**Pattern formation, scaling and intermittency in the deformation and
fracture of the sea ice cover**

** **

**Jérôme
Weiss**^{}

**Laboratoire
de Glaciologie et Géophysique de l’Environnement, UMR CNRS 5183**

**54
rue Molière, BP 96, 38402 St Martin d’Hères Cedex, France**

weiss@lgge.obs.ujf-grenoble.fr

The sea ice cover, which insulates
the ocean from the atmosphere, plays a fundamental role in the Earth’s climate
system. This cover deforms and fractures under the action of winds, ocean
currents and thermal stresses. Except during summer when melting takes place,
this deformation and fracturing controls the amount of open water within the
ice cover and the distribution of ice thickness, two parameters of high
climatic importance. Here we present a scaling analysis of sea ice deformation
and fracture that allows to characterize the heterogeneity of fracture patterns
and of deformation fields, as well as the intermittency of stress records. We
discuss the consequences of these scaling properties, particularly for sea ice
modelling in global climate models. We show how multifractal scaling laws can
be extrapolated to small scales to learn about the nature of the mechanisms
that accommodate the deformation. We stress that these scaling properties
preclude the use of homogenisation techniques (i.e. the use of mean values) to
link different scales, and we discuss how these detailed observations should be
used to constrain sea ice dynamics modelling.

** **

**Scaling and Anomalous
Scaling in Brittle Fracture**

**Alex
Hansen**

**Department
of Physics**

**Norwegian
University of Science and Technology**

**N-7491
Trondheim**

## Norway

We review the connection between self affinity and
fractality in correlated surfaces with an emphasis on the connection to brittle
fracture surfaces. We list a number of
different methods to measure Hurst exponents and fractal dimensions making the
important distinction between intrinsic or "window" methods and
extrinsic og "global" methods.
The former class of methods are favored by experimentalists while the
latter are favored by theorists.

Anomalous scaling occurs in self affine surfaces when the
intrinsic and extrinsic methods do not give the same result. We review how this
problem has been treated in the literature and then go on to show how this
strange behavior may be understood in terms of double scaling of wavelet
coefficients.

This leads to a very powerful method to determine whether a
curve is just self affine or anomalously self affine.

We then go on to reanalyze data from the literature which
earlier have been used to claim self affinity of brittle fracture
surfaces. We find that none of the
cases we have analysed hint at anomalous scaling.

**Cracks, crack propagation and thin films: an experimental
perspective.**

**Willem-Pier Vellinga**

*University of
Groningen / Eindhoven University of Technology*

Experimental
results on cracks and crack propagation for two types of thin films will be
presented: brittle elastic thin films (diamond like carbon) and plastic polymer
films (Poly-Ethylene-Terephthalate) both on metal substrates. The experiments
consist of simple tensile loading or loading in an Assymetric Double Cantilever
Beam set-up.

In
the brittle elastic thin film interaction between cracking, delamination
(interface cracking) and buckling is observed. A breakeable spring model has
been used to interpret these results. Disorder and energy release interact in
this case.

In
the plastic thin film in a similar experiment extensive shear banding
occurs, caused by the roughening of the
metal as well as the intrinsic softening of the polymer. Effects of roughness
on the propagation of interface cracks in experiment is illustrated.

**Microstructure, Fracture Morphology and **

**Mechanical Properties of Polymers**

**Fabrice Lapique**

Det Norske
Veritas, Høvik - Norway

Fabrice.Lapique@dnv.com

**Paul Meakin, Jens Feder and Torstein Jøssang**

University
of Oslo, Norway

The fracturing of four different polyolefins materials (a
polypropylene homopolymer, a propylene-ethylene copolymer, a polyethylene
homopolymer and an ethylene-hexene copolymer) was studied with the objective of
developing a better understanding on the relationships between the morphology
of semi crystalline polymers, the morphology and growth kinetics of their
fracture surfaces. Samples were injection moulded or hot pressed to generate
different microstructures and fracture experiments (impact) were performed at r
oom temperature, 0C and -20C.

