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This book contains contributions presented at the IUTAM Symposium "Fracture Phenomena in Nature and Technology" held in Brescia, Italy, 1-5 July, 2012.The objective of the Symposium was fracture research, interpreted broadly to include new engineering and structural mechanics treatments of damage development and crack growth and also large-scale failure processes as exemplified by earthquake or landslide failures, ice shelf break-up and hydraulic fracturing (natural or for resource extraction or CO2 sequestration), as well as small-scale rupture phenomena in materials physics including, e.g. inception of shear banding, void growth, adhesion and decohesion in contact and friction, crystal dislocation processes and atomic/electronic scale treatment of brittle crack tips and fundamental cohesive properties. Special emphasis was given to multiscale fracture description and new scale-bridging formulations capable to substantiate recent experiments and tailored to become the basis for innovative computational algorithms.




This Special Issue of the International Journal of Fracture contains selected papers presented at the IUTAM Symposium Fracture Phenomena in Nature and Technology that was held at the School of Engineering, University of Brescia, Italy, during the week of July 1–5, 2012.
D. Bigoni, A. Carini, M. Gei, A. Salvadori

Modeling fracture by material-point erosion

The present work is concerned with the verification and validation of an implementation of the eigenfracture scheme of Schmidt et al. (SIAM J Multi-scale Model Simul 7:1237–1266, 2009) based on material-point erosion, which we refer to as eigenerosion. Eigenerosion is derived from the general eigenfracture scheme by restricting the eigendeformations in a binary sense: they can be either zero, in which case the local behavior is elastic; or they can be equal to the local displacement gradient, in which case the corresponding material neighborhood is failed, or eroded. When combined with a material-point spatial discretization, this scheme gives rise to material-point erosion, i. e., each material point can be either intact, in which case its behavior is elastic, or be completely failed—or eroded—and has no load bearing capacity. We verify the eigenerosion scheme through convergence studies for mode I fracture propagation in three-dimensional problems. By way of validation we apply the eigene-rosion scheme to the simulation of combined torsion- traction experiments in aluminum-oxide bars.
A. Pandolfi, B. Li, M. Ortiz

Crack front perturbations revisited

The problem of the in-plane dynamic perturbation of a crack propagating with a front that is nominally straight is solved, to second order in the perturbation. The method of approach is a streamlined and generalized version of that previously applied to first order by the author and co-workers. It emerges, however, that the analysis at second order requires for its consistency the introduction of a new singular term, of a type not present at first order. The analysis is restricted to the case of Mode I loading, for clarity of exposition. It is carried out at a level of generality that incorporates viscoelastic response as well as propagation in a “vertically stratified” medium including, as a special case, propagation in a slab of finite thickness. For illustration, the general solution is specialized to the case of a stationary crack in an infinite elastic medium and agreement with a solution recently developed by methodology that is specific to the static case is confirmed.
J. R. Willis

Localisation near defects and filtering of flexural waves in structured plates

The paper deals with localisation of flexural waves within gratings composed of either pinned points or rigid inclusions of finite radius in a structured plate. We study the filtering and resonant action of such systems. The effect of the finite size of inclusions on the dynamic localisation is analysed for the range of frequencies where only zeroth grating orders propagate. The structure of the resonant modes within gratings of inclusions is of special interest. In particular, we consider the circumstances under which such gratings can deliver for flexural waves a phenomenon similar to Electromagnetically Induced Transparency, where a resonant maximum of transmission is cut in two by a resonant minimum. We identify system designs which yield very high concentration of flexural fields within the interface that may lead to a further structural failure.
S. G. Haslinger, R. C. McPhedran, N. V. Movchan, A. B. Movchan

