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Timetable (SIPW04)

Ice fracture and cracks

Monday 4th December 2017 to Friday 8th December 2017

Monday 4th December 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
10:00 to 11:00 Chris Petrich
Growth process and structure of refrozen cracks in sea ice
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Sze Dai Pang
Probabilistic Fracture Mechanics and Its Implications on Ice
Quasibrittle materials are materials in which the fracture process zone (FPZ) is not negligible as compared with the cross section dimension and encompass a wide variety of materials such as concrete, mortar, rocks, toughened ceramics, frozen sand, and ice also belongs to this class of materials. The size of the FPZ is typically 5-50 times the size of the dominant material inhomogeneity and for ice, it could be the grain size. For type 1 size effect which occurs in positive geometry structures failing at macrocrack initiation and is typical of flexural failures, the size effect is governed by Weibull statistics when the structure size dwarfs the size of the FPZ. When the structure size is comparable to the size of the FPZ, the probability distribution of the quasibrittle fracture can be described by a Gaussian core with a far-left Weibull tail. This is concluded from scaling laws derived from a hierarchical model of chains and bundles of representative volume elements starti ng from the atomic scale and include the effects of loading rate and temperature. The implications of probabilistic fracture mechanics on the strength of ice are investigated for different size of the ice sheet, varying strain rates and temperature effect.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Chris Borstad
Continuum damage models for fracturing and weakening of Antarctic ice shelves
Most of the Antarctic ice sheet drains to the ocean through floating ice shelves.  In many sectors of Antarctica, ice shelves are thinning due to oceanic or atmospheric warming, making them more susceptible to fracturing and even collapse.  Here I outline the different modes of ice shelf fracturing, including partial-thickness crevassing, through-thickness rifting, shear margin weakening, and the dominant mechanisms of iceberg calving. In the absence of a unifying damage framework capable of representing the diverse spatial and temporal scales of these mechanisms, I highlight two end-member damage models for particular cases.  The first is a fully elastic damage model, appropriate for representing the propagation of through-thickness rifts and tabular iceberg calving.  The second is a fully viscous damage model, appropriate for gradual weakening (especially in shear margins) of an ice shelf.  For the latter case, I present an adjoint-based inverse met hod for assimilating remote sensing data to infer a scalar damage variable over an ice shelf.  Results from a case study of the Larsen B ice shelf on the Antarctic Peninsula are used to inform the development of a damage evolution framework for application in ice sheet models.
14:30 to 15:30 Roiy Sayag
On the Formation and Evolution of Rifts in Ice Shelves
Rifts that form at the fronts of floating ice shelves are fractures that cut through the entire thickness of the ice. They are believed to be the precursor of calving, which accounts for a significant part in the mass loss of present ice sheets. Here we investigate the formation of rifts in ice shelves and their evolution by combining laboratory-scale experiments of ice sheets together with theoretical modeling. Experimentally we model the deformation of ice using a thin film of non-Newtonian fluid that is driven axisymmetrically by buoyancy. The viscous fluid intrudes a bath of an inviscid, denser fluid that represents the ocean. Consequently, the circular symmetry of the propagating front breaks up near the grounding line into a set of tongues with a characteristic wavelength that coarsens over time, a pattern that is reminiscent of some ice rifts. Theoretically, we model the formation of rifts as a hydrodynamic instability of a power-law fluid. Our model resolves the formation of rifts and the coarsening of the characteristic wavelength, and predicts coarsening transition times that are consistent with our experimental measurements. We discuss the instability mechanism and its implications.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Erkan Oterkus
Peridynamic Modelling of Ice Fracture
