Equivalent plastic strain gradient crystal plasticity – Enhanced power law subroutine

Abstract

A gradient crystal plasticity theory is presented including a defect energy based on the gradient of an equivalent plastic strain measure. Preserving the single crystal slip kinematics, the gradient hardening contribution models dislocation long range interactions adding a back-stress term to the flow rule, similar to other gradient crystal plasticity theories. Owing to a reduced number of four nodal degrees of freedom the finite element implementation can handle systems consisting of an increased number of grains (compared to other theories) without elaborate or costly computer systems. Emphasis is put on the enhancement of the power law material subroutine. The associated implicit Euler scheme is optimized based on an improved starting value for the Newton scheme. Three-dimensional simulations illustrate that the proposed algorithm facilitates significantly larger time steps compared to the standard Newton scheme. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Full text

GAMM-Mitt. 36, No. 2, 134 –148 (2013) / DOI 10.1002/gamm.201310008
Equivalent plastic strain gradient crystal plasticity –
Enhanced power law subroutine
Stephan Wulfinghoff∗and Thomas B¨
ohlke∗∗
Institute of Engineering Mechanics, Chair for Continuum Mechanics,
Karlsruhe Institute of Technology (KIT), Kaiserstr. 10, 76131 Karlsruhe, Germany
Received 28 September 2012, revised 13 December 2012, accepted 23 January 2013
Published online 07 October 2013
Key words Equivalent plastic strain, Gradient plasticity, Power law
A gradient crystal plasticity theory is presented including a defect energy based on the gradi-
ent of an equivalent plastic strain measure. Preserving the single crystal slip kinematics, the
gradient hardening contribution models dislocation long range interactions adding a back-
stress term to the flow rule, similar to other gradient crystal plasticity theories. Owing to a
reduced number of four nodal degrees of freedom the finite element implementation can han-
dle systems consisting of an increased number of grains (compared to other theories) without
elaborate or costly computer systems. Emphasis is put on the enhancement of the power law
material subroutine. The associated implicit Euler scheme is optimized based on an improved
starting value for the Newton scheme. Three-dimensional simulations illustrate that the pro-
posed algorithm facilitates significantly larger time steps compared to the standard Newton
scheme.
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1 Introduction
Classical continuum mechanical plasticity theories (e.g. von Mises [1], Hill [2]) without in-
ternal length scale are of strong practical interest. Although the theory of large plastic defor-
mations (e.g. Lubliner [3]) and the associated numerics (Weber and Anand [4], Simo [5]) are
still subject of current research (Caminero et al. [6]), their predictability has reached a quite
reliable state for many important macroscopic engineering applications like metal forming
processes or crash simulations. Contrary, these theories fail on the meso- and microscale,
where the material properties differ from the macroscopically observed behavior. For exam-
ple, the strength of micro specimens increases compared to their macroscopic counterparts.
Examples of size-dependent behavior are given, e.g., by thin films loaded by bending (St¨olken
and Evans [7]), compressed pillars (Greer et al. [8]) or thin wires (Fleck et al. [9], Yang et
al. [10]). Moreover, fine grained metals have superior mechanical properties concerning their
strength (Hall [11], Petch [12]). These size effects are neither fully understood nor can they be
generally predicted quantitatively to a satisfying extent by existing theories. Therefore, cur-
rent research focuses on new theories and the extension of existing models in order to improve
the understanding of existing materials on the one hand and to facilitate the dimensioning of
∗Corresponding author E-mail: stephan.wulfinghoff@kit.edu, Phone: +49 721 60848133
∗∗ E-mail: thomas.boehlke@kit.edu
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GAMM-Mitt. 36, No. 2 (2013) 135
micro components and micro systems as well as the design of new synthetic materials on the
other hand. One of the most popular ideas to physically explain the experimentally observed
size effects is to associate the higher strength with (possibly higher order) plastic strain gra-
dients (Aifantis [13], M¨uhlhaus and Aifantis [14]) and the closely related increased amount
of geometrically necessary dislocation (GND) density (Fleck and Hutchinson [15], Fleck et
al. [9]). This quantity, often measured by Nye’s dislocation density tensor (Nye [16]), is
closely related to the plastic strain gradients. Those are the larger the smaller the size of the
body under consideration and thereby introduce a physically motivated internal length scale
into the theory. However, the uncertainty of the relation between GNDs and the material
behavior has led to an extended number of strain gradient theories. For example, various
thermodynamical theories are based on the postulate of an increase in the free energy (e.g.
