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Proc. R. Soc. A (2012) 468, 2682–2703
doi:10.1098/rspa.2012.0073
Published online 19 April 2012
Equivalent plastic strain gradient enhancement
of single crystal plasticity: theory and numerics
BY STEPHAN WULFINGHOFF*
AND
THOMAS BÖHLKE
Institute of Engineering Mechanics, Karlsruhe Institute of Technology,
Karlsruhe, Germany
We propose a visco-plastic strain gradient plasticity theory for single crystals. The
gradient enhancement is based on an equivalent plastic strain measure. Two physically
equivalent variational settings for the problem are discussed: a direct formulation and an
alternative version with an additional micromorphic-like field variable, which is coupled
to the equivalent plastic strain by a Lagrange multiplier. The alternative formulation
implies a significant reduction of nodal degrees of freedom. The local algorithm and
element stiffness matrices of the finite-element discretization are discussed. Numerical
examples illustrate the advantages of the alternative formulation in three-dimensional
simulations of oligo-crystals. By means of the suggested formulation, complex boundary
value problems of the proposed plastic strain gradient theory can be solved numerically
very efficiently.
Keywords: strain gradient plasticity; crystal plasticity; finite elements
1. Introduction
The continuum mechanics of single crystal models (Hill 1966; Lee 1969;
Rice 1971; Ortiz & Stainier 1999) are based on a well-established physical
basis. The discovery of dislocations as the fundamental carriers of plasticity
justifies the mesoscopic kinematical assumptions concerning the plastic slip rates
(Orowan 1934) and the slip system geometry associated with these models.
Experiments (Schmid & Boas 1935) provide evidence of the correlation between
the projected shear stresses and the slip system activity. Also non-Schmid effects
can be taken into account within this setting (Kocks 1987; Yalcinkaya et al. 2008).
In contrast, single crystal hardening models and the prediction of the closely
related dislocation micro-structure are still subjects of current research. Besides
purely phenomenological hardening laws (Taylor 1938; Koiter 1953; Hill 1966)
local dislocation density evolution or balance equations (Gillis & Gilman 1965;
Mecking & Kocks 1981; Franciosi & Zaoui 1982; Estrin 1996) allow to estimate
the evolution of the total dislocation line length per unit volume and to model the
hardening behaviour of crystals more physically. These models are successfully
applied at scales and specimen dimensions, where no size effect is observed.
*Author for correspondence (wulfi[email protected]).
Received 7 February 2012
Accepted 21 March 2012
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However, the inability to predict the size-dependent behaviour at smaller scales
observed in experiments, e.g. the Hall–Petch effect (Hall 1951; Petch 1953) or the
size-dependent strength of micro-specimens (Fleck et al. 1994; Stölken & Evans
1998; Xiang & Vlassak 2006; Dimiduk et al. 2007; Gruber et al. 2008) motivated
many authors to enrich the classical local theory, for example, based on Nye’s
dislocation density tensor (Nye 1953; Bilby et al. 1955; Kröner 1958) or related
measures associated with the gradients of the plastic slips. Nye’s tensor can be
computed from the spatial gradients of the plastic variables and introduces an
internal length into the theory. It can be interpreted as the total Burgers vector
per unit area, i.e. it has a clear physical meaning and must be distinguished from
the above-mentioned dislocation density (the total line length per unit volume).
Nye’s tensor or familiar gradient-type quantities (Fleck & Hutchinson 1993;
Liebe & Steinmann 2001; Liebe 2004; Gurtin et al. 2007) provide the physical
motivation for size-dependent hardening models. However, a generally accepted
correlation between the gradients of the variables and the hardening behaviour of
the crystal could not be established, yet. Several thermodynamic gradient theories
for polycrystals have been proposed associating the gradients of plastic quantities
with focus on an increase of the free energy (Fleck et al. 1994; Steinmann 1996;
Menzel & Steinmann 2000) or dissipation (Gurtin & Anand 2005; Fleck & Willis
2009). In the case of single crystals, a refined kinematical theory allows to compute
geometrically necessary screw and edge dislocation densities for the different slip
systems, closely related to Nye’s tensor (Gurtin 2002), which has been generalized
to the context of large deformations by Gurtin (2006). Associated thermodynamic
theories have been proposed by e.g. Cermelli & Gurtin (2002), Berdichevsky
(2006), Ohno & Okumura (2007) or Ekh et al. (2007). The dislocation field
theory of Evers et al. (2004), see also Geers et al. (2006) and its generalization
(Bayley et al. 2006) have been compared with thermodynamical approaches by
Ertürk et al. (2009) and Bargmann et al. (2010). Similar non-local theories are,
for example, based on incompatibility-dependent hardening moduli (Acharya &
Bassani 2000) or other non-work-conjugate formulations (Kuroda & Tvergaard
2008). In the micromorphic approach of Forest (2009), a local inelastic variable
is energetically coupled to an additional degree of freedom. A strong energetical
coupling can formally be interpreted as penalty approximation of, e.g. a strain
gradient plasticity theory. However, the coupling parameter allows for a specific
adjustment of the scaling behaviour of the model, which was investigated by
Cordero et al. (2010) and Aslan et al. (2011) for single crystal laminates and by
Cordero et al. (2012) for periodic single crystals. Based on a closely related largedeformation theory for isotropic elastic–plastic materials, Anand et al. (2011)
regularized strain-softening phenomena by means of the gradient of an additional
variable related to the equivalent plastic strain. Hochrainer (2006) generalized
Nye’s concept to a higher-dimensional continuum dislocation theory, which was
implemented by Sandfeld et al. (2010).
The fact that size-dependent theories usually require the computation of
additional gradients makes the numerical treatment by the finite-element method
more complex and computationally expensive (compared with local theories)
since the variables have to be approximated by (at least piecewise) differentiable
shape functions. One of the first works on this issue was published by de
Borst & Mühlhaus (1992). Shu et al. (1999) presented and compared different
element formulations for the Toupin–Mindlin framework of strain gradient theory.
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S. Wulfinghoff and T. Böhlke
They introduced the displacement gradients as additional nodal degrees of
freedom (associated with C 0 -continuous shape functions) and coupled them to
the true gradients by Lagrange multipliers. Liebe & Steinmann (2001) proposed
a thermodynamic framework, where the yield condition is treated in a weak
sense, leading to a distinction between plastically active and inactive nodes.
A multi-field incremental variational framework for gradient-extended standard
dissipative solids, including a generalization of this framework, can be found in
Miehe (2011). Becker (2006) and Han et al. (2007) propose a projection of the
plastic variables from the integration points to the nodes which subsequently
allows the computation of plastic strain gradients and an implementation close to
local theories. Forest (2009) proposes a generalized micromorphic approach with
a local variable and a micromorphic variable. In addition, Wieners & Wohlmuth
(2011) propose a primal–dual finite-element approximation. Both approaches
preserve a local evaluation of the yield function and facilitate the application
of the radial return algorithm.
One drawback of gradient plasticity theories is that a unique and generally
accepted correlation between the gradients of the variables and the hardening
behaviour is missing. Another disadvantage of implementations, including the
plastic variables as additional nodal variables, is the significant increase in
the computational effort owing to the extended number of degrees of freedom,
especially in single crystal computations. Consequently, the application of the
theory to three-dimensional problems is numerically extraordinarily expensive.