Despite of its advantages, utilization of the Arctic region for sailing brings new challenges due to its harsh environment. Therefore, ship structures must be designed to withstand ice loads in case of a collision between a ship and ice takes place. Although experimental studies can give invaluable information about ship-ice interactions, full scale tests are very costly to perform. Therefore, computer simulations can be a good alternative. Ice-structure interaction modelling is a very challenging process. First of all, ice material response depends on many different factors including applied-stress, strain-rate, temperature, grain-size, salinity, porosity and confining pressure. Furthermore, macro-scale modeling may not be sufficient to capture the full physical behaviour because the micro-scale effects may have a significant effect on macroscopic material behaviour. Hence, it is necessary to utilize a multi-scale methodology. In order to capture the macro-scale behaviour of ice, well-known Finite Element Method (FEM) has been used in various previous studies. Within FEM framework, various techniques can be used to model crack propagation such as cohesive zone models (CZM) and extended finite element method (XFEM). However, a universally accepted CZM failure model is not currently available and the crack propagation may have mesh dependency. Although, the mesh dependency problem can be overcome by XFEM, enrichment process may lead to an algebraic system with billions of unknowns which is difficult to solve numerically. Furthermore, FEM is based on classical continuum mechanics which does not have a length scale parameter and is incapable of capturing phenomenon at the micro-scale. Hence, other techniques should be utilized at the micro-scale and linked to FEM simulation. However, it is not straightforward to obtain a smooth transition between different approaches at different scales. By taking into account all these challenging issues, a state-of-the-art technique, peridynamics can be utilized for ice fracture modelling. Peridynamics is a non-classical (non-local) continuum mechanics formulation which is very suitable for failure analysis of materials due its mathematical structure. Cracks can occur naturally in the formulation and there is no need to impose an external crack growth law. Furthermore, due to its non-local character, it can capture the phenomenon at multiple scales. 
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 5th December 2017
09:00 to 10:00 Kaj Riska
Ice edge failure process
In the paper results from ice indentation tests are described. The tests were carried out to clarify the failure process of ice and also the loads caused by indentation into ice. The test results showed for the first time the nature of the brittle ice failure in caused a narrow high pressure zone (a line-like feature) transmitting the high pressures. The test results are described in some detail.

After looking at the test results, the implications on modeling ice failure are discussed. Especially the results of more recent tests are discussed and a phenomenological model for ice failure is given.
10:00 to 11:00 Mao See Wu
Crack nucleation in ice – a historical review and research challenges
This presentation focuses on crack nucleation in polycrystalline freshwater and saline ice. A historical review of the plausible nucleation mechanisms is provided, followed by a discussion of outstanding research challenges.
Since the late 1990s, several nucleation mechanisms have been investigated. These include: (i) dislocation pileup against obstacles such as grain boundaries, (ii) grain boundary sliding leading to displacement incompatibility at triple junctions, and (iii) elastic anisotropy of the hexagonal ice crystals giving rise to microstructural stresses which can nucleate cracks from precursors. For saline and sea ice, pressurized brine pockets are stress concentrators and likely crack nucleation sites.
The basal slip system is dominant in ice, and cracks may nucleate to relieve the strain heterogeneity arising from the strong plastic anisotropy of ice deformation. Furthermore, the interaction of various crack nucleation mechanisms under different conditions, e.g., temperature, grain size and texture, and the effect of time and loading rate on crack nucleation in viscoelastic ice, are issues that have received little attention. These are therefore research challenges that can be investigated in the future. 
