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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Particle Strengths A holistic and straightforward analysis framework for understanding particle strength distributions In Particle Strengths: Extreme Value Distributions in Fracture, distinguished researcher Dr. Robert F. Cook delivers a thorough exploration of the science and related engineering of fracture strength distributions of single particles tested in diametral compression. In the book, the author explains particle strengths in the broader context of material strengths to permit readers to design with particles in systems in which mechanical properties are crucial to application, manufacturing, and handling. Particle Strengths compiles published data on particle strengths into a common format that includes over 140 materials systems and over 270 published strength distributions derived from over 13000 individual particle strength measurements. It offers physically consistent descriptions of strength behavior, including the strength threshold, using simple polynomial distribution functions that can easily be implemented. Readers will also find: * A thorough introduction to particles and particle loading, including discussions of particle failure and human activity * Comprehensive explorations of stochastic scaling of particle strength distributions, including concave and sigmoidal stochastic distributions * Practical discussions of agglomerate particle strengths, including those relevant to pharmaceuticals, foods, and catalysts * Detailed treatments of applications and scaling of particle strengths, including particle crushing energy and grinding particle reliability Perfect for materials scientists and engineers, mining and construction engineers, and environmental scientists, Particle Strengths: Extreme Value Distributions in Fracture will also benefit anthropologists, zoologists, pharmaceutical scientists, biomaterials scientists and engineers, and graduate students studying materials science, and chemical, mechanical, and biomedical engineering.
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Particle Strengths
Extreme Value Distributions in Fracture
Robert F. Cook
Copyright © 2023 by Robert F. Cook. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.
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Library of Congress Cataloging-in-Publication Data:
Names: Cook, Robert F. (Independent scientist), author.
Title: Particle strength : extreme value distributions in fracture / Robert F. Cook.
Description: Hoboken, New Jersey : Wiley ; American Ceramic Society,
[2023] | Includes bibliographical references and index.
Identifiers: LCCN 2022057645 (print) | LCCN 2022057646 (ebook) | ISBN
9781119850939 (hardback) | ISBN 9781119850946 (adobe pdf) | ISBN
9781119850953 (epub) | ISBN 9781119850960 (ebook)
Subjects: LCSH: Strength of materials. | Particle dynamics. | Materials science.
Classification: LCC TA405 .C844 2023 (print) | LCC TA405 (ebook) | DDC
620/.43–dc23/eng/20230105
LC record available at https://lccn.loc.gov/2022057645
LC ebook record available at https://lccn.loc.gov/2022057646
Cover Design: Wiley
Cover Image: Courtesy of Robert F. Cook
Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
To Michelle,
who always said this should be done,
and was never-endingly encouraging.
Contents
Cover
Title page
Copyright
Dedication
Preface
Abbreviations and Symbols
1 Introduction to Particles and Particle Loading
1.1 Particle Failure and Human Activity
1.1.1 Particles as Structural Components
1.1.2 Particle Loading
1.1.3 Particles in Application
1.2 Particle Shapes and Sizes
1.3 Summary: Particle Loading and Shape
References
2 Particles in Diametral Compression
2.1 Extensive and Intensive Mechanical Properties
2.2 Particle Behavior in Diametral Compression
2.2.1 Force-Displacement Observations
2.2.2 Force-Displacement Models
2.3 Stress Analyses of Diametral Compression
2.4 Impact Loading
2.5 Strength Observations
2.6 Strength Empirical Distribution Function
2.7 Outline of
Particle Strengths
2.7.1 Individual Topics
2.7.2 Overall Themes
References
3 Flaw Populations
3.1 Flaw Sizes and Strengths
3.2 Populations of Flaws and Strengths
3.2.1 Population Definitions
3.2.2 Population Examples
3.3 Samples of Flaws and Strengths
3.3.1 Sample Definitions
3.3.2 Sample Examples
3.4 Heavy-Tailed and Light-Tailed Populations
3.5 Discussion and Summary
References
4 Strength Distributions
4.1 Brittle Fracture Strengths
4.1.1 Samples of Components
4.1.2 Analysis of Sample Strength Distributions
4.2 Sample Strength Distributions
4.2.1 Sample Analysis Verification
4.2.2 Sample Examples
4.3 Discussion and Summary
References
5 Survey of Extended Component Strength Distributions
5.1 Introduction
5.2 Materials and Loading Survey
5.2.1 Glass, Bending and Pressure Loading
5.2.2 Alumina, Bending Loading
5.2.3 Silicon Nitride, Bending Loading
5.2.4 Porcelain, Bending Loading
5.2.5 Silicon, Bending and Tension Loading
5.2.6 Fibers, Tensile Loading
5.2.7 Shells, Flexure Loading
5.2.8 Columns, Compressive Loading
5.2.9 Materials Survey Summary
5.3 Size Effects
5.3.1 Stochastic
5.3.2 Deterministic
5.3.3 Size Effect Summary
5.4 Discussion and Summary
References
6 Survey of Particle Strength Distributions
6.1 Introduction
6.2 Materials Comparisons
6.2.1 Alumina
6.2.2 Quartz
6.2.3 Limestone
6.2.4 Rock
6.2.5 Threshold perturbations
6.3 Size Comparisons
6.3.1 Small Particles
6.3.2 Medium Particles
6.3.3 Large Particles
6.4 Summary and Discussion
References
7 Stochastic Scaling of Particle Strength Distributions
7.1 Introduction
7.2 Concave Stochastic Distributions
7.2.1 Alumina
7.2.2 Limestone
7.2.3 Coral
7.2.4 Quartz and Quartzite
7.2.5 Basalt
7.3 Sigmoidal Stochastic Distributions
7.3.1 Fertilizer
7.3.2 Glass
7.4 Summary and Discussion
References
8 Case Study: Strength Evolution in Ceramic Particles
8.1 Introduction
8.2 Strength and Flaw Size Observations
8.3 Strength and Flaw Size Analysis
8.4 Summary and Discussion
References
9 Deterministic Scaling of Particle Strength Distributions
9.1 Introduction
9.2 Concave Deterministic Distributions
9.2.1 Alumina
9.2.2 Quartz
9.2.3 Salt
9.2.4 Rock
9.2.5 Coal
9.2.6 Coral
9.3 Sigmoidal Deterministic Distributions
9.3.1 Glass
9.3.2 Rock
9.4 Linear Deterministic Distributions
9.4.1 Cement
9.4.2 Ice
9.5 Deterministic Strength and Flaw Size Analyses
9.5.1 Linear Strength Distributions
9.5.2 Concave Strength Distributions
9.6 Summary and Discussion
References
10 Agglomerate Particle Strengths
10.1 Introduction
10.2 Pharmaceuticals
10.2.1 Porosity
10.2.2 Shape
10.2.3 Distributions
10.3 Foods
10.4 Catalysts
10.5 Discussion and Summary
References
11 Compliant Particles
11.1 Introduction–Hydrogel Particles
11.2 Deformation
11.2.1 Axial
11.2.2 Transverse
11.3 Strength
11.4 Summary and Discussion
References
12 Fracture Mechanics of Particle Strengths
12.1 Introduction
12.2 Uniform Loading
12.2.1 Work and Elastic Energy
12.2.2 Mechanical Energy and Surface Energy
12.2.3 The Griffith Equation
12.2.4 Configurational Forces: 𝒢 and
R
12.3 Localized Loading
12.3.1 Analysis
12.3.2 Examples
12.4 Spatially Varying Loading
12.4.1 Stress-Intensity Factor and Toughness
12.4.2 Crack at a Stressed Pore
12.4.3 Crack at a Misfitting Inclusion
12.4.4 Crack at an Anisotropic Grain or Sharp Contact
12.5 Combined Loading
12.5.1 Strength of Post-Threshold Flaws
12.5.2 Strength of Sub-Threshold Flaws
12.6 Long Cracks in Particles
12.6.1 Polymer Discs
12.6.2 Microcellulose Tablets
12.6.3 Ductile-Brittle Transitions
12.6.4 Agglomerate Compaction
12.7 Discussion and Summary
References
13 Applications and Scaling of Particle Strengths
13.1 Introduction
13.2 Particle Crushing Energy
13.3 Grinding Particle Reliability
13.4 Mass Effects on Particle Strength
13.5 Microstructural Effects on Particle Strength
13.6 Discussion
References
Index
End User License Agreement
List of Illustrations
CHAPTER 01
Figure 1.1 Schematic diagrams of loading ...