In the PP homopolymer the fracture tends to pass
preferentially through the nucleation point of the spherulites and seems
therefore to follow the lamellae orientation. In the case of the PP copolymer,
the fracture was shown to propagate through spherulites but in that case it did
not follow the lamellae orientation because of the presence PE inclusions which
deviate the crack path. In both materials no trace of crazing or fibrillation
was observed.

PE materials behaved in a totally different way. Patterns formed by crazing and fibrillation
were observed. In the homopolymer,
spherulites were broken and crazed matter were drawn out of them, forming the
surface patterns. The tough PE
copolymer exhibited a continuous active zone and numerous shear bands
underneath the fracture surface. Surface patterns similar to the ones observed
on the homopolymers were observed.

The structure surface can be described in terms of
self-affine fractal models, and the Hurst (or roughness) exponent(s) and
roughness measurements can be used to describe quantitatively the fracture
surface topograghy. Fracture surfaces generated in homopolymers can be
described by a single Hurst exponent, which differs for PE and PP. For
copolymers with PE and PP matrices the Hurst exponent measured at small length
scales was the same as that obtained for the matrix material, but a crossover
to a second regime characterized by a higher Hurst exponent was found at longer
length scales. The crossover was related in to the average distance between
rubber particles for the PP/PE rubber phase specimen (PP-copo). A self affine
fractal behaviour is not correlated with a two components composition, but it
appears that the introduction of a second chemical entities (introduction of
heterogeneity) modifies the crack propagation at long-length scales, the
propagation at shorter length scales remaining unchanged. Environmental stress
cracking experiments performed on HDPE indicate that each regime can the
related to voids nucleation/cavitation at short-length scales and to
coalescence at longer length scales

**Cohesive and Adhesive Failure in Soft Solids**

** **

**Anand Jagota**

**Lehigh University**

This talk
discusses some questions concerning cohesive and adhesive failure in soft
solids. Soft solids are defined as
those for which the intrinsic stress for separating an interface approaches the
elastic modulus. The talk starts with
an introduction to the cohesive zone approach to fracture, specifically as
applied to soft solids. It follows by
presenting several examples of fracture experiments along with their modeling
and interpretation using cohesive zone models.
These include cleavage of mica, peeling of viscoelastic strips, and a
compressive shear test for elastomers.
In each case we show that the cohesive zone approach models fracture
well, but the extracted cohesive stress is often much lower than one would
expect. We show that many such
counter-intuitive results in the fracture of soft solids are related to the
fact that their modulus is small. We
develop the idea of elastic crack blunting and show that if the cohesive stress
exceeds the modulus of the material, large deformations will blunt the crack
tip. Thus, the cohesive stress is
limited by large elastic deformation to be not much larger than elastic modulus. The notion of cohesive stress itself in soft
solids poses some unresolved paradoxes, and it is suggested that the statistics
of individual molecule pull-off is a way to approach this problem.