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Robert Gagnon
The Physics of Ice Crushing Associated with Indentation and Impact
Ice crushing occurs in many contexts such as ice interaction with bridges, piers, ship hulls, offshore structures, rock beds under glaciers and ice-on-ice sliding/crushing interaction within glaciers and extraterrestrial ice masses (on Saturn’s moon Enceladus). In the cases of skate blades, sled runners and curling stones local crushing on ice asperities and/or small-scale ice unevenness, and due to gouging/plowing, occurs. In-situ imaging records from small and medium scale ice-crushing experiments show that repetitive spallation of ice from a relatively-intact hard zone in the central contact region produces a sawtooth load pattern, and most of the actual ice indentation occurs during the associated sharp drops in load. At least half of the load is borne on the hard zone, where the interface pressure is ~ 20-70 MPa. The rest of the load is borne on surrounding shattered spall debris, where the pressure is ~ 0-10 MPa. Spalling influences the evolution of hard-zone size and shape during the tests. The hard zones are regions where a thin squeeze-film slurry layer of pressurized melt and ice particles is present between the ice and the contacting surface. The viscous flow of the layer generates heat that accounts for the rapid-melting component of the removal of ice from the hard zones during ice crushing. A similar process occurs at ice-on-ice contact of fragments in the surrounding crushed-ice matrix as it extrudes away from the high-pressure zones. The melting accounts for the bulk of the energy dissipation and partly explains how an indentor can rapidly move forward on hard zones. The slurry layer thickness in small-scale lab tests is ~ 0.02 - 0.17 mm, where its liquid fraction is about 16%. The layer acts as a self-generating squeeze film that is powered by the energy supplied by the loading system. When ice crushing includes a sliding component the layer’s flow characteristics and high lubricity lead to very low friction coefficients, even on rough surfaces.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Jukka Tuhkuri
Deformation and failure of sea ice cover
Sea ice cover is moved by winds and currents. This motion leads to deformation and failure; though formation of leads in tension, and through ridging and rafting in compression. This presentation concentrates on the latter process. Ridges are elongated accumulations of broken sea ice. Rafting is another form of deformed ice. During rafting, one ice sheet overrides another ice sheet. Several models of ridging and rafting have been proposed: energy based models, models concentrating on the kinematics of the processes, and numerical models. Also laboratory experiments have been conducted to study ridging and rafting. In essence, all ridging and rafting initiates when two ice sheets are pushed together. In rafting, one ice sheet slides under another ice sheet, and we get two (or more) overlapped ice sheets. In ridging, ice blocks sequentially break of the ice sheets and form a ridge. Our experimental and numerical work suggests that all ridging initiates as rafting. Also, as the ice-ice friction coefficient is larger than zero, a rafting process will eventually turn into the formation of a pile of broken ice blocks, a ridge. Therefore, rafting and ridging should not be considered as two totally different deformation processes, but rather different stages of one process.
14:30 to 15:30 Shunying Ji
Breaking Pattern of Ice Cover during its Collision with Ship/Offshore Structures
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Hayley Shen
Integrating Elastic and Viscous Properties of Ice for Ocean Wave Propagation
As a material, ice covers are mixtures of various forms of solid, water (with or without salt), and air. They are heterogeneous under all scales. A number of mathematical models have been proposed to describe the dispersive/dissipative property of an ice cover. In this talk we examine one of such models that integrates both the energy storage and dissipation capability of an ice cover. The mathematical complexity resulted in this very simple looking model, and its skill of predicting spectral dissipation will be discussed. 
Wednesday 6th December 2017
09:00 to 10:00 Steven Daly
River Ice Breakup
Breakup transforms an ice-covered river into an open river. Two ideal forms of breakup bracket the types of breakup commonly found throughout most of the globe. At one extreme is thermal breakup. During an ideal thermal breakup, the river ice cover deteriorates and melts in place, with no increase in flow and little or no ice movement. At the other extreme is the more complex and less understood mechanical breakup (also referred to as a dynamic breakup). Mechanical breakup requires no deterioration of the ice cover, but rather results from an increase in flow entering the river. The increase in flow induces stresses in the cover, and the stresses in turn cause cracks and the ultimate fragmentation of the ice cover into pieces that are carried by the channel flow. Ice jams take place at locations where the ice fragments stop; severe and sudden hydraulic transients can result when these ice jams form or when they release. This presentation will focus on mechanical breakup and the historical evolution of our understanding of this topic. The presentation will include discussions of ice cover formation and the typical resulting ice structure, wave-ice interaction, the physics of the cracking, and the current status of our understanding of breakup.