Figure 1.2 Schematic diagrams of the ...
Figure 1.3 Images of groups of particles,...
Figure 1.4 Schematic diagrams of the ...
Figure 1.5 Schematic diagrams of ...
Figure 1.6 Schematic diagrams of the ...
Figure 1.7 Schematic cross-section diagrams ...
Figure 1.8 Schematic diagrams of the ...
Figure 1.9 (a) Image of irreversible...
Figure 1.10 Schematic diagrams of...
Figure 1.11 Schematic diagrams of complete...
Figure 1.12 Schematic exploded view ...
Figure 1.13 Images of the failure sequence ...
Figure 1.14 Images of failed plaster particles...
Figure 1.15 Diagram of a particle 2-D section...
Figure 1.16 An array of 2-D particle shapes...
Figure 1.17 An array of 2-D particle shapes...
Figure 1.18 An array of 2-D particle...
Figure 1.19 An array of 2-D particle...
Figure 1.20 A 2-D digitized particle...
Figure 1.21 Schematic 2-D diagrams...
CHAPTER 02
Figure 2.1 A schematic diagram...
Figure 2.2 Schematic diagrams...
Figure 2.3 Schematic diagram...
Figure 2.4 Schematic diagram...
Figure 2.5 Linear force-displacement,...
Figure 2.6 Convex + linear force-displacement,...
Figure 2.7 Variations on smooth...
Figure 2.8 Simulated shapes of...
Figure 2.9 Plots of force...
Figure 2.10 Schematic cross-section...
Figure 2.11 (a) Schematic cross...
Figure 2.12 Plots of hysteretic...
Figure 2.13 Plots of simulated...
Figure 2.14 Plots of force...
Figure 2.15 Plot of simulated...
Figure 2.16 Plot of force...
Figure 2.17 Plots of the...
Figure 2.18 Schematic cross-section...
Figure 2.19 Filled contour maps...
Figure 2.20 Plots of the...
Figure 2.21 Schematic cross-section...
Figure 2.22 Plots of the...
Figure 2.23 Plot demonstrating correlation...
Figure 2.24 Plot of strength...
Figure 2.25 Plot of strength...
Figure 2.26 Logarithmic plots...
Figure 2.28 Strength edf...
CHAPTER 03
Figure 3.1 Images of flaws...
Figure 3.2 A series of...
Figure 3.3 (a) A schematic...
Figure 3.4 (a) Plot of...
Figure 3.5 Plots of probability...
Figure 3.6 Plots of probability...
Figure 3.7 Plots of probability...
Figure 3.8 Plot of the...
Figure 3.9 Plot of the...
Figure 3.10 Schematic diagram of...
Figure 3.11 Plots of empirical...
Figure 3.12 Plots of empirical...
Figure 3.13 Plots of empirical...
Figure 3.14 Schematic diagram of...
Figure 3.15 Plots of empirical...
Figure 3.16 Plot of relative...
Figure 3.17 Simulated empirical distribution...
Figure 3.18 Plots of probability...
Figure 3.19 Plot in semi...
Figure 3.20 Schematic diagram illustrating...
Figure 3.21 Relative strength empirical...
CHAPTER 04
Figure 4.1 Smoothly varying fitting...
Figure 4.2 Sigmoidal strength edf...
Figure 4.3 Estimated crack length...
Figure 4.4 Concave strength edf...
Figure 4.5 Estimated crack length...
Figure 4.6 Sigmoidal strength edf...
Figure 4.7 Strength edf ...
Figure 4.8 Estimated crack length...
Figure 4.9 Estimated crack length...
Figure 4.10 Strength edf ...
Figure 4.11 Strength edf ...
Figure 4.12 Estimated crack length...
Figure 4.13 Variations of crack...
Figure 4.14 Variations of crack...
Figure 4.15 Schematic diagram illustrating...
CHAPTER 05
Figure 5.1 Plots of strength...
Figure 5.2 Plots of crack...
Figure 5.3 Plots of strength...
Figure 5.4 Plots of crack...
Figure 5.5 Plots of strength...
Figure 5.6 Plot in logarithmic...
Figure 5.7 Plots of strength...
Figure 5.8 Plot in logarithmic...
Figure 5.9 Plots of strength...
Figure 5.10 Plot in logarithmic...
Figure 5.11 Plots of strength...
Figure 5.12 Plot in logarithmic...
Figure 5.13 Plots of strength...
Figure 5.14 Schematic cross-section...
Figure 5.15 Plot in logarithmic...
Figure 5.16 Plots of strength...
Figure 5.17 Plot in logarithmic...
Figure 5.18 Plot in logarithmic...
Figure 5.19 Plots of strength...
Figure 5.20 Plot in logarithmic...
Figure 5.21 Plots of strength...
Figure 5.22 Plot in logarithmic...
Figure 5.23 Plot of number...
Figure 5.24 Plots of strength...
Figure 5.25 Plot in logarithmic...
Figure 5.26 Plots of strength...
Figure 5.27 Plot in logarithmic...
Figure 5.28 Plot of maximum...
Figure 5.29 Schematic diagram illustrating...
Figure 5.30 Schematic diagram illustrating...
Figure 5.31 Schematic diagram illustrating...
Figure 5.32 Schematic diagram illustrating...
Figure 5.33 Plots of strength...
