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Localization phenomenain geological settings
Martin Konrad Paesold (MSc, BSc)
This thesis is presented for the degree of Doctor of Philosophy
of The University of Western Australia
School of Mathematics and Statistics
2015
Abstract
This thesis explores localization in geological applications in the context of multi-physics,
nonequilibrium systems. The term multi-physics refers to systems that contain a mix of
physical processes that are allowed to operate simulateneously. The thesis begins with
an investigation of purely mechanical systems and reviews localized structures stemming
from material nonlinearities. In later chapters further processes are considered and nonlin-
ear effects due to feedback mechanisms are studied and we describe how these feedback
mechanisms can lead to fascinating spatio/temporal structures.
The simplest model considered here is that of a strut on a Winkler type foundation and it
is well known that depending on the foundation characteristics various localized buckling
patterns can emerge if the strut is under compression. Owing to the steady movement of
tectonic plates the strut is deformed at a constant rate. We investigate how the buckling
pattern evolves over time and offer qualitative interpretations of the observed behaviour.
Other localization patterns encountered in geological applications are the formation of
shear zones. The chapter on energy based localization criteria introduces a generalization
to the slip line field theory for application in such problems. The new formulation pro-
poses a thermodynamic continuum-mechanic framework that fulfils the energy balance
under consideration of the second law. The energy balance admits multiple steady states
and, thus, this approach allows us to identify a critical mechanical dissipation parame-
ter, here called the Gruntfest number, which separates loading conditions that lead either
to homogeneous or localized plastic deformation. The geometry of the localized failure
follows the generalized slip line pattern and this is used to simplify the semi-analytical
solution of localization problem using the method of characteristics. The result is veri-
fied in numerical studies that are performed with the newly developed code REDBACK
that is capable to handle thermo-mechanical coupling simultaneously. The structure of
REDBACK is detailed here as well.
vii
Acknowledgements
First of all I would like to thank my parents and my wife for their continuing support and
encouragement. Without them this thesis would not have been possible.
Discussions with Andrew Bassom, Manolis Veveakis, Thomas Poulet, Klaus Regen-
auer-Lieb, Tim Dodwell and Giles Hunt have been invaluable throughout the course of
my studies. Further, I am indebted to Bruce Hobbs, Alison Ord, Ali Karrech, Nev Fowkes,
Thomas Stemler and Des Hill. Shannon Algar volunteered to proof read the manuscript.
I am grateful for financial support through the International Postgraduate Research
Scholarship, the Australian Postgraduate Award, the CSIRO top-up scholarship and the
UWA safety-net top-up scholarship. The use of the computer infastructure at iVEC is
gratefully noted.
ix
Statement of candidate contribution
This thesis contains work prepared for publication, some of which has been co-authored.
I am the sole author of Chapters 1, 2, 3 and 6.
A modified version of Chapter 4 has been submitted as co-authored work to the Journal
of Mechanics of Material and Structures. I am the primary author and wrote 80% of the
manuscript. In addition, I produced all analytical and numerical results.
Chapter 5, which details the implementation and capabilities of the code REDBACK, is
in preperation for publication to be submitted as co-authored work. I am the secondary
author and contributed 50% of the manuscript. In addition, I am one of the three main
developers of REDBACK and contributed regularily to the source code of REDBACK.
The permission to use this co-authored work in this thesis has been granted by the
authors and the confirmations are attached.
xi
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Dynamic mode interaction in the Swift-Hohenberg equation . . . . . . . . . . . . . . 7
2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Static equilibrium solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Evolution of transient folding patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Evolutionary Galerkin procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Self-similar blow-up solutions of the nonlinear Schrodinger equation . . . . . . 31
3.1 Blow-up solutions of the nonlinear Schrodinger equation . . . . . . . . . . . . . . . . 32
3.2 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Perturbation analysis around individual peak . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Matching of subsequent peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Energy based criteria for the onset of localized plastic deformation . . . . . . . . 41
4.1 Fundamental principles of continuum thermo-mechanics . . . . . . . . . . . . . . . . 45
4.2 Localization criterion and patterns of plastic failure . . . . . . . . . . . . . . . . . . . . . 54
4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Multi-physics simulations of rock mechanics using REDBACK . . . . . . . . . . . . . 75
5.1 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Behaviour of diatomaceous mudstone with increasing confinement . . . . . . . 88
5.4 Episodic Tremor and Slip events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
xvii
xviii Contents
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A Reduction of order technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B Variation of parameters technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
List of Figures
1.1 Two examples of folded structures, MacDonnell Ranges, Northern
Territory, Australia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Physical and chemical processes and their actions on material parameters. . 3
2.1 Strut supported by nonlinear springs and linear dashpots in parallel. . . . . . . 9
2.2 Foundation force u−u3 +αu5 and strain energy u2/2−u4/4+αu6/6 of
the nonlinear springs in the foundation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Snakes and ladders bifurcation diagram of the Swift-Hohenberg equation. . 14
2.4 Results of finite element analysis that highlight the dependance on
deformation rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Bifurcation point as function of deformation rate. . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Dynamical pitchfork bifurcation of Eq. (2.27). . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Energy surfaces for different values of foundation stiffness. . . . . . . . . . . . . . 23
2.8 Energy cross-section of the energy surface constructed from the
two-mode ansatz for α = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.9 Cross-terms of energy functional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.10 Routes over energy surface for rates between 10−5 ≤ R≤ 10−1. . . . . . . . . . 26
2.11 Load–end-shortening curves for the routes through the energy landscape
compared against the stationary equilibrium states. . . . . . . . . . . . . . . . . . . . . 27
2.12 Runs with identical initial conditions but at incrementally different rates
remain either in mode ψs or transition to ψa. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.13 Snakes and ladders for quadratic–quintic foundation with a restoring
force fe = u−u2 +0.23u3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Heat lines in a mild steel specimen at high temperatures. . . . . . . . . . . . . . . . . 43
xix
xx List of Figures
4.2 Rate sensitivity of the yield stress and Eyring plot for temperature
sensitivity of the yield stress for various materials. . . . . . . . . . . . . . . . . . . . . . 53
4.3 Bifurcation diagram and stability plot of two dimenstional heat equation. . . 60
4.4 S-curve bifurcation diagram and spatial pattern of solution. . . . . . . . . . . . . . 61
4.5 Dependence of bifurcation diagram on aspect ratio λ . . . . . . . . . . . . . . . . . . . 61
4.6 Approximation coefficients Gri (i = 1,2,3) as a function of centre
temperature θc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Results of perturbation analysis of heat equation. . . . . . . . . . . . . . . . . . . . . . . 65
4.8 Geometry of the problem configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.9 Evolution of ideal visco-plastic material deformed under constant force
boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.10 Evolution of ideal visco-plastic material deformed under constant force
boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.11 Dissipation profiles for the final state of a run that converges towards the
low temperature branch AB and a run that converges towards branch CD. . 70
4.12 Heat lines under fast constant velocity loading. . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 Structure of REDBACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Structure of simulation step in REDBACK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Numerical model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Yield surfaces for modified Cam–Clay and original Cam–Clay compared
to experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Matching experiments CD1-CD6 with simulation results. . . . . . . . . . . . . . . . 90
5.6 Activation volume as a function of confining pressure. . . . . . . . . . . . . . . . . . 91
5.7 Distribution of deviatoric (a) and volumetric (b) plastic strain for
numerical experiment CD1 at 20% strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.8 Model geometry (not to scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.9 Evolution of shear zone during the numerical simulation . . . . . . . . . . . . . . . . 95
5.10 Time evolution of the temperature and displacement (exaggerated 20×)
across the model. The geometry of the simulated column was repeated 5
times in the X– and 3 times in the Y –direction for visualisation purposes. . . 96
List of Tables
4.1 Rate sensitivity parameters for the flow law: ε in = ε0 (σY/σ0)n (c.f.
Fig. 4.2(a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1 Dimensionless parameters used in REDBACK. The coefficient δ is
defined such that T ? = (T −Tre f )/(δTre f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Mapping of REDBACK kernels implementing their respecting terms in
Eq. 5.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Confinement pressures used in experiments from (Oka et al, 2011) . . . . . . . 88
5.4 Simulation parameters (see definitions in Tab. 5.1) . . . . . . . . . . . . . . . . . . . . . 92
5.5 Simulation parameters (see definitions in Tab. 5.1) . . . . . . . . . . . . . . . . . . . . . 94
xxi
Chapter 1
Introduction
Geological structures come in great variety and commonly display complex, intricate
and beautiful features. Price and Cosgrove (1990) survey structures ranging from faults,
folded multi-layer assemblies or pinch–and–swell structures and Fig. 1.1 shows two ex-
amples of folds. The task of the geologist is to describe and map these structures and
subsequently to deduce the geological history of a region and the physical conditions dur-
ing the formation of the observed structures. This is an exiting intellectual challenge in
itself but also has important economic applications during the discovery and exploration
of profitable resources. Hence, the mechanisms that can produce geological structures
have been the focus of intensive research and as an example we give a brief review of the
reserch on folding.
The first systematic study of folds in geomaterials is probably due to Sir James Hall
(1815) who conducted analogue experiments with wet cloth to understand the structures
he observed on the Berwickshire coast in Scotland. Our modern understanding of fold
formation is mainly based on the seminal work by Biot (1961, 1965). He considered a
layer of a material embedded in a matrix of another material and realized that the con-
trast in rheological properties between the embedded material and embedding material is
the parameter that principally governs the formation of folds. For viscous materials this
Fig. 1.1 Two examples of folded structures, MacDonnell Ranges, Northern Territory, Australia.
1
2 1 Introduction
competency contrast is the ratio between the respective viscosities. Biot’s analysis shows
that different wavelengths of a fold are amplified at different rates – a process known as
wavelength selection – and the wavelength with the maximum amplification is denoted
the dominant wavelength, λD, that is given by
λD
h= 2π
(µL
6µM
)1/3
, (1.1)
where h is the thickness of the embedded layer and µL, µM the viscosity of the layer and
the embedding matrix, respectively. The above relation is important to the geolegist in the
field because the wavelength of a fold and the thickness of the folded material are easily
measured and properties of the materials during folding can be readily inferred.
Sherwin and Chapple (1968) tested Biot’s prediction on over 800 single layer fold spec-
imens and though they found that Biot’s theory was not directly applicable, it could be
modified to take into account the shortening and thickening of the layer. A further re-
finement of Biot’s theory is the extension to nonlinear viscous materials (Smith, 1975,
1977; Fletcher, 1974). The validity of these various theories is restricted to small deflec-
tions where the assumptions in a linear stability analysis remain valid and in an effort to
overcome this problem, Schmalholz and Podladchikov (2000) formulated a theory that
is valid beyond this limit and which describes the formation of large amplitude folds in
linear materials.
A common issue of the aforementioned wavelength-selection theories appears to be
that in order to predict observed wavelengths unnaturally high competency contrasts are
required and the theories only predict periodic folds which disagrees with the common
observation that folds are aperiodic (Hobbs and Ord, 2012). Further, these theories are
unable to explain the finite extend of folds and the fact that most folds are localized. In
order to amend the wavelength selection theories intial pertubations are commonly pro-
posed on the grounds that no geological system is free of such pertubations (Abbassi and
Mancktelow, 1990) and numerical studies show good agreement between the simulated
and observed fold shapes (Mancktelow, 1999).
Another explanation of localized folds is given by Hunt et al (1989) who studied a
strut on a nonlinear elastic foundation and found that the buckling of the strut localizes
if the foundation softens as it is deformed. Such softening behaviour could be a conse-
quence of nonlinearities inherent to the materials or a result from the interaction between
1 Introduction 3
different physical processes. Hobbs et al (2008) discuss thermo-mechanical coupling and
Regenauer-Lieb et al (2009) study chemo-mechanical coupling in relation to folding and
show that these approaches are viable alternatives to the classial folding theories and are
able to explain the aperiodicity and localization of folds. Further, their findings indicate
that structures on different length scales form due to different physical mechanisms and
as an example chemo-mechanical coupling leads to folding on a centimeter scale wheras
thermo-mechanical coupling yields features on a 100 m to 1 km scale. Hence, this multi-
physics approach offers the possibility to relate field observations to the processes that
could have been active during the formation of a structure and gain a deeper understand-
ing of the geological history of a region.
Subsequently, Hobbs et al (2011) generalized and extended their multi-physics ap-
proach to geology and reviewed the processes that can be active during rock deformation
in the Earth’s crust. They presented a generalized thermodynamic framework that can
treat thermal, mechanical, chemcial and hydrological processes in a unified manner and
describe their possible inter-relations. Following this formalism, this thesis is concerned
with thermodynamical systems that are in a nonequilibrium configuration and contain
ongoing deformations, chemical reactions, heat or fluid fluxes because in such nonequi-
librium settings spatial and/or temporal localized structures are likely to emerge (Cross
and Greenside, 2009) and the aim of this work is to understand localization phenomena
in geological settings.
Deformation
HeatChange in
temperature
Mineralreactions
+ diffusionFluid source
Change inpore pressure
Change ofviscosity,friction
Change ineffective
stress
Fig. 1.2 Physical and chemical processes (yellow box) and their actions on material parameters (whitebox). The arrows indicate the (inter-connected) influence among the processes.
4 1 Introduction
As an example of the systems that are of interest here, we consider the map of physi-
cal processes and their inter-relations in Fig. 1.2 that could be encountered in tectonical
subduction zones (Alevizos et al, 2014). There, the relentless creep of the tectonic plates
produces heat due to friction and this heat production immediately changes the tempera-
ture of the material and can possibly activate a chemical reaction. The heat change could
directly alter material properties such as viscosity or frictional angle and would lead to
an expansion of the material and hence a change in pressure. Similarily, the chemical
reaction could corrode the material and also alter material properties via a change in the
chemical composition of the material, it would produce (exothermic reaction) or consume
(endothermic reaction) heat and also could be a source or drain of fluids. All these effects
influence the mechanical properties and after the system went through the aforementioned
cycle once the deformation could be amplified which in turn could have dramatic conse-
quences such as earthquakes. This example shows how multi-physics approaches might
reveal material instabilities and explain localization phenomena and we present simula-
tion results of those episotic tremor and slip events in Chapter 5.
This thesis provides a survey of localization phenomena in geological nonequilibrium,
multi-physics systems. We start with nonlinear elastic models as the simplest models that
exhibit localized solutions. Then we continue by introducing increasingly complex mate-
rial nonlinearities, namely plasticity, that are subsequently coupled with thermal effects.
For thermo-mechanical processes that act on rate-dependent elasto-visco-plastic materi-
als we formulate a localization criterion based on energy considerations and we conclude
with a numerical investigation of fully coupled thermo-hydro-mechanical-chemical mod-
els. The thesis is divided into four main parts.
In Chapter 2, a model of geological folding comprising a thin elastic beam supported by
a nonlinear viscoelastic (Kelvin-Voigt) material is subjected to a slow rate of applied com-
pressive end-shortening. This compressive thrust is supposed to resemble the relentless
movement of tectonic plates and constitutes a constraint on the possible folding solutions.
The mathematical description reduces to the nonlinear Swift-Hohenberg equation which
is well known for its localized folding solutions and has been put forward as a simple
model that would allow the geologist to infer material parameters and ambient conditions
under which geological structures such as folds have formed (Ord and Hobbs, 2013).
In order to solve the Swift-Hohenberg equation a finite element method is implemented
and the effect of the deformation rate is investigated. We find that the transient solutions
1 Introduction 5
are close to the stationary ones if the rate is low, but as the rate increases the system
diverges from its equilibrium configuration and buckling states with higher energy than
compared to the stationary ones are attainable. In order to understand this behaviour a
low-dimensional Gelerkin method is developed which allows us to construct the energy
surface, on which the folding solution lie, such that this approach offers a convenient vi-
sual tool for the interpretation of folding dynamics of materials embedded in visco-elastic
materials.
After having discussed localization in nonlinear elastic systems we turn to a phe-
nomenon often observed in relation to localization: self-similar blow-up, which describes
the unbounded local growth of a physical quantity in space and/or time. Geological ex-
amples for blow-up are thermal runaway due to heat dissipation by plastic deformation
(Veveakis et al, 2010) or the formation of fluid pathways in shale (Veveakis et al, 2015).
In Chapter 3 we investigate the nonlinear Schrodinger equation because it is a mathemat-
ical model that admits blow-up solutions. Budd et al (1999) presented detailed numerical
results and showed that there is a countably infinite set of solutions and each solution
is characterized by the number of maxima that it possesses. It is interesting to know the
location and spacing of these maxima and Budd (2001) presented formal asymptotic re-
sults. The aim of our work is to confirm Budd’s results and to find an expression for the
distance between neighbouring peaks.
From plastic, isothermal models we move on to models of thermo-mechanical coupling
in Chapter 4 and present a localization criterion for this class of models. The onset of lo-
calization is determined via an energy bifurcation problem, providing that visco-plastic
materials admit a critical (mechanical) energy input above which deformation becomes
unstable and plastic localization ensues. Related to the question of the onset of local-
ization is the question of the spatial pattern of localization which is determined here by
the slip line field theory of plasticity. In analogy to the classical concepts of mechanics,
the conditions for the onset of localization in temperature-sensitive visco-plastic materi-
als are reached at a critical stress. However, it is shown that in visco-plastic materials a
material bifurcation occurs when the heat supply through mechanical work surpasses the
diffusion capabilities of the material. This transition from near-isothermal stable evolu-
tion to near-adiabatic thermal runaway is the well-known concept of shear heating. Here,
it is generalized and the correspondence between this runaway instability and the local-
ization of plastic deformation in solid mechanics is detailed. The obtained phase space
6 1 Introduction
controlling the localization is shown to govern the evolution of the system in the post-
yield regime. These results suggest that the energy balance essentially drives the evolu-
tion of the plastic deformation and therefore constitutes a physics-based hardening law
for thermo-viscoplastic materials.
In order to verify the results of Chapter 4 numerically the physical behaviour of temper-
ature sensitive and rate-dependant materials was implemented in a finite element solver,
called REDBACK, that is able model multi-physics Rock mEchanics with Dissipative feed-
BACKs in a tightly coupled manner. In Chapter 5 we detail this novel numerical simulator.
REDBACK provides both the prototyping flexibility to investigate more complex physics
and non-linear feedbacks as well as the computational scalability to tackle three dimen-
sional scenarios. We demonstrate the approach by modelling laboratory experiments on a
diatomaceous mudstone and identify the activation enthalpy dependency on pressure and
temperature that matches the confinement dependency of the experiments. We also extend
the approach to include chemistry and model in three dimensions the behaviour of a fluid-
saturated fault under shear, where fluid-release chemical reactions occur in a rock display-
ing rate- and temperature-dependent frictional behaviour. Those results demonstrate the
importance of a physics-based approach in a multi-scale framework, where one can aim
at extrapolating results outside the range of laboratory experiments based on the under-
standing of the underlying physical processes, when traditional engineering approaches
are often limited to interpolation within the scope of sparse and expensive experiments.
In Chapter 6 our findings are summarized and discussed.
Chapter 2
Dynamic mode interaction in the Swift-Hohenberg equation
Spatially localized patterns have been reported in a wide array of fields including struc-
tural engineering, fluid mechanics, nonlinear optics, gas discharge systems, granular me-
dia and many others (Dawes, 2010). A standard model, which has been widely studied
and exhibits localized solutions, is the Swift–Hohenberg equation (Swift and Hohenberg,
1977)
u = ru−(1+∂
2x)2
u+N(u;s) (2.1)
where u(x, t) ∈ R and r < 0 < s are parameters and N(u;s) denotes nonlinear terms and
popular choices include quadratic–cubic or cubic–quintic nonlinearities. Much of the re-
search interest focused on the static solutions of Eq. (2.1) (i.e. u = 0) and their stabil-
ity. Under certain restricted symmetry conditions and specifically chosen nonlinear terms
that initially soften (destabilize) and subsequently stiffen (re-stabilize), two alternative
forms of localized snaking equilibrium solutions have been found to bifurcate from the
perfectly flat state: one symmetric about some point on the spatial axis and the other anti-
symmetric (e.g. Woods and Champneys (1999); Hunt et al (2000); Burke and Knobloch
(2007a)). Apart from at the trivial state these paths are disconnected, but are linked by
sets of non-symmetric equilibrium states known as ladders (Burke and Knobloch, 2007b;
Dawes, 2010).
In the case of structural engineering a generalized Swift–Hohenberg equation appears
in relation with beams that rest on a foundation and are exposed to compressive forces.
The fourth order term in Eq. (2.1) is related to the bending of the beam, the second order
one to the axial load and the nonlinearity models the foundation. The pioneering contribu-
tions of Potier-Ferry (1983) and Hunt et al (1989) gave the insight that localized solutions
occur if the foundation has softening characteristics. Closely linked to the beam on a foun-
dation is a problem in structural geology, namely that of localized folding of a layer of a
7
8 2 Dynamic mode interaction in the Swift-Hohenberg equation
stiff material which is embedded in a softer matrix (Hunt et al, 1996a; Hobbs and Ord,
2012; Ord and Hobbs, 2013).
Here, motivated by the application of the Swift-Hohenberg equation in structural ge-
ology and wanting to model the slow but relentless movement of tectonic plates (Biot,
1965; Hunt et al, 1996b), we impose a slowly-growing axially-compressive displacement
to one end of a beam and investigate the transient evolution of the folding pattern over
time. Hence, our model is comprised of a generalized Swift-Hohenberg equation sup-
plemented by a constraint equation that takes into account the progressively increasing
end-shortening.
The model studied here that consists of a thin elastic strut that rests on a Winkler type
foundation is detailed and the governing equation is derived in Sec. 2.1. Subsequently,
the static equilibrium solutions of the Swift-Hohenberg equation and the complex snakes
and ladders behaviour are reviewed in Sec. 2.2. The Swift-Hohenberg equation exhibits a
free parameter, namely the axial load, which could either be prescribed as a continuation
parameter or it could be seen as a Lagrange multiplier that follows the end-shortening of
the strut. Here, we choose to prescribe an end-shortening that increases at a constant de-
formation rate and the axial load is a free parameter. In Sec. 2.3, the transient solutions of
the Swift-Hohenberg equation subject to the time-varying end-shortening constraint are
computed using a finite element technique which is described in detail and the results are
summarized and compared to the static solutions. The main result is that at low deforma-
tion rates the transient solutions are close to their static counterparts and exhibit dynamic
bifurcations from a symmetric to an anti-symmetric set of solutions and vice versa. This
dynamic bifurcation point is dependant on the deformation rate and in order to investigate
this rate-dependance a two-mode Galerkin procedure is proposed in Sec. 2.4. This reduc-
tion in degrees of freedom allows us to visualize the dynamics of the transient solutions
of Sec. 2.3 on an energy surface and to interpret the previous findings in terms of sim-
ple energy arguments. Finally, the results and the applicability of the proposed Galerkin
procedure are discussed in Sec. 2.5.
2.1 The model 9
x PP
u
Fig. 2.1 Strut supported by nonlinear springs and linear dashpots in parallel. Axial load P would normallybe accompanied by bending moments and shear forces at the points of application, but these are omitted asthey make no appearance in the formulation.
2.1 The model
In this chapter, an inextensible linear elastic beam of length L which is supported by
a nonlinear visco-elastic foundation is studied and the model is shown in Fig. 2.1. The
deformation of the beam is characterized by the vertical displacement of its centreline
u(x, t), where x is the arc-length measured along the beam, and t > 0 measures time. An
axial, compressive load P is supplied such that one end of the beam is shifted horizontally
by a distance
∆ = L−∫ L
0
√1− (u′)2dx = L−
∫ L
0
(1− 1
2(u′)2 +O
((u′)4))dx, (2.2)
where the primes denoted differentiation with respect to x.
We first formulate the governing equation of the beam in absence of the visco-elastic
foundation and the derivation follows Thompson and Hunt (1973). The curvature of the
beam is defined as
χ =ddx
arcsinu′ =u′′√
1− (u′)2, (2.3)
and the bending energy of a linear elastic strut reads
Eb =B2
∫ L
0χ
2dx =B2
∫ L
0
(u′′)2
1− (u′)2 dx =B2
∫ L
0(u′′)2 (1+(u′)2 +O
((u′)4))dx, (2.4)
where B is the bending stiffness of the strut. If we assume moderately-large deflections
and only retain quadratic terms in Eqs. (2.2) and (2.4) the potential energy of the strut is
V = Eb−P∆ =∫ L
0
(B2(u′′)2− P
2(u′)2
)dx≡
∫ L
0L (u′,u′′)dx, (2.5)
10 2 Dynamic mode interaction in the Swift-Hohenberg equation
0 0.5 1 1.5 20
0.5
1
1.5
2
Vertical displacement u
Foundation forc
e
0 0.5 1 1.5 20
0.5
1
1.5
2
Vertical displacement u
Foundation s
train
energ
y
α = 0
α = 0.25
α = 0.3
α = 0.5
α = 0
α = 0.25
α = 0.3
α = 0.5
Fig. 2.2 Foundation force u−u3 +αu5 and strain energy u2/2−u4/4+αu6/6 of the nonlinear springs inthe foundation. We note only α ≥ 0.25 yields physically admissible forces as otherwise tensile forces wouldoccur. For α = 0.25,0.3 the softening and re-stiffening properties of the foundation are very distinctive.
where L is the Lagrangian of the system. An energy minimum has to satisfy the Euler-
Lagrange equationd2
dx2∂L
∂u′′− d
dx∂L
∂u′= 0, (2.6)
and we find
Bu′′′′+Pu′′ = 0, (2.7)
which can be thought of as a vertical force balance.
Now, the contribution of the foundation is added and for the Winkler-type foundation
we assume a resistive force which is strictly vertical and local and of the form
f (u) = fe(u)+ fv(u) = k1u− k2u3 + k3u5 +η u, (ki,η > 0) (2.8)
where (˙) denotes differentiation with respect to time t, fe(u) = k1u− k2u3 + k3u5 rep-
resents the force contribution of the nonlinear elastic springs and fv(u) = η u is the con-
tribution of the viscous element. This foundation is of a softening and re-stiffening type
due to the cubic–quintic nonlinearity, which is shown in Fig. 2.2. As the vertical displace-
ment increase the cubic term gains importance and softens the foundation such that it is
energetically favourable to deflect the strut further where the foundation is soft. As this
deflection progresses the quintic term competes with the cubic one and eventually this
quintic term stiffens the foundation. This leads to a trade-off between either growing a
fold at the current position or to establish a new fold close by in a softer region. Hence,
the softening and re-stiffening of the foundation leads to so-called snaking where a fold
localizes and expands sequentially due to this ongoing competition of softening and stiff-
2.1 The model 11
ening and we explain this behaviour in more detail in Sec. 2.2. There are other choices
of foundation forces and we discuss these and their effect on the buckling solutions in
Sec. 2.2.1.
