = 0.

Under conditions of plastic flow f (�αdis) ≡ 0 and, from

Eq. (18), |�αdis| = Y α . Assuming plastic flow, the flow rule

(19) can be inverted to obtain

�αdis = Y α Ŵ

α

|Ŵα|

= Y α Ŵα

Ŵαacc

. (20)

The rate-independent theory presents various numerical

challenges due to the indeterminacy of plastic slip (see e.g.

[39,40] for further details). To circumvent these problems, a

regularized effective dissipation function Dαvis is proposed of

the form

Dαvis =

1

m + 1

[Ŵα

acc

Ŵ0

]m+1

Ŵ0,

where Ŵ0 is the reference value for the slip rate and m > 0

is the rate sensitivity (see [34] for additional details). Dαvis

is positively homogeneous of degree k, if Dαvis(aŴ) =

ak Dαvis(Ŵ) for positive k. Then Euler’s theorem on positively

homogeneous functions states that Ŵα

• [Dαvis(Ŵ

α)/Ŵ

α] =

k Dαvis(Ŵ

α). The regularized dissipative stress follows as

�αdis := Y α ∂ Dα

vis

∂Ŵα = Y α

[Ŵα

acc

Ŵ0

]mŴ

α

Ŵαacc

. (21)

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760 Comput Mech (2015) 55:755–769

Remark Recall that the energetic non-recoverable general-

ized stress is defined by

�αnr = Fα

cw Aα Ŵα

|Ŵα|. (22)

The similarity in the structure of �αnr and �α

dis (see

Eq. (20)) motivates the definition of a regularized energetic

non-recoverable generalized stress in the spirit of Eq. (21) as

�αnr := Fα

cw Aα ∂ Dαvis

∂Ŵα = Fα

cw Aα

[Ŵα

acc

Ŵ0

]mŴ

α

|Ŵα|. (23)

In the regularized theory, the generalized stress can be

expressed as

�α = �αnr + �α

dis = [Fαcw Aα + Y α]

∂ Dαvis

∂Ŵα

=[

Fαcw Aα + Y α

][Ŵα

acc

Ŵ0

]mŴ

α

|Ŵα|. (24)

The rate of change of the free energy (9) can thus be rewrit-

ten as

� =∂�e

∂ Ee : Ee +∂�

∂ϑϑ +

∑

α

Fαcw Aα[m + 1]Dα

vis(Ŵα)

−∑

α

∂�p

∂ραRα.

Remark The structure of the defect flow relation (13) relates

an increase in net dislocations (glide and GND) to an increase

in the generalized accumulated slip rate. This fundamental

assumption leads to the form of the energetic non-recoverable

generalized stress in Eq. (22) and its regularized counterpart

in Eq. (23). The regularized, generalized dissipative and ener-

getic non-recoverable stress measures are both aligned in the

direction ∂ Dαvis/∂Ŵ

α.

Remark In order to circumvent problems when Ŵαacc = 0,

a small positive value of the order of machine precision is

added to the computed value. An alternative option would be

to evaluate the yield function.

6 Weak form of the problem

The weak form of the governing equations is now derived.

We assume henceforth that the generalized stress is obtained

from the regularized dissipation function as per Eq. (24). The

weak form provides the point of departure for the numerical

implementation using the finite element method.

The spaces of displacement V , slips Q and temperature

W are defined by

V =

{

u : ui ,∂ui

∂x j

∈ L2(V), δu = 0 on ∂Vu

}

,

Q =

{

γ α : γ α,∂γ α

∂xi

∈ L2(V), γ α = 0 on ∂Vγ

}

,

W =

{

ϑ : ϑ,∂ϑ

∂xi

∈ L2(V), δϑ = 0 on ∂Vϑ

}

.

For the sake of simplicity, we assume micro-free condi-

tions on ∂V�.

6.1 Macroscopic force balance

The weak form of the balance of macroscopic forces,

obtained by testing (2) with an arbitrary displacement δu ∈

V , integrating the result over V and using the divergence

theorem, is given by

0 =

∫

V

δE : T dV −

∫

∂Vt

δu · t⋆ dA =

∫

V

δE :∂�e

∂ Ee dV

−

∫

∂Vt

δu · t⋆ dA, (25)

where 2δE = ∇δu + [∇δu]T.