10:00 to 11:00 Dany Dumont
Sea ice break-up in the marginal ice zone
Sea ice is a granular material composed of interacting elements, called floes, of different and evolving sizes and shapes. Contemporary numerical models of sea ice, although incorporating aspects of granular material dynamics, do not realistically represent the evolution of floe size distribution, which is affected by and upon which depend a myriad of dynamical and thermodynamical processes. Although this problem is theoretically, numerically and experimentally challenging, there has been significant progress over the past decade thanks to many international collaborative efforts, especially through revisiting marginal ice zone dynamics where surface gravity waves exert a strong control on floe size. This presentation will provide an overview of the theoretical and experimental knowledge on sea ice break-up and floe size distribution and will discuss how this topic is handled analytically in models.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Hung Tao Shen
Issues on modeling river ice dynamics
The presence of ice in rivers is an important phenomenon to be considered in the development of water resources in cold regions. Ice formation can affect the design, operation, and maintenance of hydraulic engineering facilities in cold regions. River ice phenomena involve complex interactions between hydrodynamic, mechanical, and thermal processes, as well as the ambient hydro-meteorological conditions and channel morphology. This presentation will discuss the current state of knowledge and unresolved issues on modeling river ice dynamics and the associated thermal processes. These issues include ice jam formation and release during freeze up and breakup, ice cover breakup, frazil ice evolution, and anchor ice formation and release. The possible similarities between these river ice phenomena and sea ice will be discussed.

Keywords: River ice, freeze up, ice jams, breakup, hydrodynamics, mathematical modeling
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Christ's College
Thursday 7th December 2017
09:00 to 10:00 Victor Tsai
Hydrofracture Propagation from Supraglacial Lake Drainage
Seasonal melt that forms at the surface of the Greenland Ice Sheet often eventually finds its way to the bed of the ice sheet, where it can have a significant effect on ice sheet dynamics. However, the way in which meltwater pathways from the surface to the bed are formed and maintained is not well understood. In this presentation, I will discuss the mechanics of hydrofracture, through which initial surface-to-bed connections are thought to be made. Hydrofracture of liquid water through its solid phase has unique mechanics, partly due to the low viscosity and turbulence of water, the higher density of water than ice, the melting of ice that occurs with viscous dissipation of turbulent energy, and the viscoelastic deformation of ice. A simplified model will be presented that describes the essential aspects of such hydrofracture, and the implications for glacier dynamics will be explained.
10:00 to 11:00 Peter Sammonds
Micromechanics of sea ice frictional slip from test basin scale experiments
Co-authors: Daniel Hatton and Daniel Feltham

We have performed high-resolution double-direct shear friction experiments on saline ice floes in the HSVA environmental test basin. The frictional motion was predominantly stick-slip.  Shear stresses, normal stresses, local strains and slip displacement were measured along the sliding faults, and acoustic emissions were monitored. High resolution measurements during a single stick- slip cycle at several positions along the fault allowed us to identify two phases of frictional slip: a nucleation phase, where a nucleation zone begins to slip before the rest of the fault, and a propagation phase when the entire fault is slipping. We employed a constitutive relation for frictional slip derived from the physics of asperity-asperity contact. We find our experimental data conform reasonably with this frictional law once slip weakening is introduced.  We deduce the interfacial faults failed in the stick-slip cycle through the process of brittle failure of asperities in shear, and at higher velocities, frictional heating, localized surface melting and hydrodynamic lubrication.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Erland Schulson
Cracks in ice and their role in brittle compressive failure
Cracks--new and old, short and long-- are ubiquitous features within the arctic sea ice cover. How they form and the role they play in mechanical behavior are important questions in ice mechanics. In this presentation, emphasis will be placed on brittle compressive failure. There, cracks preferentially oriented with respect to the applied stress state can slide intermittently across opposing surfaces in contact, activating in the process deformation mechanisms that can account for a number of observations/characteristics of brittle compressive failure on scales small and large. One such scale-independent mechanism is the wing-crack cum comb-crack mechanism: it can account for the axial splitting and shear faulting modes of terminal failure, conjugate faulting, brittle compressive strength and, upon consideration of crack-tip creep, the transition from brittle to ductile behavior. Application of confining stress above a critical level, set solely by the coefficient of kinetic friction, suppresses frictional sliding and activates, given a sufficiently high strain rate and triaxial confinement, a brittle-like mode of plastic failure governed by the different mechanism of adiabatic heating and dynamic recrystallization. These mechanisms will be described and questions arising addressed.