Steinmann [17], Gurtin et al. [18], Ohno and Okumura [19], Forest [20], Wulfinghoff and
B¨ohlke [21]) or dissipation (Fleck and Willis [22]) associated to plastic strain gradients or
closely related quantities. A generalized variational framework for gradient-extended stan-
dard dissipative solids is proposed by Miehe [23]. In the aforementioned theories, higher
order boundary conditions associated to the plastic variables naturally arise and may serve as
a tool to model interfaces like grain boundaries (Fredriksson and Gudmundson [24], Aifantis
and Willis [25], Fleck and Willis [22]). The dislocation theory of Evers et al. [26] and its
generalization by Bayley et al. [27] have been shown to be consistent with thermodynamical
approaches by Ert¨urk et al. [28] and Bargmann et al. [29]. A more direct approach without
higher order boundary conditions can be achieved if the yield stress is assumed to depend
directly on strain gradient measures (Becker [30], Han et al. [31]).
Additionally, the high interest in dislocation based hardening led to an increased research
activity in different dislocation continuum theories (Hochrainer [32], Sedl´aˇcek [33]) and as-
sociated numerical implementations (Sandfeld [34], Wulfinghoff and B¨ohlke [35]).
The implementation and computation of numerical solutions of gradient plasticity theories
is a nontrivial task. The numerical treatment of highly nonlinear constitutive laws and the
augmentation of nodal degrees of freedom are only some of the challenges making gradient
plasticity solutions computationally expensive and complex by finite elements or closely re-
lated discretization schemes like the discontinuous Galerkin scheme (Djoko et al. [36, 37]).
Another non-trivial numerical challenge is the determination of global plastically active zones
as treated in Liebe and Steinmann [38] and, for single crystal models, in Reddy et al. [39]. The
latter constitutes one of the seldom publications concerning three-dimensional single crystal
gradient plasticity simulations and solves the active set problem by tracing it back to a local
problem based on an element-wise return mapping algorithm. If the full dislocation density
tensor is to be computed the increase in degrees of freedom is particularly pronounced. The
effort is even larger if the gradients of all plastic slips are to be computed in the case of single
crystal theories. For example, compared to a classical continuum mechanical description of
solids, the number of nodal degrees of freedom can be increased from 3 to, e. g., 15 in the case
of a face-centered cubic (FCC) crystal if a gradient model is applied (in three dimensions).
Moreover, the mesh resolution must, in general, be higher due to thin boundary layers (Ekh
et al. [40]). Consequently, three-dimensional numerical solutions are rare. However, despite
the substantially increased computational effort the predictions of gradient extended plasticity
models are rather qualitatively than quantitatively correct. Hence, the massively increased
computational cost of those theories contrasts with their accuracy.
In order to reduce the degrees of freedom in crystal simulations, Wulfinghoff and B¨ohlke [21]
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136 S. Wulfinghoff and T. B¨ohlke: Gradient crystal plasticity subroutine
proposed an equivalent plastic strain gradient extended crystal plasticity theory which pre-
serves the basic crystal slip kinematics. The extension is based on the gradient of an equiva-
lent plastic strain measure (which is well known from classical hardening approaches) instead
of exact GND densities. The physically simplified approach leads to a significantly reduced
number of four nodal degrees of freedom. Hence, it facilitates three dimensional simulations
of several grains, even on standard stand-alone computers.
This paper deals with a penalty approximation of the theory developed in Wulfinghoff and
B¨ohlke [21] and focuses on the implementation details of the associated power-law material
subroutine. Preserving the single crystal slip kinematics, the dislocation long range interac-
tions are modeled by a hardening contribution based on the gradient of the equivalent plastic
strain which adds a back stress term to the flow rule, similar to other gradient crystal plas-
ticity theories. It turns out that the gradient enhancement supports the improvement of the
convergence properties of the power law subroutine based on a reasonable guess of the start-
ing solution for the local Newton iteration scheme. As a result, the local material subroutine
always converged in the illustrated simulations, even in the case of strongly nonlinear mate-
rial properties, large time steps and two million degrees of freedom. An exemplary perfor-
mance comparison of the material subroutine with and without improved initial solution guess
demonstrates the computational benefit of the approach.
Notation. A direct tensor notation is preferred throughout the text. Vectors and 2nd-order
tensors are denoted by bold letters, e.g. aor A. The symmetric part of a 2nd-order tensor A
is designated by sym(A). A linear mapping of 2nd-order tensors by a 4th-order tensor is
written as A=C[B]. The scalar product and the dyadic product are denoted, e.g. by A·B
and A⊗B, respectively. The composition of two 2nd-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. ˆε. The transpose and the inverse of a matrix are indicated by ˆ
DTand
ˆ
D−1, respectively. The Macaulay brackets are defined by x=(x+|x|)/2.