6.2 Microscopic force balance

The weak form of the balance of microscopic force, obtained

by testing (3) with an arbitrary slip δγ α ∈ Q, integrating the

result over V , and using the divergence theorem, is given by

0 = −

∫

V

δŴα : �αdV +

∫

V

δγ αταdV

= −

∫

V

δŴα :

[∂�p

∂ραAα + Y α

]∂ Dα

vis

∂Ŵα dV

+

∫

V

δγ α ∂�e

∂ Ee : Sαsym dV,

(26)

where δŴα = {δγ α, l∇δγ α}.

6.3 Energy balance

In order to derive the weak form of the energy balance we

employ the following relation:

τα γ α + div(γ αξα) = �α • Ŵα

,

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Comput Mech (2015) 55:755–769 761

where we have used the strong form of the micro-force bal-

ance relation (3).

Isotropic thermal conductivity is assumed (see Eq. (16)).

The weak form of the energy balance follows from testing (4)

with an arbitrary temperature δϑ ∈ W , integrating the result

over V , and using the divergence theorem, the defect flow

relation (13), and the definition of the dissipative generalized

stress, yielding

0 =

∫

V

δϑcϑ dV +

∫

∂Vq

δϑq⋆dA +

∫

V

∇δϑ · k∇ϑ dV

−

∫

V

δϑq dV −

∫

V

M : EedV

+∑

α

∫

V

δϑ Fαcwρα

dV −∑

α

∫

V

δϑ�α • Ŵα

dV

=

∫

V

δϑcϑ dV +

∫

∂Vq

δϑq⋆dA +

∫

V

∇δϑ · k∇ϑ dV

−

∫

V

δϑq dV −

∫

V

ϑ∂2�e

∂ϑ∂ Ee : EedV

+∑

α

∫

V

δϑ

[[

�αnr − �α

]

︸ ︷︷ ︸

−�αdis

•Ŵα

−∂�p

∂ραRα

]

dV

=

∫

V

δϑcϑ dV +

∫

∂Vq

δϑq⋆dA +

∫

V

∇δϑ · k∇ϑ dV

−

∫

V

δϑq dV −

∫

V

ϑ∂2�e

∂ϑ∂ Ee : EedV

−∑

α

∫

V

δϑ∂�p

∂ραRα

dV

−∑

α

∫

V

δϑY α[m + 1]Dαvis(Ŵ

α) dV .

(27)

6.4 The incremental problem

The time interval of interest 0 ≤ t ≤ T is partitioned into N

subintervals as 0 = t0 < t1 < · · · < tN = T , with �t =

tn+1 − tn = T/N . Note, uniform time-steps are assumed

in the derivation of the incremental problem for notational

simplicity; an adaptive time-stepping algorithm is used in

the numerical implementation. The value of a quantity w at

time tn is denoted by wn . The rate of change of a quantity

is approximated using an Euler-backward difference scheme

as w ≈ �w/�t . The incremental problem is obtained by

evaluating relations (25)–(27) at tn as

0 =

∫

V

δE :∂�e

∂ Ee

∣∣∣∣n

dV −

∫

∂Vt

δu · t⋆n dA , (28)

0 = −

∫

V

δŴα :

[∂�p

∂ρα

∣∣∣∣n

Aα + Y α

]∂ Dα

vis

∂Ŵα

∣∣∣∣

�Ŵα

�t

dV

+

∫

V

δγ α ∂�e

∂ Ee

∣∣∣∣n

: Sαsym dV , (29)

0 =

∫

V

δϑc[ϑn − ϑn−1]

�tdV +

∫

∂Vq

δϑq⋆n dA

+

∫

V

∇δϑ · k∇ϑn dV −

∫

V

δϑqn dV

−

∫

V

δϑ ϑ∣∣n

∂2�e

∂ϑ∂ Ee

∣∣∣∣n

:[Ee

n+1 − Een]

�tdV

−∑

α

∫

V

δϑ∂�p

∂ρα

∣∣∣∣n

Rαn dV

−∑

α

∫

V

δϑY αn [m + 1]

Dαvis(�Ŵα)

�tdV, (30)

7 Constitutive relations

The form of the constitutive relations and parameters sug-

gested in Anand et al. [1] are generally adopted here. These

relations are extended to account for cubic anisotropy and

include thermoelastic coupling in the balance of energy.

