Under stress, many crystalline materials exhibit irreversible plastic deformation caused by the motion of lattice dislocations. In plastically deformed microcrystals, internal dislocation avalanches lead to jumps in the stress-strain curves (strain bursts), whereas in macroscopic samples plasticity appears as a smooth process. By combining three-dimensional simulations of the dynamics of interacting dislocations with statistical analysis of the corresponding deformation behavior, we determined the distribution of strain changes during dislocation avalanches and established its dependence on microcrystal size. Our results suggest that for sample dimensions on the micrometer and submicrometer scale, large strain fluctuations may make it difficult to control the resulting shape in a plastic-forming process.

https://science.sciencemag.org/content/sci/318/5848/251.full.pdf

Dislocation Avalanches, Strain Bursts, and the Problem of Plastic Forming at the Micrometer Scale

Disorder and long-range interactions are two of the key components that make material failure an interesting playfield for the application of statistical mechanics. The cornerstone in this respect has been lattice models of the fracture in which a network of elastic beams, bonds, or electrical fuses with random failure thresholds are subject to an increasing external load. These models describe on a qualitative level the failure processes of real, brittle, or quasi-brittle materials. This has been particularly important in solving the classical engineering problems of material strength: the size dependence of maximum stress and its sample-to-sample statistical fluctuations. At the same time, lattice models pose many new fundamental questions in statistical physics, such as the relation between fracture and phase transitions. Experimental results point out to the existence of an intriguing crackling noise in the acoustic emission and of self-affine fractals in the crack surface morphology. Recent advances ...

Statistical models of fracture

INTRODUCTION TO STATCALC Introduction of StatCalc PRELIMINARIES Random Variables and Expectations Moments and Other Functions Some Functions Relevant to Reliability Model Fitting Methods of Estimation Inference Random Number Generation Some Special Functions DISCRETE UNIFORM DISTRIBUTION Description Moments BINOMIAL DISTRIBUTION Description Moments Computing Table Values Test for the Proportion Confidence Intervals for the Proportion A Test for the Difference between Two Proportions Fisher's Exact Test Properties and Results Random Number Generation Computation of Probabilities HYPERGEOMETRIC DISTRIBUTION Description Moments Computing Table Values Point Estimation Test for the Proportion Confidence Intervals and Sample Size Calculation A Test for the Difference between Two Proportions Properties and Results Random Number Generation Computation of Probabilities POISSON DISTRIBUTION Description Moments Computing Table Values Point Estimation Test for the Mean Confidence Intervals for the Mean Test for the Ratio of Two Means Confidence Intervals for the Ratio of Two Means A Test for the Difference between Two Means Model Fitting with Examples Properties and Results Random Number Generation Computation of Probabilities GEOMETRIC DISTRIBUTION Description Moments Computing Table Values Properties and Results Random Number Generation NEGATIVE BINOMIAL DISTRIBUTION Description Moments Computing Table Values Point Estimation A Test for the Proportion Confidence Intervals for the Proportion Properties and Results Random Number Generation A Computational Method for Probabilities LOGARITHMIC SERIES DISTRIBUTION Description Moments Computing Table Values Inferences Properties and Results Random Number Generation A Computational Algorithm for Probabilities UNIFORM DISTRIBUTION Description Moments Inferences Properties and Results Random Number Generation NORMAL DISTRIBUTION Description Moments Computing Table Values One-Sample Inference Two-Sample Inference Tolerance Intervals Properties and Results Relation to Other Distributions Random Number Generation Computing the Distribution Function CHI-SQUARE DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Computing the Distribution Function F DISTRIBUTION Description Moments Computing Table Values Properties and Results Random Number Generation A Computational Method for Probabilities STUDENT'S t DISTRIBUTION Description Moments Computing Table Values Distribution of the Maximum of Several |t| Variables Properties and Results Random Number Generation A Computational Method for Probabilities EXPONENTIAL DISTRIBUTION Description Moments Computing Table Values Inferences Properties and Results Random Number Generation GAMMA DISTRIBUTION Description Moments Computing Table Values Applications with Some Examples Inferences Properties and Results Random Number Generation A Computational Method for Probabilities BETA DISTRIBUTION Description Moments Computing Table Values Inferences Applications with an Example Properties and Results Random Number Generation Evaluating the Distribution Function NONCENTRAL CHI-SQUARE DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Evaluating the Distribution Function NONCENTRAL F DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Evaluating the Distribution Function NONCENTRAL t DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Evaluating the Distribution Function LAPLACE DISTRIBUTION Description Moments Computing Table Values Inferences Applications Relation to Other Distributions Random Number Generation LOGISTIC DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimators Applications Properties and Results Random Number Generation LOGNORMAL DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimators Confidence Interval and Test for the Mean Inferences for the Difference between Two Means Inferences for the Ratio of Two Means Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles PARETO DISTRIBUTION Description Moments Computing Table Value Inferences Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles WEIBULL DISTRIBUTION Description Moments Computing Table Values Applications Point Estimation Properties and Results Random Number Generation Computation of Probabilities and Percentiles EXTREME VALUE DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimators Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles CAUCHY DISTRIBUTION Description Moments Computing Table Values Inference Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles INVERSE GAUSSIAN DISTRIBUTION Description Moments Computing Table Values One-Sample Inference Two-Sample Inference Random Number Generation Computational Methods for Probabilities and Percentiles RAYLEIGH DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimator Relation to Other Distributions Random Number Generation BIVARIATE NORMAL DISTRIBUTION Description Computing Table Values An Example Inferences on Correlation Coefficients Inferences on the Difference between Two Correlation Coefficients Some Properties Random Number Generation A Computational Algorithm for Probabilities DISTRIBUTION OF RUNS Description Computing Table Values Examples SIGN TEST AND CONFIDENCE INTERVAL FOR THE MEDIAN Hypothesis Test for the Median Confidence Interval for the Median Computing Table Values An Example WILCOXON SIGNED-RANK TEST Description Moments and an Approximation Computing Table Values An Example WILCOXON RANK-SUM TEST Description Moments and an Approximation Mann-Whitney U Statistic Computing Table Values An Example NONPARAMETRIC TOLERANCE INTERVAL Description Computing Table Values An Example TOLERANCE FACTORS FOR A MULTIVARIATE NORMAL POPULATION Description Computing Tolerance Factors Examples DISTRIBUTION OF THE SAMPLE MULTIPLE CORRELATION COEFFICIENT Description Moments Inferences Some Results Random Number Generation A Computational Method for Probabilities Computing Table Values REFERENCES INDEX

