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ADAPTIVE FINITE ELEMENT COMPUTATIONS
OF SHEAR BAND FORMATION
TH. BAXEVANIS*,z and TH. KATSAOUNIS*,y
*Department of Applied Mathematics,
University of Crete, Heraklion 71409, Greece
yInstitute of Applied and Computational Mathematics,FORTH, Heraklion 71110, Greece
[email protected]@tem.uoc.gr
A. E. TZAVARAS
Department of Mathematics, University of Maryland,College Park, MD 20742, USA
Institute of Applied and Computational Mathematics,
FORTH, Heraklion 71110, Greece
Received 21 January 2009
Revised 11 May 2009
Communicated by K.-J. Bathe
We study numerically an instability mechanism for the formation of shear bands at high strain-
rate deformations of metals. We use a reformulation of the problem that exploits scaling
properties of the model, in conjunction with adaptive ¯nite element methods of any order in thespatial discretization and implicit Runge�Kutta methods with variable step in time. The
numerical schemes are of implicit�explicit type and provide adequate resolution of shear bands
up to full development. We ¯nd that from the initial stages, shear band formation is already
associated with collapse of stress di®usion across the band and that process intensi¯es as theband fully forms. For fully developed bands, heat conduction plays an important role in the
subsequent evolution by causing a delay or even stopping the development of the band.
Keywords: Shear bands; localization; adaptive ¯nite elements.
AMS Subject Classi¯cation: 22E46, 53C35, 57S20
1. Introduction
Shear bands are regions of intensely localized shear deformation that appear during
the high strain-rate plastic deformation of metals. Once the band is fully formed, the
two sides of the region are displaced relatively to each other, however the material
Mathematical Models and Methods in Applied SciencesVol. 20, No. 3 (2010) 423�448
#.c World Scienti¯c Publishing Company
DOI: 10.1142/S0218202510004295
423
still retains full physical continuity from one side to the other. Shear bands are often
precursors to rupture and their study has attracted attention in the mechanics as well
as the mathematics literature (see Refs. 35, 16, 39 and 41 for surveys).
A mechanism for the shear band formation process is proposed in Ref. 15: Under
isothermal conditions metals generally strain harden and exhibit a stable response.
As the deformation speed increases, the heat produced by the plastic work triggers
thermal e®ects. For certain metals, their thermal softening properties may outweigh
the tendency of metals to harden and deliver net softening response. This induces a
destabilizing mechanism, where nonuniformities in the strain-rate result in nonuni-
form heating, and in turn, since the material is harder in cold spots and softer in hot
spots, the initial nonuniformity is further ampli¯ed. By itself this mechanism would
tend to localize the total deformation into narrow regions. Opposed to this mech-
anism stands primarily the mechanism of momentum di®usion that tends to di®use
nonuniformities in the strain rate and second thermal di®usion that tends to equalize
temperatures. The outcome of the competition depends mainly on the relative
weights of thermal softening, strain hardening and strain-rate sensitivity, as well as
the loading circumstances and the strength of thermal di®usion.
The following model, in non-dimensional form, (described in detail in Sec. 2) has
served as a paradigm for the study of the instability mechanism:
vt ¼1
r�x;
�t ¼ ��xx þ �vx;�t ¼ vx;
� ¼ ����m� nt :
ð1:1Þ
It models the balance of momentum and energy in simple shearing deformations of a
thermoviscoplastic material with yield surface described by an empirical power law.
The parameters � > 0, m > 0 and n > 0 measure the degree of thermal softening,
strain hardening and strain-rate sensitivity, respectively, while r and � stand for
dimensionless constants. The model (1.1) simulates the essential aspects of exper-
iments like the pressure-shear test, see Ref. 16 or the thin-walled tube torsion test, see
Refs. 26 and 29. Moreover, shear bands typically appear and propagate (up to the
interaction time) as one-dimensional structures, and a thorough understanding of the
formation process is lacking even in that simple case. A number of investigations
concerning (1.1) or variants thereof have appeared in the literature approaching the
subject of shear band formation fromamechanics, seeRefs. 15, 31, 32, 13, 11, analytical
seeRefs. 17, 37�39, 12, 7 or computational perspective, seeRefs. 18, 42, 40, 21, 22, 4, 20,
36, 25, 19, 33, 8 and 10. The reader is also referred to experimental, see Ref. 27, or
computational studies, see Refs. 43 and 13 of shear bands in several space dimensions.
A stability criterion was developed in Refs. 30 and 31 using a linearized stability
analysis of relative perturbations and some simplifying arguments. This analysis
leads to the conclusion that (1.1) is stable when q :¼ ��þmþ n > 0 and loses
stability in the complementary region q < 0. This criterion corroborated well with
424 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
nonlinear stability results in Refs. 17, 37 and 39 on various special cases as well as
with nonlinear criteria for instability, see Ref. 38 under prescribed stress boundary
conditions. An e®ective equation describing the asymptotic in time response is de-
rived in Ref. 28 and leads to a second-order nonlinear di®usion equation that changes
type from forward to backward parabolic across the threshold q=0. It is still an open
question whether the solution, when q ¼ mþ n� � < 0, blows up in ¯nite time or
the solution exists for all times. Our numerical computations cannot provide a con-
clusive answer. Although the question is not relevant from a practical perspective,
since real materials would fracture well ahead such a potential critical time, it is of
mathematical interest and a computational challenge.
Our goal is to draw conclusions on the physics of the onset of shear band for-
mation. In this work we perform a systematic numerical study of (1.1) based on
. an appropriate non-dimensionalization followed by a change of time scale,
. special spatial and temporal adaptive techniques.
