On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass [1st ed. 2020] 978-3-658-28205-9, 978-3-658-28206-6

Table of contents :
Front Matter ....Pages i-xxii
Introduction (Navid Pour-Moghaddam)....Pages 1-6
Glass Properties and Refinement Processes (Navid Pour-Moghaddam)....Pages 7-15
Numerical Simulation of the Thermal Tempering Process (Navid Pour-Moghaddam)....Pages 17-58
Experimental Investigations into the Fragmentation of Tempered Glass (Navid Pour-Moghaddam)....Pages 59-119
Prediction of 2D Macro-Scale Fragmentation of Tempered Glass (Navid Pour-Moghaddam)....Pages 121-181
Investigations into the Phenomenon of Crack Branching (Navid Pour-Moghaddam)....Pages 183-205
Summary and Outlook (Navid Pour-Moghaddam)....Pages 207-209
Back Matter ....Pages 211-257

Citation preview

Mechanik, Werkstoffe und Konstruktion im Bauwesen | Band 54

Navid Pour-Moghaddam

On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass

Mechanik, Werkstoffe und Konstruktion im Bauwesen Band 54 Reihe herausgegeben von Ulrich Knaack, Darmstadt, Deutschland Jens Schneider, Darmstadt, Deutschland Johann-Dietrich Wörner, Darmstadt, Deutschland Stefan Kolling, Gießen, Deutschland

Institutsreihe zu Fortschritten bei Mechanik, Werkstoffen, Konstruktionen, Gebäudehüllen und Tragwerken. Das Institut für Statik und Konstruktion der TU Darmstadt sowie das Ins­ titut für Mechanik und Materialforschung der TH Mittelhessen in Gießen bündeln die For­ schungs- und Lehraktivitäten in den Bereichen Mechanik, Werkstoffe im Bauwesen, Statik und Dynamik, Glasbau und Fassadentechnik, um einheitliche Grundlagen für werkstoffgerechtes Entwerfen und Konstruieren zu erreichen. Die Institute sind national und international sehr gut vernetzt und kooperieren bei grundlegenden theoretischen Arbeiten und ­angewandten Forschungsprojekten mit Partnern aus Wissenschaft, Industrie und Verwaltung. Die Forschungsaktivitäten finden sich im gesamten Ingenieurbereich wieder. Sie umfassen die Modellierung von Tragstrukturen zur Erfassung des statischen und dynamischen Verhaltens, die mechanische Modellierung und Computersimulation des Deformations-, Schädigungsund Versagensverhaltens von Werkstoffen, Bauteilen und Tragstrukturen, die Entwicklung neuer Materialien, Produktionsverfahren und Gebäudetechnologien sowie deren Anwendung im Bauwesen unter Berücksichtigung sicherheitstheoretischer Überlegungen und der Energieeffizienz, konstruktive Aspekte des Umweltschutzes sowie numerische Simulationen von komplexen Stoßvorgängen und Kontaktproblemen in Statik und Dynamik.

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Navid Pour-Moghaddam

On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass

Navid Pour-Moghaddam Konstruktiver Ingenieurbau Schüßler-Plan Frankfurt am Main, Germany Vom Fachbereich 13 – Bau- und Umweltingenieurwissenschaften der Technischen Universität Darmstadt zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Dissertation von Dipl.-Ing. Navid Pour-Moghaddam aus Teheran 1. Gutachten: Prof. Dr.-Ing. Jens Schneider 2. Gutachten: Assoc. Prof. Jens Henrik Nielsen Tag der Einreichung: 15.05.2019 Tag der mündlichen Prüfung: 15.08.2019 Darmstadt 2019 D17

ISSN 2512-3238 ISSN 2512-3246  (electronic) Mechanik, Werkstoffe und Konstruktion im Bauwesen ISBN 978-3-658-28205-9 ISBN 978-3-658-28206-6  (eBook) https://doi.org/10.1007/978-3-658-28206-6 Springer Vieweg © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Vieweg imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Danksagung Die vorliegende Dissertation entstand während meiner Tätigkeit als wissenschaftlicher Mitareiter in den Jahren 2014 bis 2019 am Institut für Statik und Konstruktion der Technischen Universität Darmstadt. An dieser Stelle möchte ich allen Personen, die mich in dieser Zeit unterstützt und begleitet haben und damit zum Gelingen dieser Arbeit beigetragen haben meinen herzlichen Dank aussprechen. Insbesondere danke ich Herrn Prof. Dr.-Ing. Jens Schneider für das entgegen gebrachte Vertrauen, womit ich die großartige Möglichkeit bekam an seinem Lehrstuhl zu promovieren. Ich möchte ihm auch für seine wohlwollende Förderung, Unterstützung und hervorragende Betreuung meiner Arbeit sowie für seine Ideen und Anregungen in unzähligen und stets fruchtbaren Diskussionen danken. Ebenso danke ich herzlich Herrn Assoc. Prof. Jens Henrik Nielsen von der Technischen Universität Dänemark (DTU) für seine Bereitschaft zur Übernahme des Korreferates und die stets freundliche Unterstützung sowie die daraus resultierenden zahlreichen fachlichen Diskussionen. Ich möchte allen Kollegen und Mitarbeitern des Instituts für ihre mentale Unterstützung und ihre ständige Hilfsbereitschaft danken. Ein großer Dank gilt auch den Studierenden, die mich im Rahmen von Studienarbeiten bei meiner Forschungsarbeit unterstützt haben. Ein ganz besonderer Dank gilt meinen lieben Eltern Mina und Alireza, die mir stets zur Seite standen und mich auf meinem Weg motiviert und mir diese Ausbildung überhaupt ermöglicht haben. Abschließend möchte ich meiner lieben Frau Golnaz danken, die mich stets mit ihrer Lebensfreude, Geduld und Liebe uneingeschränkt unterstützt und motiviert hat.

Darmstadt, im Mai 2019

Navid Pour-Moghaddam

Abstract Glass is a brittle material at room temperature and deformations are linear elastic until fracture. Knowledge of the fracture behaviour of glass and the morphological aspects of the fracture pattern helps, for example, to realistically evaluate and assess the postfracture behaviour of laminated glass as well as to initiate and specify the required setup parameters for the numerical simulation of the fracture and post-fracture behaviour. In the present work, different aspects of the fracture behaviour and characteristics of the fracture pattern morphology of fragmented thermally tempered soda-lime-silica glass were investigated. The state and level of residual stress respectively the stored elastic strain energy in thermally tempered glass influence the fracture pattern and thus the fragmentation behaviour. Fragmentation analyses to determine correlations between the residual stress and the morphological properties of the fracture pattern, e.g. the number of fragments within an observation field, were carried out by means of fracture tests on glass specimens of different thicknesses with different heat treatments, i.e different residual stress levels. The fragment and fracture surface area as physical quantities of the fragmentation were determined using Computer Tomography (CT). It was, i.a., also shown that there is a correlation between the elastic strain energy and the roughness of the fracture surface. The relationship between elastic strain energy and fracture pattern morphology of thermally tempered glass was shown theoretically based on the energy criterion in Linear Elastic Fracture Mechanics (LEFM) related to the initial elastic strain energy before fragmentation and the remaining elastic strain energy in the fragments after fragmentation. Furthermore, a machine learning inspired approach for the prediction of 2D macro-scale fragmentation of thermally tempered glass based on fracture mechanics considerations and statistical analysis of the fracture pattern morphology was deduced and applied. A method called ’Bayesian Reconstruction and Prediction of Glass Fracture Patterns (BREAK)’, where the fracture pattern of thermally tempered glass is predicted and simulated by Voronoi tessellation of point patterns based on Bayesian spatial point statistics fed with energy conditions in LEFM. The fracture behaviour of glass was characterized with regard to the evolution states of a propagating crack. The stress intensity factor and the associated energy release rate at crack branching were determined based on a theoretical estimate of the initial elastic strain energy density relaxation during fragmentation using Griffith’s criterion of fracture

viii

Abstract

in LEFM and by fracture surface examinations of specimens from tensile tests on onesided pre-damaged float glass sheets.