The thermoelastic response is linearized by firstly assum-

ing a quadratic free energy of the form

�e =1

2Ee : C Ee − [ϑ − ϑ0]A : C Ee +

c

2ϑ0[ϑ − ϑ0]

2 ,

where A = α I is the isotropic thermal expansion tensor at

reference temperature ϑ0, and α is the thermal expansion

coefficient. The fourth-order elasticity tensor C under condi-

tions of cubic symmetry is given by

Ci jkl = C12δi jδkl + C44

[

δikδ jl + δilδ jk

]

+ C

3∑

r=1

δirδ jrδkrδlr ,

where C = C11 − C12 − 2C44 (see e.g. [13]). For the case

of elastic isotropy we have

C11 = κ +4

3µ, C12 = κ −

2

3µ, C44 = µ,

where κ := λ+2µ/3 is the bulk modulus, and λ and µ are the

Lamé constants. The second assumption is that ϑη ≈ ϑ0η,

123

762 Comput Mech (2015) 55:755–769

resulting in a constant, temperature-independent specific heat

c (see e.g. [2]).

The plastic part of the free energy is given by

�p =∑

α

Ecw(ρα) =C44b2

2

∑

α

ρα, (31)

where Ecw(ρα) is the stored energy of cold work, and b is

the length of the Burgers vector. The slip resistance Y α = Y

is given by

Y = Y0 +C44b

2

[∑

α

ρα

] 12

, (32)

where Y0 is the initial slip resistance.

The dislocation-accumulation modulus and the recovery

rate in the defect-flow relation (8) are given by

Aα = A0

[

1 −ρα

ραsat

]p

and Rα = R0 exp

(

−Qr

kBϑ

)

⟨

ρα − ραmin

⟩q, (33)

where p > 0, q > 0 are constants, A0 and R0 are reference

accumulation and recovery rates, ρsat is a saturation threshold

for dislocations, Qr is an activation energy for static recov-

ery, kB is Boltzmann’s constant, ραmin is the minimum net

dislocation density for recovery, and < x > is the unit step

function with a value 0 for x < 0 and x for x ≥ 0. The

recovery rate is in the form of an Arrhenius-type law. The

defect-flow relation is thus non-linear and implicit.

8 Finite element approximation

The finite element method is used to solve the highly nonlin-

ear and coupled system of governing equations. The domain

V is decomposed into a set of non-overlapping hexahedral

elements. The primary variables (displacement, slip strains

and temperature) are all approximated from the values at the

nodes of the elements using a standard Lagrangian Q1 inter-

polation in R3. The vector of primary variables at the nodes

is denoted by d. The net dislocation density ρ is stored at the

quadrature points associated with the elements.

The state of the system is assumed known at time tn . The

spatially discrete form of the residual equations (28)–(30) are

respectively denoted [Ru, Rγ , R

θ ] =: R. A global (outer)

iterative Newton–Raphson scheme is used to determine the

solution dn+1 such that Rn+1 ≈ 0. Let (k) denote the cur-

rent iteration. The approximate solution at iteration (k) of

time-step n is denoted d(k). The resulting linearised prob-

lem for the incremental change in the solution is denoted

δd := d(k+1) − d

(k) is given by

∂R

∂d

∣∣∣∣(k)

δ d = −R(k),

where the tangent matrix is given by A(k) := ∂R

(k)/∂d. The

approximate solution d(k+1) is then computed and the norm

of the incremental change in the solution vector checked to

see if the scheme has converged. Further details are given in

Fig. 1.

An inner Newton algorithm is employed to solve the time-

discrete form of the (implicit) defect flow relation (8) for ρ

using a backward-Euler scheme. The discontinuous unit step

function is approximated using the smooth logistic function;

that is

Fig. 1 Algorithm for the finite element approximation

123

Comput Mech (2015) 55:755–769 763

〈x〉 ≈1

1 + exp(−r x),

where r ≡ 1 controls the sharpness of the transition at x = 0.