2 Kinematical Assumptions
Let the gradient of the displacement field u(x,t)of an aggregation of several grains
B=
i
Bi(1)
be small, i.e. ∇u1. Accordingly, the additive decomposition
ε=sym(∇u)=εe+εp(2)
of the strain tensor into elastic and plastic parts can be assumed, where the plastic part reads
εp=
α
λαMS
α.(3)
The tensors MS
α=sym(dα⊗nα)represent the crystal geometry based on the slip direc-
tions dαand the slip plane normals nα. Two slip parameters λαwith ˙
λα≥0are introduced
per slip system to account for positive and negative shear increments. Consequently, the slip
system index αruns, for example, from 1 to N=24for an FCC-crystal with 12 slip systems.
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GAMM-Mitt. 36, No. 2 (2013) 137
One advantage of this approach is the simple form of the equivalent plastic strain based on the
slip parameters ˆ
λ=(λ1,λ
2, ..., λN)
γeq (ˆ
λ)=
α
λα,(4)
which is a phenomenological measure of the local plastic deformation.
3 Energetic Assumptions
The stored energy per unit volume is assumed to be given by
W=We(ε,εp)+Wh(ζ)+Wg(∇ζ)+1
2Hχ(ζ−γeq (ˆ
λ))2,(5)
with elastic energy We=εe·C[εe]/2=(ε−εp)·C[ε−εp]/2.
The last term of Eq. (5) couples the (a priori independent) variables ζand γeq(ˆ
λ).Thevari-
able ζmight be termed micromorphic counterpart of γeq (ˆ
λ). If the penalty parameter Hχ
is chosen sufficiently large the difference of the two variables is negligibly small. The ap-
proach (5) can be considered as penalty approximation of the theory of Wulfinghoff and
B¨ohlke [21], who weakly enforce the equality of ζand γeq(ˆ
λ)by a Lagrange multiplier ˇp
and an associated energy contribution ˇp(ζ−γeq (ˆ
λ)) instead of the last term of the stored
energy (5). This quite strong coupling scheme is softened here through the finite penalty
parameter Hχ(compare the micromorphic theory of Forest [20]). The introduction of two
variables γeq (ˆ
λ)and ζwith the same physical meaning is numerically motivated (similar to
established theories, e.g. Simo, Taylor and Pister [41]). Here, it facilitates the reduction of
additional nodal degrees of freedom (three for uand one for ζ).
The isotropic hardening energy Wh(ζ)roughly models the hardening of the crystal due to dis-
location short range interactions and slightly differs from that of Wulfinghoff and B¨ohlke [21].
The choice of its argument ζ(and not γeq(ˆ
λ)) simplifies the material subroutine algorithm (as
will be shown). Apart from these differences the approach (5) and that of Wulfinghoff and
B¨ohlke [21] coincide.
The gradient contribution Wg(∇ζ)phenomenologically introduces an internal length scale
into the theory which thereby becomes size-dependent, i.e. the inability of classical theories
to account for the above mentioned experimentally observed size effects is partially remedied.
Physically, the gradient contribution is supposed to model elastic long range dislocation in-
teractions in a phenomenological manner yielding an overall model behavior close to other
gradient plasticity theories (as shown in Wulfinghoff and B¨ohlke [21]). The idea is to es-
pecially capture the interaction of the material with surfaces acting as dislocation obstacles,
since these can have a highly significant influence on the material strength. Examples are
given by grain boundaries or the interface between incongruent particles and the matrix. If
the material is deformed, the plastic part of the deformation is caused by dislocations travel-
ing through the lattice. A measure of the total plastic deformation at a material point is the
equivalent plastic strain γeq ≈ζsince it is directly proportional to the number of dislocations
which passed through that point. The dislocation motion is hindered in front of the aforemen-
tioned interfaces where dislocations pile up and stop shearing the lattice. It is important to
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138 S. Wulfinghoff and T. B¨ohlke: Gradient crystal plasticity subroutine
note that the amount of plastic deformation inside of the bulk exceeds that at the interface.