Many researchers gave numerical results that faithfully followed the process of
localization up to a point where resolution is lost as a result of insu±cient mesh
re¯nement. Wright and Walter42,40 provide early computational studies of the
evolution of an adiabatic shear band from the initial, nearly homogeneous stage to
the fully localized stage. We refer to Refs. 4, 18, 22, 21 and 20 for numerical com-
putations of various one-dimensional models and to Refs. 2 and 3 for two- and three-
dimensional computational studies. In a di®erent approach, discontinuities of the
displacement ¯eld (strong discontinuities) are embedded into the ¯nite element
spaces, see Refs. 36, 25, 19, 33, 8, 10.
In this work we use mesh adaptivity to resolve the shear band. We use continuous
¯nite element spaces to approximate the unknown variables but their spatial
derivatives are approximated by discontinuous elements. We propose special spatial
and temporal re¯nement techniques to be able to capture correctly the multiscale
nature of the shear band formation. The proposed adaptive mesh re¯nement tech-
niques are motivated by works devoted to the resolution of blow-up problems, see
Refs. 1, 9 and 34 for semilinear parabolic or Schr€odinger equations, in one or two
space dimensions. These methods are well suited to handle shear bands and o®er very
good resolution up to full development. It should, however, be noted that the present
computational context is more challenging as it involves resolution of a quasilinear
hyperbolic�parabolic system. The spatial adaptation is based on a local inverse
inequality satis¯ed by the ¯nite element spaces in any space dimension, while the
time step selection process is based on preserving an energy related to the system.
While adaptive techniques for computing shear band have been used by several
authors, the present methodology has certain novel features:
. detects in an automatic way the location of the shear band,
. it resolves the singularity by placing relatively few extra spatial grid points using
an error estimator based on properties of the model,
Adaptive Finite Element Computations of Shear Band Formation 425
. the mesh sizes around the singularity can be of order 10�13 of the size of the
specimen, thus essentially overcoming the mesh dependency problem on resolving
the shear band, while the discretization is coarser away from the singularity,
. due to the local nature of the spatial adaptive technique only one or two elements
are re¯ned at each re¯nement step, thus the extra computational cost for com-
puting the new ¯nite element space as well as the related sti®ness matrices, is
rather small,
. it uses a physical property of the model as selection mechanism for the time step.
Regarding past work, we note the moving grid method developed in Ref. 18 to
re¯ne the mesh locally and reduce the discretization error, the local mesh re¯nement
technique in Refs. 4 and 5 that distributes uniformly over the domain certain scaled
residuals of the solution, and the mesh re¯nement strategy of Ref. 20 based on a
posteriori error estimation. Finally, for computations of two-dimensional shear band
models, the reader is referred to Ref. 6 (where the area of an element is taken
inversely proportional to the value of the deformation measure at its center) and to
Ref. 24, where a combination of a front tracking method and error indicators are used
to re¯ne the mesh.
We perform computations ¯rst for the adiabatic model � ¼ 0 and then study the
e®ects of heat conduction � > 0. The e®ect of the heat di®usion on shear band
formation was reported initially in Ref. 40, where based on numerical evidence, after
a critical time the pro¯le of the solution remains quasi-constant. Our results, for the
adiabatic case, indicate strain localization and shear band formation for parameter
values in the region q < 0. It is shown that shear band formation is associated with a
collapse of the stress-di®usion mechanisms across the cross-sectional location of the
band (see Fig. 6) which is in accordance with the theoretical analysis in Ref. 28.
Regarding the e®ect of heat di®usion, systematic computational runs indicate the
following: heat di®usion does not a®ect much the time that localization initiates, it
a®ects considerably the rate of strain-rate growth inside the band, and it invariably
delays and can even stop the subsequent evolution of the localization process.
However, the stopping e®ect takes a long time to materialize, and in practical situ-
ations it could well be that the material will break before this e®ect is observed.
This paper is organized as follows: in Sec. 2 we introduce the mathematical model.
The ¯nite element approximations are described in Sec. 3 while in Sec. 4 several
numerical results are presented.
2. Description of the Model
From a mathematical perspective, shear band formation provides a fascinating
example of competition between the nonlinear instability mechanism induced by net
softening response, with the stabilizing e®ects of dissipative mechanisms (momentum
and thermal di®usion). Its mathematical aspects are connected to resolving parabolic
regularizations of ill-posed problems, see Ref. 39, which of course presents associated
426 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
computational challenges. The mathematical model (2.1) is ¯rst non-dimensionalized
leading to the non-dimensional form (2.6). All material related characteristics are
integrated in the two non-dimensional parameters, r and �, in (2.6). This system
admits the special class of the uniform shearing solutions. We next introduce a
reformulation of the problem, motivated by the form of the uniform shearing sol-
utions and a time rescaling of exponential type, see (2.14). The resulting rescaled
system (2.15) captures the scaling characteristics of the problem. Its use allows us to
achieve a much longer time in the numerical computations with considerably smaller
computational e®ort, and enables us to draw conclusions on the physics of the onset
of localization and reveal physical features of the model in particular in the non-
adiabatic case.
As shear bands appear and propagate as one-dimensional structures (up to
interaction times), most investigations have focused on one-dimensional, simple
shearing deformations. An in¯nite plate located between the planes x ¼ 0 and x ¼ d
is undergoing a simple shearing motion with the upper plate subjected to a prescribed
constant velocity V and the lower plate held ¯xed. The equations describing the
simple shear,
�vt ¼ �x;�t ¼ vx;
c��t ¼ k�xx þ ��t;
ð2:1Þ
describe the balance of momentum, kinematic compatibility and balance of energy
equations, the quantities appearing in them are the velocity in the shearing direction
v(x, t), plastic shear strain �ðx; tÞ, temperature �ðx; tÞ, shear stress �ðx; tÞ, while the
constants �, c, k standing respectively for the density, speci¯c heat and thermal
conductivity. Assumptions that have been factored in the model are that elastic
e®ects as well as any e®ect of the normal stresses are neglected, that all the plastic
work is converted to heat, and the heat °ux is modeled through a Fourier law. The
plates are assumed thermally insulated, that is �xð0; tÞ ¼ �xðd; tÞ ¼ 0.