Similar approaches for smoothly approximating the unit step

function in the context of gradient crystal plasticity have been

adopted by Miehe et al. [32].

It is possible, in the numerical scheme, for the general-

ized accumulated slip rate to approach zero. This renders the

computation of Ŵαacc in Eq. (11) meaningless. A number of

the order of machine precision is therefore added to Eq. (1)

to circumvent this issue.

The software library AceGen [30] is used to describe the

finite element interpolation, and to compute the parts of the

residual R that derive from a potential, and the (algorith-

mically consistent) tangent A exactly using automatic dif-

ferentiation, at the level of the quadrature point. A detailed

presentation of the automatic differentiation tools used in

AceGen can be found in [43]. The use of automatic differen-

tiation instead of classical analytical derivation to construct

the tangent for highly-nonlinear problems greatly reduces the

possibility of error (see for example [38]). Furthermore, auto-

matic differentiation introduces no significant computational

overhead when compared to the construction of an analyti-

cally derived tangent. Various authors have used numerical

differentiation to compute the tangent for problems in gradi-

ent plasticity. Numerical differentiation, while efficient and

straightforward, is inaccurate and can lead to instabilities

for highly nonlinear problems. For additional information

on the use of AceGen for problems in plasticity, including

the source code, the reader is referred to the examples and

extensive documentation provided in [28]. The structure of

the inner and outer Newton schemes employed here is based

on the aforementioned reference.

The resulting non-symmetric matrix problem is solved

using the PARDISO library [29]. A time-step duration con-

trol procedure is used to optimise the time-step size in an

attempt to ensure quadratic convergence of the Newton–

Raphson scheme.

9 Numerical examples

Two three-dimensional numerical example problems are pre-

sented to illustrate the key features of the theory summarised

and developed in the previous sections. The first example

is motivated by the solution of the defect-flow equation (8)

presented by Anand et al. [1] for a zero-dimensional, rigid

system with a prescribed temperature evolution. The numer-

ical example presented here illustrates the thermal anneal-

ing process after the application of a mechanical load in a

problem with two slip systems. The rate of recovery dur-

ing the annealing process is relatively slow requiring 5000 s

Fig. 2 The problem of a double slip system specimen subject to shear

loading

to be simulated. The longer term response is also assessed.

The second example illustrates thermomechanical coupling

effects for a face-centred cubic (FCC) crystal structure. The

rate of applied mechanical loading is chosen to be moder-

ate (relative to the first example) to emphasise the coupling

effects. The FCC example also demonstrates the good per-

formance of the numerical implementation for a complex

crystal structure and a realistic geometry.

The ability of gradient crystal plasticity formulations to

capture size effects is well documented in the literature (see

e.g. [10]) and is not explored in any significant detail here.

The numerical examples are thus performed on specimens

that are small but with a characteristic length several order

of magnitude larger than the length scale l in Eq. (1). The

influence of loading rate on the thermomechanical response

has been investigated by Bargmann and Ekh [4], and is not

explored here.

9.1 Shear of a double slip system specimen

Consider the [L]3 = [1]3 mm double slip system crystal sub-

ject to shear-type loading shown in Fig. 2. The two slip sys-

tems are constrained in the e1 − e2 plane. The vectors s1 and

s2 are inclined at 60◦ and 60◦ to the e1 axis, respectively.

The elastic response is assumed isotropic and the reference

temperature is specified by ϑ0 = 293 K. The system is ini-

tially loaded mechanically at a fixed prescribed temperature

(i.e. it is cold worked) causing plastic deformation to occur

and dislocations to accumulate. The mechanical loading is

then terminated and the temperature on the upper and lower

faces of the domain increased linearly for a fixed duration to

allow for annealing. The long term response is then assessed

by fixing both the external temperature and the mechanical

loading.

