adaptive finite element computations of shear band...

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

[email protected]

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

http://dx.doi.org/10.1142/S0218202510004295

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


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


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


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


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 < � � � <


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


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


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


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.


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


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


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.


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.


(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).


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.


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).


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 �.


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.


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.


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.


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adaptive finite element computations of shear band...

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