For the shear stress we set
� ¼ fð�; �; �tÞ; ð2:2Þ
where f is a smooth function with fð�; �; 0Þ ¼ 0 and fpð�; �; pÞ > 0, for p 6¼ 0. The
relation (2.2) may be viewed as describing the yield surface for a thermoviscoplastic
material or (by inverting it) as a plastic °ow rule, and we require that the material
exhibits thermal softening, f�ð�; �; pÞ < 0, and strain hardening f�ð�; �; pÞ > 0. The
di±culty of performing high strain-rate experiments causes uncertainty as to the
speci¯c form of the constitutive relation (2.2). In the experimental and mechanics
literature on high strain-rate plasticity, see Refs. 15, 16 and 29, there is extensive use
of empirical power laws,
� ¼ �r�
�r
� ��� �
�r
� �m �t
�:r
� �n
¼ �����m� nt ; ð2:3Þ
Adaptive Finite Element Computations of Shear Band Formation 427
obtained by ¯tting experimental data to the form (2.3). The parameters �r; �r; �:r are
the reference values of the temperature, strain and strain-rate associated to the
determining experiment, while the material constant �r and the �, m, n are obtained
by ¯tting the data. The ¯tted values of �; m, n serve as measures of the degree of
thermal softening, strain hardening and strain rate sensitivity for the material at
hand. According to experimental data for most steels we have � ¼ Oð10�1Þ; m ¼Oð10�2Þ and n ¼ Oð10�2Þ. In the case of the cold-rolled steel: AISI 1018, for example,
with a constitutive law of the form (2.3), the following set of parameters are deter-
mined in Ref. 15:
� ¼ 436MPa; � ¼ 7800 kg=m3; c ¼ 500 J kg�1C�1;
k ¼ 54 W=ðmCÞ m ¼ 0:015; n ¼ 0:019; � ¼ 0:38:ð2:4Þ
We refer to Refs. 15, 39 and 41 for further details on the model, and to Ref. 29 for the
experimental setup and an outline of how the data are ¯tted to power laws. The
following non-dimensionalization is also often employed:
x̂ ¼ x
d; t̂ ¼ t�
:0; v̂ ¼ v
V; �̂ ¼ �
�0=�c; �̂ ¼ �
�0; ð2:5Þ
where d is the specimen size, �0 stands for a reference stress to be appropriately
selected, �0 ¼ �0�c is a reference temperature and �
:0 ¼ V
d is a reference strain rate. One
then obtains the non-dimensional form
vt ¼1
r�x;
�t ¼ ��xx þ �vx;�t ¼ vx
ð2:6Þ
and
� ¼ 1
�0f
�0�c�; �; �
:0�t
� �;
for ðx; tÞ 2 ½0; 1� � ½0;1Þ and where we dropped the hats. This system contains two
non-dimensional numbers:
r ¼ �V 2
�0; � ¼ k
�cVd; ð2:7Þ
where r is a ratio of stresses and depends on the choice of the normalizing stress �0and � the usual thermal di®usivity. The reference stress �0 is useful in normalizing the
constitutive function f. For the empirical power law (2.3), by selecting �0 ¼�r
�0�c =�r
� ���ð 1�rÞm �
:0
�:r
� �n, we obtain the non-dimensional form
� ¼ ����m� nt : ð2:8Þ
The parameters � > 0, m > 0 and n > 0 serve as measures of the degree of thermal
softening, strain hardening and strain-rate sensitivity. Another commonly used
428 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
constitutive relation is the Arrhenius law
� ¼ e���� nt : ð2:9Þ
In the form (2.9), the Arrhenius law does not exhibit any strain hardening and the
parameters � and n measure the degree of thermal softening and strain-rate sensi-
tivity, respectively. The system (2.6)�(2.8) is supplemented with boundary con-
ditions, in the non-dimensional form
vð0; tÞ ¼ 0; vð1; tÞ ¼ 1; ð2:10Þ�xð0; tÞ ¼ 0; �xð1; tÞ ¼ 0 ð2:11Þ
and initial conditions
vðx; 0Þ ¼ v0ðxÞ; �ðx; 0Þ ¼ �0ðxÞ > 0; �ðx; 0Þ ¼ �0ðxÞ > 0; ð2:12Þfor x 2 ½0; 1�. We take v0x > 0, in which case a maximum principle shows that �t ¼vx > 0 at all times and thus all powers are well-de¯ned.
2.1. Uniform shearing solutions
System (2.6) admits a special class of solutions describing uniform shearing. These
are:
vsðx; tÞ ¼ x;
�sðx; tÞ ¼ �sðtÞ :¼ tþ �0;
�sðx; tÞ ¼ �sðtÞ;where
d�sdt
¼ fð�s; tþ �0; 1Þ;
�sð0Þ ¼ �0
and �0; �0 are positive constants, standing for an initial strain and temperature. The
resulting stress is given by the graph of the function
�sðtÞ ¼ fð�sðtÞ; tþ �0; �:0Þ;
whichmay be interpreted as stress versus timebut also as stress versus (average) strain.
This e®ective stress-strain curve coincides with the �s � t graph, and the material
exhibits e®ective hardening in the increasing parts of the graph and e®ective softening
in the decreasing parts. The slope is determined by the sign of the quantity f�f þ f� .