The governing equation of the system, which can be viewed as a force balance between
the strut and foundation, is the combination of Eqs. (2.7, 2.8),
Bu′′′′+Pu′′+ k1u− k2u3 + k3u5 +η u = 0. (2.9)
We subject this equation to the constraint
∆ =12
∫ L
0(u′)2dx = ρt, (2.10)
where ρ is a deformation rate and this time-dependent constraint is chosen because we
wish to investigate the transient evolution of localized buckling patterns. Given the above
constraint the present model possesses similarities to geological scenarios such as plate
tectonics.
To generalize the problem to a smaller parametric group, (2.9) is rescaled using
x 7→(
Bk1
)1/4
x, u 7→√
k1
k2u, and t 7→ η
k1t, (2.11)
yielding the non-dimensional form,
u =−(
u′′′′+ pu′′+u−u3 +αu5)
and12
∫∞
−∞
(u′)2 dx = Rt, (2.12)
where
p(t) =P(t)√
Bk1, α =
k1k3
k22
and R =ρηk2
k21
(k1
B
)1/4
. (2.13)
We note that the final system is in fact a two parametric group in (α,R), as the load
parameter p(t) is a free variable directly imposed by the shortening constraint.
We conclude by mentioning that the total strain energy of strut and foundation is
E(t,u) =∫ L
0
(12
u′′2 +12
u2− 14
u4 +α
6u6)
dx, (2.14)
and this energy functional is used to construct the energy surfaces in Sec. 2.4.2.
12 2 Dynamic mode interaction in the Swift-Hohenberg equation
2.2 Static equilibrium solutions
In this section the symmetries of Eq. (2.12) are reviewed that lead to a branch for sym-
metric and one for anti-symmetric solutions in the p/∆–bifurcation diagram and we show
that these branches form the famous snakes and ladders structure.
2.2.1 Reversibility and symmetry considerations
Before seeking solutions of the dynamical system represented by Eq. (2.12), it is useful
first to review the associated stationary solutions, as expressed by the reduced fourth-order
reversible ordinary differential equation in the spatial variable x
u′′′′+ pu′′+u−u3 +αu5 = 0. (2.15)
We have chosen to give the foundation a ‘cubic-quintic’ nonlinearity and the resistive
force of the nonlinear elastic springs is fe(u) = u− u3 +αu5. Other examples found in
the literature are ‘quadratic’ ( fe(u) = u−u2) and ‘quadratic-cubic’ ( fe(u) = u−u2+αu3)
foundations. All these forms are reversible in that no odd derivatives of u with respect to
x appear in either the static (2.15) or the dynamic (2.12) governing equation and solutions
are then the same whether x runs forwards or backwards. Consequently, those emerg-
ing from the symmetric section, where u′(0) = u′′′(0) = 0 holds, must be symmetric if
reflected on the axis x = x0. This is expressed by invariance under the transformation
RS : x 7→ −x, u 7→ u (2.16)
and allows for the appearance of symmetric homoclinic (localized) solutions that start
close to the flat state, pass through the symmetric section, and reflect back to the flat state
again.
There is, however, a further symmetry in our chosen cubic-quintic foundation, which
is absent from the alternatives described. The nonlinear foundation characteristic has no
even powers of u, and so equal-and-opposite deflections into the foundation are met with
equal-and-opposite forces. Now, solutions running forwards and backwards from the anti-
symmetric section, where u(0) = u′′(0) = 0 holds, must again be the same, reflected about
both the u- and x-axes. This further symmetry condition is expressed by invariance under
2.2 Static equilibrium solutions 13
the transformation
RA : x 7→ −x, u 7→ −u. (2.17)
The possibility then arises of anti-symmetric homoclinic solutions, unavailable to beams
with either the quadratic or quadratic-cubic types of nonlinear foundation, which start
from close to the flat state, pass through the anti-symmetric section and return to flat.
These symmetric and anti-symmetric solution form the characteristic snakes and ladders
bifurcation diagram shown in the next section.
2.2.2 Snakes and ladders
It is well-known (Hunt et al, 1989; Champneys and Toland, 1993) that, at p = 2 the gov-
erning equilibrium equation (2.15) gives rise to a Hamiltonian-Hopf bifurcation from the
trivial unbuckled state u(x) = 0, into a periodic buckling mode. Depending on the bound-
ary conditions, this could either be written as us = ±qs cosx and considered symmetric,
or ua = ±qa sinx and be anti-symmetric. We compute both these two primary solutions
branches in AUTO (Doedel et al, 2007) over the half-space x ∈ [0,L]. The arc-length con-
tinuation is started close to the bifurcation point and seeded with either the symmetric or
anti-symmetric eigenmode. At x = 0, symmetric (u′(0) = u′′′(0) = 0) or anti-symmetric
(u(0) = u′′(0) = 0) boundary conditions are imposed and at x = L the beam is clamped
(u(L) = u′′(L) = 0). An alternative choice of boundary conditions at x = L would be ho-
moclinic boundary conditions (Hunt et al, 2013) that implement the fact that the localized
solution converges exponentially fast to the flat state as x→ ∞. The choice of clamped
boundary conditions is therefore only justifiable if L is sufficiently large. During the arc-
length continuation, we treat the load p in Eq. (2.15) as a free parameter and choose the
end-shortening ∆ as continuation parameter.
Fig. 2.3 shows a typical load–end-shortening bifurcation diagram and corresponding
solution shapes as ∆ increases. The solution branches form the classic snakes-and-ladders
configuration of a pair of snaking equilibrium solutions, one symmetric and the other
anti-symmetric, with each solution appearing with fluctuating load as end-shortening in-
creases. The equilibrium shapes themselves are all homoclinic, with a central region that
grows with ∆ – these are the initial stages of a heteroclinic connection to a periodic state
14 2 Dynamic mode interaction in the Swift-Hohenberg equation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
1.2
1.4
1.6
1.8
2
∆
p
Symmetric
Anti−symmetric
0 1 2 3 4 5 6 7 8
1.2
1.4
1.6
1.8
2
p
∆
α = 0.4 α = 0.5
α = 0.3
(a)
(b)
Fig. 2.3 (a) Path of the static solutions in p-∆–space for different values of the foundation stiffness, namelyα = 0.3,0.4,0.5. The symmetric and anti-symmetric branches exhibit the familiar snaking behaviour. (b)Snakes and ladders for α = 0.3. Black and grey denote stable and unstable static states and we observe thatthe symmetric and anti-symmetric modes alternate in stability. The inset plots show the spatial configura-tion of the folded patterns and we notice that the evolution of the buckling pattern is a trade-off betweengrowing individual buckles and adding additional buckles. If one buckle grows up to the point where itis energetically unfavourable to grow any further because the foundation becomes too stiff the symmetrychanges and an additional buckle evolves.
at the Maxwell load (Hunt et al, 2000). Along the snakes the folded profiles are each a
function of both x and ∆ and they change shape but also grow in amplitude as ∆ increases.
The snaking paths are connected at bifurcation points by the rungs of a ladder (Burke
and Knobloch, 2007b; Dawes, 2010), comprising states of transition between symmetry
and anti-symmetry and the ladder itself is neither symmetric or anti-symmetric. For a
recent account of such behaviour see Kao et al (2014). Unlike the response under con-
2.3 Evolution of transient folding patterns 15
trolled load, where limit points have a part to play, stability here is only lost or gained at
bifurcation points (Thompson, 1979).
2.3 Evolution of transient folding patterns
For transient states in the evolution of Eq. (2.12), solutions cannot be constrained to be
either symmetric or anti-symmetric. The dynamics gives rise to possible interactions be-
tween these two standard forms and this leads to intriguing behaviour not observed in the
stationary counterpart. First, we detail the integration of the gradient flow problem (2.12)
employing finite element techniques and subsequently we discuss observations made dur-
ing the simulation of the aforementioned mode interactions.
2.3.1 Finite element procedure for constrained gradient flow
In this section, we present a finite element procedure that allows us to solve Eq. (2.12)
and the method follows closely Peletier (2001). Firstly, we discretize Eq. (2.12) over the
large but finite domain X := [−L,L] by computing its weak form and impose the boundary
conditions u(±L) = u′′(±L) = 0. The weak form of Eq. (2.12) is found by multiplying it
by a suitable test function v and integrating over the domain X giving
∫X
uvdx =−∫
Xu′′v′′ dx+ p
∫X
u′v′ dx−∫
Xfe(u)vdx. (2.18)
The domain X is discretized into N nodes xi = ih−L where h = 2L/N and i = 0,1, . . . ,N
where each node possesses two degrees of freedom (ui and u′i) and the boundary condi-
tions require that u0 = uN = 0. Then, the finite element solution over x ∈ [xi,xi+1] can be
approximated as
uh(t) = ui(t)φi +ui+1(t)φi +u′i(t)ψi +u′i+1(t)ψi+1 (2.19)
where φi, (i = 1, . . . ,N−1) and ψi, (i = 0, . . . ,N) are cubic shape functions defined by
φi(x j) = ψ′i (x j) = δi j and ψi(x j) = φ
′i (x j) = 0 (i, j = 0, . . . ,N). (2.20)
16 2 Dynamic mode interaction in the Swift-Hohenberg equation
Hence, the shape functions read
φi = 2s3−3s2 +1, φi+1 = s3−2s2 + s, (2.21)
ψi =−2s3 +3s2, ψi+1 = s3− s2, (2.22)
where s = (x− xi)/(xi+1− xi), x ∈ [xi,xi+1]. The nodal degrees of freedom are collected
in a single vector U(t) = (u1(t), . . . ,uN−1(t),u′0(t), . . . ,u′N(t))
T , so that the finite element
solutions of (2.18) are solutions to the system of equations
AU =−(BU− pCU +D), (2.23)
where the 2N×2N matrices are defined as
Ai j =∫
Xφiφ j dx, Bi j =
∫X
φ′′i φ′′j dx, Ci j =
∫X
φ′i φ′j dx and Di =
∫X
fe(U)φidx.
(2.24)
Since Di is the integral of a polynomial of order 15, a ten-point Gaussian integration is
used to calculate all integrals exactly (Cook et al, 1989). The shortening constraint is
included, first by differentiating the non-dimensional form of Eq. (2.10) with respect to
time, then rewriting in matrix form
UTCU = R. (2.25)
Thus the numerical reduction of the constrained gradient flow equation (2.12) to a system
of differential algebraic equations of index–1UTC 0
A 0
U
p
=
R
−(B− pC)U−D
, (2.26)
is achieved and such systems of differential algebraic equations can be solved using MAT-
LAB’s inbuilt function ode23s (Shampine et al, 1999) or ode15s and this is the ap-
proach chosen here. Alternatively, the Fortran based code DDASSL (Petzold, 1982) could
be employed.
2.3 Evolution of transient folding patterns 17
2.3.2 Numerical experiments
Here, the dynamic behaviour of Eq. (2.26) for different rates of loading R is investigated
and all calculations presented were carried out over a uniform mesh with N = 196 and
L = 100. The stiffness parameter of (2.12) was taken as α = 0.3, allowing comparisons
with the stationary solutions of Fig. 2.3 as computed in Sec. 2.2.2. Fig. 2.4 shows a series
of numerical solutions for increasing rates of loading ranging from R = 10−5 to R = 10−1.
To avoid the system becoming locked in the trivial equilibrium state, all runs are seeded
by an incremental displacement into the symmetric localised modeshape.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
1.2
1.4
1.6
1.8
2
∆
p
Fig. 2.4 The deformation rate R was varied from 10−5 to 10−1 and the arrow indicates the trend of theload-displacement curve as the rate increases. We observe that the load-displacement curve follows thestatic equivalent more closely for lower R and can be considered quasi-static. As R increases the bifurcationfrom the symmetric to anti-symmetric mode (or vice versa) is delayed until the strut is deformed too quicklyand the transition vanishes entirely. Further, the curves drift away from the static solution branches and theevolution is governed by the dynamics.
In Fig. 2.4 the grey lines represent the load–displacement curves of the static states
and the black lines are the load–displacement curves of the transient solution. The arrow
indicates the trend of the curves as R increases and we make some general observations:
• For low deformation rates (R < 10−4) the system is quasi-static. Jumps between near-
equilibrium states occur immediately or soon after stability is lost.
• Increasing the deformation rate means that the solution tends to drift further from the
static solution. It also tends to delay the jumps so they can occur with increasing as
well as decreasing load.
18 2 Dynamic mode interaction in the Swift-Hohenberg equation
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
1.2
1.4
1.6
∆
p
(a)
10−4
10−3
10−2
10−1
2
2.5
3
3.5
4
R
∆∗
(b)
Fig. 2.5 (a) The deformation rate R was varied from 5 · 10−5 to 5 · 10−2 and the bifurcation point ∆ ∗ wasrecorded (black squares). In order to locate the bifurcation the point of maximum curvature along the p-∆–curve is found. (b) For each rate R the bifurcation point ∆ ∗ is shown. The solid line represents a fitin the least squares sense and the fitted expression is ∆ ∗ = m ·Rn +∆0, where m = 9.021, n = 0.515 and∆0 = 2.134. Hence, ∆ ∗ ∝
√R.
• Behaviour at high rates is dominated by the dynamics. Significantly, a higher rate can
lead to a jump being by passed, so the system remains in a symmetric or anti-symmetric
state even when its stationary counterpart has become unstable.
These general observations are confirmed by the results presented in Fig. 2.5 where the
rate is varied between 5 ·10−5 ≤ R≤ 5 ·10−2 and the bifurcation point ∆ ∗ was recorded.
In order to find the bifurcation point we locate the point of maximum curvature along
the p-∆–curve and the results are presented in Fig. 2.5. The expression ∆ ∗ = m ·Rn +
∆0 was fitted in the least squares sense to our results and we find that m = 9.021, n =
0.515 and ∆0 = 2.134 and it appears that ∆ ∗ ∝√
R. There are slight deviations from
this trend around R ∼ 4 · 10−3 which can be attributed to the fact that the bifurcation
point is close to the intersection of the two snakes and our curvature based determination
of ∆ ∗ fails. This square root is very intriguing and has been observed in other systems
with dynamical bifurcations (Gaeta, 1995; Berglund and Gentz, 2002). An example is the
pitchfork bifurcation of the dynamical system
y =(λ − y2)y, (2.27)
λ = µ, (2.28)
2.3 Evolution of transient folding patterns 19
where y ∈ R, λ is a bifurcation parameter and λ = µt such that µ can be seen as defor-
mation rate analogous to R. The pitchfork bifurcation is located at (y,λ ) = (0,0), but this
bifurcation is delayed depending on the value of µ . If λ > 0 the stationary state y? = 0 is
unstable and Eq. (2.27) can be linearised around this stationary state. The solution of the
linearised system reads y(t) = exp(µt2/2
)= exp
(λ 2/(2µ)
). Hence, if the bifurcation
point is operationally defined as the value of λ at which y(λ = µt) ≥ δ , where δ is a
small threshold, then this condition is satisfied when λ = O(√
µ). This can be verified
numerically and results are presented in Fig. 2.6.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
λ = µ t
y
10−4
10−3
10−2
0
0.2
0.4
0.6
λ∗
µ
(a)
(b)
Fig. 2.6 (a) Dynamical pitchfork bifurcation of Eq. (2.27). The solid black line denotes the steady state inthe stationary case, y? =
√λ , and the grey lines represent numerical results at various rates µ . The squares
mark the delayed dynamical bifurcations and the bifurcation point is defined via y(λ = µt) ≥ δ = 10−2.(b) Bifurcation point λ ∗ as function of µ . The solid line denotes a least-square fit of λ ∗ = Aµn +λ0, whereA = 6.37, λ0 = 0.02 and n = 0.534.
The observed behaviour is complex and very interesting. The most important feature
we wish to consider here is the clear dynamical interplay between ‘symmetric’ and ‘anti-
symmetric’ which appears critical at slow rates of loading R< 5 ·10−3 and the delay of the
bifurcation. We propose that at low rates the solution can be written as a superposition of
the symmetric and anti-symmetric static solutions. This assumption reduces the number
of degrees of freedom to two and leads to a Galerkin procedure which is described in the
following section.
20 2 Dynamic mode interaction in the Swift-Hohenberg equation
2.4 Evolutionary Galerkin procedure
As can be seen from the finite element analysis, the buckling pattern stays close to the
stationary states if the deformation rates are reasonably slow. Hence, it is interesting to
see whether the buckling pattern could be expanded in terms of the symmetric and anti-
symmetric stationary states for any given time and whether this Galerkin procedure yields
satisfying results. With this reduction to two degrees of freedom a visual interpretation of
the results from the finite element analysis is available that is based on an energy surface,
which is constructed via the stationary states.
2.4.1 Overview and formulation
As the applied deformation rate increases from zero, the dynamical solutions of the partial
differential equation (2.12) might be expected to diverge from their statical counterparts.
A key question is then: how well can a description based solely on the static mode shapes
represent the dynamical behaviour? Closely related to this is a second question: can such
a description reveal any underlying behaviour or phenomena that is left more obscure in
the larger dynamical context? We therefore look to tease apart the dynamical solutions,
by decomposing into linear combinations of the symmetric ψs and anti-symmetric ψa
stationary mode at the given ∆ = Rt,
u(x, t) = qa(t)ψa(x, t)+qs(t)ψs(x, t). (2.29)
The stationary modes are readily available from the arc-length continuation in Sec. 2.2.2,
but these calculation were only performed over the half-space [0,L] and therefore the
symmetric and anti-symmetric modes have to be reflected appropriately such that the
corresponding shapes over [−L,L] can be constructed and the comparison to the finite
element technique in Sec. 2.3 is meaningful.
As the nonlinearity in the governing equation is likely to manifest itself in a way that
cannot be fully represented by the proposed linear combination, this Galerkin procedure is
likely to be subject to some error but for slow evolution rates this error might be expected
to be small.
In this reduced system, at any time t the end-shortening ∆ = Rt is given by,
2.4 Evolutionary Galerkin procedure 21
∆ =12
∫X
u′2dx =12
q2s
∫X
ψ′2s dx+
12
q2a
∫X
ψ′2a dx, (2.30)
where X = [−L,L] and the cross-term involving∫
X ψ ′sψ′a vanishes because the integrand
is odd. Since 1/2∫
ψ ′2s dx = 1/2∫
ψ ′2a dx = ∆ , we conclude q2s + q2
a = 1. Thus, the end-
shortening constraint is satisfied as long the mode amplitudes, qs and qa, lie on the unit
circle and we choose to write
qs = cosθ , qa = sinθ . (2.31)
Therefore, the description reduces to a single degree of freedom allowing the dynamical
response to be visualized on the energy surfaces discussed later (c.f. Fig. 2.7).
In order to find the governing equation of this reduced system, Eqs. (2.29) and (2.31)
are substituted into (2.12) which gives
−sinθθψs + cosθψs + cosθθψa + sinθψa = (2.32)
−(cosθψ
′′′′s + sinθψ
′′′′a + p(cosθψ
′′s + sinθψ
′′a )+ fe (cosθψs + sinθψa)
),
where fe = u−u3+αu5 is the elastic part of the foundation force in nondimensional form
(c.f. Eq. (2.8)). An evolution equation for θ is found by multiplying the above expression
by ψs and ψa and integrating over the domain X . The two equations derived this way can
be solved for p and θ simultaneously and
Aa cosθθ =
(−1
2Aa−Ba + pCa
)sinθ −Da (2.33)
(cos2
θCaAs + sin2θCsAa
)p = cos2
θ
(Bs +
12
As
)Aa + sin2
θ
(Ba +
12
Aa
)As (2.34)
+ cosθDsAa + sinθDaAs
where
Ai =∫
ψ2i , Bi =
∫ψ′′2i , Ci =
∫ψ′2i , Di =
∫fe(u)ψi. (i = a,s) (2.35)
Alternatively, the ansatz (2.29) can be substituted into the energy functional
E[u] =∫
X
12(u′′)2− p
2(u′)2
+12
u2− 14
u4 +α
6u6 dx (2.36)
22 2 Dynamic mode interaction in the Swift-Hohenberg equation
and calculating the force −(∂E/∂qs,∂E/∂qa) yields equivalent expressions. Hence, the
Galerkin procedure can be seen as a gradient flow on the energy surfaces as detailed in
Sec. 2.4.2.
The system of Eqs. (2.33, 2.34) constitutes a system of differential algebraic equations
of index–1 which could be solved with the MATLAB routine ode23s or ode15s as be-
fore. However, it was found that when the system sits in an energy minimum that evolves
into a maximum, it tends to remain at that maximum during the subsequent evolution.
Exceptions are if R < 10−6, but then the state transition is strongly delayed from the point
where the system lost stability. The reason for this behaviour could be that Eq. (2.33) is
a first order equation and, hence, does not contain inertia terms and the velocity in the
θ -direction, vθ , is always equal to the current slope of the energy surface since the system
does not accelerate as it moves down the slope. Therefore, if the system is close to a max-
imum where the energy surface is flat the side-ways velocity is marginal and the system
stays close to the maximum. In order to force the system off a maximum a small sideways
driving force is required and during the evolution the system is restrained from sitting in
either the purely symmetric or the purely anti-symmetric state. This is achieved by insist-
ing that θb ≤ θ ≤ π/2−θb, where the bias θb = 10−12 is a small positive constant. This
limits solutions to the first quadrant of the unit circle, and is implemented in the solution
code by resetting θ should it be out of bounds. Due to the requirement of the bias θb,
an Euler forward integration with a fixed time step is used to solve Eq. (2.33) instead of
MATLAB’s inbuilt routines that do not allow such control in a convenient manner.
2.4.2 Energy surfaces
The proposed two-mode ansatz u = cosθψs + sinθψa allows us to compute the energy
for any value of θ and ∆ according to Eq. (2.14) and construct an energy surface over the
(θ ,∆)-plane. Hence, the energy is defined in terms of relative amplitudes qs = cosθ and
qa = sinθ of the two known localized modeshapes of the static problem: ψs (symmetric
about some point x) and ψa (anti-symmetric about the same point). Two assumptions are
key to this reduced dimensional view: (1) that these modeshapes can usefully represent
the solutions of the related PDE at the same end-shortening ∆ , and (2) alone or in linear
combination, they capture reasonably well the mode shapes that can occur. The first as-
2.4 Evolutionary Galerkin procedure 23
End−
short
enin
g ∆
α = 0.3
0
1
2
3
4
5
End−
short
enin
g ∆
α = 0.4
1
2
3
4
5
End−
short
enin
g ∆
θ [π]
α = 0.5
0 0.25 0.5
1
2
3
4
5
Fig. 2.7 Energy landscapes constructed for the two-mode ansatz (2.29) for various values of α , the stiffnessparameter of the foundation. The mode amplitudes are expressed as qs = cosθ and qa = sinθ , θ ∈ [0,π/2].The energy of the symmetric mode ψs was taken as the reference value and darker areas mean higherpotential energy than brighter areas. The red line highlights the global energy maximum.
sumption is clearly valid since the chosen modeshapes are solutions to the governing PDE
and the second assumption should be satisfied at least at low rates as suggested by the re-
sults in Sec. 2.3.2. But the validity of the second assumption is likely to be questionable
especially as the rate is increased.
The contours of energy surfaces constructed as described above are shown in Figs. 2.7
where darker areas denote higher potential energy than lighter areas and the datum of the
energy surfaces was chosen to be the energy of ψs. We would like to emphasize that the
movement along the ∆ -axis of this energy surface is prescribed by the constraint ∆ = Rt
but the system is allowed to evolve freely in the θ -direction according to the gradient
24 2 Dynamic mode interaction in the Swift-Hohenberg equation
2 2.5 31.1
1.2
1.3
1.4
1.5
1.6
p
∆
0 0.25 0.5−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
θ [π]
Energ
y
∆ = 2.15
∆ = 2.25
∆∗∗
∆∗
(a) (b)
Fig. 2.8 (a) Enlargement of Fig. 2.3(b) around first ladder. At the bifurcation point ∆ ∗= 2.198 the stationarysolutions along the symmetric snake become unstable. At the value ∆ ∗∗ = 2.35 the symmetric snake attainsa global energy maximum in the energy surface constructed with the two-mode ansatz (2.29). (b) Energycross-section of the energy surface constructed from the two-mode ansatz for α = 0.3. We note that for∆ < ∆ ∗ the symmetric state is a global energy minimum in the two-mode approach and for ∆ > ∆ ∗ theanti-symmetric state becomes the global minimum but the symmetric one remains a local minimum and thebifurcation is delayed in the two-mode ansatz.
flow, Eq. (2.33). Hence, only the energy differences along the θ -axis are important for
any given ∆ and the presented energy surfaces contain all relevant information.
We start with some general remarks about the energy surfaces. As one follows the anti-
symmetric state (θ = π/2) along the ∆ -axis the state alternates between energy minima
and maxima which is an imprint of the alternating stability of the static equilibria observed
in Fig. 2.3. For all values of ∆ , either the symmetric or anti-symmetric static state is the
global energy minimum and the location of the global minimum changes between the
static states at the bifurcation points ∆ ∗ computed with AUTO in Sec. 2.2.2 (refer to
Fig. 2.8).
Since the Galerkin procedure described here is in effect a gradient flow along the θ -
direction the global energy maximum maxθ E(θ ,∆) is of great importance as it deter-
mines the direction of the flow and the global energy maximum, which is highlighted in
red in Fig. 2.7. A state transition ψs→ψa is only possible once ψs attains a global energy
maximum and analogously for ψa → ψs. We denote the value of end-shortening where
this occurs with ∆ ∗∗. We find that ∆ ∗∗ > ∆ ∗, where ∆ ∗ refers to the related bifurcation
point of the unrestricted static case (c.f. Fig. 2.8). As an example we consider the first bi-
furcation point along the symmetric branch in Fig. 2.3(b), which is located at ∆ ∗ = 2.198.
In Fig. 2.8, a cross-section of the energy surface is presented for ∆1 = 2.15 and ∆2 = 2.25
and we notice that the symmetric mode ψs is the global energy minimum at ∆1, but at
∆2 this mode has become a local minimum. This is the imprint of the bifurcation at ∆ ∗
2.4 Evolutionary Galerkin procedure 25
0 0.25 0.50
0.1
0.2
0.3
0.4
θ [π]
2.5 cos2θ sin
4θ
1.5 cos2θ sin
2θ
2.5 cos4θ sin
2θ
Fig. 2.9 Energy contributions that involve a mix of ψs and ψa (c.f. Eq. (2.37)). These cross terms gothrough extrema as the system transitions from ψs→ ψa (or conversely) and cause the bifurcation delay inthe two-mode approximation compared to bifurcation diagram of the static solutions.
that occurs in the unrestricted static case but a state transition is not yet possible in terms
of the two-mode ansatz. This discrepancy between the bifurcation behaviour of the un-
restricted static case and the two mode approximation can be understood if the energy
functional (2.14) is evaluated for our ansatz (2.29) and
E[qsψs +qaψa] = E[qsψs]+E[qaψa]−32
q2s q2
a
∫X
ψ2s ψ
2a dx (2.37)
+5α
2q4
s q2a
∫X
ψ4s ψ
2a dx+
5α
2q2
s q4a
∫X
ψ2s ψ
4a dx.