Zusammenfassung Glas ist bei Raumtemperatur ein spröder Werkstoff und die Verformungen sind bis zum Bruch linear elastisch. Die Kenntnis des Bruchverhaltens von Glas und der morphologischen Aspekte des Bruchbildes hilft beispielsweise, das Nachbruchverhalten von Verbundglas realistisch zu bewerten sowie die erforderlichen Rüstparameter für die numerische Simulation des Bruch- und Nachbruchverhaltens einzuleiten und zu spezifizieren. In der vorliegenden Arbeit wurden verschiedene Aspekte des Bruchverhaltens und der Eigenschaften der Bruchbildmorphologie von fragmentiertem, thermisch vorgespanntem KalkNatron-Silikatglas untersucht. Der Zustand und das Niveau der Eigenspannung bzw. die daraus resultierende gespeicherte elastische Dehnungsenergie in thermisch vorgespanntem Glas beeinflussen das Bruchbild und damit das Fragmentierungsverhalten. Fragmentierungsanalysen zur Ermittlung von Zusammenhängen zwischen der Eigenspannung und den morphologischen Eigenschaften des Bruchbildes, z.B. der Anzahl der Fragmente innerhalb eines Beobachtungsfeldes, wurden mittels Bruchversuchen an Glasproben unterschiedlicher Dicke mit unterschiedlichen Wärmebehandlungen, d.h. unterschiedlichen Eigenspannungen, durchgeführt. Die Fragment- und Bruchfläche als physikalische Größen der Fragmentierung wurden mittels Computertomographie (CT) bestimmt. Es wurde unter anderem auch gezeigt, dass es einen Zusammenhang zwischen der elastischen Dehnungsenergie und der Rauheit der Bruchfläche existiert. Der Zusammenhang zwischen elastischer Dehnungsenergie und Bruchbildmorphologie von thermisch vorgespanntem Glas wurde theoretisch basierend auf dem Energiekriterium in der Linear-Elastischen Bruchmechanik (LEBM) in Bezug auf die elastische Dehnungsenergie vor der Fragmentierung und die verbleibende elastische Dehnungsenergie in den Fragmenten nach der Fragmentierung dargestellt. Darüber hinaus wurde ein vom maschinellen Lernen inspirierter Ansatz für die Vorhersage der 2D-Makrofragmentierung von thermisch vorgespanntem Glas basierend auf bruchmechanischen Überlegungen und statistischer Analyse der Bruchbildmorphologie abgeleitet und angewendet. Ein Verfahren namens ’Bayes’sche Rekonstruktion und Vorhersage von Glasbruchbildern (BREAK)’, bei dem das Bruchbild von thermisch vorgespanntem Glas durch Voronoi-Tessellierung von Punktmustern basierend auf Bayes’schen räumlichen Punktstatistiken, die mit Energiebedingungen in LEBM gespeist werden, vorhergesagt und simuliert wird.

x

Zusammenfassung

Das Bruchverhalten von Glas wurde im Hinblick auf die Entwicklungsstufen eines sich ausbreitenden Risses charakterisiert. Der Spannungsintensitätsfaktor und die damit verbundene Energiefreisetzungsrate bei der Rissverzweigung wurden basierend auf einer theoretischen Schätzung der anfänglichen Relaxation der elastischen Dehnungsenergiedichte während der Fragmentierung mit dem Bruchkriterium von Griffith in LEBM und durch Bruchflächenuntersuchungen an Proben aus Zugversuchen an einseitig vorgeschädigten Floatglasscheiben ermittelt.

Résumé Le verre est un matériau fragile à température ambiante et les déformations sont linéaires et élastiques jusqu’à la rupture. La connaissance du comportement à la rupture du verre et des aspects morphologiques du modèle de fracture permet, par exemple, d’évaluer de manière réaliste le comportement après rupture du verre feuilleté ainsi que d’initier et de spécifier les paramètres de mise en place nécessaires pour la simulation numérique du comportement à la rupture et après rupture. Dans le cadre de ce travail, différents aspects du comportement de la fracture et les caractéristiques de la morphologie des fractures du verre silico-sodo-calcique trempé thermiquement fragmenté ont été étudiés. L’état et le niveau de contrainte résiduelle, respectivement l’énergie de déformation élastique stockée dans le verre trempé thermiquement, influencent le modèle de rupture et donc le comportement à la fragmentation. Des analyses de fragmentation pour déterminer les corrélations entre la contrainte résiduelle et les propriétés morphologiques du modèle de fracture, par exemple le nombre de fragments dans un champ d’observation, ont été effectuées au moyen d’essais de fracture sur des échantillons de verre de différentes épaisseurs avec différents traitements thermiques, c’est-à-dire différents niveaux de contrainte résiduelle. Le fragment et la surface de fracture ont été déterminés par tomodensitométrie (TDM) en tant que grandeurs physiques de la fragmentation. Il a également été démontré, entre autres, qu’il existe une corrélation entre l’énergie de déformation élastique et la rugosité de la surface de rupture. La relation entre l’énergie de déformation élastique et la morphologie du motif de rupture du verre trempé thermiquement a été démontrée théoriquement sur la base du critère d’énergie de la mécanique linéaire élastique de la rupture (MLER) relatif à l’énergie de déformation élastique initiale avant fragmentation et l’énergie de déformation élastique restante dans les fragments après fragmentation. De plus, une approche inspirée de l’apprentissage machine pour la prédiction de la fragmentation à l’échelle macro 2D du verre trempé thermiquement, basée sur des considérations de mécanique de la fracture et l’analyse statistique de la morphologie du modèle de fracture a été déduite et appliquée. Une méthode appelée ’Reconstruction bayésienne et prévision des modèles de fractures du verre (BREAK)’, où le modèle de fracture du verre trempé thermiquement est prédit et simulé par tessellation de Voronoi des modèles de points basés sur des statistiques bayésiennes de points spatiales alimentées par les conditions énergétiques dans MLER. Le comportement à la rupture du verre a été caractérisé par rapport aux états d’évolution d’une fissure de propagation. Le facteur d’intensité de contrainte et le taux de libération

xii

Résumé

d’énergie connexe à la ramification des fissures ont été déterminés à partir d’une estimation théorique de la relaxation initiale de la densité d’énergie de déformation élastique pendant la fragmentation en utilisant le critère de rupture de Griffith du MLER et en examinant la surface de rupture d’échantillons prélevés sur une face des plaques en verre float endommagé par traction.

Contents Glossaries 1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Topics and Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii 1 1 4 5

2 Glass Properties and Refinement Processes 7 2.1 Definition and Structure of Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Properties of Soda-Lime-Silica Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Fracture Mechanical Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Tempering Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Numerical Simulation of the Thermal Tempering Process 3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Thermo-Mechanical Behaviour of Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Viscoelastic Material Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Thermorheologically Simple Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Structural Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical Simulation of 1D Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Stress Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Results of Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Convergence Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Model Parameter Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 3D FE-Modelling of the Thermal Tempering Process . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Heat Transfer Coefficients for Glass Plates with Holes . . . . . . . . . . . . . . 3.5.2 FE-Model and Element Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Definition of Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Convergence Study of Mesh Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 19 20 20 21 23 24 25 32 36 39 45 47 48 49 53 55 56

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4 Experimental Investigations into the Fragmentation of Tempered Glass 4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Strain Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Heat Treatment of Glass Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Specimen Preparation for the Thermal Tempering . . . . . . . . . . . . . . . . . . . 4.3.2 Cooling Air Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Engine Power of the Tempering Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Residual Stress Measurements and Approach Assessment . . . . . . . . . . 4.4 Fracture Tests and Fragmentation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Fragmentation and Fracture Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Elastic Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Fragmentation Analysis Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Fragment Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Particle Weight and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fractographic Examinations of the Fracture Surface. . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 The Fracture Surface in Tempered Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Analysis of the Fracture Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Computer Tomographic (CT) Examination and Determination of the Fragment and Fracture Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 CT System and 3D Reconstruction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Model Processing for Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Results of the Fragment and Fracture Surface Area . . . . . . . . . . . . . . . . . . 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass 5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Geometrical Evolution of a 2D Voronoi Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Strain Energy Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Methodology and Fracture Mechanics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Basic Modelling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Fragment Size Parameter δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Fracture Intensity Parameter λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Comparison of the Prediction Method with Literature . . . . . . . . . . . . . . . 5.5 Mathematical Foundations of Point Processes and Point Pattern Analysis . . 5.5.1 Spatial Point Process and Spatial Point Pattern. . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Exploratory Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Geometrical Properties of Random 2D Voronoi Tessellations . . . . . . . . . . . . . . . . 5.6.1 Random Tessellations over Spatial Point Patterns and Statistics of their Geometrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 62 65 66 69 72 75 77 78 86 87 89 95 100 100 103 106 106 110 112 117 121 121 125 126 127 127 130 137 141 145 145 152 158 158