12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Jerome Weiss
Coulomb’s mechanics of sea ice: From geophysical evidences to experimental modeling
In 1773, Charles Augustin de Coulomb 1 proposed his celebrated failure criterion, postulating that under shear and compressive stress states failure occurs along a fault plane when the applied shear stress acting on that plane overcomes a resistance made of two parts of different nature: a cohesion τ_0, and a frictional resistance proportional to the normal pressure σ_n. The relevance of the Coulomb’s “theory” of failure for faulting and earthquake mechanics was recognized more than one century ago2, and remains nowadays a major tool of interpretation in civil engineering, the mechanics of granular media, and solid Earth geophysics. Its application to sea ice mechanics is much more recent. Satellite imagery3, 4 as well as in-situ stress data5 revealed that most of Artic sea ice pack deformation occurs through the activation of Coulomb’s faults. Many questions remain however, such as the partitioning between cohesion and friction in the resistance of sea ice faults, the competition between faulting and healing (through refreezing) in setting the long-term dynamics of faulting, or sliding velocity effects. To explore these questions, an analog experiment was recently developed in Grenoble, consisting of a thin ice layer sitting on top of a water tank and mechanically deformed at various rates with a circular Couette-like geometry6. This allowed sliding along a circular fault surface over arbitrarily large slip distances, and analyzing the competition between faulting and healing from the control of the rotation velocity and the air temperature of the cold room. The results marked out the relevance of Coulomb’s mechanics towards small normal stresses, explored the role of mechanical forcing and air temperature on ice faults long-term dynamics, and may represent a benchmark for the future development of sea ice mechanical models. 
14:30 to 15:30 Sveinung Loset
Global Ice Fracture Experiments at Spitsbergen and Its Impact on Numerical Simulation of Ice Actions
Co-author: Wenjun Lu (Norwegian University of Science and Technology)

The ice cover in the Arctic is both diminishing in areal extent and thinning. This leads to a situation where gravity waves are more prone to break up the ice cover into floe ice, and penetrate deeper into the ice fields in the Arctic. When this type of broken ice is interacting with offshore structures and ships, global fracturing of small or larger floes will be a major part of the interaction process and should be considered when either physically or numerically simulating the interaction process. An ice floe may fracture in different patterns. For example, it can be local bending failure or global splitting failure depending on the contact properties, geometry and confinement of the ice floe. Modelling these different fracture patterns as a natural outcome of numerical simulations is rather challenging. This is mainly because the effects of crack propagation, crack branching, multi fracturing modes and eventual fragmentation within a solid material are still questions to be answered by the on-going research in the Computational Mechanic community. In addition, the scale fracture properties of sea ice are still under discussions. To remedy some of these questions for ice we have conducted a number of physical fracture experiments at Spitsbergen during the winter of 2015-2017. The outcome of this research will be reported as well as the impact on numerical simulations of ice-structure interaction.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Discussion INI 1
Friday 8th December 2017
09:00 to 10:00 Veronique Dansereau
A Maxwell-Elasto-Brittle model for the drift and deformation of sea ice
In recent years, the viscous hypothesis and other underlying physical assumptions of the viscous-plastic (VP) rheology widely used in current climate and operational models have been revisited and found to be inconsistent with the observed mechanical behaviour of sea ice. Other studies have suggested that while the VP model can represent the mean global drift of sea ice with a certain level of accuracy, it fails at reproducing some key observed properties of sea ice deformation. We developed a new mechanical model, named Maxwell-Elasto-Brittle, as an alternative to the VP rheology in the view of accurately reproducing the drift and deformation of the ice cover in continuum sea ice models. The model builds on a damage mechanics framework used for ice and rocks. A viscous-like relaxation term is added to a linear-elastic constitutive relationship together with an effective viscosity that evolves with the local level of damage of the material, like its elastic modulus. This framework allows the internal stress to dissipate in large, permanent deformations along faults, or leads, once the material is highly damaged, while reproducing the small deformations associated with the fracturing process and retaining the memory of elastic deformations over relatively low damage areas. A healing mechanism counterbalances the effects of damaging over large time scales.