The present work provides a gradient theory for crystals which leads to
efficient numerics. The formulation and model behaviour are close to existing
gradient theories. We choose an equivalent plastic strain measure (which unites
the plastic strain history of all slip systems in one scalar quantity) for the
gradient enhancement aiming at a significant decrease of additional nodal degrees
of freedom compared with existing theories. One central motivation for this
proceeding is that equivalent plastic strain measures have been used successfully
in the context of phenomenological local hardening models. Most of the abovementioned works are based on geometrically necessary dislocations (GND),
given by Nye’s tensor which has a clearer physical interpretation than the
gradient of the equivalent plastic strain. However, the present theory allows the
simulation of three-dimensional systems (consisting of several grains) at reduced
computational cost, owing to the significant reduction of degrees of freedom. The
close relation to other gradient theories will be shown, and as long as the exact
dependence of the hardening behaviour on geometrically necessary dislocation
density measures (if there is any) remains unclear, even more complex gradient
theories remain imprecise.
The outline of the paper is as follows: first, we discuss the basic kinematical
and energetic assumptions. Subsequently, two different but physically identical
variational settings are introduced. Details of the finite-element implementation
are discussed before some numerical results illustrate the performance of
the alternative formulation in three-dimensional applications. Finally, we
analyse the ability of the theory to reproduce the experimentally observed
size effects.
Notation. A direct tensor notation is preferred throughout the text. Vectors
and second-order tensors are denoted by bold letters, e.g. a or A. A linear
mapping of second-order tensors by a fourth-order tensor is written as A = C[B].
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The scalar product and the dyadic product are denoted, e.g. by A · B and A ⊗ B,
respectively. The composition of two second-order tensors is formulated by AB.
Completely symmetric and traceless tensors are designated by a prime, e.g. A .
Matrices are denoted by a hat, e.g. 3̂. The transpose and the inverse of a matrix
are indicated by D̂ T and D̂ −1 , respectively.
2. Equivalent plastic strain gradient enhancement of single crystal plasticity
(a) Constitutive equations
(i) Kinematics
We describe the motion of a body B by the displacement field u(x, t), which
maps the positions x ∈ B of the particles to their associated displacements u at
time t. Here, B is open with boundary vB and closure B̄ := B ∪ vB. We restrict
ourselves to the small deformation context, i.e. H 1, where H := Vu =
vui /vxj e i ⊗ e j . Furthermore, we define the strain tensor by 3 := sym(H ) = VS u
and assume the classical relation
H p :=
N
la d a ⊗ n a
(2.1)
a=1
for the plastic part of the displacement gradient, where the scalars la are slip
parameters and N is the number of slip systems which are characterized by slip
plane normals n a and slip directions d a (orthogonal to n a ) of unit length. The
symmetric part of H p is the plastic strain
la M Sa .
(2.2)
3p := sym(H p ) =
a
of slip system a is defined by M Sa := sym(d a ⊗ n a ). In
The Schmid-tensor
order to capture the actual single crystals kinematics, the slip systems are defined
pairwise with common slip plane normal and opposite slip directions. This implies
for octahedral slip systems, e.g. N = 24. The slip parameters are constrained to
increase monotonously, i.e. l̇a ≥ 0 with initial values equal to zero. The effective
plastic slips (in each slip system) are given by the difference between two pairwise
defined slip parameters. It should be noted that alternatively 12 slip systems
with possibly positive and negative slip increments for face-centred cubic (FCC)crystals can be introduced. For most gradient theories, this leads to a reduced
number of degrees of freedom in the finite-element implementation. The elastic
strain is defined by 3e := 3 − 3p . We denote the plastic domain of a slip system a
by Baact ⊆ B (where l̇a > 0).
Additionally, the equivalent plastic strain measure is given by the expression
la ,
(2.3)
geq (l̂) :=
M Sa
a
which maps the slip parameters l̂ := (l1 , l2 , . . . , lN )T to the equivalent plastic
strain geq . Recall that (ˆ•) denotes the use of matrix notation. Note that, we choose
definition (2.3) for simplicity and that other definitions of geq (l̂) are possible.
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S. Wulfinghoff and T. Böhlke
The theory can easily be generalized to more complex (e.g. tensorial) quantities
like the plastic part of the displacement gradient H p (l̂). This generalization
allows the application of the theory presented in the following to more complex
models, which assume the free energy to depend, for example, on Nye’s dislocation
density tensor.
(ii) Free energy and dissipation
We assume the free energy per unit volume to be of the additive form (formally
similar to, e.g. Steinmann 1996)
W (3, l̂, Vgeq ) = We (3, l̂) + Wh (l̂) + Wg (Vgeq ),
(2.4)
with We (3, l̂) := 1/2(3 − 3p (l̂)) · C[3 − 3p (l̂)], where C is the elastic stiffness
tensor. The second part Wh (l̂) phenomenologically accounts for the isotopic
hardening behaviour of crystals. Physically, this observation is primarily
attributed to the multiplication of dislocations, which are mutually acting as
obstacles leading to an increase of the macroscopically observed yield stress with
proceeding plastic deformation. The third part Wg (Vgeq ) introduces an internal
length into the theory, i.e. it leads to a size-dependent mechanical response
of the model. In the case of metals, size effects are observed experimentally
for, e.g. micro specimens, steels with varying grain sizes or nanoindentation
tests. The size effect can partially be explained by geometrically necessary
dislocation configurations represented by Nye’s tensor, which is closely related
to the gradients of the plastic slips. The dissipation D (dissipated power per unit
volume) is assumed to be a superposition of the dissipation contributions of the
individual slip systems
Da =
tad l̇a ,
(2.5)
D=
a
a
where tad are dissipative forces which extend power over the plastic slip rates
(cf. Cermelli & Gurtin 2002).
(b) Variational form and strong form of the field equations
The basis for the subsequent theoretical development and for the numerical
implementation is the principle of virtual power, which states that for a given
state, described by the primary variables (u, l̂), the virtual power of the external
forces is equal to the virtual power of the internal forces. The formal treatment
is similar to that of Gurtin et al. (2007). The external forces are assumed to
be given by the traction field t̄ on the associated Neumann boundary vBt ⊂ vB.
The tractions t̄ generate power over v := u̇. The
micro-tractions X̄ are defined
analogically, i.e. X̄ generates power over ġeq = a l̇a on the Neumann boundary
vBX ⊆ vB. The primary variables (u, l̂) are assumed to be known on the Dirichlet
boundary vBu := vB \ vBt and vBgeq := vB\vBX , respectively. Here, we restrict
ourselves to micro-hard Dirichlet boundary conditions l̂ = 0̂ ∀ x ∈ vBgeq .
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Plastic strain gradient enhancement
˙
∗ T
We choose v ∗ and l̂∗ = (l̇1∗ , l̇2∗ , . . . , l̇N
) to be the independent virtual
kinematical quantities and require both of them to respect the Dirichlet boundary
conditions on vBu and vBgeq . The dependent virtual rates can be expressed by
3̇∗ := VS v ∗ ;
3̇e∗ := 3̇∗ −
l̇a∗ M Sa
and
ġ∗eq :=
vgeq
a
a
The principle of virtual power states
∗
∗
∗
(Ẇ + Pdis ) dv =
t̄ · v da +
B
vBt
vBX
vla
X̄ġ∗eq da
l̇a∗ =
l̇a∗ .
(2.6)
a
˙
∀ v ∗ , l̂∗ ,
(2.7)
with
Ẇ ∗ =
vW ∗ vW ˙ ∗
vW
· l̂ +
· 3̇ +
· Vġ∗eq
v3
v(Vg
)
eq
vl̂
and the virtual power of the dissipative forces
∗
=
tad l̇a∗ .
Pdis
(2.8)
(2.9)
a
Expression (2.8) motivates the introduction of the micro-stress
x̌ :=
vWg
,
v(Vgeq )
(2.10)
which is energetically conjugate to Vgeq .