Fracture process in cortical bone: X-FEM analysis of microstructured models

Bones tissues are heterogeneous materials that consist of various microstructural features at different length scales. The fracture process in cortical bone is affected significantly by the microstructural constituents and their heterogeneous distribution. Understanding mechanics of bone fracture is necessary for reduction and prevention of risks related to bone fracture. The aim of this study is to develop a finite-element approach to evaluate the fracture process in cortical bone at micro-scale. In this study, three microstructural models with various random distributions based on statistical realizations were constructed using the global model’s framework together with a submodelling technique to investigate the effect of microstructural features on macroscopic fracture toughness and microscopic crack-propagation behaviour. Analysis of processes of crack initiation and propagation utilized the extended finite-element method using energy-based cohesive-segment scheme. The obtained results were compared with our experimental data and observations and demonstrated good agreement. Additionally, the microstructured cortical bone models adequately captured various damage and toughening mechanisms observed in experiments. The studies of crack length and fracture propagation elucidated the effect of microstructural constituents and their mechanical properties on the microscopic fracture propagation process.
Simin Li, Adel Abdel-Wahab, Emrah Demirci, Vadim V. Silberschmidt

Minimum theorems in 3D incremental linear elastic fracture mechanics

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York, 2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam, 2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin, 2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369, 2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82, 1966) in plasticity. Following the cornerstone work of Rice (1989) on weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999; Leblond et al. in Int J Solids Struct 36:105–142,1999) proposed asymptotic expansions for stress intensity factors in three dimensions—see also Lazarus (J Mech Phys Solids 59:121–144,2011). As formerly in 2D, expansions can be given a Colonnetti’s decomposition (Colonnetti in Rend Accad Lincei 5, 1918, Quart Appl Math 7:353–362, 1950) interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103, 1999), Leblond et al. (Int J Solids Struct 36:105–142, 1999) however, symmetry of Ceradini’s theorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived.
A. Salvadori, F. Fantoni

Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach

Basalt columns, septarias, and mud cracks possess beautiful and intriguing crack patterns that are hard to predict because of the presence of cracks intersections and branches. The variational approach to brittle fracture provides a mathematically sound model based on minimization of the sum of bulk and fracture energies. It does not require any a priori assumption on fracture patterns and can therefore deal naturally with complex geometries. Here, we consider shrinkage cracks obtained during unidirectional drying of a colloidal suspension confined in a capillary tube. We focus on a portion of the tube where the cross-sectional shape cracks does not change as they propagate. We apply the variational approach to fracture to a tube cross-section and look for two-dimensional crack configurations minimizing the energy for a given loading level. We achieve qualitative and quantitative agreement between experiments and numerical simulations using a regularized energy (without any assumption on the cracks shape) or solutions obtained with traditional techniques (fixing the overall crack shape a priori). The results prove the efficiency of the variational approach when dealing with crack intersections and its ability to predict complex crack morphologies without any a priori assumption on their shape.
C. Maurini, B. Bourdin, G. Gauthier, V. Lazarus

Damage mechanisms in the dynamic fracture of nominally brittle polymers

Linear elastic fracture mechanics provides a consistent framework to evaluate quantitatively the energy flux released to the tip of a growing crack. Still, the way in which the crack selects its velocity in response to this energy flux remains far from completely understood. To uncover the underlying mechanisms, we experimentally studied damage and dissipation processes that develop during the dynamic failure of polymethylmethacrylate, classically considered as the archetype of brittle amorphous materials. We evidenced a well-defined critical velocity along which failure switches from nominally-brittle to quasi-brittle, where crack propagation goes hand in hand with the nucleation and growth of microcracks. Via postmortem analysis of the fracture surfaces, we were able to reconstruct the complete spatiotemporal microcracking dynamics with micrometer/nanosecond resolution. We demonstrated that the true local propagation speed of individual crack fronts is limited to a fairly low value, which can be much smaller than the apparent speed measured at the continuum-level scale. By coalescing with the main front, microcracks boost the macroscale velocity through an acceleration factor of geometrical origin. We discuss the key role of damage-related internal variables in the selection of macroscale fracture dynamics.
Daniel Bonamy, Davy Dalmas, Claudia Guerra, Julien Scheibert

Tight sedimentary covers for CO2 sequestration

CO2 storage at depth is a promising way to reduce the spread of greenhouse gases in the atmosphere. Obviously the sedimentary cover should ensure the sealing of reservoirs. The latter are periodically fractured to allow injection and we propose a model to predict the maximum bearable gas pressure before reinitiating these fractures in the caprock or incidentally along the interface between the reservoir and the caprock. The method is based on a twofold criterion merging energy and stress conditions. Specific conditions related to the gas pressure acting on the crack faces and the swelling of the reservoir due to the pressure rise require taking into account several terms in addition to the classical singular term that describes the state of stress at the tip of the main crack.
D. Leguillon, E. Karnaeva, A. Baroni, C. Putot