For example, if the interface is impenetrable for dislocations, the region close to the interface
is deformed almost purely elastically (i. e., the region close to the interface stores a consider-
able amount of elastic strain energy). Hence, the pile-up region in between the bulk and the
interface is characterized by an increased gradient of the plastic deformation, here described
by ∇γeq ≈∇ζ(it can usually be expected that the gradient approximately points from the in-
terface to the bulk). However, the energetic representation of the strain gradient influence by
one scalar variable, instead of a dislocation density tensor is a (mainly numerically motivated)
simplification. Consequently, not all aspects of the micro structures which are predicted with
a theory based on the full tensor (cf. e. g. [42]) can be expected to be captured by the ap-
proach at hand due to its simplifying nature. Therefore, it is emphasized that the approach
at hand focuses on the interaction of the bulk material with interfaces disturbing dislocation
motion. These interactions can lead to a significant increase of elastic strain energy and defect
energy (as well as the associated back stresses) close to the interface. This energy has to be
supplied by the macroscopic loading. Consequently, this effect is macroscopically observed as
increased strength. Moreover, this hardening mechanism implies a size effect since it clearly
depends on the ratio of the interface and the bulk.
4 Variational and Strong Form of the Field Equations
In the following, a deformation process of the crystal aggregation is considered at an arbitrary
but fixed time denoted by t. The dissipation D(per unit volume) is assumed to solely depend
on the local plastic rates ˙
λαof the slip parameters. If δurepresents an arbitrary infinitesimal
virtual displacement field with fixed variables ζand ˆ
λ, the principle of virtual displacements
can be postulated
δ
ΔB
Wdv=
ΔB
∂εW·δεdv!
=
∂ΔB
t·δuda(6)
for an arbitrary part ΔB⊆Bof the body, where tis the traction vector and δε=sym(δu)
is a virtual strain increment. Body forces and inertia effects are neglected. Application of
Gauss’ theorem and the chain rule yields

ΔB
div (∂εW)·δudv−
∂ΔB
((∂εW)n−t)·δuda=0 (7)
with the outer normal n. Consequently, the equalities
div (σ)=0,t=σn ∀x∈B,(8)
with σ=∂εW, hold for each point of the body (since ΔBwas arbitrarily chosen). Re-
lation (8)2illustrates that ∂εWrepresents the Cauchy stress. Equations (8)1and (8)2are
supplemented by Dirichlet and Neumann boundary conditions
u=¯
u∀x∈∂Bu,σn =¯
t∀x∈∂Bt=∂B\∂Bu,(9)
where ∂Buand ∂Btdenote the Dirichlet and Neumann parts of the boundary.
In this paper, the theory will be restricted to micro-hard grain boundaries, i.e. ζ=0on Γ,
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GAMM-Mitt. 36, No. 2 (2013) 139
where Γrepresents the union of all grain boundaries1. Therefore, ΔB⊂B
iis chosen to be
a subset of an arbitrary grain (with index i), i.e. no grain boundary is included in ΔBin the
following. A generalization of the principle of virtual displacements (6) is obtained if uas
well as ˆ
λare kept fixed and δζ is arbitrary

ΔB∂ζWhδζ +∂∇ζWg·δ∇ζ+Hχ(ζ−γeq(ˆ
λ))δζdv=
∂ΔB
Ξδζ da, (10)
where Ξis a micro traction, the necessity of which will be shown in the following. A formally
identical procedure as before yields

ΔB∂ζWh−div (∂∇ζWg)+Hχ(ζ−γeq(ˆ
λ))δζ dv(11)
+
∂ΔB
(∂∇ζWg·n−Ξ)δζ da=0.(12)
Introducing the stresses
ξ=∂∇ζWg,β=∂ζWh,p
χ=−Hχ(ζ−γeq (ˆ
λ)),(13)
the resulting identities
pχ=β−div (ξ),Ξ=ξ·n(14)
hold inside of the grains, but in general not on the grain boundaries. Eq. (14)2illustrates
the necessity of the introduction of the micro traction Ξ. Considering exclusively micro-hard
grain boundaries, the associated boundary conditions read
ζ=0 ∀x∈Γ.(15)
Accordingly, since ζ≈γeq (ˆγ)=αλαand λα≥0, Eq. (15) implies standard micro-hard
conditions λα≈0on Γ(for sufficiently large penalty parameter Hχ).
5FlowRule
A thermodynamically consistent flow rule is obtained from the requirement of positive dissi-
pation
Dtot(ΔB)= 
∂ΔBt·˙
u+Ξ ˙
ζda−
ΔB
˙
Wdv!
≥0(16)
with
˙
W=∂εWe·˙
ε+∂ζWh˙
ζ+∂∇ζWg·˙
∇ζ+Hχ(ζ−γeq (ˆ
λ)) ˙
ζ
+∂εpWe·˙
εp−Hχ(ζ−γeq (ˆ
λ)) ˙γeq .(17)
1More complex boundary conditions are not within the scope of the contribution at hand but will be investigated
in the future.