Based on these results, equation (2.7) can be expressed in terms of the primary
variables (u, l̂)
vWh
S
S
S ∗
∗
S
∗
d ∗
la M a +
ta l̇a dv
V v −
l̇a M a · C V u −
l̇ +
vla a
B
a
a
a
a
˙
Vl̇a∗ dv =
Vl̇a∗ da ∀v ∗ , l̂∗ . (2.11)
X̄
t̄ · v ∗ da +
+ x̌ ·
B
vBt
a
vBX
a
Using Gauss’ theorem and the chain rule, we obtain
vW
h
M Sa · C[3e ] + div(x̌) − tad +
l̇a∗ dv (2.12)
−div(C[3e ]) · v ∗ −
vl
a
B
a
˙
(C[3e ]n − t̄) · v ∗ da +
(x̌ · n − X̄)
(2.13)
l̇a∗ da = 0 ∀v ∗ , l̂∗ .
+
vBt
vBX
a
˙
For vanishing l̂∗ , this expression yields
div(s) · v ∗ dv −
B
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vBt
(sn − t̄) · v ∗ da = 0,
(2.14)
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S. Wulfinghoff and T. Böhlke
where the Cauchy stress s = C[3e ] has been introduced. Since v ∗ is arbitrary the
integrands of both integrals must vanish independently and point-wise. The first
integral in equation (2.14) implies
div(s) = 0
∀x ∈ B.
(2.15)
The second integral in equation (2.14) yields the Neumann boundary conditions
sn = t̄
∀x ∈ vBt .
(2.16)
Choosing now v ∗ = 0 and l̇b∗ = 0 ∀ b ∈ {1, 2, . . . , N } \ {a}, we find
B
M Sa
· s + div(x̌) −
tad
vWh
+
vla
l̇a∗
dv +
vBX
(X̄ − x̌ · n)l̇a∗ da = 0. (2.17)
Introducing the resolved shear stress ta := s · M Sa , equation (2.17) yields the
field equations
tad = ta −
vWh
+ div(x̌) ∀ x ∈ B.
vla
(2.18)
Note that the contribution div(x̌) is equal for all slip systems due to the special
choice of the scalar equivalent plastic strain geq . For the Neumann boundary
conditions, we obtain
x̌ · n = X̄
∀ x ∈ vBX .
(2.19)
Intending to use a visco-plastic overstress constitutive formulation, we define the
yield criteria (cf. Miehe 2011) by
fa := tad − t0c ,
with the initial yield stress t0c . From equation (2.18), it follows that
vWh
c
fa = ta + div(x̌) − t0 +
∀ x ∈ B.
vla
(2.20)
(2.21)
Rate-independent single crystal plasticity theories suffer from the problem of
possibly occurring linearly dependent constraints (Kocks 1970). In order to
circumvent this problem, we base our formulation on a visco-plastic constitutive
flow rule of the Perzyna-type (Perzyna 1971)
l̇a =
g(fa )
,
h
(2.22)
where g(x) is monotone with x ≤ 0 ⇔ g(x) = 0 and h is a viscosity parameter
(cf. Simo & Hughes (1998) or Miehe (2011) in the context of gradient theories).
We define the viscous stress by tvis (l̇a ) := g −1 (hl̇a ) ∀ x ∈ Baact , i.e. tvis (l̇a ) = fa
in the case of plastic yielding, and find from equation (2.21) the shear stress
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Box 1. Equivalent plastic strain gradient theory.
balance laws
balance of linear momentum 0 = div(s)
dissipative stress tad = ta − vWh /vla + div(x̌)
Neumann boundary conditions sn = t̄ on vBt , x̌ · n = X̄ on vBX
∀x ∈ B
∀x ∈ B
constitutive equations
free energy W = We (3, l̂) + Wh (l̂) + Wg (Vgeq )
Cauchy stress s = vWe /v3 = C[3e ]
non-local stress
x̌ = vWg /v(Vgeq )
yield criterion fa = tad − t0c
flow rule l̇a = g(fa )/h
equilibrium in the plastic zones Baact
vWh
c
ta + div(x̌) − t0 +
+ tvis (l̇a ) = 0
vla
∀ x ∈ Baact .
(2.23)
Note that the negative of div(x̌) can formally be interpreted as a back-stress (cf.
Gurtin et al. 2007), which is slip system independent, since Wg depends only on
the gradient of the equivalent plastic strain.
From the equations (2.5), (2.18) and (2.23), the dissipation can be deduced
(t0c + tvis (l̇a ))l̇a .
(2.24)
D=
a
This result formally corresponds to the result of the local theory (i.e. Wg = 0).
Hence, compared with a purely local theory, the dissipative mechanisms due to
plastic slip rates are not altered in the theory at hand. In the case of plastic
yielding, the dissipative stress tad , as obtained in equation (2.18), is
tad = t0c + tvis (l̇a )
∀x ∈ Baact .
(2.25)
The introduction of a dissipation potential (here discussed only for linear
viscosity), e.g. by
f (ťa )2
˙
f(l̂) =
,
(2.26)
sup ťa l̇a −
2h
ťa
a
allows to construct a global incremental potential of the model (cf. Miehe 2011).
However, this procedure is not part of the present work. The model equations
introduced in this section are summarized in box 1.
(c) Alternative formulation
In the following, we define a model in terms of four primary variables (u, l̂, z, p̌)
which is (concerning its mechanical behaviour) physically equivalent to the model
introduced in §2b. We introduce an additional field variable z and weakly enforce
its equivalence to geq by a Lagrange parameter p̌. This apparently complex
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approach leads to a massive reduction of the computational effort when solving
the problem by use of the finite-element method. This idea is inspired by the work
of Simo et al. (1985) (also by Forest 2009), who introduced the weak enforcement
of the equality of two (a priori different) field variables by a force-like field which
is treated like an additional field variable (see also Shu et al. 1999).
The free energy is assumed to take the form
W (3e , l̂, z, Vz, p̌) = We (3e ) + Wh (l̂) + Wg (Vz) + p̌(geq (l̂) − z).
(2.27)
We define x := vWg /v(Vz) for the subsequent developments and start from the
weak form
vWh
S
S ∗
∗
S
∗
∗
d ˙ ∗
ta la dv
V v −
l̇a M a · s(V u, l̂) +
l̇ + x · Vż +
vla a
B
a
a
a
X̄ ż∗ da.
t̄ · v ∗ da +
+ p̌˙ ∗ (geq (l̂) − z) dv + p̌(ġ∗eq − ż∗ ) dv =
B
B
vBt
vBX
(2.28)
Application of Gauss’ theorem and the chain rule yields
vW
h
(M Sa · s − p̌) − tad +
l̇a∗ dv
−div(s) · v ∗ −
vl
a
B
a
+ p̌˙ ∗ (geq (l̂) − z) dv − ż∗ (p̌ + div(x)) dv
B
B
˙
(sn − ¯t) · v ∗ da +
(x · n − X̄)ż∗ da = 0 ∀ v ∗ , l̂∗ , p̌˙ ∗ , ż∗ .
+
vBt
vBX
(2.29)
The field equations and Neumann boundary conditions associated with (2.29) are
summarized in box 2. Additionally, the yield criteria and flow rule are listed. It
should be noted that the slip parameters can alternatively be handled as internal
variables. The associated evolution rules can then be derived based on the reduced
dissipation inequality (for further details, see Forest (2009)).
Similar to the case before (equation (2.23)), we obtain the relation
(ta − p̌) −
t0c
vWh
+
+ tvis (l̇a ) = 0
vla
∀ x ∈ Baact
(2.30)
for the plastic zones.