In the above expression, the first two terms on the right hand side are the energies of
the static states modulated by the mode amplitudes such that one vanishes and the other
rises as one transitions from ψs to ψa and vice versa. The following three cross-terms
vanish at the limits θ = 0 and θ = π/2 but pass extrema for intermediate values of θ
and the dependence of these terms on θ is shown in Fig. 2.9. Therefore, the form of
the energy surface is a trade-off between the cross-terms and the energies of the static
modes. Hence, the cross-terms that stem from the nonlinearity of the foundation are the
cause for the delay of the bifurcation. As shown in Fig. 2.7 and Eq. (2.37) the spread of
the zones of bistability where both modes ψs and ψa are stable increases as α increases
and for α = 0.5 the static states never attain a global energy maximum such that no
bifurcation between the two could be observed. This also seems to indicate that if the
relative energy difference between the symmetric and anti-symmetric shape decreases
the discrepancy between the unrestricted static case and the two-mode approximation
becomes more pronounced. Hence, from now on we focus on the case α = 0.3 where
26 2 Dynamic mode interaction in the Swift-Hohenberg equation
the discrepancy between the stability of the full system and the two mode description is
the least pronounced. As a side note, in order to remove the discrepancy further mode
shapes could be incorporated into the Galerkin procedure. It was numerically verified that
the bifurcation could be triggered at the correct ∆ ∗ as compared to the full static case if
the eigenvector of the mode shape at the bifurcation point, which corresponds to the zero
eigenvalue is added (van der Veen et al, 2000). The addition of further modes, however, is
counter productive for the current study since the aim is to give a visual interpretation of
the dynamic behaviour observed earlier and the addition of further modes would require
us to draw the energy surfaces in higher dimensions.
2.4.3 Routes over the energy surface
θ [π]
∆
0 0.25 0.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Fig. 2.10 Routes over energy surface in red for rates between 10−5 ≤ R≤ 10−1. If the rate is smaller than acritical value Rc the system transitions from ψs→ ψa and back, but beyond Rc the trajectories remain closeto ψs for all times. The energy contours are equivalent to the ones in Fig. 2.7.
Eq. (2.33) was integrated for a range of rates and the routes through the energy surface
that the system takes are highlighted in Fig. 2.10. These routes can be interpreted as the
trajectories of a ball that rolls over the energy surfaces and the velocity of this ball along
the ∆ -axis is prescribed as v∆ = R, but the velocity of the ball along the θ -axis, vθ , is
determined by the gradient of the surface. This analogy is not completely suitable as
Eq. (2.33) is a first order equation and hence, does not contain inertia terms, and vθ is
2.4 Evolutionary Galerkin procedure 27
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
1.2
1.4
1.6
1.8
2
∆
p
R = 10−1
Fig. 2.11 Load–end-shortening curves for the routes through the energy landscape seen in Fig. 2.10, com-pared against the stationary equilibrium states. The arrow marks the trend of the curves as R increases.
always equal to the current slope and the system does not accelerate as the system rolls
down a slope.
The routes for 10−5 ≤ R ≤ 10−1 presented in Fig. 2.10 can be compared to the obser-
vations in Sec. 2.3.2. Figs. 2.10 and 2.11 present the effect of increasing R. As R increases
the transition ψs → ψa is delayed and in the extreme case R is so large that the system
passes the energy maximum of ψs and remains in the symmetric state permanently. Along
each trajectory the load p can be recorded and plotted over ∆ (c.f. Fig. 2.11) and we ob-
serve similar features as in Fig. 2.4, namely that the bifurcation point is delayed as R
increases and eventually a drift away from the stationary bifurcation curves occurs. Since
the two-mode Galerkin procedure is based on the static modes a drift away from those
solutions is only poorly captured which leads to the kink for the curve at R = 10−1.
The energy surface perspective presented here allows us to define a critical rate which
marks the point after which the system stays close to the symmetric mode for all time. In
Fig. 2.12 the results of a bisection search are presented and a lower bound of the critical
rate is Rc = 8.2077455491 · 10−4± 10−15 meaning that for rates beyond Rc the system
never fully transitions to ψa.
28 2 Dynamic mode interaction in the Swift-Hohenberg equation
θ [π]
∆
0 0.25 0.5
2.5
3
3.5
4
4.5
Fig. 2.12 Runs with identical initial conditions but at incrementally different rates remain either in modeψs or transition to ψa. All runs are performed around Rc = 8.208 ·10−4 and the difference in rate betweenthe final runs is ∆R = 10−15. Hence, Rc can be defined as a critical rate.
2.5 Discussion
In this chapter – motivated by observations of localized folding in geological settings
and the permanent deformation of geological structures due to mechanisms such as plate
tectonics – an analysis of localized folding patterns of the Swift-Hohenberg equation sub-
jected to continuous deformation was attempted. We chose the Swift-Hohenberg equation
because it can be seen to model an elastic beam supported by a visco-elastic foundation
and the static localized solutions are well understood, which was an ideal starting point
for our investigation. We extended the static perspective of the well known snakes and
ladders behaviour to the time-dependent regime and investigated the importance of the
deformation rate R. Secondly, the results of a finite element analysis were shown and we
noticed that for low R the evolution is quasi-static but an increase in R lead to a divergence
away from the static modes. Most notably the transient solutions follow the stability of
the static solutions for low rates and in order to understand the mode interactions at low
rates, we proposed a two-mode Galerkin procedure, which employed the static modes
along the snakes to expand the transient solution. This procedure lead to a reduction to a
system with one degree of freedom such that an energy surface could be constructed and
visualized. This approach has some interesting properties, and introduces what may prove
to be a useful visual tool in the analysis of visco-elastic behaviour. The energy perspective
allowed us to interpret the results of the finite element analysis in a convenient manner.
2.5 Discussion 29
General features observed in the static solutions and the finite element analysis could be
reproduced with the reduction to the two-mode system, but this approach possesses some
discrepancies if compared to the stability and behaviour of the unrestricted case. Hence,
more modes would be required to capture the full dynamics observed in the finite element
analysis and some modes that could remedy the discrepancies have been discussed in the
main text. However a higher number of degrees of freedom would prohibit a convenient
visual interpretation. Since this visual interpretation was the main purpose of this work
the Galerkin formulation has not been extended to higher degrees of freedom. Further,
the suitability of this extension is questionable since the finite element analysis or other
numerical techniques such as collocation methods are superior to the Galerkin procedure
especially since during the Galerkin procedure the solution is expanded in modes that are
derived from collocation techniques. Therefore, the presented procedure should only be
used for a qualitative interpretation and is not meant to compete with proven methods
such as the aforementioned ones.
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
p
∆
Fig. 2.13 Snakes and ladders for quadratic–quintic foundation with a restoring force fe = u−u2 +0.23u3.We note that the snakes are strongly skewed to the right and the Galerkin procedure presented here, whichis based on the static solutions along the snakes, would require at least four modes for most values of ∆ ,e.g. ∆ = 6, which is highlighted above.
Another shortcoming of the Galerkin procedure is that it might be hard to adapt to other
foundation types and this chapter exclusively focused on the cubic–quintic foundation and
other foundations such as quadratic–cubic ones, fe = u−u2+αu3, have not been consid-
ered. Fig. 2.13 shows the snakes and ladders bifurcation diagram for the quadratic–cubic
case and we notice that the snaking structure is strongly skewed such that for most values
30 2 Dynamic mode interaction in the Swift-Hohenberg equation
of ∆ at least four primary stationary modes would be required. As an example, a vertical
line at ∆ = 6 intersects each snaking branch twice such that at least four modes should be
included in a reduction similar to our Galerkin procedure (c.f. Fig. 2.13). Following the
arguments before, this would not permit a simple two-dimensional energy surface to be
constructed.
This chapter only offers qualitative interpretations and a quantitative study is left for fu-
ture work. Some of the questions that remain open include the characterization of the drift
away from the quasi-static solutions, which was observed in the finite element analysis.
We expect this drift to be related to a competition between the deformation rate at which
work is done on the system and the rate at which the system can dissipate energy in the
viscous elements of the foundation. Another question relates to the delay of the dynam-
ical bifurcation in the unrestricted system that appears to scale as√
R and one possible
approach might be to interpret the evolution of the buckling pattern as a slow process and
the evolution as a fast one (Kuehn, 2011). We hope that our qualitative study can yield
valuable insight for future work.
Chapter 3
Self-similar blow-up solutions of the nonlinear Schrodinger
equation
After having discussed localization in nonlinear elastic system in the previous chapter we
discuss a phenomenon related to localization in this chapter. Here we are concerned with
blow-up solutions, which have important applications in geology. Blow-up describes the
fact that a physical quantity can become unbounded in finite time and at a finite distance.
On example is thermal runaway (Veveakis et al, 2010) that can occur when dissipative
materials are deformed. To be more precise, in Chapter 4 we discuss the energy budget of
a elasto-visco-plastic material and we show that there is critical energy input below which
the material deforms homogeneously but if this critical value is exceeded heat dissipation
localizes which consequently results in localized plastic deformation.
Another example is the formation of fluid path ways in shales that is triggered by a dia-
genetic (fluid release) reaction if the material is under compression as shown by Veveakis
et al (2015). Shales are sedimentary rocks that usually absorb fluids and cement, which
renders them impermeable. But at high ambient pressures and temperatures fluid release
reactions can be activated and Veveakis et al (2015) derive in a hydro-mechanical frame-
work at isothermal conditions a governing equation for the fluid pressure p that reads
∂ p2
∂ z2 −λ p3 = 0, λ ∈ R, (3.1)
where λ is a material parameter and z measures the height of the sample. The solutions
of Eq. (3.1) have vanishing p throughout most parts of the shales except for periodic loci
at which the pressure peaks and a fluid paths emerge, which is of concern in oil drilling
operations.
In the case of λ = −1 Eq. (3.1) can be seen as the stationary limit of the Schrodinger
equation, which is discussed in the current chapter. Thus, chapter serves as a transition
31
32 3 Self-similar blow-up solutions of the nonlinear Schrodinger equation
from nonlinear elastic materials to plastic materials, which we discuss in a multi-physics
framework in later chapters.
3.1 Blow-up solutions of the nonlinear Schrodinger equation
Here, we study self-similar blow-up solutions of the nonlinear Schrodinger equation,
i∂tu+∆u+u|u|2 = 0, (3.2)
u(x,0) = u0(x), x ∈ Rd, (3.3)
and the term blow-up solution describes the phenomenon that a solution u(x, t) can grow
to infinity at a single point in finite time T and form an increasingly narrow and growing
peak. As a solution u(x, t) evolves in time the mass
L =∫Rd|u|2dx (3.4)
and the Hamiltonian
H =∫Rd
(|∇u|2− 1
2|u|4)
dx (3.5)
are invariant.
The dimension d on which the nonlinear Schrodinger equation is studied constitutes a
bifurcation parameter with a critical value d = 2. If d ≥ 2, all solutions with a negative
Hamiltonian H blow up, but this does not hold true for d < 2. As an example, if d = 1
Eq. (3.2) is integrable. Furthermore, numerical and asymptotic results suggest that the
blow-up solutions are self-similar if 2 < d < 4 and this behaviour ceases at precisely
d = 2. Hence, Budd (2001) studied Eq. (3.2) in the limit d → 2+ in great detail and he
gives a formal asymptotic description of self-similar solutions in this limit. The aim of
this study is to extends Budd’s work and confirm his asymptotic results.
3.2 Previous results
The blow-up of the nonlinear Schrodinger equation occurs at a single point x∗ and close
to this point u(x, t) is radially symmetric and, hence, spatially a function of the single
3.2 Previous results 33
variable r = |x−x∗|. Therefore, u(r, t) has to satisfy the partial differential equation
i∂tu+∂2r u+
d−1r
∂ru+u|u|2 = 0, ur(0, t) = 0, (3.6)
around x∗. Eq. (3.6) is solved by the ansatz
u(r, t) =1
2a(T − t)exp(−i2a
log(T − t))
Q
(r√
2a(T − t)
), (3.7)
where a ∈ R is a nonlinear eigenvalue that couples phase and amplitude of u(r, t). We set
y = r/√
2a(T − t) and the function Q(y) has to satisfy the differential equation
Qyy +d−1
yQy + ia(yQ)y−Q+Q |Q|2 = 0, (3.8)
subject to the boundary conditions
Qy(0) = 0, (3.9)
Q(0) = γ ∈R, (3.10)
y(
1+ia
)Q(y)y +Q(y)→ ∞, y→ ∞. (3.11)
Budd (2001) found evidence that Eq. (3.8) admits solutions with multiple peaks and in
the limit d→ 2+, a→ 0 and there exist a constant γ such that
d−2∼ 1aγ
exp(−λ
a
), λ =
2π
3−√
32
, (3.12)
and the jth peak is located at y = κ j/a, where
κ j = 1+O(a log(a)). (3.13)
On the other hand, for a solution with a single peak
d−2 =3a
exp(−λ
a
(1+O
(a2)))(1+O
(a2)) , (3.14)
where λ = 2π/3−√
3/2 and
κ = 1+O(a2) . (3.15)
At this point, an interesting question is regarding the precise form of κ j in Eq. (3.13).
34 3 Self-similar blow-up solutions of the nonlinear Schrodinger equation
3.3 Perturbation analysis around individual peak
In order to extend the asymptotic form of κ j, Eq. (3.13), we perform a perturbation anal-
ysis around an individual peak. We start by centring Eq. (3.8) around the peak and set
y = κ/a+ s which yields
Qss + iκQs−Q+Q |Q|2 =−ia(sQ)s−ad−1κ +as
Qs. (3.16)
Given our ansatz (3.7) we note that the solution u is invariant under phase transformations
u 7→ eiϕu, ϕ ∈ R, (3.17)
such that the function Q is unique up to a phase and we set
Q(s) = exp(−iκs/2)S(s). (3.18)
Hence, Eq. (3.16) becomes
Sss−(
1− κ2
4
)S+S |S|2 =−ia(sS)s−a
κ
2sS−a
d−1κ +as
(Ss− i
κ
2S). (3.19)
We proceed with constructing an asymptotic solution S and to this end we expand S and
κ as
S = S0 +a logaS1 +aS2 +a2 log2 aS3 +a2 logaS4 +O(a2) , (3.20)
κ = κ0 +a logaκ1 +aκ2 +a2 log2 aκ3 +a2 logaκ4 +O(a2) . (3.21)
From now on forward we set d = 2 as any correction to d is exponentially small (c.f.
Eq. (3.12)) and can be neglected compared to the higher order terms of the asymptotic
expansion. Substitution of Eqs. (3.20) and (3.21) into Eq. (3.19) yield a series of equations
at the various orders of the expansion:
3.3 Perturbation analysis around individual peak 35
O(a0) : S0,ss−α2S0 +S0 |S0|2 = 0, (3.22)
O(a loga) : S1,ss−α2S1 +2S1 |S0|2 +S1 |S0|2 =−
12
κ0κ1S0, (3.23)
O(a1) : S2,ss−α2S2 +2S2 |S0|2 +S2 |S0|2 =−
12
κ0κ2S0− i(sS0)s (3.24)
− 12
sκ0S0−1κ0
(S0,s− i
12
κ0S0
),
O(a2 log2 a) : S3,ss−α2S3 +2S3 |S0|2 +S3 |S0|2 =−S0S2
1−2S0∣∣S2
1∣∣ (3.25)
− 12
κ0κ1S1−14
κ21 S0−
12
κ0κ3S0,
O(a2 loga) : S4,ss−α2S4 +2S4 |S0|2 +S4 |S0|2 =−2S0S1S2−2S0S1S2 (3.26)
−2S0S1S2−12
κ0κ1S2−12
κ0κ2S1−12
κ1κ2S0−12
κ0κ4S0
−i(sS1)s−s2(κ0S1 + κ1S0)+
κ1
κ20
(S0,s− i
κ0
2S0
)− 1
κ0
(S1,s−
i2(κ1S0 + κ0S1)
),
where α2 = 1−κ20/4.
The equation at order O(a0) is readily solved by
S0(s) =√
2α sech(αs) , (3.27)
and the second solution at this order is irrelevant as it would diverge as s→±∞. For all
solutions at further orders we write S j(s) = u j(s)+ iv j(s) ( j = 1,2,3,4) and treat the real
and imaginary parts separately. Further, we define the two differential operators
K =∂ 2
∂ s2 −α2 +3|S0|2, (3.28)
L =∂ 2
∂ s2 −α2 + |S0|2, (3.29)
and note that K and L constitute the left hand sides of our asymptotic series of equations
corresponding to the real part u j and imaginary part v j, respectively. Since at all orders
of the expansion the solution will be a superposition of the homogeneous solutions and
the particular solutions and the homogeneous part is identical at all orders we wish to find
solutions uh and vh that satisfy K uh = L vh = 0 first.
Differentiating Eq. (3.22) yields K S0,s = 0 and
uh1 = S0,s =−√
2α2 tanh(αs)/cosh(αs). (3.30)
36 3 Self-similar blow-up solutions of the nonlinear Schrodinger equation
Given one homogeneous solution uh1 the second one uh2 can be found by reduction of
order which is detailed in Appendix A and
uh2 = 6α sS0,s−α (cosh(2αs)−5)S0. (3.31)
Now, we turn to the operator L and inspection of Eq. (3.22) immediately reveals that
vh1 = S0, (3.32)
and a further solutions is again constructed by reduction of order and we find
vh2 = (2αs+ sinh(2αs))S0. (3.33)
We note that uh1,vh1→ 0 as s→±∞, but the other two solutions diverge and uh2,vh2 ∼
exp(αs) as s becomes large.
Since the homogeneous solutions are now known, particular solutions can be con-
structed by variation of parameters which is detailed in Appendix B and the results are
S0 =√
2α sech(αs) , (3.34)
S1 = c1uh1−κ0κ1
4α20(sS0)s + ic2vh1, (3.35)
S2 = u2 + i(
c4vh1−s2
4S0
), (3.36)
where
u2 = c3uh1 +Cuh2 (3.37)
−(
3κ0
8α4 −2
3α2κ0+
κ0s2
8α2 +κ0κ2s4α2
)S0,s−S0
(κ2κ0
4α2 +κ0s4α2
),
+1
4α
(1
3κ0− κ0
4α2
)sinh(2αs)S0.
We note that S0,S1→ 0 as s→±∞ but u2 grows exponentially and
lims→±∞
u2 = lims→±∞
(−√
2α2C± 1
2√
2
(1
3κ0− κ0
4α2
))e±αs, (3.38)
but the choice C = 0 and κ0 = 1 gives a convergent solution. As we have established the
value of κ to leading order we can continue to state the remaining results and
3.4 Matching of subsequent peaks 37
S3 = u3 + iv3, (3.39)
where
u3 = c5uh1 +1
72[3(9c2
1−12c22−8κ3
)−18c1κ1s+κ
21(3s2−16
)]S0 (3.40)
− 118(−3c1κ1 +2κ
21 s+6κ3s
)S0,s−
118
(κ1s−3c1)2 S3
0
v3 = c6vh1 + c2
(13(3c1−κ1s)S0,s +
16
κ1S0
). (3.41)
We note that S3→ 0, s→±∞ for all choices of c1,c2,c5 and c6. On the other hand, the so-
lution S4 is divergent and we write S4 = c7uh1+c8uh2+u4,part + i(c9vh1+c10vh2+v4,part),
where the particular solutions are found by variation of parameters again. Collecting all
divergent terms of S4 and considering |s| → ∞ shows that u4 and v4 are asymptotic to
u4,div =eαs
540√
2
(6(3(4√
3ln2−7)c1 +10(√
3ln2+3)c2−270c8)
28√
3ln2−165+κ1
)(3.42)
+e−αs
540√
2
(−
6(3(4√
3ln2+7)c1 +10(√
3ln2−3)c2−270c8)
28√
3ln2+165+8π2+κ1
),
v4,div =eαs
9√
6
(c10−
427
(√3ln2−3
)c2
)(3.43)
+e−αs
9√
6
(c10−
427
(√3ln2+3
)c2
),
in the limit |s| → ∞ and the constants c1 and c2 define S1. We conclude that κ1 and c10
can be chosen such that S4 vanishes towards one side but diverges towards the other side
and therefore the distance of successive peaks is O(loga) (Budd, 2001). In order to find
exact values of the distance the constants c1, c2 and c8 that define an individual peak have
to be identified which can be achieved via matching of subsequent peaks.
3.4 Matching of subsequent peaks
Here we consider a multi-bump solution with two peaks at yL = κL/a and yR = κR/a
where L and R denote the left and right solution, respectively. As before, we centre the
respective peaks at yL and yR and introduce the independent variables sL and sR where
sR = sL−(κ
R1 −κ
L1)
loga, (3.44)
38 3 Self-similar blow-up solutions of the nonlinear Schrodinger equation
and
κL1 =
6(3(4√
3ln2+7)c1 +10(√
3ln2−3)c2−270c8)
28√
3ln2+165+8π2(3.45)
κR1 =−
6(3(4√
3ln2−7)c1 +10(√
3ln2+3)c2−270c8)
28√
3ln2−165(3.46)
and the (˜) is used to differentiate the constants that define the right peak from the ones
that refer to the left peak. Up to now the constants c1,c2,c8 and c1, c2, c8 could not be
determined and in this section these constants are found by matching the left and right
peak. In Eqs. (3.46, 3.45), κL1 and κR
1 are chosen such that the left peak grows as sL→ ∞
and the right peaks grows as sR →−∞ in accordance with Eq. (3.42) and similarly for
c10 and c10. Thus, the matching condition becomes that the growth of one peak has to
coincide with the decay of the other peak such that the solution is overall continuous.
In order to match the decay of the left peak with the growth of the right one it is required
that
exp(
iκLsL
2
)SL
0 = exp(
iκRsR
2
)a2 logaSR
4 ,
exp(
iκLsL
2
)2√
2αe−αsL = exp(
iκRsR
2
)a2 loga
(A+ iB
)e−αsR, (3.47)
where
A =1
540√
2
(−
6(3(4√
3ln2+7)c1 +10(√
3ln2−3)c2−270c8)
28√
3ln2+165+8π2+κ
R1
), (3.48)
B =− 881√
6c2. (3.49)
Eq. (3.47) requires that the magnitude and phase on both sides coincide and
1 =a4 log2 a
8α2 exp(2α(sL− sR))(A2 + B2) , (3.50)
12(κ
LsL−κRsR)= tan
BA. (3.51)
Analogously, the growth of the left peak has to match the decay of the right one and
exp(
iκLsL
2
)a2 logaSL
4 = exp(
iκRsR
2
)SR
0 ,
exp(
iκLsL
2
)a2 loga (A+ iB)eαsL = exp
(iκRsR
2
)2√
2αeαsR, (3.52)
3.5 Discussion 39
where
A =1
540√
2
(6(3(4√
3ln2−7)c1 +10(√
3ln2+3)c2−270c8)
28√
3ln2−165+κ
L1
), (3.53)
B =8
81√
6c2. (3.54)
Therefore,
1 =a4 log2 a
8α2 exp(2α(sL− sR))(A2 +B2) , (3.55)
12(κ
RsR−κLsL)= tan
BA. (3.56)
Eqs. (3.50, 3.51) and Eqs. (3.55, 3.56) constitute as system of equations that can be solved
for c1,c2,c8 and c1, c2, c8 and combining the equations yields
A2 +B2 = A2 + B2, (3.57)
BA=− B
A. (3.58)
Due to the apparent nonlinearities in Eqs. (3.50, 3.51), Eqs. (3.55, 3.56) and the possible
singularities due to the terms tan(B/A) and tan(B/A) this system is very intricate and
requires detailed analysis, which will be the topic of future work.
3.5 Discussion
In this chapter self-similar blow-up solutions were discussed and we outlined their impor-
tance in geological settings. Particularly, in the following chapter we encounter blow-up
of temperature in elasto-visco-plastic materials, which forms the basis for a localization
criterion.
The formal asymptotic description by Budd (2001) was verified and could be extended.
We were able to give closed form expressions for an asymptotic series including five
orders of the expansion. As Budd stated the distance between successive peaks in a multi-
bump setting is O(a loga). At this point, the distance between two peaks could be de-
termined by matching arguments but a solution of the system of equations (3.50, 3.51)
and (3.55, 3.56) has been omitted. This system of equations requires a detailed analysis
that is left for future work.
Chapter 4
Energy based criteria for the onset of localized plastic
deformation
So far only systems that can be seen as purely mechanical have been studied but in this
chapter we turn our attention to multi-physics and multi-scale systems. The term multi-
physics describes the fact that multiple physical processes are permitted to work concur-
rently and multi-scale refers to the fact that these processes might operate on different
length- or time-scales. For example a chemical reaction can progress very rapidly and
might only be active over a range of a few milli- or centimeters but this reaction might
influence the mechanical properties of the material on a larger scale.
The main concern of the present chapter is the localization of plastic deformation in a
rate-dependent and temperature sensitive material and we put forward the hypothesis that
the classical criterion for localization can be generalized from the isothermal and adia-
batic limit by means of an energy–based bifurcation criterion. Such energy-based criteria
have been suggested in earlier studies (Cherukuri and Shawki, 1995a,b) where the total
kinetic energy of the material was used as an indicator of inhomogeneous plastic defor-
mation. In contrast, our criterion is based on the total thermo-mechanical energy budget
and the derivation of this criterion involves an analytic analysis and characterization of
the multiple steady states of the energy budget followed by a numerical analysis of the
transient states. We showcase the development of the criterion, highlight its relation to the
characteristics of hyperbolic differential equations (slip–line fields) and present numerical
examples.
The problem of the localization of plastic deformation is commonly considered solved
once both the spatial configuration of localization and the necessary loading conditions for
its onset are extracted. The classical works of Hill (1950, 1962), Rudnicki and Rice (1975)
and Rice (1976) on the class of rate and temperature independent materials provide a solid
platform upon which modern solid mechanics is built and these works suggest that the
41
42 4 Energy based criteria for the onset of localized plastic deformation
spatial configuration as well as the critical loading conditions for the onset of localization
can be obtained from the stationary limit of a material dependent wave equation.