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140 S. Wulfinghoff and T. B¨ohlke: Gradient crystal plasticity subroutine
Equation (16) is formally similar (except for the last two terms in (17)) to the superposition
of Eqns. (6) and (10), if the arbitrary virtual fields are chosen to be equal to the corresponding
rates (except for the unit). Therefore, the associated terms can be dropped yielding the reduced
inequality
Dtot(ΔB)=−
ΔB
(∂εpWe·˙
εp+pχ˙γeq )dv!
≥0.(18)
Considering the arbitrariness of ΔBand the identity ∂εpWe=−∂εWe=−σ, the local dis-
sipation (per unit volume) can be deduced
D=σ·˙
εp−pχ˙γeq
(3),(4)
=
ασ·MS
α−pχ˙
λα=
α
(τα−pχ)˙
λα
!
≥0,(19)
with the resolved shear stresses τα=σ·MS
α. The flow rule must satisfy inequality (19). One
example of such kind of thermodynamically consistent constitutive approach is the following
generalization of the widely used power law
˙
λα=˙γ0τα−pχ−τC
0
τDp
(14)
=˙γ0(τα+ div (ξ)) −(τC
0+β)
τDp
(20)
with the reference shear rate ˙γ0, the initial yield stress τC
0, the drag stress τDand the strain
rate sensitivity p. This example illustrates the influence of the different energetic stresses on
the plastic deformation process: the elastic resolved shear stresses ταare the natural driving
forces of the shear rates as a consequence of the externally applied macroscopic loads. The
short range dislocation interactions are modeled by Wh(ζ)and increase the local yield stress
through β, i.e. they imply isotropic hardening. The energy contribution Wg(∇ζ)is supposed
to model the long range interactions and enters the flow rule through the term div (ξ)which
can formally be interpreted as (the negative of) a back stress. This stress is similar to the back
stresses of other single crystal gradient theories (e.g. Gurtin et al. [18]).
6 Numerical Implementation
6.1 Finite Element Equations
The global residuals of the finite element implementation are deduced from equations (6)
and (10) evaluated at time tn+1 (the time is discretized into a finite number of steps)
Gu=
B
σ·δεdv−
∂Bt
¯
t·δuda!
=0,(21)
Gζ=
B
((β−pχ)δζ +ξ(∇ζ)·δ∇ζ)dv−
ΓΞ
¯
Ξδζ da!
=0,(22)
where the index n+1of all stresses is dropped for simplicity. The virtual fields are chosen to
be zero on the Dirichlet boundary. The Neumann boundary term in Eq. (22) vanishes since,
in this paper, only micro-hard boundary conditions are considered, i.e. the Neumann part ΓΞ
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GAMM-Mitt. 36, No. 2 (2013) 141
of the grain boundaries represents an empty set. The term is appended for completeness.
Equations (21) and (22) are linearized in order to apply the finite element scheme. There-
fore, the standard iteration scheme (based on ideas originally proposed e.g. in Wilkins [43] or
Hughes and Taylor [44]) is applied, i.e. the nodal variables are given by uand ζ. Assuming
that an approximate solution of these fields is given, the associated stresses are subsequently
computed at all integration points and linearized with respect to uand ζas well as their spatial
derivatives εand ∇ζ(again, the index n+1is dropped). This procedure will be discussed
in detail. Subsequently, the finite element equations can be derived based on the numerical
quadrature of the linearized Eqns. (21) and (22), given by

B
(∂εσ[Δε]+∂ζσΔζ)·δεdv=−Gu,

B
(δζ ∂ζ(β−pχ)Δζ−δζ ∂εpχ·Δε+∇(δζ)·∂∇ζξ∇(Δζ)) dv=−Gζ.(23)
The derivatives denote the algorithmically consistent tangent operators, which will be subse-
quently derived. Different tangent operators, like ∂∇ζσor ∂∇ζ(β−pχ), will be shown to be
equal to zero. Based on the finite element solution of (23) the approximate solution is up-
dated, i.e. u←u+Δuand ζ←ζ+Δζ. For the application of the Finite Element Method
to Equations (23) we refer to the standard literature.
6.2 Integration Point Equations
As already mentioned the computationof the stresses in Eqns. (21) and (22) is based on a given
approximate finite element solution, i.e. β=∂ζWh(ζ)and ξ=∂∇ζWg(∇ζ)can directly be
evaluated. Based on the following implicit Euler discretization of the flow rule (20),
Δλα
Δt=λα−λα,n
t−tn
=˙γ0τα−pχ−τC
0
τDp
,(24)
the other stresses σand pχcan be computed by rearranging the equations εe(σ)=C−1[σ]
=ε−εp(σ,p
χ)and pχ=−Hχ(ζ−γeq (ˆ
λ)) in order to obtain the residuals
rσ=−C−1[σ]+ε−εp
n−Δt
α
˙γ0σ·MS
α−pχ−τC
0
τDp
MS
α
!
