Acta Polytechnica Vol. 52 No. 6/2012

Energetic Approach to Large Strain Gradient Crystal Plasticity

Jan Kratochvíl1, Martin Kružík2,1

1Czech Technical University, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague, Czech Republic
2Institute of Information Theory and Automation of the ASCR, Pod Vodárenskou věží 4, 182 08 Prague, Czech Republic

Corresponding author: kruzik@utia.cas.cz

Abstract
We formulate a problem of the evolution of elasto-plastic materials subjected to external loads in the framework of large
deformations and multiplicative plasticity. We focus on a spontaneous inhomogenization interpreted as a structuralization
process. Our model includes gradients of the plastic strain and of hardening variables which provide a relevant length
scale of the model. A simple computational experiment interpreted as a hint of a deformation substructure formation is
included.

Keywords: energetic solution, crystal plasticity, gradient plasticity, slip system.

1 Introduction

The elastic-plastic behavior of crystalline materials
poses a challenge for mathematical analysis on the mi-
croscopic, the mesoscopic, and the macroscopic scales.
Here, we study a rate-independent model arising in the
crystal plasticity. A common and successful approach
to the analysis of crystalline materials is by means of
energy minimization; see e.g. Ortiz & Repetto [30].
This is manifested for various elastic crystals, even
for those with the potential of undergoing phase tran-
sitions. The applicability of variational methods has
been broadened to include rate-independent evolution.
Typically, these models are characterized by energy
minimization of a functional including macroscopic
quantities such as the macroscopic deformation gra-
dient as well as a dissipation functional. In order to
introduce a physically relevant scale to our problem we
assume, following earlier works of Bažant & Jirásek
[3], Dillon & Kratochvíl [9], Gurtin and Gurtin &
Anand [15, 16], Mainik & Mielke [22] and others, that
our energy functional depends also on the gradient
of the plastic tensor. The gradient term models non-
local effects caused by short-range interactions among
dislocations. It is not clear, however, which function
of the gradient should be used. For further details we
refer to Kratochvíl & Sedláček [18], and to Bakó &
Groma [1] for attempts to derive it from statistical
physics revealing thus complexity of the problem, for
more details. A related approach to non-local models
in damage and plasticity was undertaken in Bažant
& Jirásek [3], see also [8, 10, 12, 23].
In this paper, we formulate the so-called energetic

solution due to Mielke et al. [28] to our problem. This
concept of a solution is based on two requirements.
First, as a consequence of the conservation law for

linear momentum, all work put into the system by
external forces or boundary conditions is spent on
increasing the stored energy or it is dissipated. Sec-
ondly, the formulation must satisfy the second law
of thermodynamics, which in the present mechanical
framework has the form of a dissipation inequality.
The last requirement enters the framework as the
assumption of the existence of a nonnegative convex
potential of dissipative forces. As a consequence, the
imposed deformation evolves in such a way that the
sum of the stored and dissipated energies is always
minimized. The main advantage of this approach is
that it allows us to exploit the theory of the mod-
ern calculus of variations and suggests a numerical
approach to this problem.
To expose the essence of the mathematical struc-

ture of the energetic approach, we first analyze a
proto-model called here a material with internal vari-
ables. It freely follows the exposition of Francfort
& Mielke [11] and we recall it here to motivate the
notion of the energetic solution. In the second step,
the framework is applied to elasto-plastic materials by
specifications of some internal variables. One of the
main results is that the described energetic approach
can be identified with crystal plasticity with strain
gradients in the version formulated by Gurtin [15].
Gurtin’s model is formulated in the mathematical
language of differential equations. From the point
of view of numerical solution of a boundary value
problem of crystal plasticity the energetic formulation
is more convenient.
Our results are closely related to [22], where the

authors proved the existence of energetic solutions
to strain gradient plasticity with polyconvex energy
density allowing even for +∞; see [2] and to Giaco-
mini & Lussardi [13] where the linear-elastoplasticity