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5.6.2 5.6.3 5.6.4

5.7

5.8

Point Pattern Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the Voronoi Cell Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the Statistical Distributions of Voronoi Cell Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology of Fracture Pattern Recognition and Generation (Method BREAK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Fracture Image Recording and Morphological Fracture Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Spatial Statistics Model Calibration and Evaluation of Candidate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Stochastic Fracture Pattern Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 163 165 168 169 171 176 179

6 Investigations into the Phenomenon of Crack Branching 6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fracture Process and Associated Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Characterization of Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Tensile Test on One-Sided Pre-Damaged Float Glass. . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Preparation of the Specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Experimental System and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Results of Crack Branching Behaviour and Fracture Velocity . . . . . . . 6.3.4 Branching Stress Intensity Factor Based on Fracture Surface Examinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Summary of Results and Comparison with Literature . . . . . . . . . . . . . . . . . . . . . . . .

183 183 184 184 188 189 191 193

7 Summary and Outlook

207

References and Regulations

211

Appendix A.1 Properties of Soda-Lime-Silica Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 1D Forced Convection: MATLAB Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 1D Forced Convection: Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Fracture Tests: Results of the Fragment Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Fracture Tests: Results of Particle Weight, Volume and Base Area . . . . . . . . . . A.6 Fracture Tests: CT Results of Fragment and Fracture Surface Area . . . . . . . . . A.7 Statistical Distributions of Voronoi Cell Characteristics . . . . . . . . . . . . . . . . . . . . . . A.8 Stochastic Fracture Pattern Simulation of Thermally Tempered Glass via Method BREAK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Tensile Tests: Parameters and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 223 225 231 237 240 243 246

199 203

250 256

Glossaries

Abbreviations AM ARN ASN

Additive Manufacturing Array of round nozzles Array of slotted nozzles

BREAK

Bayesian Reconstruction and Prediction of Glass Fracture Patterns

CDF CSR CT

Cumulative Distribution Function Complete Spatial Randomness Computer Tomography

EP

Engine power of the tempering oven

FDM FEA FEM

Finite Difference Method, Fused Deposit Modelling Finite Element Analysis Finite Element Method

HCP HPP

Hexagonal Close Packing Homogeneous Poisson process

LEFM

Linear Elastic Fracture Mechanics

MCMC MHCP

Markov Chain Monte Carlo Matérn Hard-core process

xviii

Glossaries

PDF PPDE PVT

Probability Density Function Parametrized partial differential equaions Poisson-Voronoi Tessellation

RBM

Reduced Basis Method

SCALP SP

Scattered Light Polariscope Strauss process

Symbols 0 1 ∞ (x, y, z) (r, θ )

Initial state Fragmented state Final state, infinity, environment Cartesian coordinates Cylinder coordinates (radius and polar angle)

AC Afr Ah Aη AS a aN α αT

Voronoi cell area Fracture surface of a fragment Mirror constant Energy relaxation zone of a fragment Fragment surface area Thermal diffusivity Notch length Uniformity parameter / Regularity of the Voronoi cell structure Thermal expansion coefficient

B β

The 2D domain of the glass fracture pattern Scaling factor for heat transfer coefficient identification, Intensity of the original SP

Cr cp cR cS

Weight coefficient for structural relaxation Specific heat capacity Rayleigh wave speed Shear wave speed

D

Inner diameter of a nozzle

Glossaries

δi j δ δHCP

Kronecker’s delta Fragment Size Parameter Uniform equidistance between two adjacent nuclei in HCP

E ei j εii εi j ε th

Young’s modulus Deviatoric strain tensor trace of the strain tensor strain tensor Thermal strain

F(r f ) fn (x)

Empty-space function Probability density of a point pattern x

G

Shear modulus, Energy release rate Energy release rate at crack branching Critical energy release rate Nearest-neighbour function Gravity Fracture surface energy Specific fracture surface energy, Interaction parameter of SP

Gb Gc G(rg ) g Γ γ

h hj h(r) ηˆ η

Activation energy, Distance between the nozzles and the glass surface Heat transfer coefficient Distance between the pressure chamber and the air jets Pairwise interaction function Viscosity Energy relaxation factor

θ

Point process parameters

K

Bulk modulus, Kernel function Stress intensity factor at crack branching

H

Kb

xix

xx

Glossaries

KI KIc K(r) k κeq κnoneq

Stress intensity factor (Mode I) Critical stress intensity factor Ripley’s K-function Thermal conductivity Factor for the equidistant discretization of the thickness Factor for the non-equidistant discretization of the thickness

L(r) LT λ λr

L-function to transform Ripley’s K-function Distance between the nozzles Fracture / Point Process Intensity Parameter Relaxation time for structural relaxation

Mp

Response function of the material property

ND Nn Nu nB nV ν

Number of fragments in the observation field with the length of D Number of nodes along the thickness Nusselt number Number of branching nodes Number of vertices of a Voronoi cell Poisson’s ratio

ξ

Reduced time

P PC Pr P Π

Air pressure Voronoi cell perimeter Prandtl number Probability of obtaining n points xi , i = 1...n in the domain B Potential energy

q

Heat flux

Re Rg rB rf

Reynolds number Universal gas constant Crack length at the first branching Location-event distance

Glossaries

xxi

rg rh rHC ρ ρˆ ρE

Event-event distance Distance from the fracture origin to the mist/hackle boundary Minimum Euclidean distance of points (Hard-core distance) (≡ δHC ) Glass density Roughness of the fracture surface FE mesh density

σii σi j si j σf σm σs σ (z)

Trace of the stress tensor stress tensor Deviatoric stress tensor Fracture stress Mid-plane stress Surface stress Stress function along the z-axis of the glass plate

T Tf Tf r Tg t tc th τ τ1q , τ2p

Temperature Fictive temperature Partial fictive temperature Glass transition temperature Time, Glass plate thickness Thickness of the compression layer Thickness of the mist and velocity hackle layer Shear stress Relaxation times for stress relaxation

UT U UD Uη U0 U1 UR,Rem UR,Rel

Total elastic strain energy [J] Elastic strain energy per unit surface area [J/m2 ] Elastic strain energy density [J/m3 ] Elastic strain energy in the relaxation zone Aη Initial elastic strain energy Remaining elastic strain energy Relative remaining elastic strain energy Relative released elastic strain energy

υ

kinematic viscosity

xxii

Glossaries

VT (x) vfr

Voronoi tessellation over the spatial point pattern x Fracture velocity

φ ϕB ϕN φN

Shift function, Angle of the first branching Notch oblique angle Diameter of the notch head

W w w1q , w2p

Width Cooling air velocity Weight coefficients for stress relaxation

X x

Spatial point process Spatial point pattern xi , i = 1...n

Zn

Normalizing constant in SP

1 Introduction 1.1 Motivation In recent decades, the use of glass as a construction element of the structural glazing and as a building material in various engineering sectors such as the automotive sector, the shipping industry and solar technology has increased significantly. This fact has substantially increased the demand for structural and architectural flexibility and versatility of glass components. Glass components are applied as architectural and structural elements in buildings, see Figure 1.1. However, glass is a brittle material with a linear elastic material behaviour at room temperature until fracture and lacks the capability of stressredistribution in contrast to more common building materials. The theoretical strength of soda-lime-silica glass is over 10 000 MPa (L E B OURHIS, 2008). However, the tensile strength of glass is governed by the size and the distribution of process-specific surface flaws which can occur during the glass production and handling, the load duration and the ambient conditions. These issues indicate a reduction of the actual engineering strength of annealed float glass to less than 100 MPa with a large variation in the strength value, see e.g. S CHULA (2015), H ILCKEN (2015) or S CHNEIDER et al. (2016). In order to ensure an overall transparency so-called point-fixings are used to connect the structural glass and the support structure. Due to the high sensitivity of float glass to concentrated loads and its time dependent strength (B EASON et al., 1984) float glass is not suitable for bolted connections (N IELSEN, 2009). Also used as glass furniture components e.g. doors, work surfaces or just from the architectural and artistic point of view glass plates can be provided with various holes and cut-outs which are the weak points of the components. Accordingly,