CHAPTER 06
Figure 6.1 Plot of strength...
Figure 6.2 Plot of crack...
Figure 6.3 Plot on expanded...
Figure 6.4 Plot in logarithmic...
Figure 6.5 Plots of strength...
Figure 6.6 Plot in logarithmic...
Figure 6.7 Plots of strength...
Figure 6.8 Plot in logarithmic...
Figure 6.9 Plots of strength...
Figure 6.10 Plot in logarithmic...
Figure 6.11 Plots of strength...
Figure 6.12 Plot in logarithmic...
Figure 6.13 Plot of strength...
Figure 6.14 Plot in logarithmic...
Figure 6.15 Plots of representative...
Figure 6.16 Plot in logarithmic...
Figure 6.17 Plots of representative...
Figure 6.18 Plots of representative...
Figure 6.19 (a) Plot in...
Figure 6.20 Schematic diagrams illustrating...
CHAPTER 07
Figure 7.1 Schematic diagrams illustrating...
Figure 7.2 Plot of strength...
Figure 7.3 Plot in logarithmic...
Figure 7.4 Plots of strength...
Figure 7.5 Plots in logarithmic...
Figure 7.6 Plot of strength...
Figure 7.7 Plot in logarithmic...
Figure 7.8 Plot of strength...
Figure 7.9 Plot in logarithmic...
Figure 7.10 Plot of strength...
Figure 7.11 Plot in logarithmic...
Figure 7.12 Plots of strength...
Figure 7.13 Plot in logarithmic...
Figure 7.14 Plots of strength...
Figure 7.15 Plot in logarithmic...
Figure 7.16 Plot of strength...
Figure 7.17 Plot in logarithmic...
Figure 7.18 Plot of strength...
Figure 7.19 Plot in logarithmic...
Figure 7.20 Plot of strength...
Figure 7.21 Plot in logarithmic...
Figure 7.22 Plot of strength...
Figure 7.23 Plot in logarithmic...
Figure 7.24 Plot in logarithmic...
Figure 7.25 Plot of number...
Figure 7.26 Schematic diagrams illustrating...
Figure 7.27 Plot of ranked...
CHAPTER 08
Figure 8.1 (a) Means and...
Figure 8.2 Strength distributions of...
Figure 8.3 Plots in logarithmic...
Figure 8.4 Crack length distribution...
Figure 8.5 Plot in logarithmic...
Figure 8.6 (a) Plot of...
Figure 8.7 Plot of crack...
Figure 8.8 Simulated strength distributions...
Figure 8.9 Simulated crack length...
Figure 8.10 (a) Strength distributions...
Figure 8.11 (a) Strength distributions...
Figure 8.12 Logarithmic plot of...
Figure 8.13 Schematic diagram illustrating...
Figure 8.14 Logarithmic plot of...
CHAPTER 09
Figure 9.1 Schematic diagrams illustrating...
Figure 9.2 Schematic diagrams illustrating...
Figure 9.3 Plot of strength...
Figure 9.4 Plot in logarithmic...
Figure 9.5 Plot of strength...
Figure 9.6 Plot in logarithmic...
Figure 9.7 Plot of strength...
Figure 9.8 Plot in logarithmic...
Figure 9.9 (a) Plot of...
Figure 9.10 (a) Plot of...
Figure 9.11 Plot of strength...
Figure 9.12 Plot in logarithmic...
Figure 9.13 Plot of strength...
Figure 9.14 Plot in logarithmic...
Figure 9.15 (a) Plot of...
Figure 9.16 (a) Plot of...
Figure 9.17 (a) Plot of...
Figure 9.18 (a) Plot of...
Figure 9.19 Logarithmic plot of...
Figure 9.20 Plot of strength...
Figure 9.21 Plot in logarithmic...
Figure 9.22 Plot of strength...
Figure 9.23 Plot in logarithmic...
Figure 9.24 (a) Plot of...
Figure 9.25 (a) Plot of...
Figure 9.26 (a) Plot of...
Figure 9.27 (a) Plot of...
Figure 9.28 (a) Plot of...
Figure 9.29 (a) Simulated deterministic...
Figure 9.30 Simulated crack length...
Figure 9.31 Simulated deterministic concave...
Figure 9.32 Plot in logarithmic...
Figure 9.33 Plot in logarithmic...
Figure 9.34 Logarithmic plot of...
Figure 9.35 Plot of ranked...
CHAPTER 10
Figure 10.1 Schematic diagram of...
Figure 10.2 Semi-logarithmic plots...
Figure 10.3 Schematic diagrams of...
Figure 10.4 (a) Semi-logarithmic...
Figure 10.5 Semi-logarithmic plot...
Figure 10.6 Diagrams illustrating the...
Figure 10.7 Diagrams illustrating variations...
Figure 10.8 Plots of the...
Figure 10.9 Plot of the...
Figure 10.10 Plot of strength...
Figure 10.11 Plot demonstrating correlation...
Figure 10.12 Plots demonstrating application...
Figure 10.13 Plots of microcellulose...
Figure 10.14 Replot of microcellulose...
Figure 10.15 Strength edf behavior...
Figure 10.16 Strength edf behavior...
Figure 10.17 Replot of microcellulose...
Figure 10.18 Plot in logarithmic...
Figure 10.19 Strength behavior of...
Figure 10.20 Strength behavior of...
Figure 10.21 Simulated strength edf...
CHAPTER 11
Figure 11.1 Images of arrays...
Figure 11.2 Plot of force...
Figure 11.3 Images of an...
Figure 11.4 Schematic cross-section...
Figure 11.5 Sequence of images...
Figure 11.6 Logarithmic plot of...
Figure 11.7 Plots of force...
Figure 11.8 Plots of force...
Figure 11.9 Logarithmic plot of...
Figure 11.10 Schematic cross-section...
Figure 11.11 Deformation profiles for...
Figure 11.12 Deformation profiles for...
Figure 11.13 Plot of relative...
Figure 11.14 Plot of strength...
Figure 11.15 Plot in logarithmic...
Figure 11.16 Logarithmic plot of...
Figure 11.17 Logarithmic plot of...
Figure 11.18 (a) Plot of...
Figure 11.19 Aspects of strength...
CHAPTER 12
Figure 12.1 Schematic diagrams of...
Figure 12.2 Schematic diagrams of...
Figure 12.3 Linear elastic force...
Figure 12.4 Energy vs crack...
Figure 12.5 Schematic diagram of...
Figure 12.6 Energy vs crack...
Figure 12.7 Schematic cross-section...
Figure 12.8 Crack length ...
Figure 12.9 Cone crack formation...
Figure 12.10 Schematic cross-section...
Figure 12.11 Schematic diagram of...
Figure 12.12 Plot of the...
Figure 12.13 Simulated strength edf...
Figure 12.14 Schematic diagrams of...
Figure 12.15 (a) Schematic diagram...