In particular, the slip line field theory for ideal rigid plastic materials in plane strain
has been developed in the middle of the last century (Hill, 1950) and successfully applied
to metal forming processes (Johnson et al, 1982). The theory is based on solving the
hyperbolic differential equations of mass and momentum balance (at its stationary limit)
and provides a closed form solution for the failure of such an idealized material using
the method of characteristics. The slip line field theory also forms the background for the
development of a criterion of localization of plastic deformation stemming from a material
bifurcation (Rudnicki and Rice, 1975; Rice, 1976). The study of this material bifurcation
was the subject of the early approaches of accelerating waves in rate and temperature
independent solids (Hill, 1962; Rudnicki and Rice, 1975) and shows that localization
instabilities occur at the stationary wave limit of the linear elasto-plastic wave equation
Likηk = ρc2ηi, (4.1)
which can be derived if the response of a homogeneous and homogeneously deformed
material to small wave disturbances of the form η j exp[i(kkxk−κct)] is considered where
Lik is the acoustic tensor, ηi is the vector of the jump of the wave speed, ρ the material
density, ki is the wave number, xi is the position, κ = |ki| the wave frequency, t is time and
c the wave speed. If all wave speeds are such that c2 > 0 the homogeneous deformation is
stable with respect to small disturbances and if c2 < 0 a disturbance grows exponentially
fast. Hence, the instability is marked by c = 0 and it is required that (Hill, 1962; Rudnicki
and Rice, 1975)
det [Lik] = det[ν jC
epi jklνl
]= 0 (4.2)
where Cepi jkl is the elasto-plastic stiffness matrix of the material obeying an incremental
elasto-plastic response σi j =Cepi jkl εkl and it can be seen that the properties of the rate- and
temperature-independent material are encapsulated in Eq. (4.2). The vector νi = ki/κ is
the normal unit vector of the discontinuity imposed by the propagating acceleration wave.
Condition (4.2) allows for the calculation of both the orientation of the localized plane
given by the vector νi, and the critical stress ratio for the onset of localization which is
given by the critical value of the tangent modulus.
4 Energy based criteria for the onset of localized plastic deformation 43
Although the slip line field theory has played an undeniable role in underpinning the
theory of plasticity, and has still some use in the limit analysis and design (Khan et al,
2008), it has been superseded by advanced numerical techniques that are capable of mod-
elling nonlinear, elastic, viscous and plastic materials (Needleman and Tvergaard, 1992).
The main drawback of the slip line field theory is that it cannot be used for rate and
temperature dependent constitutive laws which significantly hampered the applicability
to modern engineering applications since the importance of temperature is well known
in constitutive properties of most materials. Soil, rocks and ceramics are significantly
influenced by temperature with strain localization being strongly affected by thermal
loading (Huckel and Baldi, 1990; Huckel and Pellegrini, 2002). In polymers and poly-
carbonates temperature and strain rate are key parameters influencing the response of the
material, even at ambient conditions (Bauwens-Crowet et al, 1974). Finally, even in the
analogue rigid materials for which the slip line field theory was developed, i.e. metals,
temperature was shown to be important. This becomes obvious in particular under con-
ditions of high speed deformation or at large strain where mechanical work is dissipated
and the effects of heat become important. An excellent example is the thermal cross that
is often observed during forging of mild steel (Johnson et al, 1964) caused by localized
plastic dissipation on slip lines as shown in Fig. 4.1 for a flat punch geometry.
Fig. 4.1 Heat lines in a mild steel specimen at high temperatures (Johnson et al, 1964).
The thermal cross becomes visible as heat lines owing to reaching temperatures of
around 680 C and their pattern (Fig. 4.1) closely resembles the original slip lines cal-
culated in theoretical plasticity (Hill, 1950). The coincidence of heat lines and slip lines
44 4 Energy based criteria for the onset of localized plastic deformation
suggests a strong relationship and calls for an extension of the original theory beyond
isothermal conditions. Later studies (Benallal and Lemaitre, 1991; Benallal and Bigoni,
2004) generalized the localization concepts and extended the acoustic tensor criterion to
the realm of coupled thermo-mechanical response for rate-independent materials, involv-
ing an updated formulation of the acoustic tensor for the limits of isothermal and adiabatic
conditions.
However, when dealing with rate-dependent thermo-plastic coupling, the mathemati-
cal study of the eigenvalue problem (4.1) of plasticity breaks down. The procedure for
determining the conditions for the onset of localized deformation differs significantly
from that of rate-independent materials (Anand et al, 1987). In this regime, the mate-
rial instability leading to localization is approached through stability analyses of the field
equations, rather than through the eigenvalues of the acoustic tensor. These techniques
were first introduced for one-dimensional plastic shear deformation of nonlinear viscous
fluids by Gruntfest (1963), and later by Clifton (1980); Bai et al (1981) and Bai (1982).
The one-dimensional problem of simple shear of a temperature dependent visco-plastic
layer has also been treated semi-analytically by Chen et al (1989) and Leroy and Moli-
nari (1992). These concepts gave rise to the proposition of an energy based localiza-
tion theory in which instabilities emerge when the mechanical input rate rises signifi-
cantly leading to a departure from the near isothermal limit towards the near adiabatic
limit (Cherukuri and Shawki, 1995a,b). Since the energy equation can provide informa-
tion about the time evolution of the system, this regime has been extensively studied in
earth sciences (Regenauer-Lieb et al, 2013a,b) for one-dimensional failure patterns seen
in landslides (Veveakis et al, 2007), and fault mechanics (Veveakis et al, 2010).
The adiabatic limit of the energy equation, also known as adiabatic shear bands, was
the focus of considerable research efforts during past decades in material sciences (Grunt-
fest, 1963). It is of particular interest since the adiabatic shear limit can act as a precursor
to failure, irrespective of its mode (ductile or brittle) (Dodd and Bai, 2012). In spite of
the apparent relationship suggested by observations such as reported in Fig. 4.1 a gen-
eralized slip line field theory that extends Hill’s theory to nonlinear thermo-visco-plastic
materials is not yet developed. Here, we show that such a theory can be obtained by ex-
panding the conditions necessary for the loss of ellipticity of the momentum equations,
in the realm of coupled thermo-mechanical problems for temperature-dependent, visco-
plastic materials. The generalization of the slip line field theory is two-pronged: Firstly, at
4.1 Fundamental principles of continuum thermo-mechanics 45
the limit of stationary thermo-mechanical wave propagation (following the classical con-
cepts of mechanics) the stress equilibrium conditions define a spatial pattern of failure and
dissipation, which is the product of stress and velocity gradient. Secondly, a bifurcation
analysis of the energy balance supplies the necessary conditions for a jump in the dissi-
pation. Further on, the transient analysis of the system provides the evolution of plastic
deformation from near-isothermal to near-adiabatic conditions and verifies the results of
the stationary analysis.
In the remainder of this chapter we present a detailed mathematical approach to the
problem. In Sec. 4.1 the continuum thermo-mechanical framework considered in this
chapter is described and in Sec. 4.2 the extension of the slip line field theory from ideal
plastic to visco-plastic materials is presented and this entails a generalization of Hencky’s
and Geiringer’s equations. These results are compared to finite element simulations of
simple geometries and Johnson’s heat lines in Sec. 4.3. We conclude with a discussion of
the importance of the obtained results in Sec. 4.4.
4.1 Fundamental principles of continuum thermo-mechanics
The problem at hand consists of solving the fully coupled thermodynamical behaviour
of geomaterials under external loading and to predict the onset of plastic failure and the
pattern of this failure. In this section we detail the framework of thermodynamics and con-
tinuum mechanics that is used later on to describe the behaviour of rate- and temperature-
sensitive materials. We also show the limitations of this framework. The derivations of the
governing equations are based, firstly, on the conservation of mass, momentum and en-
ergy and, secondly, it must be ensured that the second law of thermodynamics – increase
of entropy – is obeyed.
In the following sections the shorthands
y =dydt, ∂xy =
∂y∂x
(4.3)
and Einstein summation are used.
46 4 Energy based criteria for the onset of localized plastic deformation
4.1.1 Force balance equation
Firstly, we consider the conservation of mass and momentum in an elasto-visco-plastic
material that is contained within a volume Ω . It is required that the mass remains constant
andddt
∫Ω
ρ dΩ = 0, (4.4)
where ρ denotes the material density. Since the material is compressible the volume Ω
can dilate and the evolution of a volume element dΩ is governed by 1
˙dΩ = ∂xivi dΩ , (4.5)
where vi denotes the material velocity. Substitution of Eq. (4.5) into Eq. (4.4) yields the
local form of the conservation of mass equation
ρ +ρ∂xivi = 0, (4.6)
which states that the rate of change of ρ is equal to the material flux. Now, we turn to the
conservation of linear momentum which reads
ddt
∫Ω
ρvi dΩ =∫
Γ
Ti dΓ +∫
Ω
fi dΩ , (4.7)
and the left hand side represents the change of momentum of the material in Ω due to the
surface traction Ti on the boundary Γ and body forces fi dΩ , where fi is the force density,
respectively. Employing Gauss’ theorem, the surface integral in Eq. (4.7) is rewritten as
∫Γ
Ti dΓ =∫
Γ
σi jn j dΓ =∫
Ω
∂x jσi j dΩ
where n j is normal to the control surface and σi j the stress tensor. After computing the
time derivative in an analogous manner as before and substituting Eq. (4.6) into (4.7), the
force balance is obtained
∂x jσi j + fi = ρ vi with σi jn j = Ti,d at Γσ , ui = ui,d at Γu, (4.8)
1 This can be understood if one considers a rod of length ∆x = x2− x1 where x1 and x2 are start and end ofthe rod, respectively. If the material velocity is v(x1) = v1 and v(x2) = v2 ' v1+∂xv∆x then the rod stretchesby ∆v∆t = (v2− v1)∆t = ∂xv∆x∆t during time ∆t. Hence, d(∆x)/dt = ∂xv∆x.
4.1 Fundamental principles of continuum thermo-mechanics 47
where Γσ and Γu denote the surfaces with prescribed surface tractions, Ti,d , and displace-
ments, ui,d , respectively. The force balance governs the mechanical properties of the mate-
rial and a body in equilibrium has to satisfy Eq. (4.8). The thermal properties are captured
by the energy balance which is derived in the next section.
4.1.2 Energy balance equation
The energy of a body can be separated into kinetic energy K and internal energy U and
the total energy E = K +U must satisfy the first principle of thermodynamics
E =(K +U
)= Pext +Q. (4.9)
This principle stipulates that the total energy has a rate of change that is equal to the
power due to external forces, Pext , and the rate of heat supply, Q. The kinetic and internal
energies read
K =12
∫Ω
ρvivi dΩ and U =∫
Ω
ρudΩ , (4.10)
where u is the specific internal energy and the time derivatives of the energy components
are
K =∫
Ω
ρvivi dΩ and U =∫
Ω
ρ udΩ , (4.11)
where the conservation of mass, Eq. (4.6), was used again to simplify the time derivatives.
The external power and heat supply can be written in integral form as
Pext =∫
Ω
fivi dΩ +∫
Γ
Tivi dΓ , (4.12)
Q =∫
Ω
r dΩ −∫
Γ
qini dΓ , (4.13)
where r is the rate of heat production and qi is the heat flux. The surface integrals in the
above equations can be rewritten applying Gauss’ theorem
∫Γ
Tivi dΓ =∫
Γ
σi jn jvi dΓ =∫
Ω
(vi∂x jσi j +σi j∂x jvi
)dΩ , (4.14)∫
Γ
qini dΓ =∫
Ω
∂xiqi dΩ . (4.15)
48 4 Energy based criteria for the onset of localized plastic deformation
In Eq. (4.14), we set σi j∂ jvi = σi j(∂ jvi +∂iv j
)/2 = σi jDi j where the first equality fol-
lows from the symmetry of σi j and the second one is due to the definition of the rate-of-
deformation, Di j. Substituting Eqs. (4.11)–(4.15) into Eq. (4.9) and employing the force
balance, Eq. (4.8), yields
ρ u = σi jDi j + r−∂xiqi, (4.16)
which is the local form of the energy conservation. It states that temporal changes of
energy are due to shear heating, heat production and the divergence of the heat flux.
In a laboratory experiment it is hard to control the internal energy directly and, hence,
we wish to express the energy balance in terms of temperature T which is more eas-
ily controlled in experiments and computer simulations. To this end, we introduce the
Helmholtz energy ψ = ψ(εe,T ) as a measure of the elastic and thermal energy stored in
the system and we choose temperature T and the elastic strain εe as state variables (Kar-
rech et al, 2011). Since, the specific internal energy, u, is commonly expressed in terms
of εe and specific entropy s, the specific free energy ψ and the internal energy are Legen-
dre transforms of each other and if u is Legendre transformed with respect to s, we find
u(s,εe) = ψ(T,εe)+ sT . Deriving this expression with respect to time and substituting it
into Equation (4.16) yields
ρ u = ρ
(∂εe
i jψ
)εe
i j +ρ (∂T ψ + s) T +ρT s = σi jDi j + r−∂xiqi. (4.17)
The term ρT s equals the heat dissipation density (Hobbs et al (2011)) and rearranging
gives
ρT s = σi jDi j + r−∂xiqi−ρ
(∂εe
i jψ
)εe
i j−ρ (∂T ψ + s) T . (4.18)
The significance of this equation is that it couples various processes in a thermodynami-
cally admissible manner (Hobbs et al (2011)) and the processes of interest here are mate-
rial deformations and thermal transport. At this point, it is desirable to express the partial
derivatives of ψ in Eq. (4.18) in terms of the state variables s, εe and their conjugate
variables explicitly and those relation are known as equations of state.
4.1.2.1 Equations of state
In order to deduce the equations of state we start with the second law of thermodynamics
4.1 Fundamental principles of continuum thermo-mechanics 49
ddt
∫Ω
sρ dΩ ≥∫
Ω
rT
dΩ −∫
Γ
qini
TdΓ . (4.19)
The left hand side of this equation is the rate-of-change of the entropy of the whole sys-
tem. The right hand side contains the entropy production due to prescribed heat produc-
tion, r, and external heat fluxes. The left hand side is generally bigger than the right hand
side because it contains extra contributions to the entropy production due to thermody-
namic processes. These contributions vanish for reversible processes and if the system is
in equilibrium equality holds in Eq. (4.19).
We proceed with deriving the local form of the second law. To this end, Gauss’ theorem
is applied to the surface integral in Eq. (4.19) which yields
∫Γ
qini
TdΓ =
∫Ω
∂xi
(qi
T
)dΩ =
∫Ω
(1T
∂xiqi−qi
T 2 ∂xiT)
dΩ . (4.20)
Due to the conservation of mass, Eq. (4.6), the time derivative of the entropy production
simplifies toddt
∫Ω
sρ dΩ =∫
Ω
sρ dΩ . (4.21)
Substituting Eqs. (4.20) and (4.21) into Eq. (4.19), we find the local form of the second
principle of thermodynamics
ρT s≥ r−∂xiqi +qi
T∂xiT. (4.22)
Expanding the left-hand side using Eq. (4.18), the above expression yields the Clausius-
Duhem inequality
D ≡ σi jDi j−ρ
(∂εe
i jψ
)εe
i j−ρ (∂T ψ + s) T − qi
T∂xiT ≥ 0, (4.23)
where D is the specific dissipation. If elasticity and plasticity are decoupled the strain can
be decomposed additively into elastic and inelastic components, ε ini j , εi j = εe
i j + ε ini j such
that Eq. (4.23) can be written as
D = σi j˙
ε ini j −
qi
T∂xiT +
(σi j−ρ∂εe
i jψ
)εe
i j−ρ (∂T ψ + s) T ≥ 0. (4.24)
Following Coleman and Noll (1963), the above inequality must be satisfied for all admis-
sible processes which leads to the equations of state
50 4 Energy based criteria for the onset of localized plastic deformation
ρ∂εei j
ψ = σi j and ∂T ψ =−s. (4.25)
Combining the equations of state (4.25) and the local expression of the first principle,
Eq. (4.17), results in the dissipation equation
ρT s = σi j˙
ε ini j + r−∂xiqi. (4.26)
and since s = s(T,εe)
ρT s = ρT(
T ∂T s+ εei j∂εe
i js)= ρcvT −T
(∂T σi j
)εe
i j = σi j˙
ε ini j + r−∂xiqi, (4.27)
where we used ∂ s/∂T = cv/T , the equations of state and cv is the specific heat capacity
at constant volume. We recall Fourier’s law and write qi = −κ∂xiT with κ the thermal
conductivity. This leads to the heat equation
ρcvT = ∂xiκ∂xiT +χσi j˙
ε ini j + r+T
(∂T σi j
)εe
i j with (4.28)
∂xiT = qi,d at Γq, T = Td at ΓT ,
where Γq and ΓT denote the surfaces with prescribed heat fluxes and temperatures, respec-
tively. The significance of this equation is that it couples the temperature evolution to the
mechanical properties of the material and this it is at the centre of our analysis. In the fol-
lowing sections we show that Eq. (4.28) admits multiple steady states and which steady
state the deforming body attains is specified by the external loading conditions.
The Taylor-Quinney coefficient (Taylor and Quinney, 1934b), χ ∈ [0,1], is introduced
and this coefficient quantifies the amount of mechanical work converted to heat and is
of particular importance in the field of thermodynamics with internal state variables, as
it incorporates the evolution of all the internal state variables ξ of the system, as evident
from its definition (Veveakis et al, 2010)
χ = 1− Y ξ
σi jεpi j. (4.29)
In this expression Y is a thermodynamic potential, dual in energy with the internal state
variable ξ . In conclusion, we define the mechanical dissipation of the material as
Φ = χσi j˙
ε ini j (4.30)
4.1 Fundamental principles of continuum thermo-mechanics 51
In the context of thermo-mechanical coupling the term r in Eq. (4.28) could be thought
of as a heat sink that has the effect of limiting temperature production after the onset of
localization. Possible heat sinks due to a post-localization transformation are melting or
endothermic chemical reactions (Rosakis et al, 2000). Without the heat sink uncontrolled
thermal runaway would ensue and hence r acts as a stabilizer (Veveakis et al, 2010). In the
remainder of this work r will be neglected, because the aim of this work is to find criteria
for the onset of localization and a detailed study of the post-localization regime is not of
central interest. In doing so, the post-failure evolution of temperature will be exaggerated,
since the heat-absorbing processes gathered in r do not limit the temperature evolution.
4.1.3 Constitutive modelling
We first split the strain rate into elastic (reversible) and plastic (irreversible) parts εi j =
εei j + ε in
i j . For the elastic component we adopt a linear elastic law of the form
εei j =C−1
i jklσkl, (4.31)
where Ci jkl is the elasticity tensor.
For the irreversible part, we assume that the Helmholtz free energy is invertible, such
that the evolution of the plastic strain depends on the stress and temperature through a
smooth function of the plastic potential g
εini j = ε0g(σi j,T ), (4.32)
where ε0 is a reference strain rate. We focus on temperature contributions that act inde-
pendently of the stress such that the visco-plastic flow law can be decomposed as
εini j = ε0
⟨f (σi j)
⟩e−T0/T , (4.33)
where the activation temperature is denoted by T0, and the Macaulay brackets 〈·〉 ensure
zero plastic strain before yield (Freed and Walker, 1993). This decomposition is supported
by experimental data at elevated temperatures, below the phase transition temperature of
the material (Bauwens-Crowet et al, 1974) and the two most representative constitutive re-
sponses of temperature and rate dependent materials are an Arrhenius-type dependency on
52 4 Energy based criteria for the onset of localized plastic deformation
Table 4.1 Rate sensitivity parameters for the flow law: ε in = ε0 (σY/σ0)n (c.f. Fig. 4.2(a)).
Material Type of testing σ0 [MPa] ε0 [1/s] n Reference
Steel alloy Uni. extension 1850 10−4 250 Boyce et al (2007)Glassy Polymers Uni. extension 62 5 ·10−5 50 Engels et al (2009)Polycarbonates Tensile creep 58 10−5 50 Bauwens-Crowet et al (1974)Porous Rocks Uni. compression 1.36 10−8 80 Hickman and Gutierez (2007)
temperature, with either a power-law or an exponential dependency on stress. In Fig. 4.2
and Tab. 4.1 different constitutive responses are compared and Eq. (4.33) has wide appli-
cability. The function f (σi j) is an arbitrary flow stress function, which for the example of
J2 visco-plasticity takes the form
⟨f (σi j)
⟩=
⟨√q
k−1⟩
si j
q, (4.34)
where k is the yield stress, q = 3si jsi j/2 and si j = σi j− tr [σi j]/3 (Perzyna, 1966).
The exact form of the constitutive equation is not prescribed during the analysis of the
bifurcation in order to emphasize the generic nature of the formulation, where the onset
of plastic deformation is derived from the basic assumptions of the energetics. The only
important aspect of the constitutive response of the material is that it must obey a visco-
plastic relationship linking the plastic strain-rate with the stress. This is required so that in
the steady-state limit of the equations (σi j = T = 0) the mechanical dissipation remains
non-zero.
The formulation therefore encompasses most classes of physical behaviour described
in the summary of constitutive laws for visco-plastic materials by Chaboche (2008). Since
the rate-independent plasticity case can be deduced from the equations of visco-plasticity
as a limiting case (Chaboche, 1977; Lubliner, 2008), the presented formulation can be
seen as a generic framework for temperature-sensitive plasticity. For a more detailed dis-
cussion on the constitutive concepts of viscoplasticity the reader is encouraged to consult
the review article by Chaboche (2008).
As shown in earlier studies (Leroy and Molinari, 1992; Cherukuri and Shawki, 1995a;
Veveakis et al, 2010), the choice of the form of the temperature dependence of the plastic
flow law is not central for the results of the present study. Those studies have shown
that any nonlinear temperature dependence leads to the same physical behaviour with the
4.1 Fundamental principles of continuum thermo-mechanics 53
0.95%
1%
1.05%
1.1%
1.15%
1.2%
1.25%
1.3%
1.35%
1% 1000% 1000000% 1E+09%
Porous%Rock%
Glassy%polymers%
Steel%alloy%
Polycarbonates%
0σ σΥ
( )0ln pε ε! !
Fig. 4.2 (a) Rate sensitivity of the yield stress for various materials, for tests at room temperature. The solidlines correspond to the flow law ε p = ε0 (σY/σ0)
n, with the values of the parameters given in Table 1. (b)Eyring plot for temperature sensitivity of the yield stress of polycarbonates, from uniaxial extension andtensile creep tests (Bauwens-Crowet et al, 1974). Two possible flow laws have been used to approximate thereported behaviour of the material with respect to strain-rate and temperature variations: a power law creep(solid lines), σY/σ0 = A(ε p/ε0)
n exp(Q/(RT )) (with A = 0.065, n = 0.03 and Q/R = 6.6 kJ/mol), andan Eyring-type flow law (dashed lines), σY/σ0 = A T (ln2Cε p/ε0 +Q/(RT )) (with A = 6.7 ·10−3/K, C =9 ·10−26 and Q = 207 kJ/mol). Note that in this case σ0 = 58 MPa, ε0 = 9 ·10−6/s and R = 8.31 J/mol/K.
54 4 Energy based criteria for the onset of localized plastic deformation
Arrhenius type exponential dependency being the one that allows for the most convenient
mathematical treatment.
4.2 Localization criterion and patterns of plastic failure
In this section, we study the steady-state limit of the material response, in which T = σi j =
0 and the elastic contribution to εi j can be neglected. In this limit, the problem reduces to
that of the study of the response of a rigid (visco-)plastic material and the current setting
can therefore be considered to be a direct extension of the slip line field theory to thermo-
visco-plastic materials. We note that in the present formulation the temperature equation
Eq. (4.28) yields non-trivial solutions only if dissipation is non-zero which is achieved in
the post-yield regime. Therefore, we expect that the orientation of possible localization
planes arises from the characteristics of the stress equilibrium, in accordance with the
theory of plasticity (Hill, 1950).
Further, we anticipate that given an arbitrary set of slip lines, two distinctly different
cases of material response can be identified from the energy balance equation (4.28).
The plastic material may deform homogeneously across the whole domain, and therefore
across the slip lines; alternatively the material deforms in a localized manner along the
slip lines in accordance with the velocity gradient discontinuities in the classical case.
In our formulation we distinguish these material responses via an energy–based criterion
expressed through the multiple steady states which are possible. The complete field ap-
proach of the present work consists of the identification of the generalized patterns of
slip lines given by an arbitrary yield surface and the derivation of the conditions for lo-
calized plastic deformation along these slip lines. This section first discusses the slip line
field theory for materials with arbitrary yield criteria and then turns to the possible steady
states of the temperature equation (4.28) and its bifurcation behaviour.
4.2.1 Plane strain and slip line field theory
We restrict ourselves to general plane strain conditions and in this case the stress equilib-
rium, Eqs. (4.8), reads
4.2 Localization criterion and patterns of plastic failure 55
∂σ11
∂x1+
∂σ12
∂x2= 0, (4.35)
∂σ21
∂x1+
∂σ22
∂x2= 0. (4.36)
where gravity and inertia terms are neglected. Now, the stress decomposition σi j = pδi j +
si j is employed where p is the volumetric mean stress and si j the deviatoric stress and we
formulate the governing equations in the equivalent coordinate system where the stress
tensor is rotated such that its elements are the mean stress p = I1/3 and the von Mises
stress q =√
3J2. In these expressions I1 = tr(σi j) is the first invariant of the stress tensor
and J2 = si jsi j/2. The corresponding coordinate transformation dates back to Levy (Hill,
1950) and is given through the Mohr circle transformation
σ11 = p−qsin2ϕ, σ22 = p+qsin2ϕ, σ12 = σ21 = qcos2ϕ, (4.37)
where ϕ is the rotation angle of the coordinate system.
At the point of initial yield, where the temperature equation is inactive, the response of
the system is solely governed by the stress equilibrium equations and initial plastic yield
occurs along the slip lines (Hill, 1950), which are the characteristic traces of the stress
equilibrium in a mathematical sense. Hence, in the remainder of this section the geometry
of the failure patterns is determined.