Calibration of brittle fracture models by sharp indenters and inverse analysis

In several engineering areas structural analyses concern also fracture processes of brittle materials and employ cohesive crack models. Calibrations of such models, i.e. identification of their parameters by tests, computer simulations of the tests and inverse analyses, have been investigated in the literature particularly with reference to non-destructive indentation tests at various scales. To this timely research, the following contributions are presented in this paper: a simple piecewise-linear cohesive crack model is considered for brittle materials (here glass, for example); for its calibration by “non-destructive” indentation tests novel shapes are attributed to instrumented indenters, in order to make fracture the dominant feature of the specimen response to the test; such shapes are comparatively examined and optimized by sensitivity analyses; a procedure for inverse analysis is developed and computationally tested, based on penetration versus increasing force only (no imprint measurements by profilometers) and is made “economical” (i.e. computationally fast, “in situ” by small computers) by model reduction through proper orthogonal decomposition in view of repeated industrial applications.
V. Buljak, G. Cocchetti, G. Maier

Statistics of ductile fracture surfaces: the effect of material parameters

The effect of material parameters on the statistics of fracture surfaces is analyzed under small scale yielding conditions. Three dimensional calculations of ductile crack growth under mode I plane strain, small scale yielding conditions are carried out using an elastic-viscoplastic constitutive relation for a progressively cavitating plastic solid with two populations of void nucleating second phase particles represented. Large particles that result in void nucleation at an early stage are modeled discretely while small particles that require large strains to nucleate are homogeneously distributed. The three dimensional analysis permits modeling of a three dimensional material microstructure and of the resulting three dimensional stress and deformation states that develop in the fracture process region. Material parameters characterizing void nucleation are varied and the statistics of the resulting fracture surfaces is investigated. All the fracture surfaces are found to be self-affine over a size range of about two orders of magnitude with a very similar roughness exponent of 0.56 ± 0.03 In contrast, the full statistics of the fracture surfaces is found to be more sensitive to the material microscopic fracture properties: height fluctuations are shown to crossover from a Student’s distribution with power law tails at small scales to a Gaussian behavior at large scales, but this transition occurs at a material dependent length scale. Using the family of Student’s distributions, this transition can be described introducing an additional exponent μ = 0.15 ± 0.02, the value of which compares well with recent experimental findings. The description of the roughness distribution used here gives a more complete quantitative characterization of the fracture surface morphology which allows a better comparison with experimental data and an easier interpretation of the roughness properties in terms of microscopic failure mechanisms.
Laurent Ponson, Yuanyuan Cao, Elisabeth Bouchaud, Viggo Tvergaard, Alan Needleman

Efficient pseudo-spectral solvers for the PKN model of hydrofracturing

In the paper, a novel algorithm employing pseudo-spectral approach is developed for the PKN model of hydrofracturing. The respective solvers compute both the solution and its temporal derivative. In comparison with conventional solvers, they demonstrate significant cost effectiveness in terms of balance between the accuracy of computations and densities of the temporal and spatial meshes. Various fluid flow regimes are considered.
Michal Wrobel, Gennady Mishuris

A solution to the parameter-identification conundrum: multi-scale interaction potentials

Softening is a structural property, not a material property. Any material will show softening, but in this paper the focus is primarily on cement and concrete, which show this property very clearly owing to their coarse heterogeneity (relative to common laboratory-scale specimen sizes). A new model approach is presented, based on pair-potentials describing the interaction between two neighbouring particles at any desired size/scale level. Because of the resemblance with a particle model an equivalent lattice can be constructed. The pair-potential is then the behavioral law of a single lattice element. This relation between force and displacement depends on the size of the considered lattice element as well as on the rotational stiffness at the nodes, which not only depends on the flexibility of the global lattice to which the element is connected but also on the flexural stiffness of the considered element itself. The potential Fr relation is a structural property that can be directly measured in physical experiments, thereby solving size effects and boundary effects.
J. G. M. van Mier