Figure 12.16 Plot of the...
Figure 12.17 Plot of stress...
Figure 12.18 Plot of crack...
Figure 12.19 (a) Schematic diagram...
Figure 12.20 Schematic diagram of...
Figure 12.21 Schematic cross-section...
Figure 12.22 Equilibrium stress-crack...
Figure 12.23 Plot of stress...
Figure 12.24 Schematic cross-section...
Figure 12.25 Plot of stress...
Figure 12.26 Schematic cross-section...
Figure 12.27 Toughness estimates for...
Figure 12.28 Toughness estimates for...
Figure 12.29 Schematic cross-section...
Figure 12.30 Plot of maximum...
Figure 12.31 Schematic cross-section...
Figure 12.32 Plot of induced...
CHAPTER 13
Figure 13.1 (a) Strength cdf...
Figure 13.2 (a) Strength cdf...
Figure 13.3 (a) Specific failure...
Figure 13.4 Failure energy E...
Figure 13.5 Schematic diagram of...
Figure 13.6 (a) Plot of...
Figure 13.7 (a) Strength cdf...
Figure 13.8 Overlaid applied stress...
Figure 13.9 Repeated stochastic variations...
Figure 13.10 Reliability behavior for...
Figure 13.11 (a) Schematic diagram...
Figure 13.12 Plot of strength...
Figure 13.13 (a) Schematic diagram...
Figure 13.14 Schematic diagram of...
Figure 13.15 Schematic diagram of...
Figure 13.16 Equilibrium stress-crack...
Figure 13.17 Plot of strength...
Figure 13.18 Plot of strength...
Guide
Cover
Title page
Copyright
Dedication
Table of Contents
Preface
Abbreviations and Symbols
Begin Reading
Index
End User License Agreement
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Preface
This book is concerned with strengths and strength distributions of particles. As such, it is a much-altered and expanded version of a review of the subjects published on-line several years earlier. The goal now, as then, is to present a comprehensive, unified, and objective view of particle strength measurements and the analyses required to interpret and apply the results of such measurements. The book draws from the fields of materials science and engineering, mechanics of materials, and fracture mechanics.
Particles are considered here as entities—both natural and man-made, and nearly always solid—that are of comparable extent in all three dimensions. The extents are usually, but not exclusively, small. Examples include grains of sand and crystals of table salt, but also aspirin tablets, fertilizer pellets, glass marbles, and river rocks. Attention is restricted to brittle particles, those that fail mechanically by fracture with negligible associated plastic deformation. The particle strengths of interest here are thus brittle fracture strengths, characterized by the unstable and unchecked propagation of a crack from a flaw in a component under the influence of a tensile stress. An intriguing aspect of particle failure is that cracks propagate under tensile stress—and yet the predominant form of particle loading is compression. Strictly, “diametral compression,” in which the particle is squeezed across a diameter. Diametral compression applied to the exterior of a particle leads to development of a zone within the particle of tensile stress perpendicular to the compression direction—a phenomenon quantified by Hertz in the 1880s.
Particles encounter diametral compression in a variety of ways: between displaced platens in a laboratory setting, the most common method of particle strength measurement; by impact against, or by, a hard surface; squeezed between rollers; or by the weight of other overlying particles in a packed particle array. The diversity of particle loading methods is matched by the diversity of particle materials. The majority of particles are inorganic materials—glasses and crystalline ceramics, rocks, and minerals—although there is a significant minority of organic materials—foods, pharmaceuticals, and hydrogels, and those that contain both materials—seashell fragments, coal, and coral. The variety of loading methods and diversity of materials have lead to a plethora of presentation schemes for reporting particle strengths and related failure metrics, significantly impeding scientific and engineering application of particle strength information. Thus, in order to advance applications of particle strength information, a major feature of the book is the presentation in a single, unbiased format of a large number of measured strength distributions of particles. The format enables identification of critical features of particle strength distributions and thence identification of patterns of behavior and underlying causes.
In particular, a major point made throughout the book is that it is the population of strength-controlling flaws that is the fundamental physical property that determines measured strength distributions of particles. The flaws are characterized by size and spatial distributions that are material properties. The measurements are characterized by the number and size of the particles tested. As the test parameters change, the strength distributions change. Hence, a second important feature of the book is the clear and quantitative development and application of analysis that relates fundamental properties of the invariant flaw population to observed characteristics of resultant strength distributions. Two important attributes of the strength distribution presentation method and analysis are used throughout the book. First, presentation of particle strength distributions in unbiased coordinates permits simple identification of predominant size effects. Second, fits to measured particle strength distributions are used simply as intermediate smoothing steps to infer underlying flaw populations. The commonly used linearized presentation scheme and fit to strength data in transformed coordinates are shown to be obscuring and misleading.
The idea and motivation for this book started some years ago during consideration of strength predictions for small microelectromechanical systems (MEMS) components. Progress in the MEMS area required the disentangling of much strength distribution folklore, especially associated with linearized distributions, leading to the development of a coherent (and simple) probability framework for strength and flaw distributions. Application of the developed framework to ceramic materials lead to analysis of refractory particle strengths and the idea that particles might be the end state of size effects: particles might contain only one strength-controlling flaw and thereby exhibit no variation in strength distribution from that set by the flaw population. Motivation for this book is then in part a pursuit of answers to the related questions of “What do the strength distributions of particles look like?” and “How did they get that way?” My research career prior to this had deliberately avoided such questions through experimental use of controlled-flaw strength techniques; deliberately introducing the one strength-controlling flaw by indentation. I was, however, extremely familiar with strength testing of many, small, brittle specimens under controlled conditions, and the subsequent analysis to extract underlying material properties.
The book chapters can be viewed in groups that address the above questions, although not in sequence. Chapter 1 describes the major particle types considered here, their shapes and sizes, their applications, and their qualitative failure mechanisms. Chapter 2 describes the mechanics of particle loading and diametral compression and quantitative measures of particle strength. The introductory concepts of Chapters 1 and 2 are used in a detailed outline of the book at the end of Chapter 2.
Chapters 3 and 4 provide the analytical framework for much of the book. Clear physical and mathematical descriptions are given for flaw populations, strength distributions, and how they are related. Chapter 3 describes forward analysis from flaws to strengths and Chapter 4 describes reverse analysis from strengths to flaws. Many of the subsequent chapters rely on analysis developed in Chapter 4. The analyses in these chapters are general and apply to strength distributions of all brittle components, not just particles.
Chapters 5 and 6 provide surveys of the measured strength distributions of “conventional” extended components and particles, respectively. The surveys show clear differences in the strength distribution shapes and relative widths of the two component types, and the different ways size effects are manifested in strength distributions. Both chapters make extensive use of Chapter 4 analyses to deconvolute strength behavior into underlying flaw distributions.
