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Statistical Physics of Disorder and Pattern Formation in Fracture
Pattern formation, scaling and intermittency in the deformation and fracture of the sea ice cover
Laboratoire de Glaciologie et Géophysique de l’Environnement, UMR CNRS 5183
54 rue Molière, BP 96, 38402 St Martin d’Hères Cedex, France
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
Department of Physics
Norwegian University of Science and Technology
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.
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
Det Norske Veritas, Høvik - Norway
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
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,
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 a0. In turn, a0 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 a0 ED*. It is suggested that a0 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/ Yj)1/2 c/ rj = D*, where c is the crack size, rj, 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
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