Handbook of statistical distributions with applications

Paper is a material known to everybody. It has a network structure consisting of wood fibres that can be mimicked by cooking a portion of spaghetti and pouring it on a plate, to form a planar assembly of fibres that lie roughly horizontal. Real paper also contains other constituents added for technical purposes.This review has two main lines of thought. First, in the introductory part, we consider the physics that one encounters when 'using' paper, an everyday material that exhibits the presence of disorder. Questions arise, for instance, as to why some papers are opaque and others translucent, some are sturdy and others sloppy, some readily absorb drops of liquid while others resist the penetration of water. The mechanical and rheological properties of paper and paperboard are also interesting. They are inherently dependent on moisture content. In humid conditions paper is ductile and soft, in dry conditions brittle and hard.In the second part we explain in more detail research problems concerned with paper. We start with paper structure. Paper is made by dewatering a suspension of fibres starting from very low content of solids. The processes of aggregation, sedimentation and clustering are familiar from statistical mechanics. Statistical growth models or packing models can simulate paper formation well and teach a lot about its structure.The second research area that we consider is the elastic and viscoelastic properties and fracture of paper and paperboard. This has traditionally been the strongest area of paper physics. There are many similarities to, but also important differences from, composite materials. Paper has proved to be convenient test material for new theories in statistical fracture mechanics. Polymer physics and memory effects are encountered when studying creep and stress relaxation in paper. Water is a 'softener' of paper. In humid conditions, the creep rate of paper is much higher than in dry conditions.The third among our topics is the interaction of paper with water. The penetration of water into paper is an interesting transport problem because wood fibres are hygroscopic and swell with water intake. The porous fibre network medium changes as the water first penetrates into the pore space between the fibres and then into the fibres. This is an area where relatively little systematic research has been done. Finally, we summarize our thoughts on paper physics.