Chapters 7, 8, and 9 include the majority of the particle strength distributions presented, analyzed, and discussed here. As above, all three chapters make extensive use of Chapter 4 analyses to deconvolute strength behavior into underlying flaw distributions. Similarities and differences in particle strength and flaw distributions within and between material groups are emphasized, along with the different ways particle size influences strength behavior.
Chapters 10 and 11 describe and discuss the strength behavior of agglomerate and hydrogel particles, respectively; particles very different from the “hard,” inorganic particles considered in the earlier chapters. The majority of agglomerate and hydrogel particles are “soft” and largely organic. Although strength distributions are presented in both chapters, the emphasis in Chapter 10 is on fabrication effects on strength and the emphasis in Chapter 11 is on pre-failure deformation.
Chapters 12 and 13 are the most analytical in the book with regard to fracture, addressing basic fracture mechanics of particles and applications of fracture mechanics in engineering contexts, respectively. Chapter 12 makes clear the basic physics of particle strengths, in which external work generates internal elastic energy changes that are transformed into surface energy changes on fracture, and how that physics is applied in flaw and strength analyses. Chapter 13 extends the fracture ideas to the use and manipulation of strength distributions in engineering applications, relying on fundamental scaling principles.
Overall, the book emphasizes the development and application of descriptive analyses. The analyses enable extraction of the information contained in particle strength and strength distribution measurements (or, at least, most of the information). A self-imposed test of the veracity of a particular analysis, used throughout the book, was whether simulated data could pass for experimental measurements. The reader can judge. The book contains very little in the way of theory, except perhaps the assumptions that flaws act independently (although that is relaxed in Chapter 9) and that particle materials are linear elastic (and that is relaxed in Chapter 11). In addition, the book contains almost no statistics, except the occasional provision of means and standard deviations. Fit parameters are not given; plots are intended to convey all the information. There is, however, probability analysis required to translate strength distribution characteristics into flaw distribution characteristics and vice versa. More refined next-step probability ideas such as Bayesian (conditional) probability and the related hazard concept, clearly applicable to particle reliability, are not addressed.
My thinking on the subjects in this book has developed and benefited over the years from many discussions and interactions with colleagues, including: S.J. Bennison, S.J. Burns, F.W. DelRio, E.R. Fuller Jr, B.R. Lawn, D.B. Marshall, M.L. Oyen, G.M. Pharr IV, and M.V. Swain.
Robert F. Cook
Winterville NC
August 2022
Abbreviations and Symbols
cdf
cumulative distribution function
ccdf
complementary cumulative distribution function
edf
empirical distribution function
probability density function
SIF
stress-intensity factor
contact radius, impression semi-diagonal
component dimension, deformation zone dimension
crack length
component dimension
population pdf
sample pdf, component dimension
number of flaws in component (size of component), component stiffness
component dimension
particle mass
time
displacement, contact displacement
displacement, particle or mass velocity
displacement, particle diametral displacement under load, work
crack area
proportionality factor relating strength and crack length
B()
beta function
diameter of particle (size of particle)
Young’s modulus, particle failure energy
population cdf, force applied to component, particle failure force
population ccdf
mechanical energy release rate
sample cdf, hardness
estimate of sample cdf
sample ccdf
stress-intensity factor
length of component
elastic modulus
number of components in sample (size of sample)
force applied to component, porosity
sample strength edf
radius of particle, exterior radius of convex tablet, fracture resistance
toughness
energy
particle or component volume
work
yield stress
beta function exponent
beta function exponent
surface energy, elastic-plastic geometry term
contact displacement under load, microstructural traction zone length
strain
relative width of strength distribution
component compliance, flaw spatial density, grain size
relative strength, microstructural geometry term
relative crack length, Poisson’s ratio
relative crack length
material mass density
strength, stress
viscous deformation time constant
residual stress geometry term
applied stress geometry term
fundamental volume element
population volume
“varies as”
“approximately equal to”
Subscripts, superscripts, indices, dummy variables, and less-frequent uses of these and other symbols are defined in context.
1 Introduction to Particles and Particle Loading
The importance of particles as structural components in a range of engineering applications is described, providing an introduction and motivations for the work to follow. The geometrical aspects of particles in the load-bearing configuration of diametral compression are defined and used in a description of the observed failure modes of loaded particles. Images of particles commonly and not-so-commonly encountered in everyday life are presented and discussed. The geometry of particle loading is followed by an analysis of particle shape. A discrete Fourier analysis method is used to describe a range of particle shapes—multi-lobed, rough, angular, or variable—and the effects of particle shape on estimates of particle size are considered. The similarities of the analyses of particle shape and the following analyses of particle strength are discussed.
1.1 Particle Failure and Human Activity
Particles occur in nature and are produced by human industry. In many cases, particles are placed under load. This section introduces particles, how they are loaded, and how they may fail to support loads in application.
1.1.1 Particles as Structural Components
The force required to break or crush a particle is critical in many areas of human activity. Particles—objects that are limited in extent in three dimensions and often small—are usually not thought of as structural components, for which the ability to support a mechanical force is a distinguishing property. The familiar load-bearing structural components, columns in compression, rods or bars in tension, beams in bending, and shafts in torsion are all extended in one dimension to form the component axes. For columns and bars in structural applications, the directions of the forces associated with applied loading are parallel to the component axes; inward for columns, outward for bars. For beams and shafts in structural applications, moments are associated with applied loading but differ in orientation with respect to the component axes; perpendicular for beams, parallel for shafts. The cross-sections perpendicular to the axes of all four of these simple components are usually uniform, greatly simplifying structural analyses (Mase 1970; Timoshenko and Goodier 1970; Sadd 2009). Such structural components are usually large, and their load-bearing function obvious, for example, a pillar or column supporting a building. Figure 1.1 shows schematic diagrams of loaded simple structural components: (a) column, (b) bar, (c) beam, and (d) shaft. In Figure 1.1 applied forces are shown as dark shaded arrows; unless otherwise stated, this scheme is used throughout the book.
Figure 1.1 Schematic diagrams of loading geometry for the four major forms of structural components: (a) compression of a column; (b) tension of a bar; (c) bending of a beam; (d) torsion of a shaft. Applied forces in (a) and (b) and applied moments in (c) and (d) shown in dark shading.
Particles are, however, usually load-bearing in their many applications and are best thought of as short columns with nonuniform cross-sections. The column-like loading configuration, in which a diameter of a particle is placed in compression, is known as diametral compression. (Terms in italics are definitions to be used throughout—some are specific to this book.) Figure 1.2 shows schematic diagrams of particles in diametral compression, (a) a spherical particle and (b) a cylindrical particle. Both particles are sitting on flat platens and the sphere and cylinder are loaded symmetrically by a point force and a line force, respectively (the cylinder axis is parallel to the platen). The particle dimensions parallel to the loading axes are the diameter of the sphere and the diameter of the cylindrical face. The maximum cross-sectional dimensions perpendicular to the loading axes are the diameter, in the case of the sphere, and the diameter or the thickness, for the cylinder. (Consistent with the idea that particles are distinguished by limited extent in all dimensions, the thickness of the cylinder is comparable to the face diameter). The particle cross-sections perpendicular to the loading axes are nonuniform—the cross sectional areas of the particles are larger at the center and smaller at the extremities where the forces are applied. The load-bearing capacity of particles in diametral compression is a major focus of this book.