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Acta Polytechnica Vol. 52 No. 6/2012

framework is considered. Here we allow for finite
quasiconvex stored energy density and large defor-
mations. This is motivated by relaxation theory in
the calculus of variation, where the effective macro-
scopic energy density is quasiconvexification of the
microscopic energy density. Thus, our results may
be applied to plasticity of materials with developing
microstructures, as in shape memory alloys, [4], or
[26] for instance. We refer an interested reader to
[19] for a model describing cyclic plasticity in these
materials. Another related paper is Carstensen et al.
[6], where the authors use the energetic approach to
plasticity without strain gradients.
In what follows, Ω ⊂ Rn, is an open bounded do-

main and Lβ(Ω; Rn), 1 ≤ β < +∞ denotes the usual
Lebesgue space of mappings Ω → Rn whose modu-
lus is integrable with the power β, and L∞(Ω; Rn)
is the space of measurable and essentially bounded
mappings Ω → Rn. Further, W 1,β(Ω; Rn) stan-
dardly represents the space of mappings which live in
Lβ(Ω; Rn) and their gradients belong to Lβ(Ω; Rn×n).
If f : Rn → R is convex but possibly nonsmooth we
define its subdifferential at a point x0 ∈ Rn as the set
of all v ∈ Rn such that f(x) ≥ f(x0) + v ·(x−x0) for
all x ∈ Rn. The subdifferential of f will be denoted
∂subf and its elements will be called subgradients of
f at x0.

2 Materials with internal
variables

Consider a material whose elastic properties depend
on internal variables z ∈ Z ⊂ Rm. The stored en-
ergy density is then W = W(Fe,z), where Fe ∈ Rn×n
is the elastic strain. We are interested in the rate-
independent evolution of the material. To this end,
we assume the existence of a nonnegative convex po-
tential δ = δ(ż) of dissipative forces, where ż denotes
the time derivative of z. In order to ensure rate-
independence, δ must be positively one-homogeneous,
i.e., δ(αż) = αδ(ż) for all α > 0. Finally, we define
for z ∈ Z a thermodynamic force

Q := −
∂

∂z
W(Fe,z). (1)

The evolution rule is introduced in the form

Q(t) ∈ ∂subδ(ż(t)), (2)

where ∂subδ is the subdifferential of δ. Hence, there
is ω(t) ∈ ∂subδ(ż(t)) such that Q(t) = ω(t). Maximal
monotonicity of the subdifferential implies that for
all θ ∈ ∂subδ(ξ) we have

〈ω(t) −θ, ż(t) − ξ〉≥ 0. (3)

Remark 2.1. In particular, taking ξ = 0 and realiz-
ing that the one-homogeneity of δ yields δ(ż) = 〈ω,ż〉
for all ω ∈ ∂subδ(ż) we get

δ(ż(t)) = 〈ω(t), ż(t)〉 = 〈Q(t), ż(t)〉≥ 〈θ, ż(t))〉 (4)

for all θ ∈ ∂subδ(0). Inequality (4) expresses the so-
called maximum dissipation principle (see e.g. Hill
[17] or Simo [31]), which says that thermodynamic
forces “available” in the so-called elastic domain
∂subδ(0) are not strong enough to overcome frictional
forces.

In what follows, Ω ⊂ Rn, is a bounded Lips-
chitz domain representing the so-called reference con-
figuration, ν is the outer unit normal to ∂Ω, and
∂Ω ⊃ Γ0, Γ1 which are disjoint. The deformation
will be denoted y : Ω → Rn. The evolution of the
system will be controlled by external forces. Let
f(t) : Ω → Rn be the (volume) density of the external
body forces and g(t) : Γ1 ⊂ ∂Ω → Rn be the (sur-
face) density of the surface forces. The equilibrium
equations governing the mechanical behavior of the
system are:

−div
(

∂

∂∇y
W(∇y(t),z(t))

)
= f(t) in Ω, (5)

y(t,x) = y0(x) on Γ0, (6)
∂

∂∇y
W(∇y(t),z(t))ν(x) = g(t,x) on Γ1. (7)

The full system characterizing the proto-model con-
sists of (5)–(7) supplemented by (2):

−
∂

∂z
W(∇y(t),z(t)) ∈ ∂δ(ż(t)),

z(0) = z0, z ∈ Z, (8)

where z0 ∈ Z is an initial condition for the internal
variable.