Figure 1.1 Application of glass as structural element: (a) Louvre Museum Paris, (b) Shopping Center MyZeil Frankfurt am Main, (c) Glass staircase; (image source: (c) Jens Schneider, TU Darmstadt)

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6_1

2

1 Introduction

(a)

(b)

(c)

(d)

Figure 1.2 Residual load-bearing capacity of laminated glass: (a) Canopy after impact load by stone, (b) Windscreen of a passenger car after impact load by head impact, (c) Point-fixed laminated safety glass made of annealed glass, (d) Point-fixed laminated safety glass made of tempered safety glass (image source: Jens Schneider, TU Darmstadt)

the optimisation of structural glazing with regard to structural and architectural application has always been the main motivation of glass researchers. However, the design and optimisation of structural glazing in engineering requires both knowledge of the relevant mechanisms during the fracture process in glasses and knowledge of the characteristics of the post-fracture behaviour of laminated glass in the glass-polymer composite. The methodical processing of the relevant processes and their transformation into numerical models for the reliable simulation of the fracture and post-fracture behaviour can be used for the development of novel glass structures in various fields: Architecture/construction, solar technology, automotive engineering, aviation, consumer goods (smartphones, tablets, displays, glasses). The main criteria for assessing the load-bearing behaviour and hazard potential of monolithic glasses are their fragmentation and the morphology of the fracture pattern. These depend strongly non-linearly on the distortion energy density present in the glass at the time of fracture, which can be converted into fracture energy (e.g. influence of thermally impressed inherent stresses). Analytical and empirical models exist for the approximate prediction of the geometry and number of fragments as a function of the stress state. However, these are strongly limited with regard to their meaningfulness. On the other hand, current developments of numerical methods allow a mesh-independent modelling of real fracture processes, which can be used for an improved prognosis of complex crack propagation as well as the resulting fragment morphology. While complete failure of a laminated glass structure can be prevented by the layered structure of glass and polymeric interlayer film when individual glass sheets break (see Figure 1.2), the combination of polymeric properties and characteristic fragment morphology results in large time- and temperature-dependent deformations. A reliable determination of the residual bearing capacity is currently only possible on the basis of empirical component tests on a scale of 1:1. The development of suitable numerical models requires a systematic analysis of the relevant mechanisms and influence parameters of the fracture process as well as the fracture mechanical properties and physical quantities of the fracture pattern morphology. In the case of thermally tempered glass plates, the fragmentation is the direct consequence

1.1 Motivation

3

of the elastic strain energy that is stored inside the material due to the residual stress state. Hence, the fragment size and thus the morphology of the fracture pattern depends on the amount of the energy released. Therefore, the fracture pattern morphology and thus the fracture behaviour of glass elements can be investigated very well on thermally tempered glasses, since the level of the residual stress and thus the main influence parameter for fracture, i.e. the stored elastic strain energy, is known and can be varied for the investigations. Float glass is set in to a permanent inner stress state by thermal tempering. For this purpose, glass plates are led into an oven on rollers and heated above the glass transition temperature. Directly from the oven, the plates come into a cooling section where they are rapidly cooled down from both sides by air jets to the ambient temperature. Due to a compressive residual stress at the surface balanced with an internal tensile stress the surface flaws will be in a permanent state of compression which has to be exceeded by external loading before failure will occur. Hence, thermal tempering makes the glass considerably more resistant to impact, knocks and bending stresses. However, the thermal tempering requires that subsequent treatments to the glass such as cut-out, grinding, drilling etc. must be carried out before quenching the glass. It is also known that the briefly occurring tensile stresses on the glass surface at the beginning of the quenching, which means that the glass temperature is in the glass transition temperature range (Tg = 550 ◦ C), can lead to failure of the glass in case of surface flaws, holes and unfavourable cut-out geometries at edges. However, for a quantitative assessment of glass failure during the thermal tempering process, there is a lack of precise knowledge of the soda-lime-silica glass strength considering the process-specific surface flaws for temperatures up to the glass transition range. Thus, systematic improvements of the tempering process as well as the plant technology have so far only been possible to a limited extend. This is required, for example, to select the initial temperature above the glass transition temperature, the cooling rate, the nozzle arrangement, the nozzle diameter, the distance between the nozzles and the glass surface as well as the roller distances. Thermally tempered glasses will fragmentize completely, if the residual stress state within the glass plate is disturbed sufficiently (Fig. 1.3). The fragmentation of tempered glass sets standards for the quality of the tempering process and the degree of safety. The European Standard EN 12150-1 (2015) defines the minimum number of fragments required for soda-lime safety glass on the basis of fragmentation test results. As an example, for glass thicknesses of 4 mm to 12 mm the counted number of fragments in an observation field of 50 mm × 50 mm should not be less than 40 pcs. There exists a relation between the residual stress state, glass thickness and fragment density. In Figure 1.3 the different resulting fragment density for a low (a) and a high (b) residual stress state is shown. Furthermore, the fracture structure resulting from the primary crack propagation and the secondary crack branching can also be influenced by the residual stress state but

4

Figure 1.3 state

1 Introduction

Fragmentation of tempered glass: (a) low residual stress state, (b) high residual stress

also by the intensity of the impact and the boundary conditions. Several studies on the fragmentation behaviour in tempered glasses showing the relation between the residual stress, glass thickness and fragment density have been reported, see e.g. ACLOQUE (1956a), A KEYOSHI et al. (1965), BARSOM (1968), G ULATI (1997), S HUTOV et al. (1998), and M OGNATO et al. (2017). The interrelation between the glass thickness and the fragmentation behaviour in tempered thin glasses with glass thicknesses 2.1 mm to 3.2 mm have been studied by L EE et al. (2012). It is known that the number of fragments is significantly dependent on the degree of tempering of the glass, i.e., to the tensile stress in the middle layer of the tempered glass.

1.2 Topics and Objectives The main objective of this thesis is to provide a comprehensive insight into the influencing variables and parameters of the fracture process as well as the fracture mechanical properties and physical quantities of the fracture pattern and fragment morphology of fragmented thermally tempered soda-lime-silica glass in order to characterize the fracture behaviour of glass panes. The results serve as basic research for further work, e.g. in the field of modelling fracture processes in brittle materials such as glass. For this purpose, i.a., the relationship between the fragment density as the number of fragments within a certain observation field, particle weight and volume, fragment and fracture surface area and the permanent residual stress respectively the elastic strain energy in fragmented tempered glasses of various thicknesses are presented in this work based on experimental results of fracture tests. To establish the correlations, the fracture tests are carried out on thermally tempered glass plates with different residual stress levels. The empirical equations of the heat and mass transfer coefficients required to determine the engine power of the tempering oven are shown. For the determination of the fragment density, the resulting

1.3 Outline

5

fracture patterns are scanned. The fragment and fracture surface area is determined by Computer Tomography (CT). Furthermore, a Machine Learning inspired method for the 2D macro-scale morphological reconstruction and prediction and thus a fast simulation of the glass fracture patterns of thermally tempered glass via stochastic tessellation is shown. The theoretical method is based on Linear Elastic Fracture Mechanics (LEFM) merged with specific fracture pattern statistics from the fracture tests. In order to understand the fracture behaviour and for an almost realistic modelling of the fracture processes in glass, it is essential to consider the phenomenon of bifurcation or crack branching, which occurs when a crack runs rapidly at high stress intensity. The goal of the investigations into the phenomenon of crack branching in this thesis is to define the evolution states of the fracture process from the start of crack propagation to start of crack branching. In addition, the beginning of the crack branching is characterized with a stress intensity based on elastic strain energy conditions from the fracture tests and fracture surface investigations by tensile tests on float glass panes pre-damaged on one side and compared with values from the literature. Overall, a detailed investigation of the fracture processes is aimed at in order to develop an understanding of fracture behaviour and fracture pattern morphology of thermally tempered soda-lime-silica glass.