** **

**The Fractal nature of Fracture IN BRITTLE MATERIALS**

** **

**J. J. Mecholsky, Jr. **

**Department of Materials Science & Engineering**

**University of Florida, **

**Gainesville, FL**

How do materials fracture? More importantly, how can we model the
fracture process at all length scales?
The answers to these questions have not been fully developed; thus, the
entire description of the fracture process is not yet known. This presentation will outline one path to
achieve the answers. There are characteristic features encoded on the fracture
surfaces during the fracture process for all materials that fail in a brittle
manner. The fracture surface contains
quantitative information about the stress and energy associated with a specific
fracture event. This presentation will
review the important characteristics of the fracture process as shown on the
fracture surface and discuss one approach to modeling the bond breaking process
and subsequent crack propagation path that results in the fracture surfaces we
observe. The approach presented here
is predicated upon the fact that fracture is a fractal process. Thus, we should
be able to describe the entire fracture event from the atomic to the
macroscopic length scales using relatively simple mathematical relationships.
We outline the approach to achieve an understanding of fracture on the atomic,
nanometer, microscopic and macroscopic length scales using quantum mechanics,
fracture mechanics, quantitative fracture surface analysis and fractal
geometry. The novel aspect of this research is the application of fractal
geometry to explain observed and predicted behavior during fracture. A model based on fractal geometry is
suggested as providing the atomic basis of fracture that will fit experimental
observations as a result of brittle fracture. The atomic model is based on
molecular orbital theory. Molecular dynamics provides the details of the
surface created during fracture. It is hypothesized that the fundamental unit
of fracture at the atomic scale is a quantity known as a_{0}. In turn, a_{0} can be related to the
fracture energy, g, and the elastic modulus, E, through a scaling parameter,
the fractal dimensional increment, D*, i.e., g = ½ a_{0} ED*. It is suggested that a_{0} is a
measure of the relative strain at the crack tip just before fracture and is
related to the available free volume for fracture in materials. The characteristic markings of mirror, mist
and hackle observed on the fracture surfaces of glasses, ceramics and polymers
are related to the fractal dimensional increment: (Y/ Y_{j})^{1/2}
c/ r_{j }= D*, where c is the crack size, r_{j}, is either the
mirror-mist radius (j = 1), mist-hackle radius (j = 2) or crack branching
boundary (j = 3), Y and Yj are constants related to the initial and propagating
crack geometry, respectively. The
combination of atomistic modeling, experimental measurements and the
application of fracture mechanics and fractal geometry leads to a suggested
sequence and organization of the brittle fracture process. Brittle fracture, i.e., bond breaking, is a
series of bond reconfiguration events at the crack tip dictated by minimum
energy configurations. This reconfiguration leads to an increase in free volume
all along the crack front. As the crack
moves, some of these regions will move either approximately above or below the
fracture plane. Nearest neighbor regions
of free volume will either add or annihilate.
The regions that add will grow in size.
The ones that annihilate each other will return to their approximate
original positions. The grown regions
will then become nearest neighbors to other regions and the process continues
as long as energy is supplied to the system.
The addition of regions of free volume will create what is observed on
the fracture surfaces as mirror, mist and hackle.

**Molecular Mechanisms of deformation and failure in**

**glassy materials**

Jörg Rottler, Princeton University

Mark O. Robbins, Johns Hopkins University

Understanding
the molecular origins of macroscopic mechanical properties is a

fundamental
scientific challenge. Fracture of both amorphous and crystalline

materials
involves many length scales reaching from the continuum to atomic

level
processes near a crack tip. While plasticity in crystals involves dislocation

motion
that is relatively well understood, theories of failure in disordered solids

are far
less advanced.

Using
molecular simulations of relatively simple models for amorphous glassy

materials,
we first study elastoplastic deformation at small strains and discuss the

dependence
of the shear yield stress on loading conditions, temperature, strain rate

and age of
the material. The influence of the loading state on the yield stress is

well
explained by classical yield criteria such as the pressure modified von Mises

criterion.
By contrast, no current model is able to explain the rate dependence

over the
entire range of temperatures in the glassy regime because of a complex

interplay
with the materials own intrinsic aging dynamics.

We then
focus on the deformation of glassy polymeric systems into crazes at large

strains.
Under tensile loading, many polymers such as polystyrene (PS) expand

into a
dense load-bearing network of fibrils and voids. An analysis of the

microscopic
deformation of individual chains shows how this expansion becomes

"jammed"
through polymer entanglements. These entanglements act like

permanent
chemical crosslinks and limit the deformation once some of them

become
fully stretched. We also find that the distribution of tension in the craze

develops
an exponential force tail in close analogy to compressed jammed

systems
such as granular media.

This force
distribution strongly influences the ultimate breakdown of the craze

fibrils
either through disentanglement or chain scission. The competition between

the two
failure modes is also accessible with our molecular model. In the last part

of the
talk, we combine these molecular level results with an analytic argument

for the
onset of crack propagation on the continuum scale and calculate the

macroscopic
fracture energy of glassy polymers, an important quantity in loadbearing

applications.