When this sign is negative, the combined e®ect of strain hardening and thermal soft-
ening delivers net softening. For a power law (2.8), the uniform shearing solution reads
�sðtÞ ¼ tþ u0;
�sðtÞ ¼ �1þ�0 þ 1þ �
mþ 1ðtþ �0Þmþ1 � �mþ1
0
� �� � 11þ�;
�sðtÞ ¼ ���s ðtÞðtþ �0Þm:
ð2:13Þ
Adaptive Finite Element Computations of Shear Band Formation 429
Observe that, for parameter values ranging in the region m < �, �sðtÞ may initially
increase but eventually decreases with t. We will focus henceforth in this parameter
range, that is in situations that the combined e®ect of thermal softening and strain
hardening results to net softening.
2.2. A rescaled system
Motivated by the uniform shearing solutions (2.13), we introduce a rescaling of the
variables and of time in the form:
�ðx; tÞ ¼ ðtþ 1Þmþ1�þ1�ðx; �ðtÞÞ; �ðx; tÞ ¼ ðtþ 1Þ�ðx; �ðtÞÞ;
�ðx; tÞ ¼ ðtþ 1Þm���þ1�ðx; �ðtÞÞ; vðx; tÞ ¼ V ðx; �ðtÞÞ; � ¼ lnð1þ tÞ:
ð2:14Þ
In the new variables, the system (2.6)�(2.8) becomes:
V� ¼1
re
mþ11þ���x;
�� ¼ Vx � �;
�� ¼ �e��xx þ �Vx �mþ 1
1þ ��;
� ¼ ����mV nx :
ð2:15Þ
The non-dimensional rescaled system (2.15) is the one used to produce the
numerical results reported in Sec. 4. Notice that all material related characteristics
are integrated in two non-dimensional parameters namely r and �. For the classical
torsional experiment on steels for a sample of size d ¼ 2:5mm with initial velocity
V 2 ½2:5; 500�, the initial strain rate �:0 takes values in the range of 103 to 2 � 105,
r 2 ½5 � 10�5; 2� and the di®usion coe±cient � 2 ½10�5; 2 � 10�3�.Although the early deformation can be regarded as adiabatic with no considerable
error, when localization sets and temperature gradients across a band become very
large thermal di®usion e®ects can no longer be regarded as negligible. The mor-
phology of a fully formed band in the late stages of deformation is thus in°uenced by
heat conduction balancing the heat production from plastic work. In Ref. 40, which is
probably the only extensive treatment of fully formed shear bands, Walter noticed
that, due to heat conduction, the strain rate essentially becomes independent of time
in the late stages of deformation, even though the temperature and stress continue to
evolve.
3. A Finite Element Method
We consider a ¯nite element discretization of the non-dimensional mathematical
model (2.6) coupled with the empirical power law (2.8), augmented with boundary
conditions (2.10), (2.11) and initial conditions (2.12). Let Th be a partition of � ¼½0; 1�; 0 ¼ x0 < x1 < � � � < xMh
¼ 1 consisting of sub-intervals I ¼ ½xi�1;xi� of lengthhi ¼ xi � xi�1; i ¼ 1; . . . ;Mh and let h ¼ supi hi. Further, let 0 ¼ t0 < t1 < � � � <
430 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
tk < � � � be a partition of ½0;1Þ with k ¼ tk � tk�1; k ¼ 1; 2; . . . . We consider the
classical one-dimensional C 0 ¯nite element space Sh;p � L2ð�Þ, de¯ned on partitions
Th of �
Sh ¼ Sh;p ¼ f 2 C 0ð�Þ : jI 2 PpðIÞ; I 2 Thg; dimSh ¼ pMh þ 1; ð3:1Þ
where PpðIÞ denotes the space of polynomials on I of degree at most p.
Remark 3.1. In general we allow the partitions Th to have variable size and vary
with time, Th ¼ T kh ; Mh ¼ Mk
h and accordingly Sh ¼ S kh. In the sequel, to simplify
the notation we drop the dependence on time.
Remark 3.2. Although C 0 ¯nite element spaces are used for interpolating the
unknown variables, their spatial derivatives (like vx) are approximated by completely
discontinuous ¯nite elements. This is a di®erent approach from the embedded strong
discontinuities method where the discontinuities in the displacement ¯eld are
embedded explicitly in the ¯nite element spaces and its derivatives are approximated
by Dirac masses, Refs. 36, 25, 19, 33, 8 and 10.
3.1. Semidiscrete schemes
A variational formulation for our model is as follows: we seek functions vh; �h, �h 2 Sh
such that
ðvh;t; Þ ¼ Fhvnh;xðxÞjx¼1
x¼0 � ðFhvnh;x;
0Þ; 8 2 Sh; ð3:2aÞ
ð�h;t; Þ ¼ ðHh��ah ; Þ þ ��h;x jx¼1
x¼0 � �ð�h;x; 0Þ; 8 2 Sh; ð3:2bÞ
ð�h;t; �Þ ¼ ðvh;x; �Þ; 8� 2 Sh; ð3:2cÞ
where
Fh � F ð�h; �hÞ ¼1
r��ah �m
h Hh � Hð�h; vh;xÞ ¼ �mh vnþ1
h;x
and ð�; �Þ denotes the L2 inner product. Let J ¼ dimSh, we write
vhðx; tÞ ¼XJi¼1
ViðtÞiðxÞ; �hðx; tÞ ¼Xj
i¼1
�iðtÞ iðxÞ; �hðx; tÞ ¼Xj
i¼1
�iðtÞ�iðxÞ;
with V ¼ ðV1; . . . ;VJÞ; U ¼ ð�1; . . . ; �JÞ and G ¼ ð�1; . . . ; �JÞ:
Using the boundary conditions (2.10) and (2.11), system (3.2a)�(3.2c) can be
written in a matrix form
MV:¼ FðV;U;GÞ; Vðt ¼ 0Þ ¼ vhðx; 0Þ ¼ Pv0ðxÞ; ð3:3Þ
MU:¼ HðV;U;GÞ; Uðt ¼ 0Þ ¼ �hðx; t ¼ 0Þ ¼ P�0ðxÞ; ð3:4Þ
Adaptive Finite Element Computations of Shear Band Formation 431
MG:¼ DV; Gðt ¼ 0Þ ¼ �hðx; t ¼ 0Þ ¼ P�0ðxÞ; ð3:5Þ
where M ¼ ði; jÞ;D ¼ ð 0i; �jÞ; i; j ¼ 1; . . . ; J are the mass and sti®ness-like
matrices respectively and
FðV;U;GÞ ¼ Fhvnh;xjx¼1 � ðFhv
nh;x;
0jÞ;
HðV;U;GÞ ¼ ðHh��ah ; jÞ � �ð�h;x; 0
jÞ; j ¼ 1; . . . ; J ;ð3:6Þ
where P : C 0ð½0; d�Þ ! Sh denotes either the interpolation or the L2 projection
operator.