1.3 Outline Chapter 2 deals with the essential aspects of soda-lime-silica glass with regard to structural and chemical composition, general material properties and fracture mechanical fundamentals as well as the thermal tempering process. An overall insight into the numerical simulation of the thermal quenching and tempering process is given in Chapter 3. A numerical model for the simulation of 1D forced convection under consideration of the thermo-mechanical behaviour of glass is presented. Also a general insight into the 3D modelling of the thermal tempering process using Finite Element Method (FEM) is given. Chapter 3 is regarded as an additive which, although outside the core topic of this thesis, from the author’s point of view is important for the understanding of the complex mechanisms in the thermal tempering process and its numerical simulation. Chapter 4 presents comprehensive experimental investigations on the fragmentation and fracture pattern morphology of thermally tempered soda-lime-silica glass. The fragmentation behaviour of heat treated soda-lime-silica glass plates is investigated by fracture tests on glass specimens of different thickness and also different residual stress levels respectively elastic strain energies. In order to product differently heat treated specimens used in the fracture tests, a simplified model and an experimental validation of the residual stress levels is provided for the determination of the engine power of the tempering oven. Different dependencies and correlations are shown with respect to the parameters of the fracture pattern morphology. The fracture surface of the fragments is analysed by fractographic examinations and the influence of the residual stress on the fractographic

6

1 Introduction

features of the fracture surface is outlined. The fragment and the fracture surface area are determined with the help of Computer Tomographic (CT) investigations. Chapter 5 presents a machine learning motivated stochastic point process model for the prediction of the fragmentation of thermally tempered glass. The fracture mechanical and mathematical foundation, deduction and application of the approach for the 2D morphological prediction of fracture patterns of thermally tempered glass via stochastic tessellation is shown. The presented approach is based on the combination of a LEFM energy criterion related to the elastic strain energy state before and after fragmentation and statistical analysis of the fracture pattern of tempered glass to determine features of the fracture pattern morphology within an observation field. After laying the foundations of deterministic fracture pattern and tessellation of stochastic point processes, the methodology of fracture pattern recognition and generation called ’Bayesian Reconstruction and Prediction of Glass Fracture Patterns (BREAK)’ is presented. Chapter 6 provides investigations into the crack branching during the fracture process of glass. The fracture process is characterized with respect to the phenomenon of crack branching and the associated energy release rate based on the results of the fracture tests in chapter 4 and the fracture mechanical considerations in chapter 5. Furthermore, the crack branching behaviour of soda-lime-silica glass was investigated by tensile tests on onesided pre-damaged float glass sheets. The fracture velocity was determined using highspeed observations of the crack propagation during tensile tests. Chapter 7 subsequently provides a summary of the results related to the objectives of the thesis and an outlook for future research on this topic.

2 Glass Properties and Refinement Processes 2.1 Definition and Structure of Glass Glass is an amorphous solid that is usually produced by a melting process. Thermodynamically, glass is called to be the frozen state of a supercooled liquid (S CHOLZE, 1988). If a liquid is cooled down below its melting point Tm , crystallisation usually takes place because the crystalline state is energetically more favourable than the liquid (S CHNEIDER et al., 2016). However, it takes some time for the molecules to build up a crystalline solid to reach the most favourable energetic position in the crystal lattice. If the melt cools down very quickly, crystallization can be prevented and the melt can remain liquid even at temperatures below Tm . Such a fragile state is called a supercooled melt or supercooled liquid. Figure 2.1 illustrates the temperature dependence of the glass structure using the volume as an example.

Figure 2.1 Schematic representation of the temperature dependence of the glass structure using the volume as an example

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6_2

8

2 Glass Properties and Refinement Processes

Figure 2.2 Atomic structure of (a) quartz crystal (SiO2 ), (b) quartz glass (SiO2 ) and (c) soda-limesilica glass

In contrast to crystallization, the phase transition from the melt to the solid state takes place continuously without abrupt changes of the material properties. The transformation range is often described in simplified terms by a single transformation temperature also referred to as glass transition temperature Tg . The melting point Tm is always above Tg . The gradient of the curves in Figure 2.1 can be interpreted as thermal expansion coefficient αT . The intersection of the straight lines, which describe the idealized solid state and the idealized liquid state, defines Tg , although it is a transition area. A distinction can therefore be made between glass as a solid body (below Tg ) and glass melt (above Tg ). In contrast to crystallisation of crystals (see Figure 2.2 (a)), crystalline germs also form when the glass solidifies, however, at room temperature they form an irregular network similar to a liquid (see Figure 2.2 (b)). The material therefore appears completely homogeneous and isotropic under macroscopic conditions. Such a non-crystalline solid is called an amorphous solid or simply glass. In the building industry, almost exclusively soda-lime silica glass is used. This is a non-metallic, oxidic glass that belongs to the group of silica glasses. The atomic network of this type of glass is formed almost exclusively by silicon dioxide (SiO2 ) as glass former. However, the network is ruptured by network changing components such as sodium and calcium oxide (see Figure 2.2 (c)). The atomic network of soda-lime-silica glass consists of silicon dioxide (SiO2 ), calcium oxide (CaO), sodium oxide (Na2 O), magnesium oxide (MgO) and aluminium oxide (Al2 O3 ). The typical composition of soda-lime-silica glass is given in Table 2.1. The soda-lime-silica glass used in the building industry is mostly produced in the float process, where raw materials are melted at approximately 1500 ◦ C depending on the composition. In the production of soda-lime-silica glass, these are quartz sand, soda, limestone and dolomite. The melt is then poured out on a bath of molten tin where it floats over the tin and is cooled to about 600 ◦ C before it enters the annealing oven. When the glass transition temperature is reached, the glass solidifies, is removed and then slowly cooled further down to approximately 100 ◦C. According to EN 572-1 (2016) planar, transparent,

2.2 Properties of Soda-Lime-Silica Glass

9

Table 2.1 Chemical composition of soda-lime-silica glass (EN 572-1 (2016))

Chemical element Silicon Calcium Sodium Magnesium Aluminium othersa a

Weight [%] Si Ca Na Mg Al

32 − 35 3.5 − 10.1 7.4 − 11.9 0 − 3.7 0 − 1.6 Tre f ) and decreases (Δξ < Δt) with decreasing temperature (T < Tre f ). This simulates the approach of the viscoelastic hot glass to the linear elastic, cooled solid glass.

3.4.2.3 Strain Calculation For the strain calculation the Kirchhoff-Love hypothesis can be used, which says that fibres orthogonal to the mid-plane of thin plates still do this even in the deformed state. In addition, a plane stress state is assumed. It follows that all stresses except σxx and σyy are zero. Since glass is an isotropic material, both the stresses and the strains in the x and y directions are considered to be the same (σxx = σyy , εxx = εyy ). The curvature around the z-axis is neglected in the case of a glass plate. Due to the temperature field symmetrical to the x − y plane, there is no bending moment (Mx = My = 0). Therefore, the curvatures around the x- or y-axis can also be neglected. The strains are thus constant in both directions along the thickness of the glass plate and equal to the strain in the mid-plane. εii (x, y, z,t) = εii0 (x, y, z,t)

(3.45)

3.4 Numerical Simulation of 1D Forced Convection

35

εii0 is the trace of the strain tensor in the mid-plane of the glass plate in the x- or y-direction. 0 (= ε 0 ) and ε 0 must be For the calculation of the mechanical strains, two unknowns εxx yy zz determined. The trace of the strain tensor therefore results in: 0 + εzz εii = 2εxx

(3.46)

Eq. (3.31) to (3.39) can be used to calculate the unknown mechanical strains. Two conditions are necessary for this. First, the equilibrium condition for the sum of the stresses over the thickness: Nn

∑ σxx (tn , zi ) · Δzi = 0 !

(3.47)

i=1

With σxx (tn , zi ) =

σii (tn , zi ) + sxx (tn , zi ) 3

(3.48)

The second condition to be fulfilled is the assumption of the plane stress state: !