In order to find the characteristics of the hyperbolic differential stress equilibrium equa-
tions, we substitute the Levy stress transformations, Eq. (4.37), into the stress equilibrium,
Eqs. (4.35,4.36) and proceed by solving for p, q and ϕ by means of the method of charac-
teristics. As the stress equilibrium only constitutes two equation we additionally assume a
generalized yield surface at a reference temperature of the form q = qY (p) and reduce the
number of unknowns to two. After substituting the stress components (4.37), Eqs. (4.35,
4.36) can be conveniently written in matrix form as
Ai j∂ (p,ϕ)
∂x1+Bi j
∂ (p,ϕ)∂x2
= 0, (4.38)
where
56 4 Energy based criteria for the onset of localized plastic deformation
A =
1−q′ sin(2ϕ) −2qcos(2ϕ)
q′ cos(2ϕ) −2qsin(2ϕ)
, (4.39)
B =
q′ cos(2ϕ) −2qsin(2ϕ)
1+q′ sin(2ϕ) 2qcos(2ϕ)
(4.40)
and the prime (·)′ denotes differentiation with respect to p. In order to simplify Eq. (4.38)
the two (left) eigenvectors ri and eigenvalues µ that satisfy
r(k)i Ai j = µ(k)r(k)i Bi j, (k = 1,2), (4.41)
are computed and Eq. (4.41) is substituted into Eq. (4.38) such that
r(k)i Bi j
(µ(k)√
1+µ(k)2
∂ (p,ϕ)∂x1
+1√
1+µ(k)2
∂ (p,ϕ)∂x2
)= 0. (4.42)
Instead of solving Eqs. (4.42) on the whole x1–x2 plane we restrict ourselves to one char-
acteristic trace that is defined by r(k)i and along which the independent variables x1 and x2
can be parametrized in terms of the arc-length sk. Parametrizing along the characteristic
and identifying∂x1
∂ sk=
µ(k)√1+µ(k)2
,∂x2
∂ sk=
1√1+µ(k)2
(4.43)
allows us to write Eqs. (4.42) as
r(k)i Bi j∂ (p,ϕ)
∂ sk= 0, (4.44)
and the coupled system of ordinary differential equations (4.43) and (4.44) is equivalent to
stress equilibrium equations (4.35)–(4.36) along the characteristic. We wish to emphasize
that Eq. (4.43) defines the geometry of the characteristic trace and that there are two
characteristic traces. In general, the eigenvalues µ(k) and eigenvectors r(k)i depend on ϕ
and general yield modulus q′(p) and
µ(1,2) =
∓√
1− (q′)2 + cos(2ϕ)
h+ sin(2ϕ)(4.45)
r(1,2) =
(√1−h2∓ cos(2ϕ)
±(h− sin(2ϕ)), 1
). (4.46)
4.2 Localization criterion and patterns of plastic failure 57
Note that in this expression p must be critical, i.e. equal to its yield value.
The obtained expressions for µ(k) and r(k) combined with Eqs. (4.44) yield the gener-
alized Hencky’s equations
√1− (q′)2
∂s p±2q∂sϕ = 0, (4.47)
that relate p and ϕ along a characteristic s.
For a given yield criterion the generalized Hencky equations can be integrated and as
an example we consider an incompressible von Mises material with qY = k = const. such
that Eq. (4.45) simplifies (as q′ = 0) to the familiar form
µ(1) =− tanϕ, µ
(2) = cotϕ, (4.48)
and the eigenvectors (4.46) to
r(1) = (− tanϕ,1), r(2) = (cotϕ,1). (4.49)
The corresponding traces are commonly known as α/β -slip lines. Along the slip lines
Eqs. (4.47) reduce to the classical Hencky’s equations
p±2kϕ =Cα,β . (4.50)
These relations have extensively used in computing the plastic regions during metal forg-
ing and Johnson et al (1982) give a concise review of a wide range of applications and
geometries. It should be noted that although the slip-line field theory has proven important
in praxis to find the slip-line field is in itself a formidable task. Hill (1950) and Dewhurst
and Collins (1973) give numerical methods that facilitate the construction of the slip-
line fields, but most approaches require an initial α- and β -line to start, which might be
obtained from experiments.
Equations equivalent to Hencky’s equations can be given for other yield criteria such
as Drucker-Prager or the modified Cam-Clay yield conditions. In the case of a Drucker-
Prager material qY = N p+q0 and along a trace p and ϕ are related via
p =1N
[Aexp
(±2N√1−N2
ϕ
)−q0
]. (4.51)
58 4 Energy based criteria for the onset of localized plastic deformation
For the modified Cam-Clay yield criterion q(p) = ±N√
p(p0− p) (0 ≤ p ≤ p0) and
the general Hencky equation reads√p2
0− (1+N2)(2p− p0)2
4N p(p0− p)∂s p∓∂sϕ = 0 (4.52)
which is an implicit relation between p and ϕ along the traces.
4.2.2 Generalized Geiringer’s equations
Geiringer’s equations are the equivalent to Hencky’s equations but instead of the stress
they define the velocity components along a characteristic. For a von Mises material
Geiringer’s equations are obtained from the condition of incompressibility, in conjunc-
tion with the plastic flow law and the small-strain compatibility equations and it can be
shown that the characteristics of the velocity relations coincide with the characteristics of
the stress equilibrium (Hill, 1950). An important result derived from Geiringer’s equation
is that across a characteristic trace the tangential velocity component can be discontinu-
ous and the characteristics are the potential loci of slip. Hence, the characteristic traces
are known as slip lines in the context of plane strain problems and define the geometry
of the localization pattern. One shortcoming of the classical slip-line field theory is that
it is unable to identify the width of the shear zone around the slip line and assigns a van-
ishing thickness instead. Another issue is that the slip-line field theory is formulated for
isothermal conditions, but in the present formulation we relax the isothermal assumption
and seek conditions of localization that replace the vanishing thickness by a finite width.
To this end, the inelastic strain in the expression of mechanical dissipation Φ , Eq. (4.30),
is substituted with the constitutive law of plasticity (4.33) to provide a temperature sensi-
tive dissipation term
Φ = βσi jεini j = χε0σi j
⟨f (σi j)
⟩e−T0/T ∂g
∂σi j, (4.53)
where σi j f (σi j) represents a tensorial product. The stability and bifurcation of the energy
balance equation (4.28) is characterized by the nonlinear response of its steady state. This
problem has been extensively studied in the literature for shear zones (Gruntfest, 1963;
Chen et al, 1989; Leroy and Molinari, 1992; Vardoulakis, 2002; Veveakis et al, 2010).
4.2 Localization criterion and patterns of plastic failure 59
In this work we generalize it for 2D loading conditions by first studying the response of
the energy balance in the original geometry and then rescaling it to a one-dimensional
subspace defined by the directions of the characteristics.
4.2.2.1 Steady state response of heat equation in two dimensions
The steady state of the energy balance equation is defined by
κ
(∂ 2
∂x21+
∂ 2
∂x22
)T +χε0σi j
⟨f (σi j)
⟩ ∂g∂σi j
e−T0/T = 0, (4.54)
and this expression can be brought into dimensionless form by setting
θ =T −Tb
Tb, xi =
xi
Li(i = 1,2), Ar =
T0
Tb, (4.55)
where Tb is the boundary temperature and Li is an appropriate length scale. Since we
are interested in deformation taking place under isothermal boundary conditions the final
dimensionless equation is (the superimposed hats are dropped for convenience)(∂ 2
∂x21+λ
2 ∂ 2
∂x22
)θ +Gr2D exp
(Ar θ
1+θ
)= 0, (4.56)
where λ = L1/L2 is an aspect ratio and the superscript 2D refers to the dimension of the
domain. The exponential term stems from the mechanical dissipation Φ and we define the
normalized dissipation function
φ = exp(
Ar θ
1+θ
). (4.57)
In order to solve the partial differential equation (4.56) on the domain [−1,1]× [−1,1] a
pseudo arc-length continuation (Chan and Keller, 1982) in Gr2D, the Gruntfest number
(Gruntfest, 1963), was carried out where
Gr2D =χε0L2
1κTb
σi j⟨
f (σi j)⟩
e−Ar ∂g∂σi j
, (4.58)
and Eq. (4.56) is subjected to isothermal boundary conditions
θ(±1,y) = 0, θ(x,±1) = 0. (4.59)
60 4 Energy based criteria for the onset of localized plastic deformation
In physical terms, Gr represents the ratio between heat production due to mechanical
deformation and heat loss due to thermal conduction or additional energy sinks (Var-
doulakis, 2002; Veveakis et al, 2010). In the limit Gr→ 0 the system deforms in virtually
isothermal conditions, whereas as Gr→ ∞ so the system deforms under near-adiabatic
conditions (Veveakis et al, 2010). We would like to stress that Gr is stress-dependant
through the term σi j⟨
f (σi j)⟩.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
θc
(a)
Ar = 4
Ar = 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2
−1
0
1
Gr
Larg
est E
igenvalu
e
(b)
Ar = 4
Ar = 5
Fig. 4.3 (a) Bifurcation diagram of the two-dimensional heat equation for Ar = 4,5. If the path of stationarystates is folded (Ar = 5) the solutions change stability at the fold and dotted lines denote unstable stationarystates. (b) Largest eigenvalues of the Jacobian matrix associated with Eq. (4.56). For Ar = 4 the largesteigenvalue remains negative, but for Ar = 5 stability of the stationary solutions changes at the folds wherethe largest eigenvalue changes sign.
It is well known that the steady state response of the system depends on the values of
Gr and Ar, see Fig. 4.3 (Law, 2006; Veveakis et al, 2010). In Fig. 4.3(a) we sketch the
maximum temperature at the centre of the domain, θc, as a function of Gr for two values
of Ar. We notice that between Ar = 4 and Ar = 5 the response of the system changes from
a stretched (Ar = 4) to a folded S-curve (Ar = 5) (Law, 2006). The stability of the system
is determined by the eigenvalues of the Jacobian matrix, shown in Fig. 4.3(b). Since the
maximum eigenvalue of the case Ar ≤ 4 (stretched) is negative for all values of Gr, the
stretched S-curve is stable throughout. In contrast, the folded S-curve (Ar = 5) exhibits
two points of stability change, coinciding with the turning points of the S-curve. This
means that the stationary solutions are initially stable up to the first turning point, then
unstable and after the second turning point re-stabilize. This manifests our localization
criterion
4.2 Localization criterion and patterns of plastic failure 61
log φ
Gr
(a)
A
B
C
D
s
φ/φ
max
(b)
Branch AB
Branch BC
Fig. 4.4 (a) Folded S-curve. Along the branch AB the solution of Eq. (4.56) corresponds to an isothermaltemperature profile whereas along the section between the turning points B and C the solutions localizes.(b) Examples of the one-dimensional spatial pattern of the dissipation profile for Ar = 10. The profiles arenormalized with respect to the maximum value of dissipation.
Gr > Grcr, (4.60)
where Grcr marks the first turning point of the S-curve and states that localized plastic de-
formation is only possible if Gr is larger than its critical value, which is also demonstrated
in Fig. 4.4. We emphasize that Gr depends on the stress state of the system, Eq. (4.58).
The above arguments are independent of λ and the variation of the steady state response
with respect to λ are presented in Fig. 4.5. This conclusion allows us to proceed with
rescaling the energy equation in a one-dimensional equivalent, along the characteristic
curves sk of the stress equilibrium equations, and extract some analytical results regarding
Grcr.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1
2
3
4
5
6
7
8
Gruntfest number
Norm
aliz
ed C
ore
Tem
pera
ture
λ=0 λ=0.5 λ=1 λ=1.5 λ=2
0 20 40 60 80 1000
5
10
15
20
25
λ
Numerical points
Grc = 0.2285 λ + 0.1966
(a) (b)
Grc
Grc
Fig. 4.5 A change in aspect ratio λ shifts the steady states along the Gr-axis, but has only little influenceon the shape of the S-curve and no influence on the stability of the steady states.
62 4 Energy based criteria for the onset of localized plastic deformation
4.2.2.2 Rescaling along the characteristics
As we look for the necessary conditions of localization to occur along the characteristics
of the stress equilibrium the characteristic curves sk also need to be the characteristics
of the temperature equation. This condition is satisfied in the limit where the elliptic op-
erator vanishes, which is equivalent to the adiabatic limit. Hence, the loss of ellipticity
of the temperature equation ensures that localization is achieved along the characteristics
of the momentum equation. Localization criteria can therefore be obtained through the
properties of the nonlinear response of the energy equation.
Replacing the spatial derivatives in Eq. (4.54) with Eqs. (4.43) yields
α
(1+µ2)
µ2
((1+µ
2) ∂ 2
∂ s2 +
(µ2−1
)µ
∂ µ
∂ s∂
∂ s
)T (4.61)
+χε0σi j⟨
f (σi j)⟩ ∂g
∂σi je−T0/T = 0,
and all super- and subscripts k, denoting the characteristics, have been dropped since
the above arguments hold true for both sets of characteristics. We consider the case of
infinitesimal variations of the slip line geometry, so that the derivatives ∂ µ/∂ s are small
and can be ignored.
Analogously to Eq. (4.55), Eq. (4.61) is normalized which results in
∂ 2θ
∂ s2 +Gr1D exp(
Ar θ
1+θ
)= 0, (4.62)
together with the boundary conditions
θ(0) = θc, θ′(0) = 0, θ(1) = 0. (4.63)
These boundary conditions give rise to symmetric solutions which are physically admis-
sible. The other set of solutions to Eq. (4.62) is anti-symmetric which would correspond
to a thermal state that is out of equilibrium and is hence discarded.
The one-dimensional Gruntfest number admits the following form
Gr1D =χε0L2
αTb
µ2
(1+µ2)2 e−Ar
σi j⟨
f (σi j)⟩ ∂g
∂σi j, (4.64)
where L is a length scale. This Gruntfest number is spatially dependent through µ , and
also incorporates the dimensionality of the system at hand, through L and we conjecture
4.2 Localization criterion and patterns of plastic failure 63
that the critical values of the Gruntfest number for the two dimensional and one dimen-
sional case are related via
Gr1Dcr =
L2
L21
µ2
(1+µ2)2 Gr2D
cr (4.65)
which follows upon comparing Eqs. (4.58) and (4.64). As an example, we consider the
experiment in Fig. 4.1 where the heat lines are oriented along the diagonals of a square
sample. When rescaling a von Mises material along the characteristics of a square domain
(see the next section) with dimension L1, then L =√
2L1 and the geometric correction
L2
L21
µ2
(1+µ2)2 =
12
as µ =− tanϕ or µ = cotϕ . From here on forward we do not distinct between the Gr1D
and Gr2D any longer since there is no dis-ambiguity in terms of the physics between the
one- and two-dimensional heat equation. The steady state response of this 1D equivalent
equation follows that of the 2D case and this is shown in the next section.
4.2.2.3 Perturbation analysis
The one-dimensional heat equation (4.62) can be studied analytically and we give asymp-
totic expressions for Grcr here. In order to perform a perturbation analysis a small param-
eter has to be identified which is revealed if we set θ = Arθ such that the heat equation
transforms to∂ 2θ
∂ s2 + Gr exp(
θ
1+ εθ
)= 0, (4.66)
where Gr = Gr/Ar. This suggests ε = 1/Ar as a suitable perturbation parameter. If εθ
1 the exponent in Eq. (4.66) can be expanded as a geometric series which yields
θ′′+Gr exp
(θ − εθ
2 + ε2θ
3 + . . .)= 0, (4.67)
where the tildes are dropped for convenience. In order to find the governing equations
at increasing orders of ε we expand θ = θ0 + εθ1 + . . . and Gr = Gr0 + εGr1 + . . . and
substitute into Eq. (4.67). Here, only asymptotic solutions up to O(ε2) are considered and
we find
64 4 Energy based criteria for the onset of localized plastic deformation
0 =(θ0 + εθ1 + ε
2θ2 + . . .
)′′+(Gr0 + εGr1 + ε
2Gr2 + . . .)× (4.68)
exp(θ0 + ε
(θ1−θ
20)+ ε
2 (θ2−2θ0θ1 +θ
30)+ . . .
).
After the exponential is expanded as a Taylor series the zeroth, first and second order
equations of an asymptotic series are obtained:
O(1) : θ′′0 +Gr0eθ0 = 0, (4.69)
O(ε) : θ′′1 +Gr0eθ0θ1 = Gr0θ
20 eθ0−Gr1eθ0, (4.70)
O(ε2) : θ′′2 +Gr0eθ0θ2 = Gr0eθ0
(2θ0θ1−θ
30 −
12(θ1−θ
20)2)
(4.71)
−Gr1eθ0(θ1−θ
20)−Gr2eθ0 .
Eq. (4.69) has a known solution
θ0(s) = θc−2ln
[cosh
(√Gr0
2eθc/2s
)], (4.72)
which satisfies the boundary conditions θ0(0) = θc as θ ′0(0) = 0 (Fowler, 1997). The
remaining boundary condition, θ0(1) = 0 fixes the value of Gr0 and
Gr0 = 2acosh2[eθc/2
]e−θc . (4.73)
Maximising Gr0 with respect to θc yields the critical value of Gr at leading order which
is Gr0,cr ≈ 0.878.
In order to find the first order approximations we note that the homogeneous part of
Eq. (4.70) is solved by tα = θ ′0 and tβ = sθ ′0 +2. Based on the homogeneous solutions a
particular solution tp can be constructed and we set tp = ξ tα +ψtβ . The parameters ξ and
ψ are required to satisfy
0 = ξ′tα +ψ
′tβ , (4.74)
Gr0θ20 eθ0−Gr1eθ0 = ξ
′t ′α +ψ′t ′
β. (4.75)
The even solution of Eq. (4.70) then reads
θ1 = θ20 −2θ0 +4sθ
′0 +6− Gr1
Gr0−2θ
′0
∫θ0 dx+C1
(sθ′0 +2
), (4.76)
4.2 Localization criterion and patterns of plastic failure 65
where Gr1 and C1 are fixed by the boundary conditions θ1(0) = θ1(1) = 0.
After θ1 and Gr1 have been established the second order corrections Gr2 and θ2 can
be computed analogously to the first order correction. In Fig. 4.6 the coefficients of the
expansion Gr = Gr0 +Gr1ε +Gr2ε2 are shown as function of the centre temperature θc
and Fig. 4.7(a) presents a comparison of the numerical results and asymptotic approxima-
tion. An approximation in the low temperature regime is capable of estimating the critical
Gruntfest number to a good degree (c.f. Fig. 4.7(b)).
0 1 2 3 4 5 6 7 8 9 10−1
0
1
2
3
4
5
θc
Appro
xim
ation c
oeffic
ients
Gri
Gr
0
Gr1
Gr2
Fig. 4.6 Approximation coefficients Gri (i = 1,2,3) as a function of centre temperature θc.
0 0.5 10
1
2
3
4
5
6
7
8
9
10
θc
Gr
(a)
ε = 0.01
ε = 0.1
0 0.05 0.1 0.15 0.20.8
0.85
0.9
0.95
1
1.05
1.1
1.15
ε = 1/Ar
Grcr
(b)
Fig. 4.7 (a) Position of steady states of the heat equation (4.66) for ε = 0.1,0.01. Squares and circles de-note numerical results and the solid lines represent a three-term asymptotic approximation. The asymptoticapproximation is suitable for θc < 3. (b) The critical Gruntfest number, Grcr, as a function of ε . Circlesdenote numerical results and the solid line represents a three-term asymptotic approximation.
In order to verify the results of the perturbation analysis of Eq. (4.62) need to be com-
puted and one could employ the pseudo arc-length continuation method as before or,
66 4 Energy based criteria for the onset of localized plastic deformation
alternatively, θc can be chosen as a continuation parameter which is simpler to imple-
ment. Here, the parameter continuation of θc in conjunction with a collocation method
is employed to solve the boundary value problem (BVP). The Matlab routine BVP4C is
used as BVP solver.
4.3 Numerical experiments
4.3.1 Model
The steady state limit characterises the long-term behaviour of the system, but the evo-
lution of the system given a certain initial condition can only be obtained through direct
time integration. The system of equations that governs the transient evolution of a von
Mises elasto-plastic material read
∂σi j
∂xi= 0,
ρc∂T∂ t
= α∂ 2T∂x2
i+χσi jε
ini j , (4.77)
εi j =σkl
Cei jkl
+ ε0
⟨√q
k−1⟩
si j
qe−T0/T .
Notably, although the momentum balance is at its stationary limit, the time evolution of
the system is achieved through the temperature equation. The energy balance therefore
acts as a post-failure evolution equation similar to the ad-hoc evolution laws of the hard-
ening modulus in classical plasticity.
In this section, we integrate this system of equations for the case of an ideal visco-
plastic (Cei jkl→∞) and an elasto-visco-plastic material with thermo-mechanical coupling.
To illustrate the results of the steady-state analysis an elementary problem set is chosen,
where in a plane strain setting the loading conditions shown in Fig. 4.8 are applied. A
square cell is pinned in the x, y-direction on the left hand side and in the y-direction on
the right hand side. It is loaded on the right boundary with either a constant force F or
constant velocity v. The cell is surrounded by a reservoir that has a constant temperature
Tb.
The results of the previous sections suggest that in such a configuration the slip lines
should propagate from the pinned corners of the rectangle, diagonally across the specimen
4.3 Numerical experiments 67
AF or v
x
y
Fig. 4.8 Geometry of the problem configuration. A square is deformed by applying either a constant forceF or constant velocity v at the right hand side of the sample. The sample is pinned on the left and in they-direction. At the centre ’A’ the sample is probed for various quantities. The theoretical analysis presentedin the previous sections for a rate-dependent von Mises material suggests that the slip lines of this modelrun diagonally across the specimen and are represented by dashed lines Johnson et al (1964).
as discussed by Johnson et al (1964). Owing to the pressure independent von Mises yield
envelope, the slip lines are expected to be perpendicular to each other. In addition, the
critical condition for the localization of plastic deformation is expected to be given by
the critical value of the Gruntfest number, as shown in the previous sections. In the time-
dependent case, we may follow the same rules of normalization, Eqs. (4.55), and define
the Gruntfest number as
Gr =(
χε0L2
4αTbe−Ar
)si j
(⟨qk−1⟩)m
. (4.78)
Because the Gruntfest number incorporates the mechanical work, we expect the Grunt-
fest analysis of the previous section to hold for both constant force and constant velocity
conditions. The difference between the two regimes would be that in the case of constant
force conditions Gr will obtain a constant value, once the stress equilibrates to a con-
stant value. Under constant velocity conditions, the stress, and therefore Gr continues to
increase throughout the time of loading. Hence, given enough time the system is always
able to cross the critical Gr value for the onset of localization.
In the following sections we present the results of the time-integration of the system
(4.77). The analysis was carried out using finite element techniques as implemented in
the code REDBACK that is described in Chapter 5. First, we present two cases for the
deformation of an ideal visco-plastic material and an elasto-visco-plastic material under
constant force boundary conditions. Further, the response of the elasto-visco-plastic ma-
terial under a constant velocity boundary condition is studied.
68 4 Energy based criteria for the onset of localized plastic deformation
4.3.2 Ideal visco-plastic material under constant force boundary
conditions
0 0.1 0.2 0.3 0.4 0.5
0
5
10
15
Gr
θA
C
B
Fig. 4.9 Evolution of an ideal visco-plastic material that is deformed with a constant force. The solid blackline marks steady states. The grey lines represent the trajectories through the Gr-θ space as the systemevolves in time. Initial and final conditions are marked by grey and black squares, respectively.
In the case of an ideal visco-plastic material (Cei jkl→∞) under constant force conditions
the stress exerted onto the sample is constant throughout the simulation and the sample
enters the plastic regime immediately. Hence, Gr is held constant throughout the loading.
The evolution of the deforming system is best represented by its orbits in Gr-θ space as
shown in Fig. 4.9 because this allows for a direct comparison between the S-curve that
marks the steady states in Gr-θ space and the steady state attained by the sample. As
Gr is constant these orbits are vertical lines in the case of ideal visco-plasticity. Fig. 4.9
summarizes the evolution of the deforming sample for various initial conditions (marked
by grey squares) and their final state (black squares). The results of these simulations
confirm the concept of the S-curve that is comprised of two stable branches that are linked
by an unstable branch (BC). If Gr is chosen to be larger than the critical value (turning
point B) or the initial state lies above the unstable branch BC the system converges to the
high-temperature branch. On the other hand, if Gr is smaller than the critical value the
system converges to the low temperature isothermal branch.
4.3 Numerical experiments 69
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
2
4
6
8
10
12
Gr
θA
(a)
C
B
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
θA
Time t
(b)
0 1 2 3 4 510
−3
10−2
10−1
ǫ
Time t
(c)
Fig. 4.10 (a) Evolution of an elasto-visco-plastic material deformed with a constant force. Grey and blacksquares mark initial and final states of the system of the simulation, respectively. The solid black squaresmark the final state of runs with initial conditions very close to but above the separatrix BC and we find thatthe these runs are indeed intrinsically unstable, blowing up in short times. (b) Centre temperature θA overtime for the three runs marked in (a). (c) Strain rate ε at point A over time for the three runs marked in (a).
4.3.3 Elasto-visco-plastic material under constant force boundary
conditions
When elasticity is incorporated and the system is loaded using constant force, the Grunt-
fest number (i.e. the overstress) is expected to relax to a constant value after its initial
evolution due to the elastic response of the system. This regime allows us to test the ex-
tent of the validity of the S-curve concept established in the Sec. 4.2.2, since the orbits of
the elasto-visco-plastic material are not constant in Gr. To this end, the system of equa-
70 4 Energy based criteria for the onset of localized plastic deformation
Dissipation φ
-1
1
s =
0
76420
(a) (b)
-1
1
s =
0
−1 −0.5 0 0.5 10
1
2
3
4
Dis
sip
ation φ
Arc−length s
(c)
−1 −0.5 0 0.5 10
1
2
3
4
Dis
sip
ation φ
Arc−length s
(d)
Fig. 4.11 Dissipation profiles for the final state of a run that converges towards the low temperature branchAB and a run that converges towards branch CD. The spatial distribution of the mechanical dissipation:(a) isothermal low temperature stable branch without significant localization of plastic deformation. (b)simulation for the high temperature branch with significant localized heating. The dissipation profiles wererecorded along the lines highlighted in (a) and (b) and are presented in (c) and (d), respectively.
tions (4.77) was integrated for different initial conditions and values of Gr. The results
are summarized in Fig. 4.10(a) where we observe that the phase space is indeed split into
two domains, in accordance with the concept of the S-curve. Grey squares denote initial
conditions and black squares denote the final state of each numerical experiment.
For initial conditions starting below and to the left of the unstable branch (branch BC
of Fig. 4.4) the system evolves towards a low-energy steady state, which coincides with
the stable branch AB of Fig. 4.4. Correspondingly, for initial conditions above the un-
stable branch the orbits diverge and eventually relax in a high-energy steady-state where
dissipation localizes along the characteristics of the stress equilibrium equations. This
high energy steady state corresponds to the branch CD of Fig. 4.4. Runs that started very
close to, but above, branch BC (solid black squares) diverge towards the high temperature
branch as well. The time evolution of these orbits showcase the different responses (Fig.
4.10b, c). Indeed, the low-energy orbits lead the system fast to the low-energy steady state
where plastic deformation is not localized, whereas the high-energy orbits provide an evo-
lution of the system towards the localization of dissipation along the characteristics of the
4.3 Numerical experiments 71
stress equilibrium equations. This transition is equivalent to the evolution from secondary
to tertiary creep in material science (Fig. 4.10b, c).