Note that formally, p̌ = −div(x) can be interpreted as a back stress again and
vWh /vla is a local hardening stress. The back stress p̌ is equivalent for all slip
systems due to the reduction to only one scalar variable z and the special choice
of geq (l̂).
From the set of equations in box 2, the equations in box 1 can be derived. In
this sense, the models (2.11) and (2.28) can be considered to be equivalent.
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Plastic strain gradient enhancement
Box 2. Equivalent plastic strain gradient theory, alternative formulation.
balance laws
balance of linear momentum
0
dissipative stress tad
back stress
p̌
equivalent plastic strain
z
Neumann boundary conditions sn
constitutive equations
free energy
= div(s)
= ta − vWh /vla − p̌
= −div(x)
= geq (l̂)
= t̄ on vBt ,
x · n = X̄ on vBX
∀x ∈ B
∀x ∈ B
∀x ∈ B
∀x ∈ B
= We (3, l̂) + Wh (l̂) + Wg (Vz)
+ p̌(geq (l̂) − z)
s = vWe /v3 = C[3e ]
x = vWg /v(Vz)
fa = tad − t0c
l̇a = g(fa )/h
W
Cauchy stress
non-local stress
yield criterion
flow rule
The original system of equations (box 1) includes three scalar partial
differential equations for the linear momentum balance and N (e.g. N = 24/12
in the case of FCC-crystals) partial differential equations for the micro force
balances, whereas the total number of scalar partial differential equations of the
new set of equations (box 2) is only four. It is emphasized that this massively
reduced degree of non-locality makes the new set of equations (box 2) interesting
for a numerical implementation via finite elements. In contrast, the discretization
of the first model leads to elaborate element stiffness matrices and active
set search algorithms on the nodal level which have barely been studied for
crystals, yet. Therefore, only the second model is implemented. The numerical
implementation is presented in the following.
3. Numerical implementation
In this section, the local (integration point) algorithm, the algorithmic tangent
moduli and the element stiffness matrices for the alternative formulation are
derived. It is shown that the number of degrees of freedom per node increases
only by one compared with a local theory. For the numerical implementation,
explicit expressions for the free energy are required. We assume quadratic forms
for simplicity
1
We = 3e · C[3e ],
2
Wh (l̂) =
1
a,b
2
la hab lb
and
1
Wg (Vz) = Kg Vz · Vz, (3.1)
2
with constant hardening moduli hab and Kg .
The starting point for the discretization via finite elements is the timediscretized problem (2.28), evaluated at time step tn+1 (for convenience, the index
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S. Wulfinghoff and T. Böhlke
‘n + 1’ is dropped in the following), given by the residuals associated with the
variations of the displacements
!
u
S ∗
(3.2)
t̄ · v ∗ da = 0,
G := V v · s dv −
B
of the variable z
∗
z
G :=
vBt
B
∗
(Vż · x − p̌ż ) dv −
!
vBX
X̄ż∗ da = 0,
(3.3)
of the back stress p̌
!
r p := z − geq (l̂) = 0
∀x ∈B
and of the slip parameters la
⎞
⎛
Dla
!
+ p̌⎠ = 0
r la := ta − ⎝t0c +
hab lb + tvis
Dt
b
(3.4)
∀ x ∈ Baact .
(3.5)
Here,
Dla la − la,n
=
Dt
Dt
(3.6)
is the slip rate which is assumed constant during the time interval [tn , t]. For
simplicity, we prescribe a linear-viscous behaviour
l̇a =
fa ⇒ tvis (l̇a ) = hl̇a .
h
(3.7)
(a) Local algorithm
The structure of the set of residuals allows for an algorithmically decoupled
determination of the quantities (u, z) and the local quantities (l̂, p̌), which
is close to standard iterative procedures in computational inelasticity, solving
the discretized momentum equations and updates of the local variables
separately (e.g. Simo & Hughes (1998); Miehe & Schröder (2001), among many
others). For the computation of the local quantities, we assume the solution (u, z)
to be given. The local problem, given by equations (3.4) and (3.5), is constrained
by the requirement Dla ≥ 0, which will be exploited to identify the local set of
active slip systems J (containing the indices of the active systems). For the
moment, assume an active set J = ∅ to be given. The aim here is the development
of an algorithm which allows the computation of
x̂ S := (l(1) , l(2) , . . . , l(nact ) , p̌)T = x̂ Sn + Dx̂ S ,
(3.8)
where nact is the number of active slip systems. Recall that (ˆ•) denotes the use of
matrix notation. The subscripts in the parenthesis and the superscript ‘S’ account
for the active slip parameters.
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Plastic strain gradient enhancement
To identify the solution, we compute the residuals (3.4) and (3.5) and therefore
use the following matrix notation
T Def.
R̂S := (r l(1) , r l(2) , . . . , r l(nact ) , r p ) = ((rˆl )T , r p )T ,
(3.9)
based on the state (u, z, l̂n , p̌n ). The solution x̂ S can then be obtained as the
S
solution of the linear system of equations  Dx̂ S = R̂S with the symmetric tangent
⎛
S
 := −
vrˆl
⎜
vR̂S
⎜ vl̂S
=
−
⎜
p T
S
⎝ vr
vx̂
vl̂S
⎞
vrˆl
vp̌ ⎟
⎟
⎟.
⎠
0
(3.10)
S
The explicit elements of  are given by
−
h
vr l(a)
= M S(a) · C[M S(b) ] + h(a)(b) + d(a)(b)
vl(b)
Dt
−
vr l(a) vgeq,n+1
vr p
=−
= 1.
=
vp̌
vl(a)
vl(a)
and
(3.11)
For convenience, the complete local algorithm is summarized in box 3 containing
a standard active set search procedure (see Ortiz & Stainier (1999); Miehe &
Schröder (2001), and references therein).
(b) Algorithmic tangent moduli
The local update procedure is coupled to the global algorithm via a generalized
form of the classical algorithmic tangent moduli
Ĉ s3 =
vŝ
,
v3̂
Ĉ sz =
vŝ
,
vz
Ĉ p3 =
vp̌
v3̂
and
C pz =
vp̌
.
vz
(3.12)
In order to compute the algorithmic tangent, consider the definition ŷ :=
(3̂T , z)T . The basis for the computation of the algorithmic tangent moduli is the
requirement, that the residuals R̂S remain zero, if the quantities (u, z) change
dR̂S =
vR̂S
vR̂S
!
dŷ + S dx̂ S = 0̂.
vŷ
v
x̂
=:B̂
(3.13)
−ÂS
Accordingly, the differential of the local variables dx̂ S can be reconstructed
explicitly as a function of the differential of the quantities dŷ
dx̂ S = ÂS−1 B̂dŷ = D̂dŷ.
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(3.14)
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2694
S. Wulfinghoff and T. Böhlke
Box 3. Local algorithm.
Local algorithm
(i) Get the quantities {3̂n , zn , l̂n , Jn , p̌n } from the previous time step as well as the
increments {D3̂, Dz, Dt}, compute 3̂ = 3̂n + D3̂ and z = zn + Dz. Set C s3 = Ĉ .
p
(ii) Compute the plastic strain from the previous time step 3̂n = a la,n M̂ Sa .
p
pz
(iii) Calculate the trial stress ŝtr = Ĉ [3̂ − 3̂n ] and trial back stress p̌tr = Cel (z − geq,n ).