=0,(25)
rp=−ζ+γeq (ˆ
λn)+Δt
α
˙γ0σ·MS
α−pχ−τC
0
τDp
−pχ
Hχ
!
=0.(26)
The solution (σ,p
χ)can be computed based on a standard Newton scheme by the linearization
of Eqns. (25) and (26). Note that the isotropic hardening increment β(ζ)=∂ζWh(ζ)(as
well as the back stress term div (ξ)) is accounted for implicitly (through Eq. (13)2)inthe
residuals (25) and (26) as a consequence of its definition as a function of ζ(and not γeq(ˆ
λ)),
i.e. βdoes not appear in the list of unknowns of the integration point subroutine.
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142 S. Wulfinghoff and T. B¨ohlke: Gradient crystal plasticity subroutine
6.3 Convergence Improvement
The numerical solution of the nonlinear system of equations (25) and (26) is challenging in
the case of large values of the rate sensitivity parameter p. The evaluation of the power term
is not always possible since, in practice, the result often exceeds the numerical range of a
standard computer. This problem can occur if the initial guess for the local Newton scheme
is not sufficiently close to the correct solution. If the time increment Δtis small, the solution
of the last step can be expected to be a good starting value. However, increasing the time step
rapidly leads to a loss of convergence.
The algorithm can be stabilized by improving the starting solution as follows: in a first step
the flow rule (20) is modified in the large stress range by a linear function (cf. Fig. 1). A
continuous differentiabletransition between power law and linear approximation is used at the
point indicated by the parameter ˙γL. Thereby, regularized versions of equations (25) and (26)
are obtained which are less sensitive concerning the numerical range of the computer. The
solution of the regularized problem can be expected to be an improved starting solution for
the correct (i.e. non-regularized) system of equations (25) and (26).
˙
λα
τα−pχ
Power law
Linear
approximation
˙γL
τC
0
Fig. 1 Regularization of the power law for the computation of an improved starting solution for the
Newton scheme.
The smaller the parameter ˙γLis chosen the smaller is the slope of the linear approximation
and, consequently, the better is the convergence of the regularized model. In contrast, ˙γL
should not be chosen too small since the regularized model (and improved starting value) is
supposed to be as close as possible to the correct model and the associated solution. To satisfy
both requirements at once the value of ˙γLis estimated by
˙γL=kL(|Δζ|+)
Δt,(27)
where kL>1is a factor close to 1 and is a small number. In this work, the values kL=2
and =10
−4are applied. This choice permits the solution of the regularizedproblem to be (in
most cases) within the power law range, since for large values of the penalty parameter Hχ
the following relation holds
Δλα
Δt≤Δγeq
Δt≈Δζ
Δt≤˙γL.(28)
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GAMM-Mitt. 36, No. 2 (2013) 143
At the same time the definition (27) of the parameter ˙γLis expected to be sufficiently small to
prevent the algorithm from exceeding the numerical range of the computer.
The choice of the numerator kL(|Δζ|+)in estimation (27), instead of simply using Δζ,is
motivated by the fact that the approximation Δγeq ≈Δζin Eq. (28) is not exact.
Having obtained the improved starting guess, the correct solution is obtained in a subsequent
Newton iteration by increasing ˙γLto a very large value to assure that the solution lies in the
power law (not in the linear) regime. Indeed, the numerical examples discussed in the next
section indicate significantly improved convergence properties.
6.4 Algorithmic Tangent Operators
Requiring that small perturbations of the nodal variables duand dζas well as the associated
fields dεand ∇(dζ)preserve vanishing local residuals, i.e.
drσ=∂σrσ[dσ]+∂εrσ[dε]+∂pχrσdpχ
!
=0,(29)
drp=∂σrp[dσ]+∂ζrp[dζ]+∂pχrpdpχ
!
=0,(30)
the algorithmic tangent operators can be calculated by solving the linear Eqns. (29) and (30)
for dσand dpχ. The result yields the linear operators required in Eqns. (23)
dσ=∂εσ[dε]+∂ζσdζ, (31)
dpχ=∂εpχ·dε+∂ζpχdζ. (32)
Based on the observation that ∂εrσ=I,∂ζrp=−1and ∂pχrσ=∂σrpone can easily show
that ∂εσis symmetric and ∂ζσ=−∂εpχ, i.e. the overall problem is symmetric, too (which
also holds true for the global stiffness matrix).