Following the results of Fig. 4.4, we expect the orbits leading to the low-energy steady
state to be stable without profound localization of the plastic deformation. On the con-
trary, the high-energy steady-state is expected to provide the necessary conditions for
localization. In Fig. 4.11 we plot the spatial distribution of the plastic strain and mechan-
ical dissipation for an orbit converging to a low-energy and high-energy steady-state. We
notice that in both cases the slip lines indeed propagate from the corners towards the
diagonals of the specimen as expected, but that in the low-energy case the plastic defor-
mation is not localized across the slip lines, whereas in the high energy case localization
is achieved. We may therefore conclude that the critical Gruntfest number constitutes a
material bifurcation criterion for temperature sensitive visco-plastic materials.
4.3.4 Heat lines under constant velocity boundary conditions
0 20 40
Dissipation φ
(a)
0 4 8 12 16 2260
Temperature θ
(b)
Fig. 4.12 Heat lines under fast constant velocity loading. The heat lines emerge from an initially homo-geneous temperature state. The heat lines follow the slip lines and can be observed as a localisation of (a)mechanical dissipation and (b) temperature.
When the elasto-visco-plastic material is loaded under constant velocity conditions,
the Gruntfest number initially grows rapidly and attains values beyond its critical value.
Hence, the temperature rises rapidly and the system can be assumed to transition quickly
from near-isothermal to near-adiabatic conditions.
An extreme end-member of this regime is the case of an elasto-visco-plastic material
evolving under fast loading, thus establishing directly adiabatic conditions. This regime
72 4 Energy based criteria for the onset of localized plastic deformation
which is also known as adiabatic shear banding is crucial in metal forming and cutting,
but also in geomaterials (Vardoulakis, 2002). In the present study adiabatic shear banding
constitutes just a limiting case of the analysis, when Gr→∞. Such a case can be simulated
by setting the thermal diffusion to zero in the temperature equation which is equivalent
to assuming that the diffusive capabilities of the material are negligible as compared to
the heat producing capabilities. In this case, the dimensionless form of the temperature
equation (4.77) reads∂θ
∂ tad= exp
(θ
1+ εθ
), (4.79)
where tad = Gr · t is finite only when t → 0, since Gr→ ∞. This means that the limiting
case of adiabatic shearing takes place in explosive timescales of the order t ∼ O(1/Gr)
(Gruntfest, 1963; Veveakis et al, 2007).
Under such a scenario the localization of dissipation along the slip lines is profound,
and patterns resembling the heat lines of Fig. 4.1 are obtained, as shown in Fig. 4.12. Note
that in this case the system enters directly the unstable area of the S-curve, and traces the
upper branch CD during the time of loading.
4.4 Discussion
The simulation of the temporal evolution of temperature and rate-sensitive materials under
various loading conditions verifies the results of the steady-state analysis of the energy
budget. In conclusion, the problem of mechanical deformation of a temperature-sensitive
visco-plastic material can be elegantly captured by the bifurcation curves of Fig. 4.3. We
have seen that for this class of materials the energy equation determines the conditions for
the onset of localization of plastic deformation, which occurs along the characteristics of
the stress equilibrium equations.
The results of Fig. 4.10 highlight this energy bifurcation as the main driver of the tran-
sient orbits in the system, even when elasticity is considered. It therefore provides a mate-
rial bifurcation criterion and, combined with the analysis of Sec. 4.2.1, a generalization of
the slip line field theory for thermo-visco-plastic materials. The main effect of an exten-
sion to rate and temperature dependence is that lines of velocity discontinuity do not exist
below a critical temperature and deformation rate corresponding to a critical dissipation
as captured by the Gruntfest number Gr. Above critical Gr the classical slip lines emerge
4.4 Discussion 73
in terms of velocity discontinuities and below critical Gr homogeneous deformation is
derived.
The extension of the analytical treatment of generalized slip line field theory to the tran-
sient regime using a numerical scheme has allowed us to test it for the problem of metal
forging presented in Fig. 4.1. The constant velocity boundary condition applied during
forging lead to a variation in Gr thereby crossing the stability regimes from homogeneous
deformation to the appearance of heat lines. The fast loading during this deformation can
lead the material directly to the extreme case of adiabatic deformation, as shown in Fig.
4.1.
This work has therefore presented a generalized approach of slip line field theory show-
ing that the simple, but restrictive, rheology of a rigid-plastic body can be extended to
include rate dependent and temperature dependent material behaviour. We have detailed
an energy based framework to obtain the modified Hencky’s and Geiringer’s equations
which still provide the information on the basic pattern of slip lines underpinning the
deformation process. We argue that this extension is therefore useful for gaining a basic
understanding into the more complex material behaviour from an analytical perspective.
The extension of the analytical approach reveals, however, a more complex material be-
haviour than in the classical theory.
The results of the present study highlight the role of the energy balance in material
bifurcations. Since the critical condition for material instability is retrieved from an en-
ergy bifurcation, the temperature equation can act as a hardening law and substitute the
experimentally derived hardening laws. Such a case would allow us to account explicitly
for different physics as well as for the mechanisms acting at different scales in a material
through their energy budget. We are therefore one step further to our quest on multiscale
analyses, since all the necessary mechanisms can be explicitly accounted for in the en-
ergy equation through the corresponding internal state variables of the micro-processes.
The results of the present work can provide the basis for a unified theory for material
behaviour, starting from solid mechanics and extending naturally to the fluid-like post
failure evolution of materials using explicitly the energy considerations as the link.
Chapter 5
Multi-physics simulations of rock mechanics using REDBACK
After discussing materials that exhibit thermo-mechanical coupling we now turn porous
materials that are fully saturated and in addition to thermo-mechanical coupling we wish
to also consider the hydrological and chemical properties of the material. This consi-
tutes an immensely complex problem and a rigorous analysis is omitted here. Instead we
present a software environment called REDBACK that allows for the efficient simulation
of the aforementioned processes and their respective couplings and we outline the imple-
mented physics. Due to the fact that within REDBACK all processes are tightly coupled
REDBACK provides us with the means to identify spatio-/ temporal instabilities for a wide
range of problems and we present two applications here.
Thermo-hydro-mechanical-chemical (THMC) coupling has traditionally been done in a
sequential way, where mechanical solvers are bootstrapped to fluid-flow solvers which in
turn are coupled to chemical solvers. The thermal aspect of the problem is often assumed
isothermal for the individual couplings or entirely ignored. Updates are usually performed
after a sequential solution procedure that ensures stability of the solution by ordering the
fully coupled problem in an ‘ad hoc’ way (e.g. Taron et al, 2009; Poulet et al, 2012).
While this is a formidable computational task which is useful for engineering applications
it has the drawbacks of overlooking instabilities that only arise through the tight coupling
of THMC processes and nonlinear feedbacks. An example is given in Chapter 4 where
the well known thermo-mechanical feedback (Gruntfest, 1963) in the tightly coupled heat
equation (including the shear heating term), continuity, and momentum equations delivers
instabilities that localize viscously in deforming materials into a shear failure plane. Nu-
merical tools exist to solve tightly coupled systems of equations (e.g. ABAQUS Hibbitt
et al, 2008) but they do not necessarily provide the flexibility required to easily inves-
tigate the driving physical processes. Their programming can be complex, limited (for
licensing reasons), and in general such tools do not provide direct access to the under-
75
76 5 Multi-physics simulations of rock mechanics using REDBACK
lying dimensionless groups qualifying the physical processes as they are not targeted at
modellers studying instabilities in particular. This chapter presents a numerical tool able
to provide the tight couplings between THMC processes with the goal of avoiding the
necessary fitting parameters for sequentially coupled engineering solutions. For this pur-
pose, REDBACK is based on a physics driven dimensionless formulation which captures
the observed multi-physics phenomena computationally.
In the sections to come we lay out the theoretical multi-physics model (Sec. 5.1), the
generic computational formulation (Sec. 5.2) and introduce two applied case studies to
illustrate some applications of the novel simulator. The first case study (Sec. 5.3) con-
cerns a poro-elasto-visco-plastic laboratory experiment of a diatomaceous mudstone with
thermo-hydro-mechanical (THM) coupling (Oka et al, 2011). It emphasises the impor-
tance of the dimensionless approach to compare the rates of the driving physical pro-
cesses involved, as well as the importance of a flexible numerical platform to easily pro-
totype while investigating the underlying physics. The second case study (Sec. 5.4) looks
at a complete THMC tightly coupled multi-physics application in geomechanics as pre-
sented in Alevizos et al (2014), extended to three dimensions. This example models a
fluid-saturated fault under shear where fluid-release chemical reactions occur, based on
the assumption that the material inside exhibits rate- and temperature-dependent plastic
behaviour.
5.1 Theoretical model
This section presents the theoretical framework for porous materials that consist of a solid
rock skeleton and a fluid saturating the porous space. We present a continuum description
and discuss mechanical, hydrodynamical, chemical and thermal aspects in turn. Through-
out this article Einstein’s summation rule is used and subscripts are indexes if not noted
otherwise.
5.1.1 Mechanics of dry materials
REDBACK is currently based on the principles of overstress plasticity, in line with the
seminal work of Perzyna (1966). The total strain rate εi j is decomposed into a reversible
5.1 Theoretical model 77
(elastic), εei j, and an irreversible, ε in
i j , part
εi j = εei j + ε
ini j . (5.1)
The reversible part is assumed to obey a linear elastic relationship of the form
εei j =C−1
i jklσkl, (5.2)
where σkl represents the stress tensor and Ci jkl is the elasticity tensor. In the current im-
plementation, the irreversible part of the strain rate obeys an associative visco-plastic flow
law of the form
εini j = λ
∂ f∂σi j
, (5.3)
where f is the yield function and λ is a (scalar) plastic multiplier, which follows the
relation
λ =
√(ε in
d
)2+(ε in
v )2. (5.4)
In this expression, ε ind and ε in
v are the deviatoric and volumetric parts of the strain rate
tensor, respectively, following the incremental relations
εind = ε0
⟨q−qY
σre f
⟩m
exp(−Qmech
RT
), (5.5)
εinv = ε0
⟨p− pY
σre f
⟩m
exp(−Qmech
RT
), (5.6)
where ε0 is a reference strain rate, q is equivalent stress, p is the volumetric mean stress,
qY and pY are the respective values at yield, σre f is a reference stress, R is the univer-
sal gas constant, T is temperature, m > 0 is an exponent and 〈·〉 denote the Macaulay
brackets. These expressions imply that the material is admitting thermal sensitivity ex-
pressed through the activation enthalpy Qmech. This activation enthalpy incorporates the
activation energies of all the micro-mechanical mechanisms, like frictional initiation (Rice
et al, 2001) or volumetric pore collapse. It is, in principle, expressed in a form
Qmech = E +PVact , (5.7)
where E is the activation energy and PVact the product of a given stress P times the acti-
vation volume Vact of the considered internal process.
78 5 Multi-physics simulations of rock mechanics using REDBACK
5.1.2 Chemical damage
Thermally activated chemical reactions are allowed to take place and in this work and we
concentrate on (de-)hydration reactions of the form
ν1(AB)sωFωR
ν2As +ν3B f , (5.8)
where the subscripts s and f refer to solid and fluid phases and νi (i = 1,2,3) are stoichio-
metric coefficients. The reaction equation (5.8) states that the solid A can release/bind the
component B into/from the fluid phase which increases/reduces the pore pressure.
The kinetics of the decomposition reaction (5.8) are assumed to follow a standard Ar-
rhenius dependency on temperature (Poulet et al, 2014a). As a result, the rates of the
forward, ωF , and reverse reaction, ωR, can be expressed as (Alevizos et al, 2014)
ωF =
[ρAB
MAB(1−φ)(1− s)
]ν1
kFe−QF/(RT ), (5.9a)
ωR =
[ρA
MA(1−φ)s
]ν2[
ρB
MB∆φchem
]ν3
kRe−QR/(RT ), (5.9b)
where ρi and Mi (i = A,B,AB) are the densities and molar masses of the respective con-
stituent and kF ,kR, QF ,QR are the pre-exponential factors and activation enthalpies of the
forward and reverse reaction, φ is porosity and ∆φchem denotes change in porosity due to
chemical processes. The terms in brackets are the respective concentrations of phase AB,
A and B. We define the solid ratio
s =VA
Vs=
VA
(1−φ)V, (5.10)
where V is a representative volume, VA and Vs are the volume of solid phase A and all
solid within V , respectively. The solid ratio is a measure of the extend of reaction (5.8).
We assume that ν1 = ν2 = ν3 = 1 and the total reaction rate then reads
ω =
[(1− s)− s∆φchem
ρAρB
ρ2AB
M2AB
MAMBK−1
c e∆h/(RT )](1−φ)
ρAB
MABkFe−QF/(RT ) (5.11)
where Kc = kF/kR and ∆h=QR−QF . The expressions for the dependency of the porosity
φ and solid ratio s on the reaction kinetics are described in detail in Alevizos et al (2014)
and briefly summarized here.
5.1 Theoretical model 79
We assume the following relations for the partial molar reaction rates of the species
involved
ωAB =−[
ρAB
MAB(1−φ)(1− s)
]ν1
kFe−QF/(RT ), (5.12a)
ωA =
[ρA
MA(1−φ)s
]ν2
kAe−QR/(RT ), (5.12b)
ωB =
[∆φchem
ρB
MB
]ν3
kBe(−QR/(RT ), (5.12c)
and these rates are linked by the stoichiometry of the considered reaction (5.8) as
−ωAB
ν1=
ωA
ν2=
ωB
ν3. (5.13)
From Eqs. (5.12, 5.13) and for ν1 = ν2 = ν3 = 1 we derive the poro-chemical model
∆φchem = Aφ
1−φ0
1+ ρBρA
MAMB
1s
, (5.14a)
s =ωrel
1+ωrel, (5.14b)
ωrel =ρAB
ρA
MA
MABKc exp
(∆hRT
), (5.14c)
where φ0 is the initial porosity and Aφ is a coefficient that determines the amount of the
interconnected pore-volume (porosity) created due to the reaction. We assume that all the
fluid generated contributes to the interconnected pore volume, and thus set Aφ = 1.
Following these considerations, the rates of the forward (ωF ) and reverse (ωR) first
order reactions can be equivalently expressed as
ωF =−ωAB, ωR = ωAωB, (5.15)
if we set kR =√
kAkB. Note that, for simplicity we have assumed in Eqs. (5.12) that the
two products are involving the same pre-exponential factor and activation energies. If this
is not the case the above model should be modified accordingly. The net reaction rate
would then be ω = ωF −ωRMABρAB
(the reverse reaction rate was normalized with the refer-
ence concentration ρABMAB
for dimensional purposes), which, however, would be essentially
irreversible (ωF ωR) in the case Kc = kF/kR 1.
80 5 Multi-physics simulations of rock mechanics using REDBACK
5.1.3 Poromechanics
Due to the dehydration reaction (5.8) the pore pressure is variable and the mechanical
framework presented in Sec. 5.1.1 has to be extended to take into account the presence
of the fluid saturating the pore space. Here, this poromechanical extension is summarized
briefly, but a full account can be found in Coussy (2004).
Following Terzaghi’s principle, the dependence of the stress σi j on the pore fluid pres-
sure, p f , is stated explicitly as σi j = σ ′i j+ p f δi j where σ ′i j is the effective stress. (Note that
stresses are taken to be negative in compression). The pore pressure evolution is defined
via p f = phyd +∆ p f , where phyd is the hydrostatic pressure (assumed constant) and ∆ p f
is the excess pore pressure.
The total porosity φ is expressed as the sum of its initial value, φ0, and the newly
created interconnected pore volume. Pore volume can be created by mechanical (∆φmech)
and chemical (∆φchem) processes such that the total porosity reads
φ = φ0 +∆φmech +∆φchem =VB
V, (5.16)
where VB is the volume occupied by fluid B. The evolution of mechanical porosity con-
tains two components, a plastic part ∆φplmech = ε
plV , with ε
plV the volumetric plastic strain,
and an elastic one ∆φ emech =(1−φ)
(βs∆ p f −λs∆T
), where βs and λs are compressibility
and thermal expansion coefficients of the solid phase, respectively.
The mass balance equations for the solid and fluid phases can be expressed as (Alevizos
et al (2014))
1−φ
ρs∂tρs +
1−φ
ρsvs
i ∂iρs−∂tφ − vsi ∂iφ +(1−φ)∂ivs
i = js, (5.17a)
φ
ρ f∂tρ f +
φ
ρ fv f
i ∂iρ f +∂tφ + v fi ∂iφ +φ∂iv
fi = j f , (5.17b)
where vsi and v f
i denote the velocities of the solid (consisting of AB and A) and fluid (B)
phases, respectively, and the source terms js = ωABMAB +ωAMA and j f = ωBMB. The
fluid and solid phases are considered to be compressible and their variations in density
are therefore expressed as
5.1 Theoretical model 81
dρ f
ρ f= β f d p f −λ f dT, (5.18a)
dρs
ρs= βsd p f −λsdT, (5.18b)
where β f and λ f are analogous to βs and λs.
Conservation of mass and momentum result in the momentum balance, which is com-
monly expressed as (c.f Eq. (4.8))
∂ jσ′i j +∂i∆ p f +bi = 0, (5.19)
where bi is the body force vector.
Given the length-scale of the geological applications targeted (meso- and macro-scales)
we can use Darcy’s law to relate the mass fluxes to the pore pressure gradient (not con-
sidering tortuosity or any higher order term)
(v fi − vs
i )φ =− kπ
µ f(∂i p f −ρ f gi), (5.20)
where µ f is the viscosity of the fluid, kπ the permeability and gi the gravity vector. The
permeability is considered to evolve as a function of porosity and the first dependency
considered is the Kozeny-Carman relationship
kπ = kπ0(1−φ0)
2
φ 30
φ 3
(1−φ)2 , (5.21)
where kπ0 is a reference permeability corresponding to the reference porosity φ0 (Sulem
and Famin, 2009).
5.1.4 Energy considerations
The energy balance equation, combined with the second law of thermodynamics and
Fourier’s law of heat conduction, provides the local form of the temperature diffusion-
reaction equation (c.f. Sec. 4.1.2 and Alevizos et al (2014))
ρC (∂tT + vi∂iT ) = ∂i(α∂iT )+χ σpli j ε
pli j +∆hi ωi, (5.22)
82 5 Multi-physics simulations of rock mechanics using REDBACK
where vi = (1−φ)vsi +φv f
i is the barycentric velocity of the mixture, ρC the heat capacity
of the mixture, α the thermal conductivity and χ the Taylor-Quinney coefficient (Taylor
and Quinney, 1934a).
5.1.5 Final set of equations
The final set of equations consists of the momentum (5.19), mass balance (5.17), and
energy equations (5.22), as well as the constitutive relationships, Eqs (5.14, 5.18). This
system of equations can be written in a dimensionless form by introducing the normalised
variables
t? =cth
L2re f
t, x? =x
Lre f, T ? =
T −Tre f
δTre f, ∆ p? =
∆ p f
σre f, σ
?i j =
σi j
σre f, (5.23)
Table 5.1 Dimensionless parameters used in REDBACK. The coefficient δ is defined such that T ? = (T −Tre f )/(δTre f )
Group Name Definition Interpretation
Gr Gruntfest num-ber
χσre f εre f L2re f
αδTre fRate of conversion of mechanical work intoheat over rate of diffusive processes
Daendo EndothermicDamkohlernumber
AendohendoρABL2re f
αδTre fEndothermic reaction rate over rate of diffu-sive processes
Daexo ExothermicDamkohlernumber
AexohexoρABL2re f
αδTre fExothermic reaction rate over rate of diffusiveprocesses
Ar Arrhenius num-ber
Qmech/(RTre f ) Activation energy over thermal energy
ArF Forward Arrhe-nius number
QF/(RTre f ) Activation energy of forward reaction overthermal energy
ArR Reverse Arrhe-nius number
QR/(RTre f ) Activation energy of reverse activation energyover thermal energy
Λ Thermal pres-surisationcoefficient
λmβm
δ Tre fσre f
Normalised thermal pressurisation coefficient,with λm and βm the mixture thermal expansionand compressibility
Le Lewis number cth/chy Thermal over mass diffusivity
Lechem ChemicalLewis number
cthσre f βm
L2re f Aendo
(ρBρm
)(MABMB
)Thermal over chemical diffusivity of forwardreaction
Pe Peclet number Lre f vre f /cth Advection rate over diffusion rate
5.2 Numerical implementation 83
where cth is thermal diffusivity, and Lre f , Tre f and σre f are normalization constants for
length, temperature and stress, respectively. The system in its final form, where the aster-
isks have been dropped to simplify the notation, is
0 = ∂ jσ′i j +∂i∆ p f +bi, (5.24a)
0 = ∂t∆ p f +Pe vpi ∂i∆ p f −Pe vT
i ∂iT −∂i
[1
Le∂i∆ p f
](5.24b)
−Λ∂tT +εV
β σre f− 1
LechemωF ,
0 = ∂tT +Pe vi∂iT −∂2ii T −Gr σ
pli j ε
pli j +Daendo ωF −Daexo ωR. (5.24c)
All dimensionless groups are defined in Tab. 5.1 and
β = (1−φ)βs +φβ f ,
vpi = (1−φ)
βs
βvs
i +φβ f
βv f
i ,
vTi = (1−φ)
λs
βvs
i +φλ f
βv f
i ,
vi = (1−φ)vsi +φv f
i ,
ωF = (1−φ)(1− s)exp(
ArF δT1+δT
),
ωR = (1−φ) s ∆φchem exp(
ArR δT1+δT
),
εpli j = ε0 exp
(Ar δT1+δT
)√⟨q−qY
σre f
⟩2m
+
⟨p− pY
σre f
⟩2m∂ f
∂σi j.
where βk→ βkσre f and λk→ λkTre f /σre f (k = s, f ).
We choose to implement this nondimensional set of equations into REDBACK because
this approach reduces the number of free parameters and allows for a direct comparison
between analytical results from scaling or bifurcation analyses and the numerical simula-
tions.
5.2 Numerical implementation
The final system of equations (5.24) involves the strongly coupled variables: displace-
ment u, temperature T and pore pressure p f as well as material properties, which can
also be highly nonlinear, such as the chemical porosity evolution. For those reasons, tra-
84 5 Multi-physics simulations of rock mechanics using REDBACK
ditional data-flow oriented computational frameworks which solve equations sequentially
(e.g. Poulet et al, 2012) are not well suited due to convergence issues when dealing with
instabilities (i.e. ultra-fast evolutions). We present therefore a new application to solve
this problem numerically in a tightly coupled manner in three dimensions.
5.2.1 Using the MOOSE framework
The Multi-physics Object Oriented Simulation Environment1 (MOOSE) (Gaston et al,
2009) provides a powerful and flexible platform to solve multi-physics problems implic-
itly and in a tightly coupled manner on unstructured meshes. The code is built on top of
libraries including the libMesh finite element library (Kirk et al, 2006) and the PETSc
solver library (Balay et al, 2014). It uses a Jacobian–free Newton–Krylov approach for
numerical efficiency (e.g. Knoll and Keyes, 2004) and can harness the growing power of
supercomputers as it scales well on parallel computer architectures. Its object-oriented
coding paradigm and user focus have been optimised to allow relatively effortless pro-
totyping of coupled systems of equations on a laptop and produce code that can then be
run on state of the art high-performance computational resources. This provides both the
flexibility to develop models quickly while investigating some interesting physics, and yet
allow modellers to treat increasingly more realistic scenarios, including complex geome-
tries and material properties. MOOSE lets modellers focus on their physical problem at
hand by hiding the complexity of all underlying computing structures and proposes a sim-
ple interface to all specific functionalities available. It provides a framework for (mesh)
dimension-independent equations and yet includes a wide library of mesh elements and
shape functions to solve the governing equations on. By design MOOSE aims at solving a
system of equations implicitly, but it also provides levels of decoupling through its multi–
app functionality and some granularity on the update timing for some objects, potentially
leading to explicit updates if needed. All those qualities, alongside the fact that it is open–
source code and backed up by a highly active development community make it a tool of
choice to solve numerically the problem described in Sec. 5.1.
1 http://mooseframework.org
5.2 Numerical implementation 85
5.2.2 REDBACK
Thermodynamics
REDBACK
Hydrodynamics
Mechanics Chemistry
MOOSE
Tensor Mechanics
FiniteElementMethod
LIBMESH
Mesh Input/ Output
PETSc
Solver Interface
Trilinos NOX
Fig. 5.1 Structure of REDBACK. REDBACK builds on top of MOOSE and it implements the physical be-haviour of porous materials detailed in Sec. 5.1. For large strain mechanics REDBACK can draw upon thetensor mechanics module that ships with MOOSE. MOOSE itself wraps around libMesh that deals with thediscretization of the governing equations and various solvers for the numerical integration.
MOOSE provides a programming environment as well as a range of pre-existing physics
modules to solve specific problems. We have derived a new application, REDBACK2, to
tackle the problem of Rock mEchanics with Dissipative feedBACKs and solve the system
of equations (5.24) in its dimensionless form. This allows greater flexibility to understand
2 http://github.com/pou036/redback
86 5 Multi-physics simulations of rock mechanics using REDBACK
the role of all dimensionless groups (defined in Tab. 5.1), their influence on the resulting
behaviour of simulations and the balance of all rates of processes involved. This approach
is particularly important when targeting applications involving material instabilities, as
one can identify the critical values of the dimensionless groups in order to obtain an
expected behaviour based on a previous analytical stability analysis and invert for the real
material parameters afterwards.
REDBACK is itself an open–source application and builds on MOOSE’s tensor me-
chanics module that implements finite strain (Rashid, 1993) and the architecture of
REDBACK/MOOSE is depict in Fig. 5.1. REDBACK extends this module with additional
constitutive models for the overstress plasticity formulation. This includes a modified
Cam–Clay model (Roscoe and Burland, 1968) to solve compaction problems with cap–
plasticity, where the yield surface is defined as
( qM
)2+ p(p− pc) = 0, (5.25)
and pc is the pre-consolidation pressure and M the slope of the critical state line.