(iv) Check the yield conditions based on the trial state
⎛
⎞
hab lb,n ⎠ ,
fatr = (ŝtr · M̂ Sa − p̌ tr ) − ⎝t0c +
a = 1, . . . , N .
b
(v) If fatr < tol ∀a ∈ {1, 2, . . . , N } (elastic update):
ŝ = ŝtr ;
p̌ = p̌tr ;
l̂ = l̂n ;
J = ∅;
Ĉ p3 = 0;
pz
C pz = Cel ;
fa = fatr and EXIT.
(vi) Else (plastic update) compute the residuals
⎛
⎞
S
tr
c
hab lb,n + p̌n ⎠ ,
Ra = M̂ a · ŝ − ⎝t0 +
a = 1, . . . , N ;
b
RN +1 = z − geq (l̂n ),
and the tangent components
Aab = (M̂ Sa )T Ĉ M̂ Sb + hab + dab
h
;
Dt
a, b = 1, . . . , N ,
as well as
A(N +1)a = Aa(N +1) = 1, a = 1, . . . , N and A(N +1)(N +1) = 0.
(vii) Define k : Rnact → RN , which maps the active slip indices {1, . . . , nact } to the global slip
system indices {1, . . . , N }. Activate the maximum loaded system with f = fmax .
(viii) While active set search not converged:
(a) Build submatrix ASab = Ak(a)k(b) and subresidual RaS = Rk(a) .
(b) Compute solution Dx̂ S = ÂS−1 R̂S .
(c) Identify minimum slip increment aMin = arg min Dla .
a∈J
If mina∈J Dla < tol : J ← J \ {amin }. Update k(a). CONTINUE.
(d) Set preliminary solution
l̂ = l̂n + Dl̂, p̌ = p̌n + Dp̌, 3̂p =
la M̂ Sa , ŝ = Ĉ [3̂ − 3̂p ].
a
(e) Identify maximum loaded system
⎛
∀a ∈
/J :
fa = M̂ Sa
· ŝ
− ⎝t0c
+
⎞
hab lb + p̌⎠ , amax = arg(maxa∈J
/ fa ),
b
i.e. compute fa only, if system a and its pairwise defined counterpart are
inactive.
(f) Eventually, update active set and repeat computation
famax > tol : J ← J ∪ {amax }, update k(a), CONTINUE.
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2695
Plastic strain gradient enhancement
Hence, the tangent operator D̂, containing the partial derivatives
⎞
⎛
⎛ S
⎞
vl̂S
vl̂S
vl̂
vl̂S
⎟
⎜ v3̂
vz ⎟ ⎝
D̂ = ⎜
vz ⎠ ,
⎝
vp̌ T vp̌ ⎠ = v3̂
p3
Ĉ
C pz
v3̂
vz
(3.15)
can be computed as a matrix product and allows to identify two of the required
algorithmic tangent moduli. The matrix B̂ is given by the expression
⎞
⎛
vrˆl
vrˆl
l
vrˆ
⎟
⎜ v3̂
vr l(a)
0̂
vz
v
3̂
⎟
⎜
= Ĉ M̂ S(a) .
(3.16)
=
,
B̂ = ⎝
p T
T
vr
vr p ⎠
v
3̂
0̂
1
v3̂
vz
The identification of the remaining algorithmic moduli Ĉ s3 and Ĉ sz can be
effectuated based on the upper entries of D̂, which allow the direct computation
of the increment of the plastic parameters
dl̂S =
vl̂S
vl̂S
dz +
d3̂.
vz
v3̂
(3.17)
Finally, the computation of the stress-increment
dla M̂ Sa
dŝ = Ĉ d3̂ − Ĉ
= Ĉ −
a
Ĉ M̂ Sa
a
vla
v3̂
T d3̂ + −
a
vla
Ĉ M̂ Sa
vz
= Ĉ s3 d3̂ + Ĉ sz dz
dz
(3.18)
leads to the identification of the algorithmic tangents Ĉ s3 and Ĉ sz . It should be
noted that due to the symmetry of the problem, the equivalence Ĉ p3 = −Ĉ sz can
be shown.
In the case of an elastic update, i.e. fa < 0 ∀ a ∈ {1, 2, . . . , N }, we set
Ĉ s3 = Ĉ ,
Ĉ sz = 0̂,
Ĉ p3 = 0̂
and
pz
C pz = Cel .
(3.19)
pz
Here, Cel is a large (negative) number which penalizes a deviation of z from geq .
Alternatively, the equation z = geq (l̂), evaluated at the elastic integration point,
can be interpreted as kinematical constraint on the nodal values associated with
z of the corresponding element.
(c) Element stiffness matrices
With the algorithmic moduli at hand, the element stiffness matrices associated
with the linearized global residuals G u and G z can be computed. The linearization
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2696
S. Wulfinghoff and T. Böhlke
of G u with respect to u yields
k̂ euu =
T
s3 ep
Wpe (B̂ ep
3 ) Ĉ B̂ 3 .
(3.20)
p
Here, Wpe is the weight due to the numerical quadrature scheme associated
is the standard
with integration point p of element e. The matrix B̂ ep
3
interpolation operator, projecting the nodal displacements to the strain at the
integration points.
The linearization with respect to z is incorporated by
T sz
ep T
e T
Wpe (B̂ ep
(3.21)
k̂ euz =
3 ) Ĉ (N̂ ) = (k̂ zu ) ,
p
where N̂ ep interpolates the nodal values of z to the integration points. The
symmetry of the problem was exploited again in order to compute k̂ ezu . Finally,
the linearization of G z with respect to z leads to
T
T
Wpe (Kg (B̂ ep ) B̂ ep − C pz N̂ ep (N̂ ep ) ),
(3.22)
k̂ ezz =
p
where B̂ ep interpolates the nodal values of z to its gradient in the integration
points. In the finite-element procedure, the matrices k̂ euu , k̂ euz , k̂ ezu and k̂ ezz represent
parts of the (symmetric) element stiffness matrix.
4. Numerical examples
We demonstrate the performance of the implementation of the alternative
formulation by the simulation results of micro-mechanical tension and torsion
tests of micro-components which have been carried out with the in-house
finite-element program of the Institute of Engineering Mechanics (Continuum
Mechanics).
In both cases, we use an oligo-crystal consisting of eight grains with a
simplified grain geometry as shown in figure 1 and random crystal orientations.
The hardening parameters are assumed to be given by hab = H [q + (1 − q)dab ]
(Hutchinson 1970) with q ∈ [1; 1.4] as suggested by Kocks (1970). We use the
following material parameters
C11
C12
C12
Kg
H
q
t0c
h
168 GPa 121 GPa 75 GPa 3 × 10−2 N 200 MPa 1.1 35 MPa 0.001 MPa s
The boundary conditions for the tension tests allow for free lateral contraction.
We choose micro-free boundary conditions for z, i.e. x · n = 0 ∀ x ∈ vBX . All
simulations were performed with standard linear hexahedrons and adaptive time
steps. The total simulation time is 1 s. Figure 2 shows the results for the tensile
and torsion test simulations for varying cube sizes. The model response for torsion
is normalized in order to allow a comparison of the responses of the cubes with
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2697
Plastic strain gradient enhancement
0
z
0.1
–35
p̌ (MPa)
137
Figure 1. Oligo-crystal with simplified grain shape and edge length L = 10 mm. Undeformed state,
tensile and torsion test simulation. The tensile test simulation was performed with micro-hard
grain-boundary conditions, in case of the torsion test simulation micro-free boundary conditions
were applied. (Online version in colour.)
