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: [email protected]
B. D. Reddy
e-mail: [email protected]
S. Bargmann
Institute of Continuum Mechanics and Materials Mechanics,
Helmholtz-Zentrum Geesthacht, Hamburg University of Technology
& Institute of Materials Research, Geesthacht, Germany
e-mail: [email protected]
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-
123
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
123
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
123
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)
123
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)
123
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
,
123
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
123
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-
123
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.
References
1. Anand L, Gurtin ME, Reddy BD (2015) The stored energy of cold
work, thermal annealing, and other thermodynamic issues in single
crystal plasticity at small length scales. Int J Plasticity 64:1–25
2. Armero F, Simo JC (1992) A new unconditionally stable fractional
step method for non-linear coupled thermomechanical problems.
Int J Numer Methods Eng 35(4):737–766. ISSN 0029-5981
3. Arsenlis AP, Parks DM (1999) Crystallographic aspects of
geometrically-necessary and statistically-stored dislocation den-
sity. Acta Mater 47:1597–1611
4. Bargmann S, Ekh M (2013) Microscopic temperature field pre-
diction during adiabatic loading using gradient extended crystal
plasticity. Int J Solids Struct 50(6):899–906
5. Bargmann S, Reddy BD (2011) Modeling of polycrystals using a
gradient crystal plasticity theory that includes dissipative micro-
stresses. Eur J Mech A 30(5):719–730
6. Bargmann S, Reddy BD, Klusemann B (2014) A computational
study of a model of single-crystal strain-gradient viscoplasticity
with an interactive hardening relation. Int J Solids Struct 51(15–
16):2754–2764
7. Bayley CJ, Brekelmans WAM, Geers MGD (2006) A comparison
of dislocation induced back stress formulations in strain gradient
crystal plasticity. Int J Solids Struct 43(24):7268–7286
8. Benzerga AA, Bréchet Y, Needleman A, van der Giessen E (2005)
The stored energy of cold work: predictions from discrete disloca-
tion plasticity. Acta Mater 53(18):4765–4779
123
Comput Mech (2015) 55:755–769 769
9. Bever MB, Holt DL, Titchener AL (1973) The stored energy of
cold work. Prog Mater Sci 17:833–849
10. Bittencourt E, Needleman A, Gurtin ME, van der Giessen E (2003)
A comparison of nonlocal continuum and discrete dislocation plas-
ticity predictions. J Mech Phys Solids 51:281–310
11. Cleveringa HHM, van Der Giessen E, Needleman A (1997) Com-
parison of discrete dislocation and continuum plasticity predictions
for a composite material. Acta Mater 45(8):3163–3179
12. Coleman BD, Noll W (1963) The thermodynamics of elastic mate-
rials with heat conduction and viscosity. Arch Ration Mech Anal
13(1):167–178
13. Dederichs PH, Leibfri G (1969) Elastic Green’s function for
anisotropic cubic crystals. Phys Rev 188(3):1175–1183
14. Ebobisse F, Reddy BD (2004) Some mathematical problems in
perfect plasticity. Comput Methods Appl Mech Eng 193(48–
51):5071–5094
15. Ekh M, Grymer M, Runesson K, Svedberg T (2007) Gradient crys-
tal plasticity as part of the computational modelling of polycrystals.