σzz (tn , zi ) = 0

(3.49)

With σzz (tn , zi ) =

σii (tn , zi ) + szz (tn , zi ) 3

(3.50)

Thus, the two unknown mechanical strains for each time step can be determined iteratively over the two conditions and the stress field can be calculated. Thermal strain can be calculated using the fictive and real temperature as described in Eq. (3.13).

36

3 Numerical Simulation of the Thermal Tempering Process

3.4.3 Results of Numerical Simulations The results of temperature and stress development over process time calculated from the numerical simulation of 1D forced convection for "infinite" glass plates without disturbances in the form of holes, cut-outs and edges are presented. The glass properties and the viscoelastic and structural relaxation characteristics are taken from Tables A.1 and A.2 in Appendix A.1. The calculations were carried out in the MATLAB program package (MATLAB, 2016). Figure 3.8 shows the temperature-time curve T (t) and the resulting stresses in the x-axis σxx (t) over time for the 1D forced convection of an infinite glass plate with the thickness of 10 mm. The calculation is carried out with the number of grid nodes through half the thickness Nn = 21 (bias factor = 1), the time increment Δt = 0.005 s and the heat transfer coefficient h = 150 W/m2 K. The initial temperature is T0 = 650 ◦C and the ambient temperature is T∞ = 20 ◦C. The heat transfer coefficient of h = 150 W/m2 K is chosen iteratively so that after quenching a surface compressive stress of approx. 100 MPa results. As shown in Figure 3.8(b), the surface reaches a tensile stress of approx. 10 MPa after a quenching time of about 2 seconds. This is due to the fact that the entire glass plate is in a viscous state at the high temperature of 650 ◦C, i.e. approx. 100 ◦C above the glass transition temperature. At the beginning of quenching, the two surfaces harden and thus assume an almost linear elastic material behaviour. While the material continues to cool down, the surfaces contract and tensile stresses are initially exerted on both surfaces. The result of the residual stress depends significantly on the value of the heat transfer between glass and the environment. Figure 3.9 shows the influence of the heat transfer coefficient h on the residual surface stress σs for different glass plate thicknesses. The

(a)

(b)

Figure 3.8 1D forced convection: (a) Temperature [◦C] vs. Time [s], (b) Stress σxx [MPa] vs. Time [s] calculated in the mid-plane and surface of an infinite glass plate with a thickness of 10 mm; Heat transfer coefficient h = 150 W/m2 K, T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K). Time increment Δt = 0.005 s; Number of grid nodes Nn = 21 (through half the thickness); Bias factor = 1; The dotted curve represents the temperature difference ΔT [◦C] between mid-plane and surface

3.4 Numerical Simulation of 1D Forced Convection

37

Figure 3.9 1D forced convection: Heat transfer coefficient h [W/m2 K] vs. Residual surface compressive stress σs,xx [MPa] calculated in the surface of an infinite glass plate for different thicknesses (4.0 mm to 19.0 mm); T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

higher the heat transfer coefficient h, i.e. the faster the glass is cooled, the greater the residual stress. For a 10 mm thick glass plate, for example, a residual surface compressive stress of approx. 17 MPa is generated with a heat transfer coefficient of 22 W/m2 K. With the same boundary conditions but a different heat transfer coefficient of approx. 600 W/m2 K, a residual surface compressive stress of approx. 200 MPa is generated. As can be seen in Figure 3.9, the choice of the heat transfer coefficient to achieve a certain residual stress also depends essentially on the glass thickness. In order to achieve the same residual surface compressive stress for glass plates with different thicknesses the heat transfer coefficient h varies considerably. In order to achieve the residual surface compressive stress of 100 MPa, a 19 mm thick glass plate at an initial temperature of T0 = 650 ◦C must be cooled to an ambient temperature of T∞ = 20 ◦C with a heat transfer coefficient of h = 75 W/m2 K. A 4 mm thick glass plate with the same boundary conditions, however, must be cooled with a much higher value of h = 528 W/m2 K to achieve the same residual surface compressive stress of 100 MPa. The heat transfer coefficients and the resulting residual stresses are listed in detail in Appendix A.3. The results of the numerical simulation of the 1D forced convection are compared with experimental data from literature. In earlier experiments which were carried out by G AR DON (1965), the residual mid-plane stresses were measured for different initial temperatures T0 and different cooling rates. The glass thickness for this purpose was 6.1 mm. In Figure 3.10, the numerically calculated residual mid-plane tensile stress σm,xx as a function of the initial temperature T0 and heat transfer coefficient h is compared to the experimental

38

3 Numerical Simulation of the Thermal Tempering Process

Figure 3.10 Comparison of the numerical model (lines) with experimental data (markers) (G ARDON, 1965) for residual mid-plane tensile stresses σm,xx [MPa] with different initial temperatures T0 [◦C] and heat transfer coefficients h [W/m2 K]; glass plate thickness 6.1 mm; dashed lines show the numerical results for heat transfer coefficients h = 13 W/m2 K and h = 109 W/m2 K according to G ARDON (1965)

results from G ARDON (1965). The markers represent the test data and the lines represent the results of the numerical model. The upper four curves (h = 222; 331; 444 and 553 W/m2 K) are calculated with the heat transfer coefficient h as determined by G AR DON (1965). The two lower curves (h = 40 and 130 W/m2 K) are calculated with slightly adjusted, higher values for h, because the residual mid-plane stresses would not match the experimental data with the originally determined values (dashed lines). The numerical results of the residual stress calculation agree predominantly with the experimental results of G ARDON (1965). The differences in the lower curves could be due to the heat radiation inside the glass, which is neglected in this calculation model. This might be of interest especially with low heat transfer coefficients, since it would increase the temperature difference between the centre and the surface of the glass more strongly and would thus have an effect on the residual stresses. In addition, it can be observed that the curves in Figure 3.10 are almost constant from an initial temperature of T0 = 650 ◦C, so that the residual stress hardly changes at higher initial temperatures. Especially with heat transfer coefficients higher than 109 W/m2 K, the residual stress drops steeply at initial temperatures lower than T0 = 650 ◦C. Thus 650 ◦C is an acceptable initial temperature for a numerical simulation of the thermal tempering process and the calculation of the residual stresses.

3.4 Numerical Simulation of 1D Forced Convection

39

3.4.4 Convergence Study Convergence studies are performed depending on the glass thickness for the model parameters time increment Δt and grid node distance Δz in order to obtain sufficiently accurate numerical results at the lowest possible calculation effort. A distinction is made between two objectives of the investigation, which the user of the program can pursue. One objective of the application is the investigation of the (final) residual stresses that occur at the end of the quenching process. The second objective is to study the transient residual stresses that develop during the cooling process before converging towards the final residual stresses. The surface tensile stress that builds up at the beginning of the cooling process is of particular importance here. The diagrams in Figure 3.11 (a), (b) and (c) show the surface stress development during the first three seconds of the quenching process for three different model parameters. For both types of investigation, Section 3.4.5 summarizes the proposals of all model parameters for different glass thicknesses in Table 3.1. Figure 3.11 (a) and (b) show the influence of time stepping on the tensile stress development during the first three seconds of the quenching process. The calculation is carried out for a glass thickness of 4.0 mm. The initial temperature is T0 = 650 ◦C and the ambient temperature is T∞ = 20 ◦C. The quenching is simulated with a heat transfer coefficient of h = 500 W/m2 K. The same boundary conditions and model parameters are used for two models with different time increments of Δt = 0.3 s (≡ 10 time steps) and Δt = 0.0003 s (≡ 10000 time steps). The number of grid nodes through half the thickness is Nn = 50 (Δz = 0.04 mm). It is obvious that time stepping is of great importance for the simulation of stress development. The simulation with Δt = 0.3 s overestimates the maximum stress value (29.69 MPa instead of 25.35 MPa). In addition, there is no pronounced peak in the curve. Figure 3.11 (c) shows the influence of the grid node distance Δz on the residual stress development during the first three seconds of quenching. The only difference to Figure 3.11 (a) and (b) is that here instead of Nn = 50 the number of grid nodes Nn = 5 (Δz = 0.4 mm) is chosen through half the thickness with a time increment of Δt = 0.0003 s. This underestimates the maximum stress and results in a maximum tensile stress of 17.37 MPa instead of 25.35 MPa as in the model with Nn = 50. In the following, the numerical model will be calibrated with respect to the model parameters time increment Δt and grid node distance Δz.