(b)
120
normal. torque M/L3 (MPa)
average tensile stress s = F/A (MPa)
(a)
60
0
0.005
0.010
average tensile strain e = DL/L
30
15
0
0.01
0.02
rotation angle
0.03
Figure 2. Mechanical responses in the case of tensile and torsion test simulations with micro-free
boundary conditions and for varying cube edge lengths L (blue filled squares, L = 10 mm; red
open circles, L = 15 mm; green filled circles, L = 30 mm; pink triangles, L = 100 mm; purple inverted
triangles, L = 300 mm). (Online version in colour.)
different dimensions. The results (figure 2) show that the size-dependence of the
mechanical response is stronger in the case of torsion (compared with tension).
This difference, which is also observed in experiments (Fleck et al. 1994), occurs
(here) due to increased strain gradients in the case of torsion, which are penalized
by Wg (Vz). On the integration point level, this effect is represented by the back
stress p̌ (cf. equation (2.30)). Figure 1(right) illustrates that p̌ can be considered
as a hindrance for further plastic deformation in strongly deformed regions while
it augments plastic processes in little distorted parts of the body and thereby
reduces gradients of z. For both load cases, the simulations show a strong size
effect for L 100 mm. The negligible size effect in the case of larger bodies is
due to the fact that the strain gradients decrease with increasing dimensions, i.e.
the theory at hand tends to a classical local theory for large bodies with small
gradients. An idealized (e.g. misorientation-induced) boundary slip resistance
can be introduced into the model by micro-hard boundary conditions. For the
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2698
S. Wulfinghoff and T. Böhlke
average tensile stress s = F/A (MPa)
700
350
0
0.005
average tensile strain e = DL/L
0.010
Figure 3. Tensile test simulations with micro-hard boundary conditions. For the second simulation
with L = 30 mm∗ , a larger initial time step was used, i.e. the elasto-plastic transition was not resolved
(blue filled squares, L = 10 mm; red open circles, L = 15 mm; green filled circles, L = 30 mm; yellow
filled triangles, L = 30 mm∗ ; pink triangles, L = 100 mm; purple inverted triangles, L = 300 mm).
(Online version in colour.)
(b)
130
450
120
400
110
350
100
300
280 000
0
70 000 140 000 210 000
no. degrees of freedom
micro-hard s = F/A (MPa)
micro-free s = F/A (MPa)
(a)
z
0.028
0.016
Figure 4. Dependence of the global mechanical response on the number of nodal degrees of freedom
of the model (a) and simulation of 512 grains (b) for 3̄max = 0.01. The diminution of d.f. due to
the Dirichlet boundary conditions is neglected (micro-free: blue filled squares, eight grains; open
circles, 512 grains; micro-hard: red filled circles, eight grains). (Online version in colour.)
simulation of the micro-component (cf. figures 1 and 3), we model the internal
grain boundaries (represented by G) as micro-hard, i.e. z = 0 ∀ x ∈ G. Formally,
this case is covered by the presented theory by considering the oligo-crystal as a
union of eight bodies. The implementation is effectuated by setting the prescribed
nodal values of z to zero and eliminating the associated lines and columns from
the system matrix and the residual vector. The non-local hardening modulus
Kg has not been changed to allow a comparison with the micro-free boundary
conditions (figure 2). However, for more realistic applications, Kg should be
reduced in the case of micro-hard grain boundaries, as the response shown
in figure 3 is much too hard compared with experimental results. The mesh
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Plastic strain gradient enhancement
2699
dependence of the results is depicted in figure 4. The study was performed with a
cube size of L = 30 mm. The choice of different cube sizes resulted in comparable
results. The convergence study suggests that, in the case of the oligo-crystal
with simplified grain geometry, the mesh with approx. 37 000 d.f. (20 × 20 × 20
elements), which has been used in figures 2 and 3, yields reasonable global results
close to the converged solution (especially for micro-free boundary conditions).
A more realistic (and therefore more complex) grain geometry is expected to
require an increased number of degrees of freedom per grain. Additionally, a
mesh dependence study with 512 crystals (8 × 8 × 8 crystals) was carried out for
micro-free boundary conditions and L = 30 mm (figure 4) to get a first impression
of the mechanical response of a polycrystal.
Table 1 shows the Euclidean norm of the residual for a tensile test simulation
with micro-hard boundary conditions, a cube size of L = 30 mm and a 8 × 8 × 8mesh. For time steps 3 and 4 (elasto-plastic transition with 16/13 iterations)
not all values are represented. The total CPU time for the solution was 25.89 s
(table 1) on a Pentium Dual Core PC with 3.0 GHz and 6 GB RAM (which limits
the maximum number of d.f., in the case of a direct solver, to approx. 300 000).
5. Summary
A geometrically linear single crystal model extended by an energetic hardening
term based on the gradient of the equivalent plastic strain geq has
been introduced. The direct finite-element implementation of the theory is
computationally expensive due to the need of piecewise differentiable shape
functions for the plastic slip parameters la . The introduction of an alternative
(but physically equivalent) formulation based on a Lagrange multiplier p̌ and a
micromorphic-like variable z drastically reduces the number of partial differential
equations and nodal degrees of freedom of the finite-element implementation. The
local algorithm and its linearization, represented by the extended algorithmic
tangent moduli, have been discussed in detail. Numerical simulations of
three-dimensional tension and torsion tests of oligo-crystals with a simplified
geometry illustrate the performance of the implementation. The overall model
response qualitatively mimics the size-dependent material behaviour observed in
experiments (Fleck et al. 1994; Stölken & Evans 1998; Xiang & Vlassak 2006;
Gruber et al. 2008). The experimentally found size-dependence of the yield stress
is not captured, a well-known consequence of the simple quadratic ansatz for the
strain gradient-dependent energy contribution, which can partially be remedied
by more elaborate approaches (Ohno & Okumura 2007).
Overall, the proposed geometrically linear model allows computationally
relatively cheap three-dimensional size-dependent simulations of oligocrystal models.
The theory is not restricted to the special scalar variable geq . Instead, the
presented concepts can be generalized to more complex (e.g. tensorial) quantities
like the full plastic part of the displacement gradient and thereby cover theories
taking, e.g. Nye’s tensor as argument of the free energy.
The boundary conditions for the field z remain an issue for further research.
The physical relevance of micro-free and micro-hard boundary conditions at
the grain boundaries is questionable. Both conditions seem to be applicable in
Proc. R. Soc. A (2012)
Proc. R. Soc. A (2012)
7.26 × 104
—
—
1.78 × 102
3.63 × 10−8
0.04 s
1.58 × 105
—
—
5.79
3.33 × 10−7
0.08 s
3.17 × 105
2.38 × 103
3.96 × 102
4.81 × 101
4.29
8.76 × 10−7
0.16 s
6.22 × 105
3.33 × 103
4.80 × 102
7.32 × 101
9.47
3.27 × 10−1
2.02 × 10−6
0.32 s
without resolution of the elasto-plastic transition (compare figure 3).
3.63 × 104
3.01 × 10−11
1.51 × 104
1.31 × 103
4.71 × 10−12
a CPU-time
0.02 s
6.17 × 105
1.53 × 103
1.73 × 102
1.40 × 101
2.99 × 10−1
2.29 × 10−6
0.333 s
6.55 × 104
2.97 × 101
2.06
2.32 × 10−1
2.62 × 10−7
0.037 s
8×8×8
8 × 8 × 8a
12 × 12 × 12
12 × 12 × 12a
20 × 20 × 20
20 × 20 × 20a
32 × 32 × 32a
elements
25.89
10.51
97.51
26.73
720.87
191.06
1658.22
CPU (s)
2700
Dt : 0.01 s
Table 1. Example for the Euclidean norm of the residual (left) and CPU-time (right) for L = 30 mm.