One of the strengths of MOOSE is that it allows the user to concentrate on the physical
problem without having to worry about numerical issues such as the discretization of the
governing equations, storage management, etc. The physics of the problem at hand are
implemented in MOOSE and REDBACK via so called kernels. In order to find the kernels
corresponding to the governing equations (5.24) their corresponding weak form needs to
be computed. This is demonstrated on the mass balance equation (5.24a) that is firstly
multiplied by a test function ψ and subsequently integrated over the domain Ω . This
yields
0 =−∫
Ω
∂ jψσi j︸ ︷︷ ︸kernel
dx−∫
Ω
∂iψ∆ p f︸ ︷︷ ︸kernel
dx+∫
Ω
ψbi︸︷︷︸kernel
dx+boundary terms, (5.26)
and the integrands are referred to as kernels within the context of MOOSE. The boundary
terms stem from the partial integration and constitute boundary conditions. MOOSE han-
dles the integration of the kernels automatically and uses them to assemble the residual
vector, i.e. right hand side, of Eqs. (5.24) before carrying out a simulation step. The indi-
vidual stages within an simulation step are shown in Fig. 5.2 and they are comprised of
5.2 Numerical implementation 87
Table 5.2 Mapping of REDBACK kernels implementing their respecting terms in Eq. 5.24
Kernel name Variable Equation
RedbackStressDivergenceTensors ux, uy, uz ∂ jσ′i j +∂i∆ p f +bi
RedbackMassConvection p f Pe vpi ∂i∆ p f −Pe vT
i ∂iT
RedbackMassDiffusion p f −∂i[∂i∆ p f /Le
]RedbackThermalPressurization p f −Λ∂tT
RedbackPoromechanics p f εV/(β σre f )
RedbackChemPressure p f −rF/Lechem
RedbackThermalConvection T Pe vi∂iTRedbackMechDissip T −Gr σ
pli j ε
pli j
RedbackChemEndo T Daendo ωF
RedbackChemExo T −Daexo ωR
an update of material parameters at the beginning of the integration step, followed by the
evaluation of the kernels and the subsequent time integration.
Update nondi-mensional
groups/material
parameters
Evaluatekernels
Perform inte-gration step
Fig. 5.2 Structure of simulation step in REDBACK. At the beginning of the integration step the materialparameters are updated (c.f. Tab. 5.1), then the kernels are evaluated and the residual of Eqs. (5.24) iscomputed followed by carrying out the integration. Then the cycle repeats.
MOOSE provides the user with the convenience of including/excluding kernels in/from
a simulation as desired such that various physical scenarios can be easily investigated.
MOOSE contains a library of common kernels, which were used to account for the time
derivative and diffusion terms. All other REDBACK–specific kernels are listed in Tab. 5.2
along with the variable they are applied to and the corresponding term in Eqs. (5.24) they
refer to. Note that each kernel applies to a single variable and that other variables are then
coupled to it such that all off–diagonal terms are considered.
REDBACK includes a few benchmarks implemented as unit tests (see source code) in
order to check the main functionalities. Two applications are presented in the following
sections to illustrate the interest of the approach and the use of the software.
88 5 Multi-physics simulations of rock mechanics using REDBACK
Table 5.3 Confinement pressures used in experiments from (Oka et al, 2011)
Case No. CD1 CD3 CD3 CD4 CD5 CD6
Effective confining pressure (MPa) 0.25 0.5 0.75 1.0 1.5 2.0
5.3 Behaviour of diatomaceous mudstone with increasing
confinement
Confinement plays a critical role in the orientation of localized features in rocks, taking
the form of (i) dilational–induced localization bands under dilation, (ii) shear bands with
dilatancy under low mean stress (compression), (iii) compaction shear bands under higher
mean stress, and (iv) compaction bands for higher confining pressure (Weinberg et al,
2015). This type of behaviour evolution as a function of confinement is characteristic of
the balance of all physical processes at play and represents therefore a great application
for multi-physics modelling using REDBACK.
For diatomaceous mudstone experimental data of six consolidated-drained (CD) tri-
axial tests under varying confining pressures is available (experiments CD1 to CD6, see
Tab. 5.3) (Oka et al, 2011). Some of these experimental observations could be reproduced
well with an elasto-visco-plastic model by Oka et al (2011), however they use differ-
ent values of material parameters for each of the six experiments and did not formulate
explicitly the confining pressure dependency for those parameters. We approach the prob-
lem from a different perspective and aim to identify the underlying physical processes
by using the problem formulation presented in Sec. 5.1.5 and specifically the nature of
the activation volume from Eq. (5.7). This physics based approach does not require new
material parameters to be fitted to each experiment CD1 to CD6 and all experiments can
be reproduced with a satisfying degree of accuracy by only varying the confining pressure
but keeping the material parameters fixed.
During our simulations, a block of fully saturated porous material is exposed to various
confining pressures and Fig. 5.3 shows the setup used for the numerical simulations pre-
sented here. The block has an aspect ratio of 1:2:1, all vertical sides are impermeable and
are kept at constant temperature (Dirichlet boundary condition). The simulations proceed
in two stages: firstly, a confining pressure matching one of the experiments CD1 to CD6 is
selected and the material is equilibrated under this confinement. During this initialization
stage, the bottom face is allowed to move along the X– and Z–direction while the middle
5.3 Behaviour of diatomaceous mudstone with increasing confinement 89
x
y
z
Fig. 5.3 Numerical model setup
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Mean effective stress (MPa)
0.0
0.5
1.0
1.5
2.0
2.5
Deviatoric stress (MPa
)
CD1
CD2CD3
CD4
CD5
CD6
0 Pc/2 Pc
Mean effective stress
Deviatoric stress
Dilatant regime
Contractant regime
CSL
Fig. 5.4 (left) Best fits for modified Cam–Clay (plain line) and original Cam–Clay (dashed line) yieldsurfaces are plotted with experimental stress paths and yield points (dots) digitised from Fig. 5 in (Okaet al, 2011). (right) Modified Cam-Clay yield surface highlighting the dilatant (CD1) and contractant (CD3-6) regimes. CD2 appears to be on the critical state line (CSL) of modified Cam-Clay envelope.
of the edges of the bottom face are pinned in one direction to avoid rotations. The other
five faces are subject to the selected confining pressure. When the material is equilibrated
the results are stored using MOOSE’s restart functionality. Then, the main simulations
start by restoring the position from the initialization step. Throughout the main simula-
tion, the displacements ux and uz of the top and bottom face are fixed to the equilibrated
values and in addition the bottom face is fixed in the Y–direction and a velocity boundary
condition is imposed on the top to simulate the compression at a constant rate. The four
vertical faces are kept under the same confining pressure.
The experimental data (Oka et al, 2011) with all yield points is reproduced in Fig. 5.4.
We have chosen to use a modified Cam–Clay model as it is more convenient than the
original Cam–Clay model from a numerical stability perspective. A least-square fit yields
pc = 2.26 MPa and M = 1.44 (see Eq. (5.25)). We refer the reader to Oka et al (2011)
for a detailed description of all experimental results. The main feature we focus on for
90 5 Multi-physics simulations of rock mechanics using REDBACK
0 5 10 15 20Axial strain (%)
0.00
0.25
0.50
0.75
1.00
1.25
Norm
alised deviatoric stress
CD1 (sim.)
CD1 (exp.) CD2 (sim.)
CD2 (exp.)
CD3 (sim.) CD3 (exp.)
0 5 10 15 20Axial strain (%)
0.00
0.25
0.50
0.75
1.00
1.25
Norm
alised deviatoric stress
CD4 (exp.) CD4 (sim.)
CD5 (exp.)
CD5 (sim.)
CD6 (exp.)
CD6 (sim.)
Fig. 5.5 Matching experiments CD1-CD6 with simulation results. Note that the results are shown in anormalised stress space, where the preconsolidation pressure σre f = 2.26 M Pa is used as the referencevalue of the stress used for normalising the experimental data. In the absence of information from Oka et al(2011) on the value of the pore fluid pressure during the experiments, the numerical results are normalisedusing the same value, requiring that the maximum pore fluid pressure during the experiments is around1 M Pa.
the purpose of this study is the evolution of the stress-strain relationships as confinement
increases (see Fig. 2 in Oka et al, 2011), progressing from a weakening to a hardening
behaviour.
The implementation of the dimensionless system of Eqs. (5.24) in REDBACK makes
it easy to understand the clear role of the various dimensionless groups (see Tab. 5.1) by
running a series of simulations where the physical processes are considered independently
and progressively coupled. Only two mechanisms are available for hardening and soften-
ing: strain rate hardening and thermal softening. The role of the activation volume on the
thermal sensitivity of the material has not been discussed yet and is left as a free param-
eter to be evaluated from the physics of the problem. Knowing that internal mechanisms
like pore collapse can have activation volumes that are depending on the temperature and
pressure conditions (c.f. Eq. (5.7)), we use the activation enthalpy Q as free parameter and
invert for its evolution and pressure/temperature dependency (c.f. Fig. 5.6).
In order to perform this inversion, we reduce the dimensionality of the problem by
setting Le= δ = 1, Gr exp(Ar)= 1, and fix the rheology at m= 2 and the elastic properties
at a Poisson’s ratio ν = 0.2 and Young’s modulus E = 180 MPa. Then, based on the
response of the six numerical simulations (see Fig. 5.5), we first optimize Λ = 0.45 and
then investigate the expression of Ar = Qmech/(R Tre f ). Based on the results displayed in
Fig. 5.6 that shows Vact as a function of confinement, we express the activation enthalpy
5.3 Behaviour of diatomaceous mudstone with increasing confinement 91
0 0.2 0.4 0.6 0.8 1−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Pc0
/ σref
V ∗
act / T
Fig. 5.6 The pressure dependant term V ∗act/T , where V ∗act =VactTre f /(Vre f α2), from Eq. (5.27). Each pointcorresponds to one of the experiments CD1-CD6 and an optimal value of V ast
act has been inferred manuallyto match the results in Fig. 5.5. The vanishing activation volume of CD2 is in agreement with the criticalstate concept highlighted through the results of Fig 5.4. The solid line is the fit with α3 = 0.6.
Qmech = Q0 + p fVact = E0 +α1Pc0T
Tre fVre f +α2 p f
TTre f
(1+α3 ln
Pc0
σre f
)Vre f , (5.27)
where
Vact = α2T
Tre f
(1+α3 ln
Pc0
σre f
)Vre f , (5.28)
Pc0 is the effective confining pressure (our control parameter) and E0/(R Tre f ) = 8. We
set α1Vre f /(R Tre f ) = −3.3, α2Vre f /(R Tre f ) = 67.5, and α3 = 0.6 when considering a
unitary reference activation volume Vre f . The logarithmic dependency of the activation
volume with confinement and linear with temperature in Eq. (5.27), is in agreement with
the theoretical results of the activation volume for defects in solids (see Eq. 1 of Varot-
sos and Alexopoulos, 1980). Because Pc0 < σre f in the selected example, the argument
of the logarithm in Eq. 5.27 is less than 1, hence the activation volume can be negative,
in accordance to the theoretical findings of Varotsos and Alexopoulos (1980). Notably,
zero activation volume is achieved at CD2 (see Fig. 5.6), with negative activation vol-
ume before (i.e. at CD1) and positive after (CD 3-6). This result rationalizes the existence
of the two regimes highlighted in Fig. 5.4b, implying that dilation corresponds to inter-
nal processes with negative activation volume, while contraction corresponds to internal
processes with positive activation volume. Therefore, the critical state (CD2) of geomate-
rials corresponds to the limit case of zero activation volume. This means that the critical
state does not involve additional internal volumetric processes, which is in line with the
phenomenological definition of critical state and the two regimes.
92 5 Multi-physics simulations of rock mechanics using REDBACK
Table 5.4 Simulation parameters (see definitions in Tab. 5.1)
Group Name Value
Gr Gruntfest number 1Daendo Endothermic Damkohler number 0Daexo Exothermic Damkohler number 0Ar Arrhenius number 8ArF Forward Arrhenius number 0ArR Reverse Arrhenius number 0Le Lewis number 1Lechem Chemical Lewis number 0Λ Thermal pressurisation coefficient 0.45Pe Peclet number 0ε0 Reference strain rate 1
From the values of the dimensionless groups used during the simulations (Tab. 5.4)
and their definitions in Tab. 5.1, we may invert for values of the real parameters. Setting
Le = 1 means that the thermal and hydraulic diffusivities are equal. A value of the thermal
diffusivity cth = 10−6 m2/s implies that the hydraulic one should be chy = k/(µ f βm) =
10−6 m2/s, where k is the permeability, µ f the fluid viscosity and βm the mixture’s com-
pressibility. For the permeability of diatomaceous mudstone reported by Oka et al (2011)
Fig. 5.7 Distribution of deviatoric (a) and volumetric (b) plastic strain for numerical experiment CD1 at20% strain.
5.4 Episodic Tremor and Slip events 93
k = 10−16 m2/s and an indicative water viscosity at room temperature (µ f = 10−3 Pa s),
we find that the mixture’s compressibility is of the order of βm = 10−7 Pa−1, which is the
typical compressibility for soft geomaterials like stiff clay, loose sand and mudstones (Ve-
veakis et al, 2014). From the definition of the Gruntfest number, an indicative value of the
Taylor-Quinney coefficient χ can be inferred which expresses the amount of mechanical
work converted into heat. Its value should be between 0 and 1. If we accept Tre f = 300 K,
σre f = 2.26 M Pa, α = 2 J/(mKs), Lre f = 2 cm, ε0 = 1 and Gr = 1 we obtain χ = 0.6. The
condition Λ = 0.45 prescribes the value of the mixture’s thermal expansion coefficient λm
and from Tab. 5.1 we obtain λm = 3×10−4 K−1, which is in the range of values expected
for soft geomaterials (Veveakis et al, 2014). Finally, the expression E0/(RTre f ) = 8 yields
a reference activation energy E0 = 20 kJ/mol. All real parameter values are within the
range of the accepted values for soft geomaterials like mudstones.
5.4 Episodic Tremor and Slip events
As a second application of REDBACK we introduce an example including fluid–release
chemical reactions, which are at the core of the specific formulation considered (see
Sec. 5.1). This theoretical model has already been applied to active subduction zones for
which time series of relative displacement are published (Alevizos et al, 2014; Veveakis
et al, 2014; Poulet et al, 2014a) as well as exhumed thrusts where spatial information is
available (Poulet et al, 2014b). These studies aimed at providing insight on the driving
mechanisms operating in such creeping shear zones and could therefore be analysed with
a simplified one–dimensional model, without resolving explicitly the mechanical prob-
lem as the stress was assumed constant in a post-failure regime. Those results opened
the door to a novel understanding of those geological systems and now further questions
can be answered by simulating those scenarios with a more realistic geometry in order
to investigate the variation of behaviours along such thrusts as one example. We present
therefore an application of the model of Alevizos et al (2014) in three dimensions with
a full mechanical description to demonstrate the capability of REDBACK to potentially
tackle those kind of problems. Note that the above-mentioned geological questions are
beyond the scope of this study.
94 5 Multi-physics simulations of rock mechanics using REDBACK
Table 5.5 Simulation parameters (see definitions in Tab. 5.1)
Group Name Value
Gr Gruntfest number 1.5×104
Daendo Endothermic Damkohler number 10−4
Daexo Exothermic Damkohler number 0Ar Arrhenius number 10ArF Forward Arrhenius number 20ArR Reverse Arrhenius number 10Le Lewis number 1Lechem Chemical Lewis number 10−3
Λ Thermal pressurisation coefficient 0Pe Peclet number 0ε0 Reference strain rate 1
Shear
zone
Host
rock
Applied
force
x
yz
Host
rock
0
5
-5
-6
Fig. 5.8 Model geometry (not to scale)
We consider the case of an active shear zone as described in (Alevizos et al, 2014),
where the fault thickness is small compared to its extension in the other two directions.
The model is therefore still one–dimensional but tackled in three dimensions for the rea-
sons explained above. However, for symmetry reasons the geometry is limited to a single
column of hexahedral elements with periodic boundary conditions in the X– and Y – di-
rection for temperature, excess pore–pressure and displacements as shown in Fig. 5.8.
The column has a horizontal face of 1×1 in normalized dimensions, a total height of 11,
and a shear zone of height 0.1 in its centre (see Regenauer-Lieb et al, 2013a). The shear
zone obeys Eqs. (5.24) while the footwall and hanging wall are considered to deform
purely elastically. The top face is fixed in all directions, while the bottom face is allowed
to move along the X–direction. A force boundary condition is imposed along the X–axis
5.4 Episodic Tremor and Slip events 95
0 20 40 60 80 100 120
Normalised time
0
10
20
30
Norm
alised temperature
(a) Temperature profile at the center of the shearzone
0 20 40 60 80 100 120
Normalised time
0.0
0.4
0.8
1.2
Norm
alised excess pore pressure
(b) Excess pore pressure profile at the center of theshear zone
0 20 40 60 80 100 120
Normalised time
0.0
0.4
0.8
1.2
Norm
alised stress
(c) Stress profile at the center of the shear zone
0 20 40 60 80 100 120
Normalised time
0
2
4
6
Norm
alised fault displace
ment
1e-3
(d) Shear zone displacement
Fig. 5.9 Evolution of shear zone during the numerical simulation
on the bottom part of the right hand side vertical face of the column to simulate shear.
The temperature and excess pore–pressure are kept constant on the top and bottom faces
of the whole column. The numerical parameters for all dimensionless groups are selected
to obtain an instability (see Alevizos et al, 2014) and are listed in Tab. 5.5.
Fig. 5.9 presents some simulation results at the centre of the shear zone. Fig. 5.9(c)
shows that the simulation starts with some elastic loading, leading to a purely plastic
response after initialization (for normalized time t ≈ 30) until the end of the experiment.
This result justifies the mechanical assumption of considering the fault in its post–failure
regime only in the work of Alevizos et al (2014). The displacement of the shear zone is
monitored in Fig. 5.9(d) where one can see the fault creeping at low strain rate until a
sudden slip occurs at t ≈ 110. Fig. 5.9(a) and Fig. 5.9(b) show the respective jumps in
temperature and excess pore–pressure and the subsequent relaxation at different time–
scales. After the jump, the fault continues creeping but at higher strain rate due to the
elevated pore–pressure. The simulation was stopped at t = 130 as the strain rate was
96 5 Multi-physics simulations of rock mechanics using REDBACK
(a) t = 40 (b) t = 109.3 (peak) (c) t = 112.5 (d) t = 130
Fig. 5.10 Time evolution of the temperature and displacement (exaggerated 20×) across the model. Thegeometry of the simulated column was repeated 5 times in the X– and 3 times in the Y –direction forvisualisation purposes.
increasing again for this particular set of parameters, leading the system to a regime of
damped oscillations (see Poulet et al, 2014a).
Fig. 5.10 illustrates how temperature evolves with time across the whole 3D model. At
t = 40 (Fig. 5.10(a)) the stress has equilibrated in the shear zone (see Fig. 5.9(c)) but no
displacement is noticeable and the temperature is constant everywhere. The temperature
and strain rates reach their maximum values at t = 109.3 (c.f. Figs. 5.9(a), 5.9(d)) and
Fig. 5.10(b) shows that the temperature is highly localized during the slip. Just after the
peak the temperature is absorbed by the host rock (Fig. 5.10(c)) and quickly drops back
towards its background value (Fig. 5.10(d)).
5.5 Discussion
Conventional engineering approaches can be very successful in reproducing experimental
results (e.g. Oka et al, 2011), but they are confronted to at least two principal limitations:
sequential coupling and parameter calibration.
Sequential coupling has been a tool of choice for reasons of numerical efficiency (e.g.
Poulet et al, 2012). This technique however reaches quickly its limits as the number of
variables augments and as strongly non-linear dependencies are considered since numer-
ical convergence becomes unmanageable. Therefore, tightly coupled simulators gained
popularity (Hibbitt et al, 2008) to overcome this limit for multi-physics problems as in-
creasing computational power has made those simulations possible. The use of MOOSE
follows this approach and is ideally suited to treat material instabilities in THMC prob-
lems as demonstrated in Sec. 5.4.
5.5 Discussion 97
Parameter calibration is a more fundamental problem and resides in the fact that cali-
bration of the theoretical and numerical models requires numerous expensive experiments
to cover the expected range of validity. Engineering simulators are indeed intended to be
applied within a predetermined range of conditions to ensure that results are valid by inter-
polation from previous experiments. A physics-based approach however can alleviate this
issue as a good fit of experimental data with a single model (c.f. Sec. 5.3) provides more
confidence to extend the simulation to a range of data beyond previous measurements and
eventually use the model as a predictive tool. For example, this aspect is essential when
geological processes are considered which occur at time-scales of millions of years. No
laboratory is able to reproduce such slow rates and experimental results cannot properly
be extrapolated to strain rates several orders of magnitude slower than those accessible in
the lab. A proper understanding of the underlying physics however is key to extrapolate
results wherever those processes are still relevant outside the range of existing data.
A physics-based approach is also critical when dealing with multi-physics, where the
interaction between all processes would require a prohibitively large number of experi-
ments to run in order to cover the cross-product of variable conditions to test. This can be
illustrated in the context of shale for example, where diagenetic reactions play a critical
role in sedimentary basins (Fowler and Yang, 2003). When a single test of dissolution/pre-
cipitation kinetics at constant temperature and pressure takes several months to perform
(Zhu and Lu, 2009), it becomes unrealistic to hope for a wide experimental coverage of
all basin conditions.
The flexibility and modularity of the MOOSE platform makes prototyping with RED-
BACK very efficient and enables modellers to investigate easily the importance of vari-
ous physical processes and material property dependencies. This analysis played a major
role in the identification of activation enthalpy dependency on pressure and temperature
to match the experimental results of Sec. 5.3. However, such simulations are quite ex-
pensive computationally since all variables are solved in a tightly coupled manner, and
must therefore be considered in a higher level workflow as a preliminary step to iden-
tify the driving physics. As such, the scaling ability of the code will be useful to resolve
spatial features in future work, like the width of a shear zone, or secondary phenomena
relevant to the application presented in Sec. 5.4 including spatial mixing zones or differ-
ences of shear zone thicknesses as observed in the Glarus thrust for instance (Herwegh
et al, 2008). Once the key processes are identified however it makes time to optimize
98 5 Multi-physics simulations of rock mechanics using REDBACK
the approach by decoupling where possible the secondary dependencies which can be
solved more efficiently in a sequential manner. This point highlights the major differ-
ence between REDBACK and MOOSE (with its physics modules). REDBACK focuses on
the identification step of the driving physical processes and therefore implements some
equations where the assumption of an equivalent fluid-release chemical reaction is hard-
coded (see Veveakis et al, 2015, for more details). Thanks to the open-source approach,
MOOSE can provide a more flexible engineering approach to run equivalent simulations
without any assumption hardcoded, but with all processes resolved numerically. In the
case where chemistry is involved, this means solving explicitly for the chemical system
with all chemical reactions the modellers wish to consider.
In conclusion, REDBACK aims at reducing the gap between the worlds of applied math-
ematics and geology. The full understanding of geological processes is indeed not limited
to the essential field observations and laboratory experiments but also involves theoretical
and numerical modelling to derive and test our comprehension of the observed phenom-
ena. Modelling starts with the derivation of theoretical models and the resulting work of
stability analysis. Such studies (Alevizos et al, 2014) lead to the documentation of the
dimensionless groups best suited to characterize the physical processes involved, but this
analytical work is often itself limited to the identification of different regimes for the
simplified case of homogeneous materials with the simplest geometry. While this type
of analysis is critical to understand the fundamental behaviour of rocks (Weinberg et al,
2015), it represents only the first step towards modelling reality and must therefore be fol-
lowed by some subsequent stages where modellers can build on this knowledge and inves-
tigate the role of material parameter distributions and geometrical complexity. REDBACK
was built to address those intermediate steps before using optimized simulators to model
truly realistic scenarios. This whole workflow is essential to understand the phenomena
responsible for some of the most interesting but puzzling geological (in-)stabilities and it
is hopefully now a little bit easier.
Chapter 6
Conclusion and discussion
In this thesis we investigated loaclization phenomena in geological settings and we em-
phasized how nonlinear material properties or nonlinear process feedback can lead to tem-
poral and/or spatial localization. The first type of nonlinear material behaviour that was
of interest was a nonlinear elastic one that we encountered while studying an elastic strut
resting on a Winkler type foundation, which is described by the Swift-Hohenberg equa-
tion, and exhibits localized folded solutions. This study extended the stationary view on
the snakes and ladders behaviour related to the solutions of the Swift-Hohenberg equation
and this extension was accomplished via a varying axial thrust that steadily compressed
the strut such that the strut was unable to attain its stationary state. The amount of thrust
can be seen as a measure of how far from equilibrium the system is and we were able to
offer a qualitative description on how the system drifts away from its equilibrium config-
uration and give a lower bound for a critical deformation rate above which a description
of the system behaviour in terms of its equilibrium states was inadequate and new ap-
proaches have to be found.
Via a discussion of blow-up solutions we went from nonlinear elastic materials to rate-
dependant, temperature-sensitive elasto-visco-plastic materials. We reviewed the thermo-
dynamical framework that described this class of materials and we found a localization
criterion, which is based on the trade-off between the heat conducting and heat producing
properties of the material. If the heat production supersedes the conduction plastic defor-
mation localizes along the classical slip lines along which velocity discontinuities occur
such that the heat dissipation attains a maximum. A re-normalization of the governing
equations revealed a convenient measure of the trade-off between heat conduction and
production and this measure is the so-called Gruntfest number. Numerical experiments
for the case of J2-plasticity have been presented that support our theoretical findings.
99
100 6 Conclusion and discussion
The numerical experiments were performed within the software environment RED-
BACK that offers capabilities to simulate concurrent thermal, mechanical, hydrological
and chemical processes. REDBACK solves the governing equations in a tightly coupled
manner such that temporal and spatial localization phenomena can be resolved. Conven-
tional sequentially coupled solvers might not be sensitive to such localized features. In
addition, REDBACK offers a physics based approach to multi-physics systems and gives
the user convenient means to study a wide range of process combinations at the click of a
button. This allows for quick prototyping on desktop computers but REDBACK is proven
to perform efficiently on state-of-the-art supercomputers such that large scale simulations
are possible. As examples to show-case the abilities of REDBACK we chose triaxial com-
pression tests on diatomaceous mudstone because experimental results are readily avail-
able and episodic tremor and slip events occurring in tectonic subduction zones that have
been studied analytically before (Veveakis et al, 2014). Compared to previous simulations
of the constitutive behaviour of mudstone (Oka et al, 2011) our approach in REDBACK
did not require a parametrization of individual realisations of the experiments but allowed
us to fix the material parametes for all experiments and only adjust the experimental con-
trol parameter, the confining pressure, and reproduce all experimental results with good
agreement. This emphasizes the importance of understanding the coupling and feedback
cycles in multi-physics systems.
We hope that this thesis can provide the basis for future investigations and further work
is required. Firstly, the steady state analysis presented in Chapter 4 should be extended to
systems with further processes and eventually should be implemented in REDBACK. This
would provide the modeller with easy means to find parameter values that lead to local-
ized features and structures in geological systems. Further, a knowledge transfer with the
geological community should be initiated and the nonequilibrium perspective promoted
in this thesis should be applied to large-scale problems. An examples of such efforts is
the work by Peters et al (2015) where folding and boudinage have been identified and
discussed as material instabilities in a multi-physics framework. Another interesting ex-
ample of large-scale multi-physics problems is the formation of ore deposits. There are
exiting opportunities especially in Australia where mining is such a prominent industry
and very detailed studies and a large host of experimental findings for various deposits
are available. An example is the copper deposit at Mt. Isa, Northern Territory, Australia
(Gessner et al, 2009; Kuhn and Gessner, 2009).