3.2. Fully discrete schemes
The initial value problem (3.3)�(3.5) is a coupled fully nonlinear system of ordinary
di®erential equations and is discretized by a Runge�Kutta method. Let V 0 ¼Vðt ¼ 0Þ; U 0 ¼ Uðt ¼ 0Þ;G0 ¼ Gðt ¼ 0Þ, given the solution V k;U k;Gk at time level
tk we compute intermediate values V k;i;U k;i;Gk;i as the solution of the following
nonlinear system:
MV k;i ¼ MV k þ kþ1
Xs
j¼1
aijFðV k;j;U k;j;Gk;jÞ; i ¼ 1; . . . ; s;
MU k;i ¼ MU k þ kþ1
Xsj¼1
aijHðV k;j;U k;j;Gk;jÞ; i ¼ 1; . . . ; s;
MGk;i ¼ MGk þ kþ1
Xsj¼1
aijDV k;j; i ¼ 1; . . . ; s
ð3:7Þ
and the solution at the next time level tkþ1 is updated by
MV kþ1 ¼ MV k þ kþ1
Xsi¼1
biFðV k;i;U k;i;Gk;iÞ;
MU kþ1 ¼ MU k þ kþ1
Xsi¼1
biHðV k;i;U k;i;Gk;iÞ;
MGkþ1 ¼ MGk þ kþ1
Xsi¼1
biDV k;i;
ð3:8Þ
where A ¼ ðaijÞ; b ¼ ðbiÞ; i; j ¼ 1; . . . ; s are the parameters de¯ning an s -stage
Runge�Kutta method. System (3.7) is a large fully nonlinear system and compu-
tationally very expensive to solve. Thus we propose an Implicit�Explicit RK (IERK)
method to discretize (3.3)�(3.5): in the system (3.7) the unknown variable is treated
implicitly while the other variables are treated explicitly. The proposed scheme
decouples the system and for each intermediate stage i the equations are solved
432 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
separately, possibly in parallel:
MV k;i ¼ MV k þ kþ1
Xi
j¼1
aijFðV k;j;U k;GkÞ; i ¼ 1; . . . ; s;
MU k;i ¼ MU k þ kþ1
Xi
j¼1
aijHðV k;U k;j;GkÞ; i ¼ 1; . . . ; s;
MGk;i ¼ MGk þ kþ1
Xi
j¼1
aijDV k;j; i ¼ 1; . . . ; s:
ð3:9Þ
In the simple case where s ¼ 1;A ¼ a11 ¼ 1; b ¼ b1 ¼ 1 (Implicit Euler), the system
(3.9), (3.8) is
MV kþ1 ¼ MV k þ kþ1FðV kþ1;U k;GkÞ; ð3:10aÞ
MU kþ1 ¼ MU k þ kþ1HðV k;U kþ1;GkÞ; ð3:10bÞ
MGkþ1 ¼ MGk þ kþ1DV kþ1: ð3:10cÞ
The last equation (3.10c) is linear while (3.10a) and (3.10b) are nonlinear. The
nonlinear equations are linearized using Newton's method. In particular, each of
these equations is of the form:
solve AðX ;Y;ZÞ ¼ 0; for X ; whereAðX ;Y;ZÞ ¼ M �X þ BðX ;Y;ZÞ �MX and �X ;Y;Z are given:
ð3:11Þ
Newton's method for (3.11) is
DAðW‘ÞðW‘þ1 �W‘Þ ¼ �AðW‘Þ; W ¼ ðX ;Y;ZÞ; ‘ ¼ 0; . . . ; ‘k; ð3:12Þ
where DA denotes the Frechet derivative of A. The starting values for the Newton's
iteration are taken to be: V kþ10 ¼ V k; U kþ1
0 ¼ U k and we set V kþ1 ¼ V kþ1‘k
;
U kþ1 ¼ U kþ1‘k
. In practice ‘k ¼ 2 or 3 is enough to guarantee convergence of the
iterative procedure. The discretization of system (2.15) is done in a completely
analogous fashion.
3.3. Adaptive re¯nement strategy
The width of the shearing band is of the order of at most a few micrometers (orders of
magnitude less than the overall size of the material surrounding the band), whereas
the lateral extent may be several millimeters in length. As the morphology of a shear
band exhibits such a ¯ne transverse scale, it would be costly to resolve the bands fully
in a large-scale computation. A resolution of the order of 10�m or less would be
required for many materials in order to follow the deformation across the band in
Adaptive Finite Element Computations of Shear Band Formation 433
detail. When a ¯xed mesh is used and the tip of a peak of a state variable becomes
narrower than the mesh spacing, it will be the length scale of the grid that regularizes
the calculation rather than the physical and constitutive features of the material.