In order to regularize our problem we may add the
gradient of the internal variable, i.e., for ω ≥ 1 and
ε > 0 put

W(∇y,z) +
ε

ω
|∇z|ω. (9)

The evolution rule changes to

ε div
(
|∇z(t)|ω−2∇z(t)

)
−

∂

∂z
W(∇y(t),z(t))

∈ ∂δ(ż(t)), z(0) = z0, z ∈ Z, (10)

so we have the thermodynamic force

Q(t) := ε div(|∇z(t)|ω−2∇z(t))

−
∂

∂z(t)
W(∇y(t),z(t)).

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Acta Polytechnica Vol. 52 No. 6/2012

The potential energy of our system can be written
(� := ε/ω)

I(t,y(t),z(t)) :=
∫

Ω
W(∇y(t),z(t)) dx

+ �
∫

Ω
|∇z(t)|ω dx−L(t,y(t)), (11)

where the work done by external forces is

L(t,y(t)) :=
∫

Ω
f(t) ·y(t) dx

+
∫

Γ1
g(t) ·y(t) dS (12)

and the following energy balance is satisfied

d
dt
I(t,y(t),z(t)) = L̇

(
t,y(t)

)
−

d
dt

Diss
(
z; [0, t]

)
,

(13)
where

Diss
(
z; [0, t]

)
:=
∫ t

0

∫
Ω
δ
(
ż(s)

)
dxds.

Hence, the integration with respect to time gives

I(t,y(t),z(t)) + Diss(z; [0, t])

= I(0,y(0),z(0)) +
∫ t

0
L̇(s,y(s)) ds.

We can also consider a more general form of δ which
can also depend on (x,z), i.e. δ := δ(x,z, ż).

Typically, however, we do not have enough smooth-
ness in the internal variable to compute the time
derivative on the right-hand side of (13).
Following Mielke [24], we define the dissipation

distance between two values of internal variables
z0,z1 ∈ Z as

D(x,z0,z1) := inf
z

{∫ 1
0
δ(x,z(s), ż(s)) ds;

z(0) = z0, z(1) = z1
}
, (14)

where z ∈ C1
(
[0, 1]; Z

)
, and set

D(z1,z2) =
∫

Ω
D
(
x,z1(x),z2(x)

)
dx, (15)

where z1,z2 ∈ Z := {z : Ω → RM; z(x) ∈
Z a.e. in Ω}. We assume that Z is equipped with
strong and weak topologies which define the notions
of convergence used below.

Following [11, 22], we impose the following assump-
tions on D:

(i) Weak lower semicontinuity:

D(z, z̃) ≤ lim inf
k→∞

D(zk, z̃k), (16)

whenever zk⇀z and z̃k⇀z̃.

(ii) Positivity: If {zk} ⊂ Z is bounded and at the
same time min{D(zk,z),D(z,zk)}→ 0 then

zk⇀z. (17)

2.1 Energetic solution
Suppose that we look for the time evolution of y(t) ∈
Y ⊂ {y : Ω → Rn} and z(t) ∈ Z during the time
interval [0,T]. The following two properties are the
key ingredients of the so-called energetic solution due
to Mielke and Theil [27, 28].

(i) Stability inequality: ∀t ∈ [0,T], z̃ ∈ Z,y ∈ Y:

I
(
t,y(t),z(t)

)
≤I

(
t, ỹ, z̃

)
+ D

(
z(t), z̃

)
(18)

(ii) Energy balance: ∀t, 0 ≤ t ≤ T

I
(
t,y(t),z(t)

)
+ Var

(
D,z; [0, t]

)
= I

(
s,y(0),z(0)

)
+
∫ t

0
L̇
(
ξ,y(ξ)

)
dξ, (19)

where

Var
(
D,z; [s,t]

)
:= sup

{ N∑
i=1
D
(
z(ti),z(ti−1)

)
;

{ti} partition of [s,t]
}
.

Definition 2.2. The mapping t 7→ (y(t),z(t)) ∈ Y×
Z is an energetic solution to the problem (I,δ,L) if
the stability inequality and the energy balance are
satisfied.