Idealized simulations have confirmed that the Maxwell-EB model reproduces the important characteristics of sea ice mechanics revealed by the analyses of available ice buoy and satellite data: the anisotropy of the deformation, the strain localization and intermittency, as well as the associated scaling laws. Sensitivity analyses show that the model, with few independent variables, can represent a large range of mechanical behaviours, with both the persistence of creeping leads and the activation of new leads with different shapes and orientations. Realistic simulations will be presented, in particular, simulations of the flow of ice through Nares Strait. These will demonstrate that the model reproduces the formation of stable ice bridges as well as the stoppage of the flow, a common phenomenon within numerous channels of the Arctic. In agreement with observations, the propagation of damage along narrow arch-like kinematic features, the discontinuities in the velocity field across these features, defining floes, and the eventual opening of polynyas downstream of the Strait are all represented.
10:00 to 11:00 Kara Peterson
Evaluation of an Elastic-Decohesive Rheology for Sea Ice
Satellite data obtained over the last couple decades have provided a wealth of information on sea ice motion and deformation for use in model comparison and validation. The data clearly show that ice deformation is focused along narrow linear features, which we would like to capture accurately in models. In this talk we describe an elastic-decohesive rheology that explicitly includes discontinuities in the deformation field to represent sea ice cracks. We compare results from an implementation in the UNM MPM sea ice model and a preliminary implementation in the LANL CICE model with RADARSAT Geophysical Processor System deformation data and results from the standard elastic-viscous-plastic rheology.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Han Duc Tran
An anisotropic elastic-decohesive constitutive relation for sea ice
Co-authors: Deborah L. Sulsky (University of New Mexico), Howard L. Schreyer (University of New Mexico)

When leads in sea ice expose warm underlying ocean to the frigid winter atmosphere, new ice is formed rapidly by freezing ocean water. Then convergence or closing of the pack ice forces the new ice in leads to pile up into ridges and to be forced down into keels. Together with thermodynamic growth, these mechanical processes shape the thickness distribution of the ice cover, and impact the overall strength of pack ice. Specifically, the deformation and strength of ice are not isotropic but vary with the thickness and orientation of the categories. To reflect these facts, we develop an anisotropic constitutive model for sea ice consisting an oriented thickness distribution. The model describes anisotropically mechanical responses in both elastic and failure regimes. In the elastic regime, the constitutive relation implicitly reflects the strong and weak directions of the pack ice depending on the distribution of thin ice and thicker ice. The existence of open water, a special case of thin ice, is also reflected in the elastic constitutive relation in which the free-traction condition is satisfied. In the failure regime, the model predicts when an initial failure, i.e. a microcrack, occurs and what the direction of the failure is. The evolution of a microcrack to a macrocrack, i.e. when free-traction crack surfaces are completely formed, is also modeled, and a numerical procedure is proposed to determine the width of cracks. Sample paths in stress space are used to illustrate how the model can simulate failure. Several examples of failure surfaces are presented to describe the behavior of ice when varying thickness distributions. The model predictions are also illustrated and compared with previous modeling efforts by examining regions under idealized loading.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 14:30 Daniel Feltham
Anisotropic sea ice mechanics
Observations show that the sea ice pack is heterogeneous and heavily flawed. Deformation in response to wind and ocean stresses occurs discontinuously at existing areas of weakness or through the formation of new damaged zones. The orientation of the damaged zones is affected by existing weaknesses in the pack and the orientation of the applied stresses. The orientation of these zones, in turn, affect the direction and magnitude of internal ice stresses. I describe work motivated by the need to represent the dependence of sea ice stress on orientation of existing weaknesses in continuum climate sea ice models. I will describe some discrete element simulation results that help provide understanding, and the form of an anisotropic continuum sea ice model designed to be used in climate models. I will present simulations highlighting the operation and features of this anisotropic rheology in climate-type sea ice simulations.
14:30 to 15:30 John Dempsey
Fracture Mechanics Applied To Ice Ih
The initial reluctance to adopt the fracture mechanics approach in ice mechanics and ice dynamics will be examined. The fundamentals of fracture mechanics and different fracture criteria will be reviewed. The relationship between models that idealize the crack-tip as a singular stress field and those that include a cohesive zone will be explored. Experimental values of material parameters determined in the laboratory and in the field will be examined. Time dependent and material effects on crack growth initiation and during crack propagation will be discussed. The limitations of fracture mechanics will be explored.
15:30 to 16:00 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons