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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

To my wife and parents.

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

xii Statement of candidate contribution

xiv Statement of candidate contribution

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)


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.


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


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)


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.


2.3.2 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.


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)


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.


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)


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-


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


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 ∆ ∗


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


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


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.


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


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).


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)


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)


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


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.


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.


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)


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


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


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)


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


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


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.


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


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)


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)


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).


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)


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


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.


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


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


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)


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,


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.


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.


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-


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


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


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.


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.


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.


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


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)


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-


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


5.2.2 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


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


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.


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


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.


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.


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


(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


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

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