Remark 2.3. Note that the stability inequality (i)
can be written in the form ∀t ∈ [0,T], z̃ ∈ Z, ỹ ∈ Y

I
(
t,y(t),z(t)

)
+D

(
z(t),z(t)

)
≤I

(
t, ỹ, z̃

)
+D

(
z(t), z̃

)
,

which means that y(t),z(t) always minimizes (ỹ, z̃) 7→
I(t, ỹ, z̃)+D(z(t), z̃). This means that the equilibrium
configurations are characterized by energetic minima.
Contrary to elasticity theory, the minimized energy
is not only the overall elastic energy described by I
but the dissipated energy is added.

It is convenient to put Q := Y × Z and to set
q := (y,z). Moreover, we define the set of stable
states at time t as

S(t) :=
{
q ∈ Q; ∀q̃ ∈ Q :

I(t,q) ≤I(t, q̃) + D(q, q̃)
}

(20)

and

S[0,T] :=
⋃

t∈[0,T]

{t}×S(t). (21)

Moreover, a sequence {(tk,qk)}k∈N is called stable if
qk ∈S(tk).

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Acta Polytechnica Vol. 52 No. 6/2012

3 Applications to
elasto-plasticity

Now we apply the energetic approach to an elasto-
plastic problem.

3.1 Problem statement
In what follows y : Ω → Rn will be a deformation of a
body Ω ⊂ Rn (in a fixed reference configuration) with
the deformation gradient F = ∇y. In particular, y
covers both elastic and plastic deformation. We define
the multiplicative split, F = FeFp, into an elastic
part Fe and an irreversible plastic part Fp, which
belongs to SL(n) := {A ∈ Rn×n; det A = 1}. The so-
called plastic strain Fp and the vector p ∈ Rm of the
hardening variables are internal variables influencing
elasticity. In other words, z(x) = (Fp(x),p(x)) ∈
SL(n) × Rm for almost all x ∈ Ω.
The energy functional I takes the form

I
(
t,y(t),z(t)

)
:=
∫

Ω
W(x,∇yF−1p ,Fp,∇Fp,p,∇p) dx

−L(t,y(t)), (22)

with L given by (12).
In order to ease the notation, we omit the depen-

dence of W on x. However, all the theory developed
in this paper may include nonhomogeneous W.
In what follows, we suppose that

y ∈ Y :=
{
y ∈ W 1,d(Ω; Rn); y = y0 on Γ0

}
,

where Γ0 ⊂ ∂Ω with a positive surface measure. More-
over, we suppose that Γ0 ∩ Γ1 = ∅. Further

Z :=
{
(Fp,p) ∈ W 1,β(Ω; Rn×n) ×W 1,ω(Ω; Rm);

Fp(x) ∈ SL(n) for a.e. x ∈ Ω
}
.

As q = (y,z) it will be advantageous and will cause
no confusion to write D as dependent on q, i.e.,

D(q1,q2) := D(z1,z2)

if q1 = (y1,z1) and q2 = (y2,z2). Similarly, we may
write I in terms of q = (y,z) as

I
(
t,q(t)

)
=
∫

Ω
W
(
x,∇yF−1p ,Fp,∇Fp,p,∇p

)
dx

−L
(
t,q(t)

)
,

where, obviously, L
(
t,q(t)

)
:= L

(
t,y(t)

)
.

In this situation, Q = (Q1,Q2) are conjugate plastic
stress and conjugate hardening forces, respectively.

Q1 = div
(
∂W(∇yF−1p ,Fp,∇Fp,p,∇p)

∂∇Fp

)
−
∂W(∇yF−1p ,Fp,∇Fp,p,∇p)

∂Fp

and

Q2 = div
(
∂W(∇yF−1p ,Fp,∇Fp,p,∇p)

∂∇p

)
−
∂W(∇yF−1p ,Fp,∇Fp,p,∇p)

∂p
. (23)

The elastic domain is defined as

Q(x,z) = ∂subżδ(x,z, 0). (24)

Remark 3.1. The principle of maximal dissipation
asserts that

Q1 : Ḟp + Q2 · ṗ (25)

is maximal if Ḟp and ṗ are kept fixed and (Q1,Q2) ∈
Q(x,z). This means that for all (A,B) ∈Q(x,z).