Figure 1.2 Schematic diagrams of the diametral compression loading geometry for two forms of particles: (a) sphere loaded by a point force; (b) cylinder loaded by a line force. Applied forces shown in dark shading.
Examples of the particles to be considered here are shown in the images of Figure 1.3. The Figure exemplifies the diversity of particles in load-bearing applications, natural, industrial, and domestic, and also provides examples of the fact that some particles are commonly encountered and some are not. The particles vary in material (organic and inorganic), size (tens of micrometers to hundreds of millimeters), and shape (from smooth and spherical to rough and angular), and include: (a) Cr2O3 particles dispersed in a cordierite (2MgO.2Al2O3.5SiO2) ceramic matrix (the particles are bright, the matrix is gray, and pores are dark). (b) Sand (largely quartz, crystalline silica, SiO2). (c) Fertilizer granules. (d) Salt (NaCl) crystals. (e) Peppercorns. (f) Aspirin tablets. (g) Ceramic pie weights. (h) Taconite (iron ore, Fe3O4 and Fe2O3) pellets. (i) Glass (largely amorphous silica and other oxides) spheres. (j) Plaster (gypsum, CaSO4) spheres. (k) Tumbled river rocks (largely quartz). (l) Jagged railway ballast (largely granite, a mixture of quartz and other minerals). Throughout the book, as various aspects of particle mechanical behavior are considered, details regarding specific particle applications, materials, sizes, and shapes will be given in context. However, independent of particle size and shape, an overall description of particle loading geometry in diametral compression can be developed. In the next Section, 1.1.2, this geometrical description is outlined and then applied in an overview of observations of how particles may fail in use as structural components.
Figure 1.3 Images of groups of particles, demonstrating the wide variety of particles in natural, industrial, and domestic applications: (a) Cr2O3 particles in a cordierite ceramic matrix composite (polished section); (b) Sand grains; (c) Fertilizer granules; (d) Salt grains; (e) Peppercorns; (f) Aspirin tablets; (g) Ceramic pie weights; (h) Taconite (iron ore) pellets; (i) Glass marbles; (j) Plaster (gypsum) moulded spheres; (k) River rocks; (l) Railway track ballast (wooden sleeper and metal rail visible in foreground). All images optical. Note variations in shape and size between and within groups. Source: Robert F. Cook.
1.1.2 Particle Loading
Detailed consideration of loading of particles in diametral compression requires a clear description of the loading geometry. Particles are distinct from columns in that locations and orientations within a loaded particle are specified relative to the axis of loading rather than an axis of the particle. A near equidimensional particle may take any orientation relative to that of an applied force whereas the axis of a column is nearly always colinear with the direction of the force (Figures 1.1 and 1.2). Figure 1.4 shows schematic cross sections of the particles loaded in diametral compression from Figure 1.2, illustrating important locations and directions. Both are axial cross sections (containing the loading axis) and both are discs. For such particles, the points of force application (top of particle) or displacement application (bottom of particle) are the poles (Figure 1.4a). The line joining the poles is oriented along the loading axis, in the axial direction (Figure 1.4b). Midway along the line joining the poles is the center of the particle (Figure 1.4a). Directions perpendicular to the axial direction in the axial plane are transverse directions (Figure 1.4b). The loaded spherical particle system of Figure 1.2a exhibits rotational symmetry, such that all angularly separated axial cross sections, and therefore transverse directions within those sections, are equivalent. The loaded cylindrical particle system of Figure 1.2b exhibits translational symmetry, such that all spatially separated cross sections parallel to the cylinder face (“through the thickness”) are equivalent. The terms pole, center, axial, and transverse in the cylindrical case refer to points and directions in such face sections.
Figure 1.4 Schematic diagrams of the axial sections of (a) spherical and (b) cylindrical particles loaded in diametral compression, illustrating locations and directions in the axial plane. Particle orientations as in Figure 1.2; both sections are discs.
Figure 1.5 shows schematic cross sections perpendicular to the loading axis through the centers of the particles loaded in diametral compression, illustrating important lines and planes. These are transverse sections, containing transverse directions indicated by arrows. The perimeter of the transverse cross section through the spherical particle is a circle and that for the cylinder is a rectangle. For the spherical particle, this perimeter is the equator; see Figure 1.5a (shown in bold). For the cylindrical particle, lines on the perimeter perpendicular to the axial and transverse directions are equatorial lines (Figure 1.5b) (bold). For both sphere and cylinder, the sections containing the equators are the labeled equatorial planes. Sections perpendicular to transverse directions that contain the axis are meridional planes, shown and labeled in the schematic diagrams of Figure 1.6. The perimeter of a meridional plane in the spherical particle is a circle and that in the cylinder is a rectangle. For the spherical particle, this perimeter is the meridian (Figure 1.6a) (bold), and spherical meridional planes are identical in shape to axial planes. For the cylindrical particle there is only one meridional plane, perpendicular to the cylinder face passing through the axis (Figure 1.6b). Meridional directions are indicated by arrows.
Figure 1.5 Schematic diagrams of the transverse sections of (a) spherical and (b) cylindrical particles loaded in diametral compression, illustrating locations and directions in the equatorial plane. Particle orientations as in Figure 1.2. Particle equators are indicated as bold lines and are (a) a circle and (b) a rectangle for the spherical and cylindrical particle, respectively.
Figure 1.6 Schematic diagrams of the meridional sections of (a) spherical and (b) cylindrical particles loaded in diametral compression, illustrating locations and directions in the meridional plane. Particle orientations as in Figure 1.2. Particle meridians are indicated as bold lines and are (a) a circle and (b) a rectangle for the spherical and cylindrical particle, respectively.
These geometrical specifications, Figures 1.4–1.6, are of two-fold importance: (1) Diametral compression is the most common form of loading for particles in testing and application. Individual tests and applications vary in detail, but the overall diametral compression form is maintained. A clear geometric description thus enables comparison of particle behavior in different tests and applications. (2) Particle failure in diametral compression is characterized by reversible deformation (Timoshenko and Goodier 1970; Sadd 2009), irreversible deformation (Mase 1979), and fracture (the formation of new surfaces, Lawn 1993). The deformation modes and fracture exhibit different geometrical characteristics. A clear geometric description of particle loading thus enables identification and description of the material constitutive laws governing particle failure. Points (1) and (2), test variations and particle failure behavior, respectively, are discussed in the following, drawing on the geometrical considerations.