7 Numerical Examples
7.1 Simulation Set-Up
The performance of the improved power law subroutine is compared to the standard subrou-
tine in a tensile test simulation. Periodic boundary conditions are applied, i.e. the displacement
field is assumed to be given by
u=¯
εx +˜
w,(33)
where ¯
εrepresents the macroscopic strain and ˜
wthe displacement fluctuations. The vari-
able ζis assumed periodic. Applying a tensile test with free lateral contractions, the shear
strains ¯εij ,i=jare set to zero. The tensile strain ¯ε11 =ε(t)is prescribed and ¯ε22 as well
as ¯ε33 represent additional degrees of freedom.
In order to model the gradient and isotropic hardening contributions, a quadratic defect energy
Wg(∇ζ)=1/2KG∇ζ·∇ζ,whereKGis a gradient modulus and a Voce hardening relation
Wh(ζ)=τC
∞ζ+1
θ0
(τC
∞−τC
0)2exp −θ0ζ
τC
∞−τC
0(34)
are applied. The plastic material parameters are given by
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144 S. Wulfinghoff and T. B¨ohlke: Gradient crystal plasticity subroutine
˙γ0pτ
DτC
0τC
∞θ0KGHχiii
10−3s−110 1 MPa 70 MPa 200 MPa 1 GPa 0.01N 107MPaiii
Here, τC
∞is the limit yield stress and θ0is an initial hardening modulus. The elastic constants
read C1111 = 168 GPa,C1122 = 121 GPa and C1212 =75GPa, respectively. To prevent the
standard material subroutine (i.e. the non-improved algorithm) from requiring excessively
small time steps, the rate sensitivity exponent is chosen relatively small (p=10). The toler-
ances of the global Newton scheme and the material subroutine are given by 10−9(times the
initial euclidean residual norm) and 10−10 (absolute), respectively. However, numerical ex-
periments indicate that much (several orders of magnitude) less demanding tolerances already
yield satisfactory results which can barely be distinguished from the presented findings. If no
convergence can be achieved at the global or at the integration point level, the time step is
reduced, otherwise it is increased (by a factor of ∼2).
Two periodic FCC grains (see Fig. 2) with random orientations are discretized by 32 ×32 ×32
standard linear hexahedrons with ∼130000 degrees of freedom (d.o.f.) in total. The cube
shaped domain has an edge length of 30 μm. In order to apply a simple geometric multigrid
solver, a penalty approximation of the micro-hard boundary conditions was implemented, i.e.
the boundary condition ζ=0is approximated. The applied strain rate is 0.05 s−1.
7.2 Discussion of the Results
The macroscopic tensile stress response is visualized in Fig. 3. The simulation results of the
standard and the improved integration point subroutine match qualitatively and quantitatively,
i.e. there is neither a dependence on the algorithm nor on the time step size observable. This
is underlined by a comparison of the final deformations of both simulations in Fig. 2.
xx
y
y
zz
γeq
0.034
0.017
5.1e-6
Back stress
-34.3
247.9
530.1
Fig. 2 (online colour at: www.gamm-mitteilungen.org) Comparison of the simulation results of the
standard (left) and improved (center) algorithm. Right: visualization of the back stress −div (ξ)in
MPa.
Fig. 3 shows that the standard algorithm requires significantly smaller time steps than the
improved Newton scheme. Especially, the numerically challenging elasto-plastic transition
diminishes the time step size of the standard scheme due to loss of convergence of the integra-
tion point Newton procedure. Contrary, no convergence problems are induced at the transition
in case of the improved scheme. In total, the standard algorithm required 158 time steps while
the improved scheme always converged leading to a total number of five steps. The conver-
gence properties of the non-improved algorithm can be ameliorated by decreasing the strain
rate sensitivity p, as illustrated in Fig. 3 (right). Consequently, larger deformations (here 5%
macroscopic strain) can be simulated in passable times, even with the standard routine.
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GAMM-Mitt. 36, No. 2 (2013) 145
Table 1 (left) summarizes the convergence of the improved scheme for p=10. The conver-
gence of the Newton scheme canbe improved by increasing the accuracyof the linear equation
solver (by increasing the maximum number of V-Cycles). Table 1 (right) shows exemplary
convergence rates for this case. However, since the solution of the individual linear systems
is more time consuming, a less accurate solution of the of the multigrid solver is accepted
leading to an increased number of global Newton steps.
Fig. 3 (online colour at: www.gamm-mitteilungen.org) Left: Macroscopic tensile response computed
by the standard and the improved algorithm for p=10. Right: Amelioration of the convergence prop-
erties of the non-improved algorithm for a decreased strain rate sensitivity p=5.