6 Conclusion and discussion 101
The mathematical investigation of geological multi-physics systems is commonly
based on scaling arguments (Veveakis et al, 2014, e.g.). A valuable addition to these
techniques might be singular perturbation analyses that have produced valuable insight
into biological systems with slow/fast dynamics (Kosiuk and Szmolyan, 2011). This per-
tubation analysis derives from the seminal work by Fenichel (1979) and is best suited
for low-dimensional dynamical systems and an extension to the partial differential equa-
tions encounterd here requires careful analysis but has been successfully attempted before
(Krupa et al, 1997).
Appendix A
Reduction of order technique
Given a second order differential operator of the form L [u] = u′′(x)− f (x)u(x) and a
function u1(x), such that L [u1] = 0, a second function u2(x) that satisfies L [u2] = 0 can
be constructed by reduction of order. To this end, we set u2(x) = v(x)u1(x), where v(x) is
unknown, which yields
L [u2] = v′′u1 +2v′u′1 + vL [u1]︸ ︷︷ ︸=0
!= 0. (A.1)
This only holds true if v′′u21 +2v′u′1u1 = 0 or equivalently
v′u21 = A (A.2)
where A is an arbitrary constant. Eq. (A.2) defines the factor v.
103
Appendix B
Variation of parameters technique
If a second order differential operator L [u] = u′′(x)+ f (x)u(x) and two solutions u1, u2
that satisfy L [u1] = L [u2] = 0 are given and a solution to
L [u] = g(x) (B.1)
is desired, then u can be constructed by variation of parameters. To this end, we set u =
ξ u1 +ψu2, where ξ (x) and ψ(x) are unknown, which yields
L [u] = ξ′u′1 +ψ
′u′2 = g, (B.2)
if we require that
ξ′u1 +ψ
′u2 = 0. (B.3)
Eqs. (B.2) and (B.3) constitute a system of equations that can be solved simultaneously
for ξ and ψ .
105
References
Abbassi MR, Mancktelow NS (1990) The effect of initial perturbation shape and sym-
metry on fold development. Journal of Structural Geology 12(2):273–282, DOI
10.1016/0191-8141(90)90011-M
Alevizos S, Poulet T, Veveakis E (2014) Thermo-poro-mechanics of chemically active
creeping faults. 1: Theory and steady state considerations. Journal of Geophysical Re-
search: Solid Earth 119(6):4558–4582, DOI 10.1002/2013JB010070
Anand L, Kim KH, Shawki TG (1987) Onset of shear localization in viscoplas-
tic solids. Journal of the Mechanics and Physics of Solids 35(4):407–429, DOI
Onsetofshearlocalizationinviscoplasticsolids
Bai Y (1982) Thermo-plastic instability in simple shear. Journal of the Mechanics and
Physics of Solids 30(4):195–207, DOI 10.1016/0022-5096(82)90029-1
Bai Y, Meyers M, Murr L (1981) Shock waves and high-strain-rate phenomena in metals.
Eds MA Myers and LE Murr, Plenum Press, NY 277
Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Eijkhout V, Gropp
WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zhang H (2014)
PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.5, Argonne National Lab-
oratory
Bauwens-Crowet C, Ots J, Bauwens J (1974) The strain-rate and temperature dependence
of yield of polycarbonate in tension, tensile creep and impact tests. Journal of Materials
Science 9(7):1197–1201, DOI 10.1007/BF00552841
Benallal A, Bigoni D (2004) Effects of temperature and thermo-mechanical couplings on
material instabilities and strain localization of inelastic materials. Journal of the Me-
chanics and Physics of Solids 52(3):725–753, DOI 10.1016/S0022-5096(03)00118-2
Benallal A, Lemaitre J (1991) Creep in Structures, Springer Berlin Heidelberg, chap Lo-
calization Phenomena in Thermo-Elastoplasticity, pp 223–235
107
108 References
Berglund N, Gentz B (2002) Pathwise description of dynamic pitchfork bifurcations with
additive noise. Probability Theory and Related Fields 122(3):341–388, DOI 10.1007/
s004400100174
Biot MA (1961) Theory of folding of stratified viscoelastic media and its implications in
tectonics and orogenesis. Geological Society of America Bulletin 72(11):1595–1620,
DOI 10.1130/0016-7606(1961)72[1595:TOFOSV]2.0.CO;2
Biot MA (1965) Mechanics of incremental deformations: Theory of elasticity and vis-
coelasticity of initially stressed solids and fluids, including thermodynamic foundations
and applications to finite strain. John Wiley & Sons, Ltd.
Boyce B, Crenshaw T, Dilmore M (2007) The strain-rate sensitivity of high-strength high-
toughness steels. Sandia Report SAND2007-0036
Budd CJ (2001) Asymptotics of multibump blow-up self-similar solutions of the nonlinear
schrodinger equation. SIAM Journal on Applied Mathematics 62(3):801–830, DOI
10.1137/S0036139900382395
Budd CJ, Chen S, Russell RD (1999) New self-similar solutions of the nonlinear
schrodinger equation with moving mesh computations. Journal of Computational
Physics 152(2):756–789, DOI 10.1006/jcph.1999.6262
Burke J, Knobloch E (2007a) Homoclinic snaking: Structure and stability. Chaos
17(3):037102, DOI 10.1063/1.2746816
Burke J, Knobloch E (2007b) On snakes and ladders: localized states in the Swift-
Hohenberg equation. Physics Letters A 360(6):681–688
Chaboche J (2008) A review of some plasticity and viscoplasticity constitutive theo-
ries. International Journal of Plasticity 24(10):1642–1693, DOI 10.1016/j.ijplas.2008.
03.009
Chaboche JL (1977) Viscoplastic constitutive equations for the description of cyclic and
anisotropic behavior of metals. Bulletin de l’Academie Polonaise des Sciences–Serie
des Sciences Techniques 25(1):33–42
Champneys AR, Toland JF (1993) Bifurcation of a plethora of multi–modal homoclinic
orbits for autonomous Hamiltonian systems. Nonlinearity 6(5):665–772, DOI 10.1088/
0951-7715/6/5/002
Chan T, Keller H (1982) Arc-length continuation and multi-grid techniques for nonlinear
elliptic eigenvalue problems. SIAM Journal on Scientific and Statistical Computing
3(2):173–194, DOI 10.1137/0903012
References 109
Chen H, Douglas AS, Malek-Madani R (1989) An asymptotic stability condition for in-
homogeneous simple shear. Quarterly of Applied Mathematics 47(2):247–262
Cherukuri H, Shawki T (1995a) An energy-based localization theory: I. Basic framework.
International Journal of Plasticity 11(1):15–40, DOI 10.1016/0749-6419(94)00037-9
Cherukuri H, Shawki T (1995b) An energy-based localization theory: II. Effects of the
diffusion, inertia, and dissipation numbers. International Journal of Plasticity 11(1):41–
64, DOI 10.1016/0749-6419(94)00038-7
Clifton R (1980) Material response to ultra high loading rates. NRC National Material
Advisory Board (US) Report 356
Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat con-
duction and viscosity. Archive for Rational Mechanics and Analysis 13(1):167–178,
DOI 10.1007/BF01262690
Cook RD, Malkus DS, Plesha ME (1989) Concepts and applications of finite element
method. John Wiley and Sons, New York
Coussy O (2004) Poromechanics, 2nd edn. John Wiley and Sons, Chichester
Cross M, Greenside H (2009) Pattern Formation and Dynamics in Nonequilibrium Sys-
tems, 1st edn. Cambridge University Press
Dawes JHP (2010) The emergence of a coherent structure for coherent structures: local-
ized states in nonlinear systems. Philosophical Transactions of the Royal Society of
London A: Mathematical, Physical and Engineering Sciences 368(1924):3551–3565,
DOI 10.1098/rsta.2010.0057
Dewhurst P, Collins IF (1973) A matrix technique for constructing slip-line field solu-
tions to a class of plane strain plasticity problems. International Journal for Numerical
Methods in Engineering 7(3):357–378, DOI 10.1002/nme.1620070312
Dodd B, Bai Y (2012) Adiabatic shear localization: Frontiers and advances. Elsevier
Doedel EJ, Champney AR, Fairgrieve TF, Kuznetsov YK, Sanstede B, Wang X (2007)
Auto07: Continuation and bifurcation software for ordinary differential equations (with
homcont). Technical report, Concordia University
Engels T, Schrauwen B, van Breemen L, Govaert L (2009) Predicting the yield stress of
polymer glasses directly from processing conditions: Application to miscible systems.
International Polymer Processing 24(2):167–173, DOI 10.3139/217.2224
Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equa-
tions. Journal of Differential Equations 31(1):53–98, DOI 10.1016/0022-0396(79)
110 References
90152-9
Fletcher RC (1974) Wavelength selection in the folding of a single layer with power-
law rheology. American Journal of Science 274(9):1029–1043, DOI 10.2475/ajs.274.
9.1029
Fowler A (ed) (1997) Mathematical Models in the Applied Sciences, 2nd edn. Cambridge
University Press
Fowler AC, Yang XS (2003) Dissolution/precipitation mechanisms for diagenesis in sed-
imentary basins. Journal of Geophysical Research: Solid Earth 108(B10):13.1–13.14,
DOI 10.1029/2002JB002269
Freed A, Walker K (1993) Viscoplasticity with creep and plasticity bounds. International
Journal of Plasticity 9(2):213–242, DOI 10.1016/0749-6419(93)90030-T
Gaeta G (1995) Dynamical bifurcation with noise. International Journal of Theoretical
Physics 34(4):595–603, DOI 10.1007/bf00674955
Gaston D, Newman C, Hansen G, Lebrun-Grandi D (2009) Moose: A parallel computa-
tional framework for coupled systems of nonlinear equations. Nuclear Engineering and
Design 239(10):1768–1778, DOI 10.1016/j.nucengdes.2009.05.021
Gessner K, Kuehn M, Rath V, Kosack C, Blumenthal M, Clauser C (2009) Coupled Pro-
cess Models as a Tool for Analysing Hydrothermal Systems. Surveys in Geophysics
30(3):133–162, DOI 10.1007/s10712-009-9067-1
Gruntfest I (1963) Thermal feedback in liquid flow: Plane shear at constant stress. Trans-
actions of the Society of Rheology 7:195–207
Hall J (1815) II. On the vertical position and convolutions of certain strata, and their re-
lation with granite. Transactions of the Royal Society of Edinburgh 7(1):79–108, DOI
10.1017/S0080456800019268
Herwegh M, Hurzeler JP, Pfiffner OA, Schmid SM, Abart R, Ebert A (2008) The glarus
thrust: excursion guide and report of a field trip of the swiss tectonic studies group
(swiss geological society, 14.–16. 09. 2006). Swiss Journal of Geosciences 101(2):323–
340, DOI 10.1007/s00015-008-1259-z
Hibbitt HD, Karlsson BI, Sorensen I (2008) ABAQUS/Standard – User’s Manual Version
6.7. Hibbit, Karlsson and Sorenson Inc., Pawtucket
Hickman R, Gutierez M (2007) Formulation of three-dimensional rate-dependent con-
stitutive model for chalk and porous rocks. International Journal for Numerical and
Analytical Methods in Geomechanics 31:583–605, DOI 10.1002/nag.546
References 111
Hill R (1950) The Mathematical Theory of Plasticity (Oxford Classic Texts in the Physical
Sciences). Oxford University Press
Hill R (1962) Acceleration waves in solids. Journal of the Mechanics and Physics of
Solids 10(1):1–16, DOI 10.1016/0022-5096(62)90024-8
Hobbs BE, Ord A (2012) Localised and chaotic folding: The role of axial plane structures.
Philosophical Transactions of the Royal Society of London A: Mathematical, Physical
and Engineering Sciences 370(1965):1966–2009, DOI 10.1098/rsta.2011.0426
Hobbs BE, Regenauer-Lieb K, Ord A (2008) Folding with thermal-mechanical feedback.
Journal of Structural Geology 30(12):1572–1592, DOI 10.1016/j.jsg.2008.09.002
Hobbs BE, Ord A, Regenauer-Lieb K (2011) The thermodynamics of deformed meta-
morphic rocks: A review. Journal of Structural Geology 33(5):758–818, DOI doi:
10.1016/j.jsg.2011.01.013
Huckel T, Baldi G (1990) Thermoplastic behavior of saturated clays: Experimental
constitutive study. Journal of Geotechnical Engineering 116(12):1778–1796, DOI
10.1061/(ASCE)0733-9410(1990)116:12(1778)
Huckel T, Pellegrini R (2002) Reactive plasticity for clays: application to a natural analog
of long-term geomechanical effects of nuclear waste disposal. Engineering Geology
64(2–3):195–215, DOI 10.1016/S0013-7952(01)00114-4
Hunt GW, Bolt HM, Thompson JMT (1989) Structural localization phenomena and the
dynamical phase-space analogy. Proceedings of the Royal Society of London A: Math-
ematical, Physical and Engineering Sciences 425(1869):245–267, DOI 10.1098/rspa.
1989.0105
Hunt GW, Muhlhaus HB, Hobbs B, Ord A (1996a) Localized folding of viscoelastic
layers. Geologische Rundschau (International Journal of Earth Sciences) 85(1):58–64,
DOI 10.1007/BF00192061
Hunt GW, Muhlhaus HB, Whiting AIM (1996b) Evolution of localized folding for a
thin elastic layer in a softening visco-elastic medium. Pure and Applied Geophysics
146(2):229–252, DOI 10.1007/BF00876491
Hunt GW, Peletier MA, Champneys AR, Woods PD, Wadee MA, Budd CJ, Lord GJ
(2000) Cellular buckling in long structures. Nonlinear Dynamics 21(1):3–29, DOI
10.1023/A:1008398006403
Hunt GW, Dodwell TJ, Hammond J (2013) On the nucleation and growth of kink and
shear bands. Philosophical Transactions of the Royal Society A: Mathematical, Physi-
112 References
cal and Engineering Sciences 371(1993):20120,431, DOI 10.1098/rsta.2012.0431
Johnson W, Baraya G, Slater R (1964) On heat lines or lines of thermal discon-
tinuity. International Journal of Mechanical Sciences 6(6):409–414, DOI 10.1016/
S0020-7403(64)80001-1
Johnson W, Sowerby R, Venter RD (1982) Plane-Strain Slip-Line Fields for Metal-
Deformation Processes: A Source Book and Bibliography, 1st edn. Pergamon Press
Kao HC, Beaume C, Knobloch E (2014) Spatial localization in heterogeneous systems.
Physical Review E 89(1):012,903, DOI 10.1103/PhysRevE.89.012903
Karrech A, Regenauer-Lieb K, Poulet T (2011) Frame indifferent elastoplasticity of fric-
tional materials at finite strain. International Journal of Solids and Structures 48(3–
4):397–407, DOI 10.1016/j.ijsolstr.2010.09.026
Khan I, Bhasin V, Chattopadhyay J, Ghosh A (2008) On the equivalence of slip-line
fields and work principles for rigid–plastic body in plane strain. International Journal
of Solids and Structures 45(25–26):6416–6435, DOI 10.1016/j.ijsolstr.2008.08.003
Kirk BS, Peterson JW, Stogner RH, Carey GF (2006) libMesh : a C++ library for parallel
adaptive mesh refinement/coarsening simulations. Engineering with Computers 22(3-
4):237–254, DOI 10.1007/s00366-006-0049-3
Knoll DA, Keyes DE (2004) Jacobian-free NewtonKrylov methods: a survey of ap-
proaches and applications. Journal of Computational Physics 193(2):357 – 397, DOI
10.1016/j.jcp.2003.08.010
Kosiuk I, Szmolyan P (2011) Scaling in singular perturbation problems: Blowing up a
relaxation oscillator. SIAM Journal on Applied Dynamical Systems 10(4):1307–1343,
DOI 10.1137/100814470
Krupa M, Sandstede B, Szmolyan P (1997) Fast and slow waves in the FitzHugh–
Nagumo equation. Journal of Differential Equations 133(1):49–97, DOI 10.1006/jdeq.
1996.3198
Kuehn C (2011) A mathematical framework for critical transitions: Bifurcations,
fast–slow systems and stochastic dynamics. Physica D: Nonlinear Phenomena
240(12):1020–1035, DOI 10.1016/j.physd.2011.02.012
Kuhn M, Gessner K (2009) Coupled Process Models of Fluid Flow and Heat Transfer in
Hydrothermal Systems in Three Dimensions. Surveys in Geophysics 30(3):193–210,
DOI 10.1007/s10712-009-9060-8
Law C (ed) (2006) Combustion Physics. Cambridge University Press
References 113
Leroy Y, Molinari A (1992) Stability of steady states in shear zones. Journal of the Me-
chanics and Physics of Solids 40(1):181–212, DOI 10.1016/0022-5096(92)90310-X
Lubliner J (2008) Plasticity Theory. Dover Publications
Mancktelow N (1999) Finite-element modelling of single-layer folding in elasto-viscous
materials: the effect of initial perturbation geometry. Journal of Structural Geology
21(2):161–177, DOI 10.1016/s0191-8141(98)00102-3
Needleman A, Tvergaard V (1992) Analyses of plastic flow localization in metals. Ap-
plied Mechanics Reviews 45(3S):S3–S18, DOI 10.1115/1.3121390
Oka F, Kimoto S, Higo Y, Ohta H, Sanagawa T, Kodaka T (2011) An elasto-viscoplastic
model for diatomaceous mudstone and numerical simulation of compaction bands. In-
ternational Journal for Numerical and Analytical Methods in Geomechanics 35(2):244–
263, DOI 10.1002/nag.987
Ord A, Hobbs B (2013) Localised folding in general deformations. Tectonophysics
587:30–45, DOI 10.1016/j.tecto.2012.09.020
Peletier MA (2001) Sequential buckling: A variational analysis. SIAM Journal on Math-
ematical Analysis 32(5):1142–1168, DOI 10.1137/S0036141099359925
Perzyna P (1966) Fundamental problems in viscoplasticity. Advances in Applied Mechan-
ics 9:243–377
Peters M, Veveakis M, Poulet T, Karrech A, Herwegh M, Regenauer-Lieb K (2015) Boud-
inage as a material instability of elasto-visco-plastic rocks. Journal of Structural Geol-
ogy DOI 10.1016/j.jsg.2015.06.005, (in press)
Petzold L (1982) Dassl. A Differential/Algebraic system solver. Tech. rep., Sandia Na-
tional Labs., Livermore, CA (USA)
Potier-Ferry M (1983) Amplitude modulation, phase modulation and localization of buck-
ling patterns. In: Thompson JMT, Hunt GW (eds) Collapse: the buckling of structures
in theory andpractice, Cambridge University Press, Cambridge, pp 149–159
Poulet T, Regenauer-Lieb K, Karrech A, Fisher L, Schaubs P (2012) Thermal-
hydraulic-mechanical-chemical coupling with damage mechanics using ESCRIPTRT
and ABAQUS. Tectonophysics 526-529(0):124 – 132, DOI 10.1016/j.tecto.2011.12.
005
Poulet T, Veveakis E, Regenauer-Lieb K, Yuen DA (2014a) Thermo-poro-mechanics of
chemically active creeping faults: 3. the role of serpentinite in episodic tremor and
114 References
slip sequences, and transition to chaos. Journal of Geophysical Research: Solid Earth
119(6):4606–4625, DOI 10.1002/2014JB011004
Poulet T, Veveakis M, Herwegh M, Buckingham T, Regenauer-Lieb K (2014b) Modeling
episodic fluid-release events in the ductile carbonates of the glarus thrust. Geophysical
Research Letters 41(20):7121–7128, DOI 10.1002/2014GL061715
Price NJ, Cosgrove JW (1990) Analysis of geological structures, 1st edn. Cambridge Uni-
versity Press, Cambridge, UK
Rashid MM (1993) Incremental kinematics for finite element applications. International
Journal for Numerical Methods in Engineering 36(23):3937–3956, DOI 10.1002/nme.
1620362302
Regenauer-Lieb K, Hobbs B, Ord A, Gaede O, Vernon R (2009) Deformation with cou-
pled chemical diffusion. Physics of the Earth and Planetary Interiors 172(1-2):43–54,
DOI 10.1016/j.pepi.2008.08.013
Regenauer-Lieb K, Veveakis M, Poulet T, Wellmann F, Karrech A, Liu J, Hauser J,
Schrank C, Gaede O, Trefry M (2013a) Multiscale coupling and multiphysics ap-
proaches in earth sciences: Applications. Journal of Coupled Systems and Multiscale
Dynamics 1(3):281–323, DOI 10.1166/jcsmd.2013.1021
Regenauer-Lieb K, Veveakis M, Poulet T, Wellmann F, Karrech A, Liu J, Hauser J,
Schrank C, Gaede O, Trefry M (2013b) Multiscale coupling and multiphysics ap-
proaches in earth sciences: Theory. Journal of Coupled Systems and Multiscale Dy-
namics 1(1):49–73, DOI 10.1166/jcsmd.2013.1012
Rice JR (1976) The localization of plastic deformation. In: Koiter W (ed) Theoretical and
Applied Mechanics, North-Holland Publishing Company, pp 207–220
Rice JR, Lapusta N, Ranjith K (2001) Rate and state dependent friction and the stability
of sliding between elastically deformable solids. Journal of the Mechanics and Physics
of Solids 49(9):1865–1898, DOI 10.1016/S0022-5096(01)00042-4
Rosakis P, Rosakis A, Ravichandran G, Hodowany J (2000) A thermodynamic inter-
nal variable model for the partition of plastic work into heat and stored energy
in metals. Journal of the Mechanics and Physics of Solids 48(3):581–607, DOI
10.1016/S0022-5096(99)00048-4
Roscoe K, Burland J (1968) On the generalised stress–strain behaviour of wet clay, Cam-
bridge University Press, chap Engineering Plasticity, pp 535–609
References 115
Rudnicki J, Rice J (1975) Conditions for the localization of deformation in pressure sensi-
tive dilatant materials. Journal of the Mechanics and Physics of Solids 23(6):371–394,
DOI 10.1016/0022-5096(75)90001-0
Schmalholz SM, Podladchikov YY (2000) Finite amplitude folding: transition from
exponential to layer length controlled growth. Earth and Planetary Science Letters
179(2):363–377, DOI 10.1016/S0012-821X(00)00116-3
Shampine LF, Reichelt MW, Kierzenka JA (1999) Solving index-1 DAEs in MATLAB
and Simulink. SIAM Review 41(3):538–552, DOI 10.1137/s003614459933425x
Sherwin JA, Chapple WM (1968) Wavelengths of single-layer folds: a comparison be-
tween theory and observation. American Journal of Science 266(3):167–179, DOI
10.2475/ajs.266.3.167
Smith RB (1975) Unified theory of the onset of folding, boudinage, and mullion structure.
Geological Society of America Bulletin 86(11):1601, DOI 10.1130/0016-7606(1975)
86〈1601:UTOTOO〉2.0.CO;2
Smith RB (1977) Formation of folds, boudinage, and mullions in non-Newtonian materi-
als. Geological Society of America Bulletin 88(2):312, DOI 10.1130/0016-7606(1977)
88〈312:FOFBAM〉2.0.CO;2
Sulem J, Famin V (2009) Thermal decomposition of carbonates in fault zones:
slip-weakening and temperature-limiting effects. Journal of Geophysical Research
114(B3):B03309, DOI 10.1029/2008JB006004
Swift J, Hohenberg PC (1977) Hydrodynamic fluctuations at the convective instability.
Physical Review A 15(1):319–328, DOI 10.1103/physreva.15.319
Taron J, Elsworth D, Min KB (2009) Numerical simulation of thermal-hydrologic-
mechanical-chemical processes in deformable, fractured porous media. International
Journal of Rock Mechanics and Mining Sciences 46(5):842 – 854, DOI 10.1016/j.
ijrmms.2009.01.008
Taylor G, Quinney H (1934a) The latent energy remaining in a metal after cold working.
Proc R Soc, Ser A 143:307–326
Taylor GI, Quinney H (1934b) The latent energy remaining in a metal after cold working.
Proceedings of the Royal Society of London 143(849):307–326
Thompson JMT (1979) Stability predictions through a succession of folds. Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
292(1386):123, DOI 10.1098/rsta.1979.0043
116 References
Thompson JMT, Hunt GW (1973) A General Theory of Elastic Stability. Wiley, London
Vardoulakis I (2002) Steady shear and thermal run-away in clayey gouges. International
Journal of Solids and Structures 39(13–14):3831–3844, DOI 10.1016/S0020-7683(02)
00179-8
Varotsos P, Alexopoulos K (1980) Negative activation volumes of defects in solids. Phys-
ical Review B 21:4898–4899, DOI 10.1103/PhysRevB.21.4898
van der Veen H, Vuik K, de Borst R (2000) Branch switching techniques for bifurcation
in soil deformation. Computer Methods in Applied Mechanics and Engineering 190(5-
7):707–719, DOI 10.1016/s0045-7825(99)00439-9
Veveakis E, Vardoulakis I, Toro GD (2007) Thermoporomechanics of creeping land-
slides: The 1963 Vaiont slide, northern Italy. Journal of Geophysical Research
112(F3):F03026, DOI 10.1029/2006JF000702
Veveakis E, Alevizos S, Vardoulakis I (2010) Chemical reaction capping of thermal insta-
bilities during shear of frictional faults. Journal of the Mechanics and Physics of Solids
58(9):1175–1194, DOI 10.1016/j.jmps.2010.06.010
Veveakis E, Poulet T, Alevizos S (2014) Thermo-poro-mechanics of chemically ac-
tive creeping faults: 2. transient considerations. Journal of Geophysical Research
119(6):4583–4605, DOI 10.1002/2013JB010071
Veveakis E, Poulet T, Paesold M, Alevizos S, Regenauer-Lieb K (2015) The role of dia-
genetic reactions during fluid flow through nominally impermeable shales. Rock Me-
chanics and Rock Engineering Submitted
Weinberg R, Veveakis M, Regenauer-Lieb K (2015) Compaction-driven melt segregation
in migmatites. Geology 43(6), DOI 10.1130/G36562.1, (in press)
Woods PD, Champneys AR (1999) Heteroclinic tangles in the unfolding of a de-
generate Hamiltonian Hopf bifurcation. Physica D 129(3-4):147–170, DOI 10.1016/
s0167-2789(98)00309-1
Zhu C, Lu P (2009) Alkali feldspar dissolution and secondary mineral precipitation in
batch systems: 3. saturation states of product minerals and reaction paths. Geochimica
et Cosmochimica Acta 73(11):3171–3200, DOI 10.1016/j.gca.2009.03.015