40

3 Numerical Simulation of the Thermal Tempering Process

(a)

(b)

(c) Figure 3.11 Surface stress development during the first three seconds of the quenching process for (a) Δt = 0.3 s; Nn = 50 (Δz = 0.04 mm), (b) Δt = 0.0003 s; Nn = 50 (Δz = 0.04 mm), (c) Δt = 0.0003 s; Nn = 5 (Δz = 0.4 mm); glass thickness 4 mm; Maximum and minimum values of the glass surface stress are shown; h = 500 W/m2 K, T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K); Bias factor = 1

3.4 Numerical Simulation of 1D Forced Convection

41

3.4.4.1 Convergency of Time Increment Δt For the convergence study of the time increment Δt, calculations are carried out for different glass thicknesses, changing only the time increment Δt. The initial temperature of T0 = 650 ◦C is considered. Since different glass thicknesses require different cooling times, different calculation duration times are used so that the glass plate has assumed the ambient temperature of 20 ◦C at the end of quenching along the entire thickness. The heat transfer coefficient in order to achieve a residual surface compressive stress of 100 MPa is applied for different thicknesses according to Appendix A.3. For the final residual stresses, the solutions are compared to a calculation with the time step of Δt = 0.005 s. For the simulation of the stress development at the beginning of the quenching process (Time ≤ 2 s), the solutions are compared to a calculation with the time step of Δt = 0.001 s. In Figure 3.12 (a), the time increment convergency for the residual mid-plane and surface stress of a glass plate after the tempering process is shown for glass thicknesses of 3 mm, 6 mm, 10 mm and 15 mm. The convergence study of the residual stresses shows how sensitive the numerical calculation reacts to the choice of time steps. Additionally the sensitivity increases with smaller thicknesses. The convergence curves for the residual surface compressive stresses converge slightly offset to the mid-plane stresses, causing an additional inaccuracy in the stress result, see Figure 3.12 (a). As an example, with a glass thickness of 6 mm and a time increment of Δt = 1 s there is a deviation of approx. 3% for the residual mid-plane stress and approx. 9% for the residual surface stress. The results of the convergence study for the maximum surface tensile stress in the beginning of the cooling for a calculation time of 2 s and different thicknesses of 3 mm, 6 mm, 10 mm and 15 mm is shown in Figure 3.12 (b). The maximum surface tensile stresses are much more error-prone with increasing time increments. For example, with a glass thickness of 6 mm and a time increment of Δt = 1 s there is a deviation of approx. 27% for the maximum surface tensile stress in the beginning of the cooling. Also here the sensitivity increases with smaller glass thickness. For smaller glass thicknesses, a finer time stepping must be used in order to minimize the susceptibility to errors. The reason for this is the higher cooling rate required for smaller thicknesses to achieve the same stress as with the larger thicknesses. Since the temperature-time curve is steeper, i.e. there is a greater change in temperature over time, the time increments must be smaller for the temperature and stress solution to converge and minimize the deviation.

42

(a)

3 Numerical Simulation of the Thermal Tempering Process

(b)

Figure 3.12 Time increment convergency for (a) Residual stress after the tempering process and (b) Maximum surface tensile stress in the beginning of the cooling (Time ≤ 2 s); glass thicknesses 3 mm, 6 mm, 10 mm and 15 mm; Heat transfer coefficient according to Appendix A.3 for a residual surface compression of approx. 100 MPa; Deviation from a model using the time increment (a) Δt = 0.005 s and (b) Δt = 0.001 s; T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

3.4.4.2 Convergency of Equidistant Grid For the convergence study of the grid node distance of an equidistant grid, calculations are performed for different glass thicknesses and only the grid node distance Δz or the total grid node number Nn through half the thickness is changed. For the residual stresses, those deviations of the stresses of the glass mid-plane and glass surface are determined that arise in comparison to a calculation with a grid node distance of Δz = 0.0025 mm. In Figure 3.13 (a), the convergency of grid node number Nn through half the thickness for the residual mid-plane and surface stress of a glass plate after the tempering process is shown for different thicknesses of 3 mm, 6 mm, 10 mm and 15 mm. It is observed that in contrast to time increment Δt, the sensitivity of the numerical model to grid node distance Δz or grid node number Nn increases with larger thicknesses. The curves in Figure 3.13 (a) show that at least 30 grid nodes should be selected in an equidistant grid for glass thicknesses larger than 12 mm, 20 grid nodes for glass thicknesses between 12 mm and 6 mm and at least 10 grid nodes for glass thicknesses smaller than 6 mm to keep the inaccuracy of the stress results below a deviation of 3%. The results of the convergence study for the maximum surface tensile stress in the beginning of the cooling for a calculation time of 2 s and different thicknesses of 3 mm, 6 mm, 10 mm and 15 mm is shown in Figure 3.13 (b). For the consideration of the maximum surface tensile stresses, the results for a varying number of grid nodes are compared with the results of a grid node distance of Δz =

3.4 Numerical Simulation of 1D Forced Convection

43

0.001 mm. Also here, the discretization sensitivity of the numerical model for solving the maximum surface tensile stress at the beginning of cooling is considerable. As an example, the discretization of a 10 mm thick glass plate with a grid node number of Nn = 10 through half the thickness results in an inaccuracy of the maximum surface tensile stress in the beginning of the cooling (Time ≤ 2 s) of about 21%. 3.4.4.3 Convergency of Non-equidistant Grid In order to get a higher accuracy and to use fewer grid nodes at the same time, the convergence of the number of grid nodes on non-equidistant grids is investigated. The residual stresses and the maximum surface tensile stresses are considered separately and only the bias factor is changed. The results for the final residual stresses are compared for different glass thicknesses with an equidistant grid with a grid node distance of Δz = 0.0025 mm. Figure 3.14 (a) shows an example of the comparison for a thickness of 6 mm. The heat transfer coefficient of h = 279.26 W/m2 K according to Appendix A.3 is applied in order to achieve a residual surface compressive stress of 100 MPa. For the investigation of the maximum surface tensile stresses in the beginning of the cooling process, numerical simulations for different glass thicknesses with varying bias factors are carried out and compared with the results obtained on an equidistant grid with a grid node distance of Δz = 0.001 mm. The curves shown in Figure 3.14 (b) show the convergence of the number of grid nodes for a glass thickness of 6 mm for different bias factors. It is shown that the convergence curves do not converge to zero with increasing bias factor, but a residual error remains. In Figure 3.15 (a), the convergency of the bias factor for residual stresses after the tempering process is shown for different thicknesses. The calculations are carried out for the total number of grid nodes Nn = 10 through half the thickness. Figure 3.15 (b) shows the convergency of the bias factor for the maximum surface tensile stress in the beginning of the cooling. Here, the total number of grid nodes Nn = 20 through half the thickness is applied. The heat transfer coefficient for each thickness is considered according to Appendix A.3 in order to achieve a residual surface compressive stress of 100 MPa. A bias factor of 4 causes a deviation in the corresponding residual stresses of less than 3% (Figure 3.15 (a)). However, in order to achieve a deviation of less than 3% for the maximum surface tensile stresses in the beginning of cooling a minimum bias factor of 8 is required for glass thicknesses larger than 10 mm (Figure 3.15 (b)).