Figure 1.7 shows schematic cross-sections of a range of particle mechanical tests, illustrating variations on the diametral compression form for spherical and cylindrical particles. Particles are shown in light shading, test fixture elements are shown hatched, and velocities are shown as solid lines with arrow heads; unless otherwise stated, this scheme is used throughout the book. The loading variation implemented most frequently is shown in Figure 1.7a, in which an upper platen is displaced by a test machine toward a particle resting on a fixed lower platen. The test machine generates the requisite force to maintain an imposed displacement rate and compress the particle. Typically, the imposed displacement rate is slow, such that the test is regarded as quasi static. Examples of such tests extend back many decades and continue into recent times (Kapur and Furstenau, 1967; Hooper 1971; Wynnyckyj 1985; Vallet and Charmet 1995; Antonyuk et al. 2005; Wang et al. 2019).
Figure 1.7 Schematic cross-section diagrams of commonly implemented mechanical tests of particles leading to diametral compression: (a) quasi-static loading between displaced platens—the most common; (b) quasi-static loading between a displaced probe and a platen; (c) crushing between two counter-rotating rollers; (d) impact by a dropped spherical weight onto a fixed particle; (e) impact against a platen of a moving particle impelled by a gas gun; (f) impact against a platen of a moving particle swung by a pendulum; (g) vibration by acoustic waves generated in a fluid; (h) impact by a striker impelled by a gas gun of a fixed particle; (i) quasi-static loading between two perpendicular sets of displaced platens leading to biaxial loading.
Other test variations include: (b) concentrated loading (Hiramatsu and Oka 1966), (c) passing particles between rotating rollers (Brecker 1974; Huang et al. 1993, 1995), (d) instrumented ball drop (King and Bourgeois 1993; Tavares and King 1998; Tavares 1999; Tavares et al. 2018), (e) impact on hard surfaces of moving particles impelled by a gas gun (Salman et al. 2002, 2003) or (f) by gravity controlled by a pendulum (Kantak et al. 2005), (g) ultrasonic fragmentation (Knoop et al. 2016), (h) ballistic impact of moving surfaces onto particles (Gustafsson et al. 2017; Xiao et al. 2019b), and (i) biaxial loading (Satone et al. 2017). Test configurations Figures 1.7a, 1.7b, and 1.7i are quasi-static and displacement controlled; configuration Figure 1.7c is rapid and displacement controlled; configurations Figures 1.7d, 1.7e, 1.7f, and 1.7h are regarded as impact-based and are rapid—d and h utilize deceleration of moving test fixture masses to generate compressive forces on stationary particles and e and f utilize deceleration of moving particles by stationary platens to generate compressive forces; configuration Figure 1.7g applies cyclic compressive forces to particles via acoustic waves transmitted through fluids. Variations on some of these configurations include: particles freely dropped onto a platen (Kantak and Davis 2004; Kantak et al. 2005) rather than impelled; contact is made with another single particle, rather than an element of the test fixture, leading to particle-particle collisions (Foerster et al. 1994; Chandramohan and Powell 2005); and contact is made between large probes or spheres and a bed of fine particles, leading to multi-particle contacts (Studman and Field 1984; Huang et al. 1998). A brief compilation of test configurations is given by Antonyuk et al. (2010).
The diversity of test configurations in Figure 1.7 reflects a diversity of mechanical measurement goals: Figures 1.7a and 1.7b enable careful observation of deformation and fracture during loading and failure of large scale particle components; Figures 1.7c, 1.7d, 1.7e, and 1.7g enable high-throughput testing of large numbers of small particle components; Figures 1.7e, 1.7f and 1.7h enable measurement of rate effects in deformation and fracture; and Figures 1.7e and 1.7j enable measurement of mixed-mode loading effects. The various test configurations, however, all generate a diametral compression loading geometry, whether the loading axis is vertical (e.g. Figure 1.7a), horizontal (e.g. Figure 1.7h), or inclined (e.g. Figure 1.7e). In addition, the various test configurations all seek to replicate specific loading cycles imposed on particles as structural components in application: By overlying weights, including those of other particles, by crushing jaws or rollers, or by colliding surfaces, including those from other particles.
Here and throughout, the terms load and loading are used as broad descriptors to refer to any configuration that generates applied forces on components, here in particular on particles. Loading may arise from direct application of a force, as in an overlying mass giving rise to a gravitational weight (a “load”). Loading may arise from an imposed displacement, in which case the force arises indirectly as a reaction to the displacement, as in platen, jaw, or roller configurations. Loading may also arise from an imposed change in displacement rate, in which case the force arises indirectly as a consequence of deceleration of a moving mass, as in impact configurations. A loading cycle is thus composed of load-unload segments, separated by the peak force generated in the cycle.
The force required to cause a structural component to fail in its load-bearing function is the component failure force (sometimes referred to as the supportable force, Ashby 1999). Thus, from performance and reliability perspectives, loading cycles applied to structural components can be characterized qualitatively by simple comparison of the peak force in a cycle to the failure force of a component. In this scheme, peak forces are conveniently classified into three domains: (i) small peak forces that do not exceed the component failure force; (ii) moderate peak forces that may exceed the component failure force; and (iii) large peak forces that definitely exceed the component failure force. Obviously, component design seeks to avoid domain (iii), often by identifying material responses in domains (i) and (ii) that lead to component failure. Observations of fracture and deformation during particle loading, also extending back many years, map onto this three domain classification: (i) At small peak forces, localized regions of irreversible deformation form and particles remain intact. (ii) At moderate peak forces, the regions of irreversible deformation expand and one or two fracture surfaces form. The fracture surfaces do not extend through particles and particles are thus partially fractured or cracked. (iii) At large peak forces, the regions of irreversible deformation continue to expand, significantly altering the shape of particles, and multiple fracture surfaces form. The fracture surfaces extend through particles and particles are thus completely fractured or cracked. These observations are now discussed in more detail, using the three domains of peak force as a framework. Attention is focused initially on the symmetric platen-particle-platen quasi-static loading configuration (Figure 1.7a), as this is the most common. The symmetric configuration provides a basis for consideration of the less common asymmetric particle-platen impact loading configuration (Figure 1.7e).
(i) Following a load-unload cycle to a small peak force, flat areas are observed at particle poles, indicative of localized irreversible deformation. The deformation flats are parallel to the equatorial plane, centered on particle axes, and, for spherical and cylindrical particles respectively, are circular or rectangular in outline. Inspection of the flats suggests that they are the surface faces of sub-surface regions of compacted material. Schematic diagrams of the flats formed at the poles of spherical and cylindrical particles are shown in Figure 1.8. An optical image of the flat formed at the pole of a spherical plaster particle is shown in Figure 1.9a. A similar flat is shown in an optical image of a sodium benzoate (food additive) particle (Antonyuk et al. 2005).
Figure 1.8 Schematic diagrams of the irreversible deformation flats formed on the poles of particles loaded between displaced platens in diametral compression as in Figure 1.7a. (a) Spherical and (b) cylindrical particle.