123

764 Comput Mech (2015) 55:755–769

The lower surface ∂Vy− is fully mechanically constrained,

i.e. u(x ∈ ∂Vy−) = 0. The upper surface ∂Vy+ is dis-

placed horizontally (i.e. in the e1 direction) at a rate of

L/500 mm s−1 until t = tA = 50 s. The temperature on

the upper and lower faces is fixed at ϑ0 = 293 K dur-

ing the mechanical loading phase. The heat flux q⋆ ≡ 0

on all remaining faces. Micro-hard boundary conditions are

assumed on the top and bottom faces, and micro-free con-

ditions on the remaining. The micro-hard conditions cor-

respond to a situation where a crystal is in contact with

a rigid surface, such as a loading platen, which acts as an

impenetrable obstacle to dislocations. The upper surface is

then prevented from displacing further, and the tempera-

ture is increased linearly to 423 K on the upper and lower

surfaces for a further 4950 s to allow for recovery due to

thermal annealing. The longer term response response for

t > tB = 5000 s is assessed by fixing both the external tem-

perature and the prescribed mechanical boundary conditions,

and continuing the simulation until t = tD = 10,000 s,

The domain is discretized using 35×35×10 Q1 elements.

A relatively coarse discretization was used in the e3 direction

as the problem is essentially planar. This reduces the compu-

tational expense of the problem which is important given the

duration of the simulation. The target maximum time-step

duration is 1 s. The constitutive parameters, chosen to match

approximately those of Cu, are given in Table 1.

The rates of thermal and mechanical loading are chosen as

relatively slow to emphasise the thermal annealing process.

The symmetry of the slip systems about the loading axis

ensures that �2 has a similar response to �1. The net dislo-

cation density ρ1 (glide and GND) increases rapidly during

the mechanical loading phase 0 < t ≤ tA, attaining a value

of approximately ρsat throughout the domain at t = 50 s, as

shown in Fig. 3. The evolution of the net dislocation density

ρ1 at a point in the center of the specimen is shown in Fig. 4a.

From Eq. (31), the stored energy of cold work Ecw(ρ1) is pro-

portional to ρ1. The distribution and evolution of the stored

energy of cold work can therefore also be obtained from the

distribution and evolution of the net dislocation density. The

net dislocation density in the vicinity of the upper and lower

boundaries at the end of the mechanical loading phase (load

point A in Fig. 3) is lower than in the rest of the domain due

to the prescribed micro-hard conditions on the slip.

The evolution of the slip γ 1 during the mechanical loading

phase is also shown in Fig. 3.

The rate of accumulation modulus A1 (see Eq. (33)1) is

approximately zero at t = tA as ρ1 ≈ ρsat. The distribution of

the generalized accumulated slip rate Ŵ1acc over the domain

evolves fairly rapidly to the distribution corresponding to

load point A in Fig. 3.

For t > tA, there is no prescribed mechanical loading and

the rate of thermal loading is slow. Thus, as shown in the

plots of Ŵ1acc during the thermal annealing phase in Fig. 3,

Ŵacc is nearly constant and small resulting in a low accu-

mulation rate. The recovery rate R is dependent on both the

temperature and the net dislocation density. The contribu-

Table 1 Constitutive

parameters for a Cu-like

materialParameter Symbol Value Unit

Young’s modulus E 115 × 103 N/mm2

Poisson ratio ν 0.35

Thermal expansion coefficient α 17 × 10−6 K−1

Burgers vector length b 2.5 × 10−7 mm

Length scale l 1 × 10−2 mm

Reference slip rate Ŵ0 1 s−1

Initial slip resistance Y0 81 N/mm2

Dissipation function exponent m 1

Thermal conductivity k 385 N/s K

Specific heat c 3.45 N/mm2 K

Reference temperature ϑ0 293 K

Reference accumulation rate A0 3 × 1010 mm−2

Reference recovery rate R0 1 × 10−5 s−1

Accumulation rate exponent p 1

Recovery rate exponent q 2

Net dislocation saturation ρsat 1 × 109 mm−2

Initial dislocation density ρ0 1 × 106 mm−2

Minimum dislocation density for recovery ρmin 0 mm−2

Ratio of activation energy for static recovery to kB Qr /kB 5773 K

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Comput Mech (2015) 55:755–769 765

(a) (b) (c) (d)

Fig. 3 The deformed specimen at various stages of loading, denoted (a–d), with the net dislocation density ρ1, the generalized accumulated slip

rate Ŵ1acc, and the slip γ 1 superimposed. The scale shown applies to the plots to the left of the scale

(a) (b)

Fig. 4 The variation in the net dislocation density ρ1 (a) and the recov-

ery rate R1 (b) at a point in the centre of the domain with time. The

markers (A–D) correspond to the various stages of the loading defined in

Fig. 3. The inflection point in the evolution of ρ1 and the corresponding

turning point in the evolution of R1 is denoted by a star

tion of these two fields to recovery varies during the thermal

annealing phase. Figure 4b shows the variation in R1 with

time for a point at the centre of the specimen. The recov-

ery rate increases initially with the increasing temperature

during the thermal annealing phase causing ρ1 to decrease.