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S. Wulfinghoff and T. Böhlke
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Plastic strain gradient enhancement
2701
order to reproduce the global size-dependent behaviour. However, their validity
is restricted to certain ranges of the deformation and size. Therefore, more
elaborated grain boundary models should be examined.
The authors acknowledge the support rendered by the German Research Foundation (DFG) under
grant BO 1466/5-1. The funded project ‘Dislocation-based gradient plasticity theory’ is part of the
DFG Research Group 1650 ‘Dislocation-based plasticity’.
References
Acharya, A. & Bassani, J. L. 2000 Lattice incompatibility and a gradient theory of crystal plasticity.
J. Mech. Phys. Solids 48, 1565–1595. (doi:10.1016/S0022-5096(99)00075-7)
Anand, L., Aslan, O. & Chester, S. A. 2011 A large-deformation gradient theory for elastic–plastic
materials: strain softening and regularization of shear bands. Int. J. Plast. 30–31, 116–143.
(doi:10.1016/j.ijplas.2011.10.002)
Aslan, O., Cordero, N. M., Gaubert, A. & Forest, S. 2011 Micromorphic approach to single crystal
plasticity and damage. Int. J. Eng. Sci. 49, 1311–1325. (doi:10.1016/j.ijengsci.2011.03.008)
Bargmann, S., Ekh, M., Runesson, K. & Svendsen, B. 2010 Modeling of polycrystals with
gradient crystal plasticity: a comparison of strategies. Philos. Mag. 90, 1263–1288. (doi:10.1080/
14786430903334332)
Bayley, C., Brekelmans, W. & Geers, M. 2006 A comparison of dislocation induced back stress
formulations in strain gradient crystal plasticity. Int. J. Solids Struct. 43, 7268–7286.
(doi:10.1016/j.ijsolstr.2006.05.011)
Becker, M. 2006 Incompatibility and instability based size effects in crystals and composites at
finite elastoplastic strains. PhD thesis, Universität Stuttgart, Stuttgart, Germany.
Berdichevsky, V. 2006 On thermodynamics of crystal plasticity. Scr. Mater. 54, 711–716.
(doi:10.1016/j.scriptamat.2005.10.027)
Bilby, B. A., Bullough, R. & Smith, E. 1955 Continuous distributions of dislocations: a new
application of the methods of non-Riemannian geometry. Proc. R. Soc. Lond. A 231, 263–273.
(doi:10.1098/rspa.1955.0171)
Cermelli, P. & Gurtin, M. E. 2002 Geometrically necessary dislocations in viscoplastic single
crystals and bicrystals undergoing small deformations. Int. J. Solids Struct. 39, 6281–6309.
(doi:10.1016/S0020-7683(02)00491-2)
Cordero, N. M., Gaubert, A., Forest, S., Busso, E., Gallerneau, F. & Kruch, S. 2010 Size effects
in generalised continuum crystal plasticity for two-phase laminates. J. Mech. Phys. Solids 58,
1963–1994. (doi:10.1016/j.jmps.2010.06.012)
Cordero, N. M., Forest, S., Busso, E., Berbenni, S. & Cherkaoui, M. 2012 Grain size effects on
plastic strain and dislocation density tensor fields in metal polycrystals. Comput. Mater. Sci.
52, 7–13. (doi:10.1016/j.commatsci.2011.02.043)
de Borst, R. & Mühlhaus, H.-B. 1992 Gradient-dependent plasticity: formulation and algorithmic
aspects. Int. J. Numer. Meth. Eng. 35, 521–539. (doi:10.1002/nme.1620350307)
Dimiduk, D. M., Uchic, M. D., Rao, S. I., Woodward, C. & Parthasarathy, T. A. 2007 Overview
of experiments on microcrystal plasticity in FCC-derivative materials: selected challenges for
modelling and simulation of plasticity. Model. Simul. Mater. Sci. Eng. 15, 135–146. (doi:10.1088/
0965-0393/15/2/009)
Ekh, M., Grymer, M., Runesson, K. & Svedberg, T. 2007 Gradient crystal plasticity as part of the
computational modelling of polycrystals. Int. J. Numer. Meth. Eng. 72, 197–220. (doi:10.1002/
nme.2015)
Ertürk, I., van Dommelen, J. & Geers, M. 2009 Energetic dislocation interactions and
thermodynamical aspects of strain gradient crystal plasticity theories. J. Mech. Phys. Solids
57, 1801–1814. (doi:10.1016/j.jmps.2009.08.003)
Estrin, Y. 1996 Dislocation-density related constitutive modelling. In Unified constitutive laws of
plastic deformation, pp. 69–106. San Diego, CA: Academic Press.
Proc. R. Soc. A (2012)
Downloaded from http://rspa.royalsocietypublishing.org/ on July 12, 2017
2702
S. Wulfinghoff and T. Böhlke
Evers, L. P., Brekelmans, W. A. M. & Geers, M. G. D. 2004 Non-local crystal plasticity model
with intrinsic SSD and GND effects. J. Mech. Phys. Solids 52, 2379–2401. (doi:10.1016/j.jmps.
2004.03.007)
Fleck, N. A. & Hutchinson, J. W. 1993 A phenomenological theory for strain gradient effects in
plasticity. J. Mech. Phys. Solids 41, 1825–1857. (doi:10.1016/0022-5096(93)90072-N)
Fleck, N. A. & Willis, J. R. 2009 A mathematical basis for strain-gradient plasticity theory—part
I: scalar plastic multiplier. J. Mech. Phys. Solids 57, 161–177. (doi:10.1016/j.jmps.2008.09.010)
Fleck, N. A., Muller, G. M., Ashby, M. F. & Hutchinson, J. W. 1994 Strain gradient plasticity:
Theory and experiment. Acta Metall. Mater. 42, 475–487. (doi:10.1016/0956-7151(94)90502-9)
Forest, S. 2009 Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng.
Mech. 135, 117–131. (doi:10.1061/(ASCE)0733-9399(2009)135:3(117))
Franciosi, P. & Zaoui, A. 1982 Multislip in f.c.c. crystals; a theoretical approach compared with
experimental data. Acta Metall. 30, 1627–1637. (doi:10.1016/0001-6160(82)90184-5)
Geers, M. G. D., Brekelmans, W. A. M. & Bayley, C. J. 2006 Second-order crystal plasticity: internal stress effects and cyclic loading. Model. Simul. Mater. Sci. Eng. 15, 133–145. (doi:10.1088/
0965-0393/15/1/S12)
Gillis, P. P. & Gilman, J. J. 1965 Dynamical dislocation theory of crystal plasticity II. Easy glide
and strain hardening. J. Appl. Phys. 36, 3380–3386. (doi:10.1063/1.1702999)
Gruber, P. A., Bohm, J., Onuseit, F., Wanner, A., Spolenak, R. & Arzt, E. 2008 Size effects on
yield strength and strain hardening for ultra-thin Cu films with and without passivation: a study
by synchrotron and bulge test techniques. Acta Mater. 56, 2318–2335. (doi:10.1016/j.actamat.
2008.01.027)
Gurtin, M. E. 2002 A gradient theory of single-crystal viscoplasticity that accounts for
geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32. (doi:10.1016/S0022-5096
(01)00104-1)
Gurtin, M. E. 2006 The burgers vector and the flow of screw and edge dislocations in finitedeformation single-crystal plasticity. J. Mech. Phys. Solids 54, 1882–1898. (doi:10.1016/j.jmps.