Q1 : Ḟp + Q2 · ṗ ≥ A : Ḟp + B · ṗ. (26)

Finally, we conclude with an example covered by
our approach.

Example 3.2 (Simple shear carried by a single slip).
Consider a single slip system defined by two orthonor-
mal vectors a,b ∈ R3 such that a is the glide direction
and b is the slip-plane normal. Further suppose that
we have a particular case of a so-called separable ma-
terial where:

W(x,Fe,z) = W1(Fe) + α
∣∣Fp∣∣2 + ε2∣∣∇Fp∣∣2,

where Fp(t) = I + γ(t)a ⊗ b where γ is the plastic
slip. The slip system is generally not fixed in the
reference configuration. The slip-plane normal b̃ in
the reference configuration has the form b̃ = (Fp)>b.
However, in this special case we have that b̃ = b, so
that the slip-plane normal is kept constant during the
process [15].

Due to the special case of Fp we may identify z :=
(γ,p) because Fp depends only on γ.

Choose the dissipation metric:

δ(z, ż) = δ(γ,p, γ̇, ṗ),

δ(γ,p, γ̇, ṗ) =

{
p|γ̇| if ṗ ≥ H|γ̇|,
+∞ otherwise,

where H is the so-called hardening function.
The evolution rule reads:

ε∆γ ∈ ∂sub
(
p|γ̇|

)
The elastic domain ∂subδ(γ,p, 0, 0) = [−p,p] if ṗ ≥
H|γ̇| and (−∞,∞) otherwise. The boundary of the
elastic domain ±p − ε∆γ = 0 defines the yield sur-
face. Thus, the energetic approach recovers Gurtin’s
calculations on shear bands in single-slip, see Gurtin
(2000).

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Acta Polytechnica Vol. 52 No. 6/2012

The dissipation distance is:

D(γ1,p1,γ2,p2)

=

{
p2|γ2 −γ1| if p2 −p1 ≥ H|γ2 −γ1|,
+∞ otherwise.

If H > 0 we get that the optimal choice is p2 =
p1 +H|γ2−γ1| which gives D(γ1,p1,γ2,p2) = p1|γ2−
γ1| + H(γ2 −γ1)2.

The following finite element computation illustrates
the performance of our model. We take n = 2,
Ω = (0, 1) × (0, 3), t ∈ [0, 1]. Further, we apply a
zero Dirichlet boundary condition on (0, 1)×{0} and
u(t,x) = −0.6t for x = (0, 1) ×{3}. As to mate-
rial properties we consider a homogeneous isotropic
material with a Young modulus 200 GPa, Poisson’s
ratio 0.3, a = (−1, 1)/

√
2, b = (1, 1)/

√
2 and the

W1(E) = λ2 tr|E|
2 + µ|E|2 where λ and µ are Lamé

constants corresponding to the used Young modulus
and to the Poisson’s ratio. The other model constants
were set as follows: ε = 40 GPa · m2, α = 2 MPa, and
H = 0.2 MPa.
The initial condition was chosen γ = 0 (no plastic

deformation) and initial yield stress p = 200 MPa.
Note that the constant defining the energy stored
in defects, i.e., α, is much smaller than the elastic
constants of the material. The specimen was dis-
cretized using piecewise affine finite elements for the
displacement as well as for Fp. A sequence of result-
ing minimization problems was solved by a Fortran
77 routine described in [5], which is based on a quasi-
Newton method. It is designed to solve large-scale
nonlinear optimization problems with box constraints.
We first observe the development of 45-degree bands
in the vicinity of the boundary where the Dirichlet
boundary conditions are applied. At the final stage
a large plastic deformation appears in the middle of
the specimen (see e.g. [20] for similar calculations).
The spontaneously-formed inhomogeneity provides a
hint of a deformation structuralization process. The
characteristic length scale introduced through the
higher gradients guarantees the intrinsic size of the
inhomogeneity independent of the FEM mesh size.

Acknowledgements

This work was supported by GAČR through projects
P107/12/0121, P201/10/0357, and by the research
project VZ-MŠMT 6840770003.

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