Int J Numer Methods Eng 72(2):197–220
16. Ekh M, Bargmann S, Grymer M (2011) Influence of grain boundary
conditions on modeling of size-dependence in polycrystals. Acta
Mech 218(1–2):103–113 ISSN 0001–5970
17. Ertürk I, van Dommelen JAW, Geers MGD (2009) Energetic dislo-
cation interactions and thermodynamical aspects of strain gradient
crystal plasticity theories. J Mech Phys Solids 57(11):1801–1814
18. Evers LP, Parks DM, Brekelmans WAM, Geers MGD (2002) Crys-
tal plasticity model with enhanced hardening by geometrically nec-
essary dislocation accumulation. J Mech Phys Solids 50(11):2403–
2424
19. Evers LP, Brekelmans WAM, Geers MGD (2004a) Scale depen-
dent crystal plasticity framework with dislocation density and grain
boundary effects. Int J Solids Struct 41:5209–5230
20. Evers LP, Brekelmans WAM, Geers MGD (2004b) Non-local crys-
tal plasticity model with intrinsic SSD and GND effects. J Mech
Phys Solids 52(10):2379–2401
21. Gurtin ME (2000) On the plasticity of single crystals: free
energy, microforces, plastic-strain gradients. J Mech Phys Solids
48(5):989–1036
22. Gurtin ME (2002) A gradient theory of single-crystal plasticity that
accounts for geometrically necessary dislocations. J Mech Phys
Solids 50(1):5–32
23. Gurtin ME (2006) The Burgers vector and the flow of screw and
edge dislocations in finite-deformation single-crystal plasticity. J
Mech Phys Solids 54(9):1882–1898
24. Gurtin ME (2008) A finite-deformation, gradient theory of single-
crystal plasticity with free energy dependent on densities of geo-
metrically necessary dislocations. Int J Plasticity 24(4):702–725
25. Gurtin ME (2010) A finite-deformation, gradient theory of single-
crystal plasticity with free energy dependent on the accumulation of
geometrically necessary dislocations. Int J Plasticity 26(8):1073–
1096
26. Gurtin ME, Anand L (2005) A theory of strain-gradient plasticity
for isotropic, plastically irrotational materials, Part I: small defor-
mations. J Mech Phys Solids 53(7):1624–1649
27. Gurtin ME, Needleman A (2005) Boundary conditions in small-
deformation, single-crystal plasticity that account for the burgers
vector. J Mech Phys Solids 53(1):1–31
28. http://www.fgg.uni-lj.si/Symech/Index.htm
29. Intel. Math Kernel Library (2014) http://developer.intel.com/
software/products/mkl/
30. Korelc J (2002) Multi-language and multi-environment generation
of nonlinear finite element codes. Eng Comput 18(4):312–327
31. Lele SP, Anand L (2008) A small-deformation strain-gradient
theory for isotropic viscoplastic materials. Philos Mag 88(30–
32):3655–3689
32. Miehe C, Mauthe S, Hildebrand FE (2014) Variational gradient
plasticity at finite strains, Part III: local-global updates and regu-
larization techniques in multiplicative plasticity for single crystals.
Comput Methods Appl Mech Eng 268:735–762
33. Niordson CF, Kysar JW (2014) Computational strain gradient crys-
tal plasticity. J Mech Phys Solids 62:31–47
34. Reddy BD (2011a) The role of dissipation and defect energy in vari-
ational formulations of problems in strain-gradient plasticity, Part
2: single-crystal plasticity. Contin Mech Thermodyn 23(6):551–
572
35. Reddy BD (2011b) The role of dissipation and defect energy in vari-
ational formulations of problems in strain-gradient plasticity, Part
1: polycrystalline plasticity. Contin Mech Thermodyn 23(6):527–
549
36. Reddy BD, Wieners C, Wohlmuth B (2012) Finite element analysis
and algorithms for single-crystal strain-gradient plasticity. Int J
Numer Methods Eng 90(6):784–804
37. Rosakis P, Rosakis AJ, Ravichandran G, Hodowany J (2000) A
thermodynamic internal variable model for the partition of plastic
work into heat and stored energy in metals. J Mech Phys Solids
48(3):581–607
38. Rudraraju S, van der Ven A, Garikipati K (2014) Three-dimensional
isogeometric solutions to general boundary value problems of
Toupin’s gradient elasticity theory at finite strains. Comput Meth-
ods Appl Mech Eng 278:705–728 ISSN 0045–7825
39. Schmidt-Baldassari M (2003) Numerical concepts for rate-
independent single crystal plasticity. Comput Methods Appl Mech
Eng 192:1261–1280
40. Schröder J, Miehe C (1997) Aspects of computational rate-
independent crystal plasticity. Comput Mater Sci 9:168–176
41. Taylor GI, Quinney H (1934) The latent energy remaining in a
metal after cold working. Proc R Soc Lond A 143:307–326
42. Taylor GI, Quinney H (1937) The latent energy remaining in a
metal after cold working. Proc R Soc Lond A 163:157–181
43. Wriggers P (2008) Nonlinear finite element methods. Springer,
Berlin
44. Wulfinghoff S, Böhlke T (2012) Equivalent plastic strain gradient
enhancement of single crystal plasticity: theory and numerics. Proc
R Soc Lond A 468(2145):2682–2703
123