Figure 1.9 (a) Image of irreversible deformation flat formed on the pole of a spherical plaster particle loaded between displaced platens in diametral compression. (b) Image of irreversible deformation flat and pre-failure meridional cracks formed in a cylindrical plaster particle loaded between displaced platens in diametral compression. Source: Robert F. Cook.
(ii) Deformation and fracture features formed during or following a load-unload cycle to a moderate peak force have been observed frequently. In situ observations have shown flats forming and expanding during loading at the poles of a variety of particles: epoxy cylinders and irregular plates (Hiramatsu and Oka 1966), plaster spheres and cylinders (Tsoungui et al. 1999), spherical fertilizer granules (Salman et al. 2003), plaster and cellulose cylinders (Procopio 2003), a sodium benzoate sphere (Pitchumani et al. 2004), a zeolite catalyst sphere (Antonyuk et al. 2005), and plaster cylinders (Wong and Jong 2014). Many observations following load-unload cycles to moderate peak forces also show flats: at the poles of a spherical limestone pellet (Kapur and Furstenau 1967), a plaster sphere (Tsongui et al. 1999), a polystyrene sphere (Schönert 2004), plaster spheres (Wu et al. 2004), metal coated polymer spheres (Zhang et al. 2007), concrete spheres (Khanal et al. 2008), cellulose cylinders and rounded tablets (Shang et al. 2013a, 2013b), silica spheres (Paul et al. 2014, 2015), and ceramic spheres (Pejchal et al. 2018). An obvious feature in these moderate force observations is the appearance of cracks such that the particles are partially fractured. The in situ observations show that the cracks form and extend during loading (Tsongui et al. 1999; Procopio 2003; Salman et al. 2003; Pitchumani et al. 2004; Antonyuk et al. 2005; Wong and Jong 2013). The cracks are visible as surface traces of crack opening displacement along particle meridians, frequently as traces from one polar flat through the particle equator to the other polar flat. The particles are only partially fractured as these meridional cracks do not extend completely through the particle. Schematic diagrams of meridional cracks and polar flats in spherical and cylindrical particles, similar to an earlier diagram (Kapur and Furstenau 1967), are shown in Figure 1.10. An optical image of the pre-failure flat and cracks formed in a cylindrical plaster particle is shown in Figure 1.9b.
Figure 1.10 Schematic diagrams of irreversible deformation flats and meridional cracks generated in (a) a spherical and (b) a cylindrical particle loaded in diametral compression. The surface traces of the cracks are shown in white and the internal crack fronts are shown as dashed lines. Note that as the cracks do not extend through the particles, the particles are only partially fractured.
(iii) By definition, loading a structural component to a large force leads to component failure and the loading cycle is truncated at the failure force of the component. For example, rods or bars in tension often fail by fracture, columns in compression often fail by buckling. The predominant feature of loading a particle to a large peak force is failure of the particle by complete fracture. In a large number of particle failure observations, fracture on meridional planes extends to the edges of a particle in the absence of observable irreversible deformation: irregular rock pieces (Hiramatsu and Oka 1966), marble cylinders (Jaeger 1967), iron ore spheres (Wynnyckyj 1985), glass and mineral spheres (Ryu and Saito 1991), glass and sapphire spheres (Shipway and Hutchings 1993), ceramic cylinders with milled flats (Fahad 1996), glass cylinders (Schönert 2004), ceramic spheres (Luscher et al. 2007), mannitol (a drug) spheres (Adi et al. 2011), rock cylinders (Erarslan and Williams 2012; Erarslan et al. 2012), iron ore pellets (Gustafsson et al. 2013), irregular rock particles (Wang and Coop 2016), ceramic spheres (Satone et al. 2017), and glass spheres and irregular rock pieces (Silva et al. 2019). In many of these cases, fracture is on a single meridional plane and a single crack propagates through the particle to generate two separated hemispherical fragments from spherical particles (e.g. Ryu and Saito 1991; Luscher et al. 2007; Silva et al. 2019), or two semicircular fragments from cylindrical particles (Fahad 1996). Figure 1.11 shows schematic diagrams of separated fragments from particles that failed in this simple manner. (Fragments are pieces generated by fracture—they can be large or small fractions of components, e.g. Cook 2021.) Similar failure by simple meridional fracture was observed in early experiments on short, dried, plaster columns (Fairhurst 1960). In some cases, particle fragments remain attached to each other after particle failure (e.g. Wynnyckyj 1985; Gustafsson et al. 2013) (sphere) or (Jaeger 1967; Erarslan and Williams 2012) (cylinder) such that the particle retains its initial shape. In some cases, simple fragmentation is accompanied by minor irreversible deformation at the poles (e.g. Wynnyckyj 1985; Ryu and Saito 1991; Wu et al. 2004). An example is shown in Figure 1.9b.
Figure 1.11 Schematic diagrams of complete fracture by a single meridional crack in (a) a spherical and (b) a cylindrical particle loaded in diametral compression. The cracks faces or fracture surfaces are shown in white. The cracks extend through particles leading to the generation of two fragments.
In the more striking cases of particle failure, fracture is on multiple meridional planes, and multiple cracks propagate through the particle to generate many separated fragments (e.g. Ryu and Saito 1991; Zhang et al. 2007). This failure mode is restricted to spherical particles. Failure is preceded by significant irreversible deformation such that large flats are formed at the poles and the incipient failure geometry is thus a barrel. The fragments generated on failure are of three typical forms. The largest are wedge shaped: near spherical wedges bound by meridional planes that terminate at the perimeter of the flat, typically three to six in number. The second largest are cone shaped: axially oriented deformed spherical sectors that terminate on the faces of the flats, typically two. The smallest are very small randomly shaped and oriented fragments that are extremely numerous. The largest fragments originate from the exterior of the particle, the smallest fragments originate from the interior of the particle. Figure 1.12 shows a schematic diagram of the separated fragments of a spherical particle that failed in this multiple fracture manner. Similar patterns are shown in images of a fragmented glass particle (Ryu and Saito 1991) and a fragmented sand particle broken in constrained impact loading, Figure 1.7h, (Xiao et al. 2019a). X-Ray tomography images of failed particles in an aggregate array suggest that both single and multiple fracture modes can be operative simultaneously (Zhao et al. 2020).
Figure 1.12 Schematic exploded view diagram of complete fracture by multiple meridional cracks in a spherical particle loaded in diametral compression. The cracks faces or fracture surfaces are shown in white. The cracks extend through the particle leading to the generation of many fragments, including wedges and truncated cones topped by irreversible deformation flats (lower cone not visible).
Optical images of the large force failure sequence of a spherical plaster particle are shown in Figure 1.13. In this case, loading by platen displacement was halted and reversed immediately after the formation of visible surface traces of meridional cracks (Figure 1.13