Thus, there is a strong dependence between the peaks of the state variables and the
mesh size h (¯xed mesh). We believe that the adaptive computation is much more
accurate than any of the computations on a ¯xed mesh. Capturing numerically this
singular behavior is not a trivial task. Special adaptive techniques in space as well as
in time have to be used to capture correctly the underlying phenomenon. We describe
now the adaptive mechanisms that are used.
3.3.1. Spatial mesh re¯nement
The adaptive mesh re¯nement strategy presented here is motivated by the work in
Refs. 9, 1 and 34 where a similar strategy is used for the numerical simulation of blow-
up solutions of the generalized KdV and nonlinear focusing Schr€odinger equations
respectively. The proposed method is able to capture the growth of the solution using
only, at the ¯nal mesh, about 40% extra spatial grid points to the initial
discretization.
In the unstable regime q ¼ ��þmþ n < 0, as time increases, vx; �; � grow near a
point xH 2 �. The proposed spatial adaptive mechanism acts in two ways: ¯rstly
locates the singular point xH automatically and secondly increases locally the resol-
ution of the ¯nite element scheme by re¯ning the mesh in a small neighborhood
around xH. The adaptive mesh re¯nement strategy is motivated by a local L1 � L1
inverse inequality that elements of Sh satisfy on I 2 Th, Ref. 14. We assume that the
partition Th satis¯es (inverse assumption): 9 c0 > 0 2 R such that h � c0 mini hi.
Then for � 2 Sh we have
jj�jjL1ðIÞ < Ch�1i jj�jjL1ðIÞ; with C :¼ Cð�; p; c0Þ: ð3:13Þ
For � ¼ vkh;x, (3.13) shows that the growth of the L1 norm of vh;x is bounded by the
size of hi and the L1 norm of vh;x on I, which does not change, (2.10). Thus at time
level k an interval I of the partition Th is re¯ned, its spatial size hi is halved, if the
following criterion is satis¯ed
jjvkh;xjjL1ðIÞhi > �hjjvk
h;xjjL 1ðIÞ; ð3:14Þ
where �h � 1. The interval I satisfying (3.14) contains xH and we allow vh;x to grow
by re¯ning the mesh, in one or at most two intervals, around xH. Using this tech-
nique, mesh sizes of the order of 10�13 of the size of the specimen, are used to resolve
the singularity, Fig. 2. On the other hand, due to the local nature of the adaptive
technique, very few elements are a®ected, thus the new ¯nite element spaces
and related sti®ness matrices, can be computed with small extra computational
e®ort. Finally, any function on the old mesh is embedded in the new one by linear
interpolation.
434 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
3.3.2. Time adaptivity
In time the solution grows rapidly near the singularity point xH, thus controlling the
size of the time step k is necessary. Choosing k adaptively is a very delicate issue in
these types of simulations. Our time step selection strategy is motivated by preser-
ving a physical property of the system, in particular the energy E(t):
EðtÞ :¼Z 1
0
r
2v2ðx; tÞ þ �ðx; tÞ
� �dx: ð3:15Þ
Using (2.6) and the boundary conditions we get
@E
@t¼ �jx¼1: ð3:16Þ
Thus to preserve the energy in time up to ¯rst-order, we de¯ne the time step size kþ1
as
kþ1 ¼ min �jEnþ1
h � Enh j
�kþ1h jx¼1
; 0
( ); ð3:17Þ
where Enh ¼
R d
0½r2 v
2hðx; tnÞ þ �hðx; tnÞ�dx, 0 a given time step threshold and � � 1.
4. Numerical Experiments
In the sequel, we present the results of several numerical experiments that illustrate
the various features and properties of the model (2.6). All the experiments were
performed on a Pentium 4 work station running Linux at 2.4 GHz, using a code that
implements in double precision the fully discrete schemes for the equivalent rescaled
system (2.15). Working with (2.15) has the advantage that we can easily access long
times, due to the relation � ¼ lnð1þ tÞ between the original time t and the rescaled
time � . In all numerical runs, unless otherwise mentioned, linear ¯nite element are
used for the spatial discretization and the implicit Euler method (3.10) for time
stepping.
4.1. Numerical results
We present a series of numerical results illustrating the features of system (2.6). The
numerical results presented here are obtained by discretizing (2.15). The material
parameters �, m, n are selected in two representative regions:
. Region I (Stable case): ��þmþ n > 0,
. Region II (Shear band): ��þmþ n < 0.
Initial conditions. There are di®erent choices of initial conditions throughout the
literature for (2.6), the majority of them are perturbations of the uniform shearing
solution (2.13). One type consists of introducing, at the center of the slab, a small
initial temperature perturbation of Gaussian form, added to the homogeneous initial
Adaptive Finite Element Computations of Shear Band Formation 435
temperature �I , as used elsewhere in the literature, see Refs. 40 and 4. Another type
of initial perturbation consists of introducing a small perturbation to the uniform
shearing solutions in all three variables v; �; �:
v0ðxÞ¼
tan�
q
� �x 0� x� 1
2� g;
1
21� tan
�
q
� �� �1� cos
�ðx� 12þ gÞ
2g
� �� �þx tan
�
q
� �1
2� g< x<
1
2þ g;
1� tan�
q
� �ð1�xÞ 1
2þ g� x� 1;
8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:
�0ðxÞ¼1:01þ 0:01cos
�ðx� 12Þ
g
� �1
2� g< x<
1
2þ g;
1 otherwise;
8<:
�0ðxÞ¼0:011þ 10�3 cos
�ðx� 12Þ
g
� �1
2� g< x<
1
2þ g;
0:01 otherwise;
8<:
ðICAÞ
where g ¼ 0:02; q ¼ 4:0001. A third commonly used perturbation is that of a geo-
metric imperfection, like the symmetric groove in the center of the slab. All these
choices, in the unstable case (Region II), will give rise to a single shear band at the
center of the slab. In all the experiments reported here we use initial conditions of the
type (ICA).