2006.03.003)
Gurtin, M. E. & Anand, L. 2005 A theory of strain-gradient plasticity for isotropic,
plastically irrotational materials. I. Small deformations. J. Mech. Phys. Solids 53, 1624–1649.
(doi:10.1016/j.jmps.2004.12.008)
Gurtin, M. E., Anand, L. & Lele, S. P. 2007 Gradient single-crystal plasticity with free
energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878. (doi:10.1016/
j.jmps.2007.02.006)
Hall, E. O. 1951 The deformation and aging of mild steel. III. Discussion and results. Proc. Phys.
Soc. Lond. 64, 747–753. (doi:10.1088/0370-1301/64/9/303)
Han, C.-S., Ma, A., Roters, F. & Raabe, D. 2007 A finite element approach with patch projection
for strain gradient plasticity formulations. Int. J. Plast. 23, 690–710. (doi:10.1016/j.ijplas.
2006.08.003)
Hill, R. 1966 Generalized constitutive relations for incremental deformation of metal crystals by
multislip. J. Mech. Phys. Solids 14, 95–102. (doi:10.1016/0022-5096(66)90040-8)
Hochrainer, T. 2006 Evolving systems of curved dislocations: mathematical foundations of a
statistical theory. PhD thesis, Universität Karlsruhe (TH), Shaker Verlag, Aachen.
Hutchinson, J. W. 1970 Elastic–plastic behaviour of polycrystalline metals and composites. Proc.
R. Soc. Lond. A 319, 247–272. (doi:10.1098/rspa.1970.0177)
Kocks, U. F. 1970 The relation between polycrystal deformation and single-crystal deformation.
Metall. Trans. 1, 1121–1143.
Kocks, U. F. 1987 Constitutive behavior based on crystal plasticity. In Unified constitutive equations
for creep and plasticity (ed. A. K. Miller), pp. 1–88. London, UK: Elsevier Applied Science.
Koiter, W. T. 1953 Stress–strain relations, uniqueness and variational theorems for elasto–plastic
material with a singular yield surface. Q. Appl. Math. 11, 350–354.
Kröner, E. 1958 Kontinuumstheorie der Versetzungen und Eigenspannungen. Berlin, Germany:
Springer.
Kuroda, M. & Tvergaard, V. 2008 A finite deformation theory of higher-order gradient crystal
plasticity. J. Mech. Phys. Solids 56, 2573–2584. (doi:10.1016/j.jmps.2008.03.010)
Proc. R. Soc. A (2012)
Downloaded from http://rspa.royalsocietypublishing.org/ on July 12, 2017
Plastic strain gradient enhancement
2703
Lee, E. H. 1969 Elastic–plastic deformation at finite strains. J. Appl. Mech. 36, 1–6. (doi:10.1115/
1.3564580)
Liebe, T. 2004 Theory and numerics of higher gradient inelastic material behavior. PhD thesis,
Universität Kaiserslautern, Kaiserslautern, Germany.
Liebe, T. & Steinmann, P. 2001 Theory and numerics of a thermodynamically consistent
framework for geometrically linear gradient plasticity. Int. J. Numer. Meth. Eng. 51, 1437–1467.
(doi:10.1002/nme.195)
Mecking, H. & Kocks, U. F. 1981 Kinetics of flow and strain hardening. Acta Metall. 29, 1865–1875.
(doi:10.1016/0001-6160(81)90112-7)
Menzel, A. & Steinmann, P. 2000 On the continuum formulation of higher gradient plasticity
for single and polycrystals. J. Mech. Phys. Solids 48, 1777–1796. (doi:10.1016/S0022-5096(99)
00024-1)
Miehe, C. 2011 A multi-field incremental variational framework for gradient-extended standard
dissipative solids. J. Mech. Phys. Solids 59, 898–923. (doi:10.1016/j.jmps.2010.11.001)
Miehe, C. & Schröder, J. 2001 A comparative study of stress update algorithms for rate-independent
and rate-dependent crystal plasticity. Int. J. Numer. Meth. Eng. 50, 273–298. (doi:10.1002/10970207(20010120)50:2<273::AID-NME17>3.0.CO;2-Q)
Nye, J. F. 1953 Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162.
(doi:10.1016/0001-6160(53)90054-6)
Ohno, N. & Okumura, D. 2007 Higher-order stress and grain size effects due to self energy of
geometrically necessary dislocations. J. Mech. Phys. Solids 55, 1879–1898. (doi:10.1016/j.jmps.
2007.02.007)
Orowan, E. 1934 Zur Kristallplastizität III. Z. Phys. 89, 634–659. (doi:10.1007/BF01341480)
Ortiz, M. & Stainier, L. 1999 The variational formulation of viscoplastic constitutive updates.
Comput. Methods Appl. Mech. Eng. 171, 419–444. (doi:10.1016/S0045-7825(98)00219-9)
Perzyna, P. 1971 Thermodynamic theory of viscoplasticity. Adv. Appl. Mech. 11, 313–354.
(doi:10.1016/S0065-2156(08)70345-4)
Petch, N. J. 1953 The cleavage strength of polycrystals. J. Iron Steel Inst. 174, 25–28.
Rice, J. R. 1971 Inelastic constitutive relations for solids: an internal-variable theory and its
application to metal plasticity. J. Mech. Phys. Solids 19, 433–455. (doi:10.1016/0022-5096(71)
90010-X)
Sandfeld, S., Hochrainer, T., Zaiser, M. & Gumbsch, P. 2010 Numerical implementation of a 3D
continuum theory of dislocation dynamics and application to micro-bending. Philos. Mag. 90,
3697–3728. (doi:10.1080/14786430903236073)
Schmid, E. & Boas, W. 1935 Kristallplastizität. Berlin, Germany: Springer.
Shu, J. Y., King, W. E. & Fleck, N. A. 1999 Finite elements for materials with strain gradient
effects. Int. J. Numer. Meth. Eng. 44, 373–391. (doi:10.1002/(SICI)1097-0207(19990130)44:3<
373::AID-NME508>3.0.CO;2-7)
Simo, J. C. & Hughes, T. 1998 Computational inelasticity. New York, NY: Springer.
Simo, J., Taylor, R. & Pister, K. 1985 Variational and projection methods for the volume
constraint in finite deformation elasto-plasticity. Comput. Meth. Appl. Mech. Eng. 51, 177–208.
(doi:10.1016/0045-7825(85)90033-7)
Steinmann, P. 1996 Views on multiplicative elastoplasticity and the continuum theory of
dislocations. Int. J. Eng. Sci. 34, 1717–1735. (doi:10.1016/S0020-7225(96)00062-6)
Stölken, J. S. & Evans, A. G. 1998 A microbend test method for measuring the plasticity length
scale. Acta Mater. 46, 5109–5115. (doi:10.1016/S1359-6454(98)00153-0)
Taylor, G. I. 1938 Plastic strain in metals. J. Inst. Metals 62, 307–324.
Wieners, C. & Wohlmuth, B. 2011 A primal-dual finite element approximation for a nonlocal model
in plasticity. SIAM J. Numer. Anal. 49, 692–710. (doi:10.1137/100789397)
Xiang, Y. & Vlassak, J. 2006 Bauschinger and size effects in thin-film plasticity. Acta Mater. 54,
5449–5460. (doi:10.1016/j.actamat.2006.06.059)
Yalcinkaya, T., Brekelmans, W. A. M. & Geers, M. G. D. 2008 BCC single crystal plasticity
modeling and its experimental identification. Model. Simul. Mater. Sci. Eng. 16, 085007 (16 pp).
(doi:10.1088/0965-0393/16/8/085007)
Proc. R. Soc. A (2012)