As the recovery rate increases so the net dislocation density

decreases as shown. At t ≈ 2600 s the recovery rate begins

to decrease as the net dislocation density has decreased to a

point where an increase in temperature no longer implies an

increase in recovery rate. The turning point in the recovery

rate with time plot coincides with the inflection point in the

plot of ρ1 with time (denoted by a ⋆ in both plots).

It is worth noting that even though Ŵ1acc is relatively small

during the thermal annealing phase it is not negligible. This is

due to the coupling of mechanical and thermal fields. Thus, as

ρ decreases during the thermal annealing phase so the accu-

mulation rate will begin to increase. The minor evolution

of slip due to coupling effects during the thermal anneal-

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766 Comput Mech (2015) 55:755–769

Fig. 5 The problem of tensile loading in a FCC dog bone specimen (dimensions in mm)

Table 2 The orientation of the

slip planes relative to the unit

cell and the fixed Cartesian basis

α sα mα α sα mα α sα mα

1 [0 1 1] (1 1 1) 5 [1 0 1] (1 1 1) 9 [1 1 0] (1 1 1)

2 [1 0 1] (1 1 1) 6 [1 1 0] (1 1 1) 10 [0 1 1] (1 1 1)

3 [1 1 0] (1 1 1) 7 [0 1 1] (1 1 1) 11 [1 1 0] (1 1 1)

4 [0 1 1] (1 1 1) 8 [1 0 1] (1 1 1) 12 [1 0 1] (1 1 1)

(a) (b) (c)

Fig. 6 The distribution of the temperature field across the deformed domain at various points in the loading history

ing phase is shown in Fig. 3. Recall that thermal annealing

is associated with the dissipation of non-recoverable elastic

energy and not plastic slip.

The longer term response of the problem is investigated by

fixing the prescribed temperature ϑ⋆ = 423 K on the upper

and lower surfaces at the end of the thermal annealing phase

and simulating another 10,000 s. The dislocation density con-

tinues to decrease for t > tC but the rate of change decreases.

The abrupt change in the recovery rate at t = tC is clear in

Fig. 4b.

The results of the finite element simulations compare well

qualitatively with those presented by Anand et al. [1]. The

amount of mechanical deformation at the beginning of the

annealing phase assumed by Anand et al. [1] was less than

what was simulated here. Thus the dislocation density at the

onset of annealing is higher in the simulation. The results of

Anand et al. [1] show no inflection in the variation of ρ with

time. The long term behaviour of the models compare well.

Given the additional complexity in the results presented here,

one should not expect an identical response.

123

Comput Mech (2015) 55:755–769 767

(a) (b)

Fig. 7 The evolution of the temperature at the point ym with time is shown in (a). The labels (A–C) correspond to the stages of loading shown in

Fig. 6. The variation in the temperature across the line y− − y+ at the end of the loading cycle is shown in (b)

(a) (b)

(c) (d)

Fig. 8 The deformed specimen with Ep22,

∑

α ρα and βϑ are shown in (a–c), respectively. The variation in the resultant force on the upper surface

∂Vt with time is shown in (d)

9.2 Tensile loading of a FCC dog bone specimen

In contrast to the previous example, the current example

emphasises thermomechanical coupling effects for a dog

bone specimen with a FCC crystal structure subject to tensile

loading. The dimensions of the specimen and the computa-

tional mesh are shown in Fig. 5. The dog bone example also

demonstrates the ability of the proposed numerical formula-

tion to efficiently solve reasonably complex problems.