Remarks on Application of Different Variables for the PKN Model of Hydrofracturing: Various Fluid-Flow Regimes

The problem of hydraulic fracture for the PKN model is considered within the framework presented recently by Linkov (Doklady Phys 56(8):436–438, 2011). The modified formulation is further enhanced by employing an improved regularized boundary condition near the crack tip. This increases solution accuracy especially for singular leak-off regimes. A new dependent variable having clear physical sense is introduced. A comprehensive analysis of numerical algorithms based on various dependent variables is provided. Comparison with know numerical results has been given.
P. Kusmierczyk, G. Mishuris, M. Wrobel

Prediction of grain boundary stress fields and microcrack initiation induced by slip band impingement

Slip localization is widely observed in metallic polycrystals undergoing cyclic deformation or post-irradiation tensile deformation, whatever their crystallographic structure. Hence, strong strain localization occurs in thin slip bands (SBs) inducing by the way local stress concentrations at their intersections with grain boundaries (GBs). Many GB stress field formulae based on the dislocation pile-up theory have been proposed since the pionnering work of Stroh and others. These allow the use of the Griffith criterion for prediction GB fracture initiation. However, recent observations show that assuming that slip is localized on a single atomic plane leads to unrealistic results. In fact, a large number of slip planes are plastically activated and then finite slip band thickness should be accounted for. Numerous crystalline finite element (FE) computations have been carried out using considering a slip bands with low critical resolved shear stress embedded in an elastic matrix. The computed GB normal and shear stress fields:
  • are considerable lower than the pile-up ones and exhibit strong dependency on the slip band thickness close to the SB corner
  • but are in fair agreement with the solution predicted by the pile-up theory far away
Since the pile-up theory leads to the overestimation of the local GB stress fields, the main goal of the current paper is to perform analytical model of GB stress components based upon FE calculations. The effect of various parameters can be understood in the framework of matching asymptotic expansions which is usually applied to cracks with V notches of finite thickness. Finally, the predicted remote stresses to GB fracture in pre-irradiated austenitic stainless steels subjected to tensile loading in various environment are compared to experimental data and the pile-up based predictions.
Maxime Sauzay, Mohamed Ould Moussa

Modeling the heterogeneous effects of retained austenite on the behavior of martensitic high strength steels

The effects of pockets of retained austenite on the behavior of martensitic steels have been investigated. A dislocation-density based crystalline plasticity and specialized finite-element formulation were used to investigate how f.c.c. austenite pockets interact with b.c.c. martensitic laths. Quasi-static and dynamic analyses were undertaken to investigate how the effects of the orientations of parent austenite grains and different crystallographic interfaces affect shear strain localization, strength, and toughness. It is shown that the orientations of the parent austenite grain have a significant effect on the dominance of specific interfacial slip systems, and this subsequently affects whether the retained austenite has incompatible slip with martensitic laths, for low austenite Euler angles, or compatible slip with martensitic laths, for high values of austenite Euler angles.
Q. Wu, P. Shanthraj, M. A. Zikry

Crack nucleation from a notch in a ductile material under shear dominant loading

We examine the nucleation of a crack from a notch under a dominant shear loading in Al 6061-T6. The specimen is loaded in nominally pure shear over the gage section in an Arcan specimen configuration. The evolution of deformation is monitored using optical and scanning electron microscopy. Quantitative measurements of strain are made using the 2nd phase particles as Lagrangian markers which enable identification of the true (logarithmic) strains to levels in the range of two. Electron microscopy reveals further that the 2nd phase particles do not act as nucleation sites for damage in the regions of pure shear deformation. The initial notch is shown to “straighten out”, forming a new, sharper notch and triggering failure at the newly formed notch. Numerical simulations of the experiment, using the conventional Johnson–Cook model and a modified version based on grain level calibration of the failure strains, reveal that it is necessary to account for large local strain levels prior to the nucleation of a crack in order to capture the large deformations observed in the experiment.
A. Ghahremaninezhad, K. Ravi-Chandar
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