Remark 4.1. The equations that are numerically resolved are in non-dimensional
form. The parameters �; m, n are selected in either of the regions (I) or (II), and we
systematically study certain aspects of the dependence in the dimensionless numbers
r and �. In the ¯gures below \TIME" stands for the physical non-dimensional time t̂
(2.5), while � is de¯ned in (2.14).
4.1.1. The stable case (Region I)
First we consider adiabatic deformation � ¼ 0 with parameters in the stable region:
q ¼ ��þ nþm > 0. We compute the solution of (2.15) for � ¼ 0:5; n ¼ 0:75;
m ¼ 0:25. The solution variables can be seen in Fig. 1 at di®erent time instances. The
computed solution approaches very fast the uniform shearing solution (2.13).
4.1.2. Adiabatic shear band (Region II)
We consider next adiabatic deformations (� ¼ 0) with parameters taking values in
the unstable region: ��þ nþm < 0 where we expect formation of shear bands;
speci¯cally, we take � ¼ 2; m ¼ 0:25; n ¼ 0:2. To be able to capture the singular
behavior of the solution, the use of the adaptive techniques (3.14), (3.17) is
436 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
mandatory. A typical behavior of the adaptive spatial and time re¯nement method is
shown in Fig. 2. In particular, Fig. 2(a) shows the evolution in time of the smallest
spatial length hmin and the time step k produced by the adaptive strategy. The
proposed spatial adaptive process (3.14) reduces the mesh size appropriately,
reaching a value of hmin � 10�13, to capture the immense growth of the solution
(vx; �; �) at the shear band point. Furthermore using (3.17), the time step k is re¯ned
appropriately to follow the rapidly changing solution with k � 10�9 at ¯nal time. On
the other hand, Fig. 2(b) shows the distribution of hi's over the unit slab at ¯nal time
of the simulation. We notice that the mesh re¯nement technique (3.14) locates the
singular point correctly and reduces the mesh size where necessary.
We compute the solution of system (2.15) with � ¼ 2; m ¼ 0:25; n ¼ 0:2. The
choice of initial condition (ICA) will yield a shear band located at x ¼ 12. The com-
puted solution of the system, up to time t � 2:62, is shown in Figs. 3�7. The singular
behavior of the solution is fully resolved using the adaptive techniques (3.14), (3.17)
with �h ¼ 1:0025 and 0 ¼ 10�5. In Fig. 8, we show the evolution in time of vx; �; �; �
at the shear band point x ¼ 12. The strain rate vxð�tÞ has reached a value of about 108,
the temperature has risen about 3000 times the initial one, the strain has grown
about 6 orders in magnitude and the stress almost collapsed and reached a value of
0.00056. As can be seen from Fig. 6, (bottom right) the stress reaches a very small
value thus the di®usion process collapses and no information gets across the shear
X
v x
-0.2 0 0.2 0.4 0.6 0.8 1 1.20.9995
1
1.0005
1.001
1.0015
1.002
t=0t=500
X
θ
-0.2 0 0.2 0.4 0.6 0.8 1 1.2-50
0
50
100
150
200
250
t=0t=100t=500
X
γ
-0.2 0 0.2 0.4 0.6 0.8 1 1.2-100
0
100
200
300
400
500
600
t=0t=100t=500
X
σ
-0.2 0 0.2 0.4 0.6 0.8 1 1.20.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
t=0t=100t=500
Fig. 1. Region I: Solution at t=0, t=100 and t=500.
Adaptive Finite Element Computations of Shear Band Formation 437
band. Further the oscillations are not a numerical artifact and are caused by the very
nature of the underlying instability which is of a backward parabolic type, see
Ref. 28.
We also study the e®ect of the non-dimensional parameter r on the solution of
system (2.15). As already mentioned in the classical torsional experiment on a steel
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++
TIME0 0.2 0.4 0.6 0.8 1
10-13
10-11
10-9
10-7
10-5
10-3
hmin
δk+
(a)
x
h i
0 0.2 0.4 0.6 0.8 1
10-13
10-11
10-9
10-7
10-5
10-3
(b)
Fig. 2. Adaptivity: (a) Evolution of hmin; k, (b) ¯nal distribution of hi.
(a) (b)
Fig. 3. Region II: Strain rate vxð�tÞ in log-scale: (a) evolution, (b) ¯nal time.
438 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
(a) (b)
Fig. 4. Region II: Temperature � in log-scale: (a) evolution, (b) ¯nal time.
(a) (b)
Fig. 5. Region II: Strain � in log-scale: (a) evolution, (b) ¯nal time.
(a) (b)
Fig. 6. Region II: Stress �: (a) evolution, ¯nal time (top right), zoom (bottom right).
Adaptive Finite Element Computations of Shear Band Formation 439
sample of size d ¼ 2:5mm, Ref. 15, for typical values of initial velocity V, the par-
ameter r takes values in the interval ½5 � 10�5; 2�. Our results, away from the shear
band point, are in excellent agreement with those presented in Ref. 15, while at the
shear band point our results exhibit better resolution than in Ref. 15. In Fig. 9, for
di®erent values of the parameter r, we compare the evolution in time of the stress
values at the shear band point. Notice that the initiation time of localization is almost
identical for all values of the parameter r. On the other hand, as the value of r
(a) (b)
Fig. 7. Region II: Velocity v: (a) evolution, (b) ¯nal time.