The upper surface of the specimen ∂Vy+ is displaced ver-

tically (i.e. in the e2 direction) a distance of 1.8 mm in a

time of 0.5 s. The lower surface ∂Vy− is fully fixed. Micro-

hard boundary conditions are assumed top and bottom and

micro-free conditions on all other faces. The temperature

on the top and bottom faces is set to the initial tempera-

ture of ϑ0 = 293 K. Cubic symmetry is assumed for the

elastic response with C11 = 153 × 103, C12 = 119 × 103

and C44 = 49 × 103 N/mm2. The initial dislocation density

ρ = 100 mm−2. The remaining material properties are as

listed in Table 1. The orientation of the slip planes relative

to the unit cell (and, for this example, the fixed Cartesian

basis) are given in Table 2. The domain is discretized using

13,470 Q1 elements with 16 degrees of freedom per node (3

displacement, 12 slip, 1 temperature).

The temperature distribution at various times during the

loading process is superimposed upon the deformed domain

123

768 Comput Mech (2015) 55:755–769

in Fig. 6. The variation in the temperature at the midpoint of

the specimen (denoted ym in Fig. 6) with time is shown in

Fig. 7a. The temperature at point ym decreases in the early

stage of loading when elastic effects dominate. Thereafter,

the temperature increases with time due to increased plastic

flow. The temperature distribution across the vertical line

y− − y+ passing through the center of the specimen at the

end of the loading cycle is shown in Fig. 7b. The maximum

heating in the specimen occurs in the neck region and is due

to plastic deformation. The temperature change due to elastic

effects is negligible once plastic flow is developed. This was

confirmed by performing the simulation with a zero stress-

temperature modulus.

The distribution of the plastic strain Ep22 and the cumu-

lative net dislocation density∑

α ρα are shown in Fig. 8a,

b. The distribution of the classical measure of the amount

of plastic energy that goes into heating βϑ is plotted over

the deformed domain in Fig. 8c. Up to 43 % of the plas-

tic energy goes to heating in the neck region of the speci-

men. This corresponds to the region where the amount of

plastic deformation is the highest. It should be emphasised

that the classical measure neglects gradient terms, thermo-

mechanical coupling, and assumes insulated boundaries. The

variation in the net reaction force on the upper face during

the loading is plotted in Fig. 8d. The response is near lin-

ear initially. The response deviates from linear with increas-

ing plastic flow and evolving slip resistance. The slip resis-

tance on the individual slip systems is the same and evolves

according to Eq. (32) which is a function of the sum of the

slip on all slip system (see Fig. 8b for a plot of this mea-

sure).

The problem was solved in 103 time steps with an average

number of five outer Newton iterations per step. Quadratic

convergence of the scheme was observed. The parallelized

solution of the linear system occupied 70 % of the computa-

tional time while the assembly of the linear system occupied

11 %. A summary of the performance of the solution scheme

is given in Fig. 9.

Fig. 9 Summary of the analysis for the FCC dog bone specimen

10 Discussion and conclusion

A recent theory of gradient single-crystal thermoplasticity

due to Anand et al. [1] has been summarized. The numer-

ical implementation of the theory using the finite element

method has been presented. A regularized viscous dissipation

was introduced as a potential for the generalized dissipative

and energetic non-recoverable stresses. The theory was eluci-

dated using two numerical example problems. The numerical

examples demonstrated the robustness and efficiency of the

finite element formulation.

The extension of the Anand et al. [1] theory to the geomet-

rically nonlinear case should be relatively straightforward.

The extension of the mechanical problem is well documented

(see e.g. [24]) and a geometrically nonlinear gradient ther-

moplasticity theory has been considered by Bargmann and

Ekh [4].

In terms of future work, a detailed comparison of the con-

tinuum theory implemented here with the discrete dislocation

plasticity models presented by Benzerga et al. [8] and exper-

iment are crucial. Previous work comparing the mechanical

gradient plasticity theory of Gurtin [22] to discrete disloca-

tion plasticity (see e.g. [10,11]) would serve as a point of

departure.

Acknowledgments A.M. and B.D.R. acknowledge the support pro-

vided by the National Research Foundation through the South African

Research Chair in Computational Mechanics. A part of this work was

undertaken while A.M. was visiting the Hamburg University of Tech-

nology. A.M. also acknowledges the support provided by the University

Research Committee of the University of Cape Town.

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