http://iopscience.iop.org/article/10.1088/0034-4885/69/3/R03/pdf

The physics of paper

The deformation and flow of disordered solids, such as metallic glasses and concentrated emulsions, involves swift localized rearrangements of particles that induce a long-range deformation field. To describe these heterogeneous processes, elastoplastic models handle the material as a collection of 'mesoscopic' blocks alternating between an elastic behavior and plastic relaxation, when they are too loaded. Plastic relaxation events redistribute stresses in the system in a very anisotropic way. We review not only the physical insight provided by these models into practical issues such as strain localization, creep and steady-state rheology, but also the fundamental questions that they address with respect to criticality at the yielding point and the statistics of avalanches of plastic events. Furthermore, we discuss connections with concurrent mean-field approaches and with related problems such as the plasticity of crystals and the depinning of an elastic line.

/pdf/deformation-and-flow-of-amorphous-solids-insights-from-31hhogb2t0.pdf

Deformation and flow of amorphous solids: Insights from elastoplastic models

We report tensile failure experiments on paper sheets The acoustic emission energy and the waiting times between acoustic events follow power-law distributions This remains true while the strain rate is varied by more than 2 orders of magnitude The energy statistics has the exponent beta approximately 125+/-010 and the waiting times the exponent tau approximately 10+/-01, in particular, for the energy roughly independent of the strain rate These results do not compare well with fracture models, for (brittle) disordered media, which as such exhibit criticality One reason may be residual stresses, neglected in most theories

/pdf/acoustic-emission-from-paper-fracture-3sr0q5u1d9.pdf

Acoustic emission from paper fracture.

We analyze 6500 mm long fracture lines of paper as an example of crack propagation involving disorder. The cracks are asymptotically self-affine, with a roughness exponent close to 0.6. Systematic deviations from the power-law-scaling exist below a lengthscale related to the microscopic heterogeneities and possibly to a cross-over from 3d to 2d crack propagation. Several analysis methods are discussed, including first return analysis and the detection of correlated trends.

Analysis of long crack lines in paper webs

Acoustic emission or crackling noise is measured from an experiment on splitting or peeling of paper. The energy of the events follows a power law, with an exponent β ~ 1.8 ± 0.2. The event intervals have a wide range, but superposed on scale-free statistics there is a time scale, related to the typical spatial scale of the microstructure (a bond between two fibers). Since the peeling takes place via steady-state crack propagation, correlations can be studied with ease and shown to exist in the series of acoustic events.

https://iopscience.iop.org/article/10.1209/epl/i2005-10365-x/pdf

Crackling noise in paper peeling

We study the statistics of crack pinning in two dimensions by experiments and simulations of directed polymers in random media. Mode I tensile tests on paper samples show a delocalization phenomenon as the notch length is varied if the fraction of cracks pinned to the notch is monitored. This is compared with the behavior of directed polymers in the presence of both an energetically favorable localized pinning center and bulk disorder. An analysis of the crack interface roughness indicates self-affine behavior with a roughness exponent ζ in the proximity of the minimum energy surface value 2/3.

Pinning of cracks in two-dimensional disordered media

The deformation of wood is analyzed using the finite element method to quantify the phenomena in wood cells and cell walls. The deformation curves of computed microstructures are compared to experimental observations in two different loading cases: compression and combination of shear and compression. Simulated and experimental shapes of deformation curves match qualitatively and the deformation shapes exhibit a similar response to change in the loading mode. We quantify the intra-cell-wall stresses to understand the effects of the different layers during the deformation. The results benefit the development of energy efficient mechanical and chemo-mechanical pulping processes for pulp, board, and composite manufacture. In addition, the aspects of cell deformation can be exploited to dismantle the wood to accelerate chemical reactions in biorefinery.

L. I. Salminen

Papers

Acoustic emission from paper fracture.

Analysis of long crack lines in paper webs

Crackling noise in paper peeling

Pinning of cracks in two-dimensional disordered media

A 3D micromechanical study of deformation curves and cell wall stresses in wood under transverse loading