In the following, a physical interpretation of the results is summarized.
•The back stress plays an important role close to the grain boundaries (Fig. 2), where it
significantly reduces the plastic deformation. Physically, this result represents the strong
dislocation interaction forces in dislocation pile-ups. This behavior is typical of strain
gradient plasticity theories.
•The macroscopic stress-strain response (Fig. 3) is comparable to the prediction of other
gradient plasticity theories. Especially, the quadratic defect energy leads to an additional,
approximately linear hardening contribution (for a detailed discussion cf. Wulfinghoff
and B¨ohlke [21]).
7.3 Towards Polycrystal Simulations
In the following, the performance of the improved material subroutine is tested in the case of
a large strain rate sensitivity exponent (p= 200) as well as an elevated number of grains (with
random orientations) and elements.
Additionally, the ultimate tensile strain is increased to ¯ε11 (1s) = 0.05. The edge length of the
cube shaped domain is 60 μm. Figs. 4 and 5 illustrate the results of a periodic micro structure
consisting of 27 grains which are discretized by 80 ×80 ×80 elements. Consequently, the
total number of degrees of freedom is ∼2 million. Except for these changes, all model param-
eters remain unchanged. The total number of integration points is ∼4 million. The associated
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146 S. Wulfinghoff and T. B¨ohlke: Gradient crystal plasticity subroutine
Step 1 Step 2 Step 3 Step 4 Step 5
1.00e+00 1.00e+00 1.00e+00 1.00e+00 1.00e+00
3.95e-03 2.79e-02 1.56e-02 2.76e-02 2.95e-02
1.89e-05 1.01e-02 8.15e-03 1.27e-02 6.75e-03
2.80e-07 2.97e-02 3.08e-03 2.35e-03 1.50e-03
4.18e-09 4.52e-04 2.03e-03 5.53e-04 4.44e-04
6.36e-11 1.84e-04 1.16e-03 1.33e-04 6.92e-05
1.04e-04 3.82e-04 1.14e-05 5.00e-06
4.10e-05 5.20e-05 5.03e-07 2.62e-07
1.44e-05 1.36e-06 9.31e-09 2.09e-09
8.16e-06 6.71e-09 8.47e-11 1.72e-11
7.42e-06 6.11e-11
1.72e-07
1.14e-09
1.70e-11
Step 1 ··· Step 5
1.00e+00 1.00e+00
4.33e-07 6.68e-04
2.40e-13 3.86e-05
2.57e-06
3.34e-08
8.98e-12
Tabl e 1 Left: Euclidean norm (normalized) of the residual of the improved algorithm. The values be-
long to the results in Fig. 3. Right: exemplary convergence rates illustrating that increasing the accuracy
of the linear equation solver tends to decrease the number of necessary Newton steps (Steps 2,3 and 4
required 14, 10 and 7 Newton iterations, respectively).
γeq
0.232
0.116
4.46e-5
Back stress
4136.5
2018.3
-99.9
Fig. 4 (online colour at: www.gamm-mitteilungen.org) Deformed periodic micro structure at 5%
macroscopic tensile strain (Back stress in MPa).
Fig. 5 (online colour at: www.gamm-mitteilungen.org) Macroscopic tensile response of the model
in Fig. 4.
material subroutines converged in all time steps and all iterations.
However, the global Newton scheme did not always converge leading to a reduction of the
time step size, especially in the elasto-plastic transition phase (cf. Fig. 5). The simulation was
done on a Pentium Dual Core PC with 3.0GHz and 6GBRAM. Neither the multigrid solver,
nor the integration point subroutine evaluation was parallelized. The global and local conver-
gence tolerances of 10−9and 10−10 (see above) were not changed. The total simulation time
was ∼41 hours.
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GAMM-Mitt. 36, No. 2 (2013) 147
8 Summary
A penalty type equivalent plastic strain gradient plasticity theory has been proposed. Based
on the principle of virtual work the field equations have been derived, and a viscoplastic
power law type flow rule has been motivated based on the requirement of positive dissipation.
The numerical implementation has shortly been summarized on the global and integration
point level and a strategy to improve the convergence of the power law subroutine has been
proposed. Three-dimensional finite element simulations have demonstrated the numerical
gain of the improved subroutine. The combination of the equivalent plastic strain gradient
theory and the improved subroutine facilitates gradient plasticity simulations of increased
numbers of grains using large rate sensitivity exponents, even on standard computers.
Acknowledgments
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”.
Moreover, thanks go to Eric Bayerschen and Felix Fritzen for their support concerning the
numerics.
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