TIME
σ
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
TIME
v x
0 0.5 1 1.5 2 2.5100
102
104
106
108
TIME
θ
0 0.5 1 1.5 2 2.5100
101
102
103
TIME
γ
0 0.5 1 1.5 2 2.5
100
102
104
106
Fig. 8. Region II: Evolution of vx; �; �; � at x ¼ 12. vx and � are in log-scale.
440 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
decreases, the localization is more abrupt thus diminishing the time when the
localization takes place. This type of behavior is also observed for vx; � and �.
4.2. E®ect of heat di®usion
Next, we compute the solution of system (2.15) in the non-adiabatic case � > 0. The
heat di®usion coe±cient is taken as � ¼ 10�6 while the thermomechanical par-
ameters �; m and n are as in the previous runs in the unstable case of Region II. The
evolution of the state variables is presented in Figs. 10�14. The behavior depicted by
the numerical simulations holds true for other values of the heat di®usion coe±cient
that we tested but do not present here. Namely, after a transient period of localiz-
ation, the inhomogeneities of the state variables eventually dissipate and the
underlying uniform shearing solution (2.13) emerges.
TIME
σ
0 0.5 1 1.5 2 2.5
0
0.1
0.2
0.3
0.4
0.5
0.6 r = 0.5r = 0.1r = 0.05r = 0.01r = 0.001r = 0.0005
Fig. 9. Evolution of � at x ¼ 12 for various values of r.
Fig. 10. Non-adiabatic case: � ¼ 10�6, strain rate vx; ð�tÞ in log-scale: evolution (left), ¯nal time (right).
Adaptive Finite Element Computations of Shear Band Formation 441
Fig. 12. Non-adiabatic case: � ¼ 10�6, evolution of strain � in log-scale.
Fig. 11. Non-adiabatic case: � ¼ 10�6, evolution of temperature � in log-scale.
Fig. 13. Non-adiabatic case: � ¼ 10�6, evolution of stress �.
442 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
The e®ect of the value of the heat di®usion coe±cient is restricted in the local-
ization stage as one may observe in Figs. 15�18. The smaller the heat di®usion
coe±cient is, the stronger the rate of localization. The e®ect of the heat di®usion on
the initiation of the localization process is negligible, and thus the early stages of
deformation that can be regarded as adiabatic. By contrast, heat di®usion has an
e®ect on the subsequent evolution and development period of the band. Finally, heat
di®usion not only delays but even stops the development of the band and returns the
solution to uniform shear, but the latter process takes a very long time and might not
be observable in practice.
Fig. 14. Non-adiabatic case: � ¼ 10�6, evolution of velocity v.
log(TIME)
log(
v x)
10-3 10-1 101 103 105 107 109 1011
100
101
102
103
k = 10-6
k = 10-5
k = 10-4
Fig. 15. Non-adiabatic case: di®erent heat di®usion coe±cients �, strain rate �t in log-scale at the centerof the slab.
Adaptive Finite Element Computations of Shear Band Formation 443
5. Discussion
Adaptive ¯nite elements computations are employed to tackle the multiscale
response of rate-dependent materials subjected to simple shearing deformations.
Numerical solutions are a necessity in order to fully describe the singular phenom-
enon associated with localization. The adaptive mesh re¯nement strategy, as well as
log(TIME)
log(
γ)
10-5 10-3 10-1 101 103 105 107 109 1011
10-1
101
103
105
107
109
1011
k = 10-6
k = 10-5
k = 10-4
Fig. 17. Non-adiabatic case: for di®erent heat di®usion coe±cients �, strain � in log-scale at the center of
the slab.
log(TIME)
log(
θ)
10-5 10-3 10-1 101 103 105 107 109 1011
100
101
102
103
104 k = 10-6
k = 10-5
k = 10-4
Fig. 16. Non-adiabatic case: di®erent heat di®usion coe±cients �, temperature � in log-scale at the center
of the slab.
444 Th. Baxevanis, Th. Katsaounis & A. E. Tzavaras
the time step control are discussed and the accuracy of the resulting schemes is tested
with the expected numerical order of accuracy observed in all cases. The constitutive
law used is of the power law type that has been used extensively in the literature and
allows considerable °exibility in ¯tting experimental data over an extended range.
The numerical runs indicate unstable response for thermomechanical parameters
taking values in the region ��þ nþm < 0. The early stages of deformation can be
regarded as adiabatic, because high strain rates do make heat conduction unim-
portant at that stage. At this stage, shear band formation occurs with a simultaneous
collapse of di®usion across the band.
Once localization sets in, heat conduction plays an important role in the sub-
sequent evolution. Actually there exists a critical time t above which the local rate of
heat generation is less than the rate of heat di®usion from the band to the outside
colder material. Beyond this critical time the inhomogeneities in the solutions are
dissipated and the outcome (assuming that the material has not already fractured, or
changed phase) would be for the deformation to dissipate and approach an under-
lying uniform shearing solution. The value of thermal di®usivity � a®ects materially
the localization rate and the critical time t de¯ned above, while its e®ect on the
initiation time for shear band formation is negligible. Namely, the smaller the value of
thermal di®usivity, the stronger the localization rate and the longer the critical time.
Acknowledgments
This work was partially supported by the National Science Foundation, and by the
program \Pythagoras" of the Greek Secretariat of Research. The authors would like
to thank the anonymous referees for their valuable comments and suggestions.
log(TIME)
σ
10-6 10-4 10-2 100 102 104 106 108 1010
0
0.1
0.2
0.3
0.4
0.5
0.6
k = 10-6
k = 10-5
k = 10-4
Fig. 18. Non-adiabatic case: di®erent heat di®usion coe±cients �, stress � at the center of the slab.
Adaptive Finite Element Computations of Shear Band Formation 445
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