44

(a)

3 Numerical Simulation of the Thermal Tempering Process

(b)

Figure 3.13 Equidistant grid: Convergency of grid node number Nn through half the thickness for (a) Residual stress after the tempering process and (b) Maximum surface tensile stress in the beginning of the cooling (Time ≤ 2 s); glass thicknesses 3 mm, 6 mm, 10 mm and 15 mm; Heat transfer coefficient according to Appendix A.3 for a residual surface compression of approx. 100 MPa; Deviation from a model using grid node distance (a) Δz = 0.0025 mm and (b) Δz = 0.001 mm; T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

(a)

(b)

Figure 3.14 Non-equidistant grid: Convergency of grid node number Nn through half the thickness for (a) Residual stress after the tempering process and (b) Maximum surface tensile stress in the beginning of the cooling (Time ≤ 2 s); bias factors 1, 2, 4, 6 and 8; glass thickness 6 mm; h = 279.26 W/m2 K; Deviation from a model using grid node distance (a) Δz = 0.0025 mm and (b) Δz = 0.001 mm; T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

3.4 Numerical Simulation of 1D Forced Convection

(a)

45

(b)

Figure 3.15 Convergency of the bias factor for (a) Residual stress after the tempering process (Nn = 10) and (b) Maximum surface tensile stress in the beginning of the cooling (Time ≤ 2 s) (Nn = 20); glass thicknesses 3 mm, 6 mm, 10 mm and 15 mm; Heat transfer coefficient according to Appendix A.3 for a residual surface compression of approx. 100 MPa; Deviation from a model using grid node distance (a) Δz = 0.0025 mm and (b) Δz = 0.001 mm; T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

3.4.5 Model Parameter Proposal From the results of the convergence studies, it is now possible to propose minimum values for the model parameters with which sufficiently high accuracy can be achieved. Table 3.1 shows minimum values of model parameters for different glass thicknesses in order to achieve a residual surface compressive stress of 100 MPa in each case. The given number of grid nodes Nn applies to half the thickness. The ambient temperature is always 20 ◦C. A distinction is made between the calculation of the final residual stress and the temporary transient maximum surface tensile stress in the beginning of the quenching process (Time ≤ 2 s). The heat transfer coefficients are determined using the corresponding tables in Appendix A.3. Investigations have shown that very high heat transfer coefficients (h > 400 W/m2 K) result in a greater error than low ones. This applies to glass thicknesses of 4 mm and smaller, where the time increment Δt should be adjusted. For this purpose, it is sufficient to halve the proposed time steps from the convergence study when calculating the residual stresses. However, when calculating the maximum surface tensile stress, they should be reduced by a power of ten. The deviation from a model using the time increment Δt = 0.005 s and the grid node distance Δz = 0.0025 mm for the residual stresses and the time increment Δt = 0.001 s and the grid node distance Δz = 0.001 mm for the maximum surface tensile stress is considered. For the given discretization and the bias factors, the deviation for the respective stresses is less than 3%. To further reduce the deviation, higher

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3 Numerical Simulation of the Thermal Tempering Process

bias factors must be selected, which is more effective than increasing the number of nodes Nn along the thickness. Table 3.1 Minimum values of model parameters for different glass thicknesses and a residual surface compressive stress of 100 MPa; (a) Residual stress (b) Maximum surface tensile stress in the beginning of the cooling (Time ≤ 2 s); Dev.≤ 3%

(a) Residual stress Glass thickness [mm]

Nn

Time steps

Total time [s]

Δt [s]

Bias factor

h [W/m2 K]

T0 [◦C]

1.8 3 4 6 8 10 12 15 19

10 10 10 10 10 10 10 10 10

1500 3000 2000 3000 1250 2000 2000 3000 4000

30 60 100 300 500 800 1200 2100 2800

0.020a 0.020a 0.050a 0.10 0.40 0.40 0.60 0.70 0.70

4 4 4 4 4 4 4 4 6

979.73 802.23 526.43 279.26 196.65 152.06 124.00 97.38 76.07

680 650 650 650 650 650 650 650 650

(b) Max. surface tensile stress Glass thickness [mm]

Nn

Time steps

Total time [s]

Δt [s]

Bias factor

h [W/m2 K]

T0 [◦C]

1.8 3 4 6 8 10 12 15 19

20 20 20 20 20 20 20 20 20

500 500 500 500 500 500 250 250 250

1 1 1 10 10 10 10 10 10

0.002b 0.002b 0.002b 0.020 0.020 0.020 0.040 0.040 0.040

4 4 4 4 4 4 6 8 14

979.73 802.23 526.43 279.26 196.65 152.06 124.00 97.38 76.07

680 650 650 650 650 650 650 650 650

a b

Dev.≈ 5%: the time increment should be halved to reach Dev.≤ 3%. Dev.≈ 5%: the time increment should be reduced by the power of ten to reach Dev.≤ 3%.

3.5 3D FE-Modelling of the Thermal Tempering Process

47

3.5 3D FE-Modelling of the Thermal Tempering Process The increasing application of tempered glass as a construction element of the structural glazing has substantially increased the demand for structural and architectural flexibility and versatility of glass components. Hence, holes and cut-outs in glass plates as architectural and structural elements in buildings or as furniture components are unavoidable (Figure 3.16). Since float glass is not suitable for bolted connections (N IELSEN, 2009) and depending on the processing status the edges represent weak points due to their low strength, glass plates with holes or cut-outs has to be thermally tempered. This makes the glass considerably more resistant to impact, knocks and bending stresses. However, regarding the complete fragmentation by disturbing the equilibrated residual stress state in thermally tempered glass, drillings or cut-outs must be done before quenching the glass. Due to the multi-dimensionality of the cooling process with respect to the inner surface of the holes, the chamfer and the edge, the numerical model of 1D forced convection described in Section 3.4 cannot be applied. However, the numerical model of 1D forced convection is a good basis for the simulation of cooling in far-field areas (regions unaffected by holes and cut-outs or edges) as well as for convergence criteria of the model parameters time step and discretization of plate thickness. This section presents a methodology for 3D FE-simulation of the thermal tempering process using the Finite Element Method (FEM) in order to calculate the residual stresses in the area of the holes or cut-outs of a tempered glass plate. A viscoelastic material behaviour of the glass according to Section 3.3 is considered for the simulation of the tempering process. The structural relaxation is taken into account using Narayanaswamy’s model. Due to different cooling rates of the convection areas such as edge, chamfer, hole’s

(a)

(b)

Figure 3.16 Examples of glass plates (a) with holes for point-fixings (image source: Jens Schneider, TU Darmstadt) and (b) with cut-outs (shown through linearly polarized light)

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3 Numerical Simulation of the Thermal Tempering Process

inner surface and far-field area, heat transfer coefficients are estimated using experimental data from the literature. The objective is to demonstrate the simulation of the residual stresses in tempered glasses with holes or cut-outs and to quantify the amount of temper stresses based on a variation of different geometrical parameters and the local heat transfer coefficient. The results of the investigations with variations of geometrical parameters for typical geometries of holes and cut-outs as well as the influence of hole, edge and corner distances have been presented in P OURMOGHADDAM et al. (2016) and P OURMOGHADDAM et al. (2018a).

3.5.1 Heat Transfer Coefficients for Glass Plates with Holes The 3D Finite Element simulation of the thermal tempering process of glass plates with holes, edges or cut-outs and the calculation of the residual stresses require the simulation of the heat transfer in the cooling process. Hereby the essential influence parameter is the heat transfer coefficient h between glass and the cooling medium. During quenching of the processed glass plate, the cooling medium air hits differently positioned surfaces such as the edge, the chamfer and the inner surface of the hole in addition to the glass plate surface. This causes the differently positioned convection surfaces to cool at different rates. If the processed glass plate is to be modelled as a volume, the different cooling of the convection surfaces is defined by the value of the heat transfer coefficient. The heat transfer due to radiation is neglected to simplify the calculations. Therefore, the heat transfer coefficients of the convection surfaces has to be determined for the FE-simulation, which is experimentally very difficult. The convection coefficients in the different convection surfaces of perforated plates (far away from edges (far-field), hole, edge and chamfer) have been identified experimentally by B ERNARD et al. (2009) using a hollow aluminium model. In the present work, the experimentally identified heat transfer coefficients are linearly factorized with a scaling factor β , as shown in Figure 3.17, in order to determine the heat transfer coefficients of glass plates with a hole.

Figure 3.17 Heat transfer coefficients for an "infinite" plate with a hole; factor of the different convection surfaces from the experimental results (B ERNARD et al., 2009)

3.5 3D FE-Modelling of the Thermal Tempering Process

49

The tables in Appendix A.3 list the heat transfer coefficients determined for the farfield area of the surface as a function of residual surface stress and glass thickness using the numerical model of 1D forced convection (Section 3.4). The values for the scaling factor β can be used in order to determine the heat transfer coefficients for the different convection surfaces applying the different factors according to Figure 3.17. In this way, the heat transfer coefficient for the different convection surfaces can be determined in relation to the experimentally determined values in B ERNARD et al. (2009). However, with the scaling factor β a simple relation of the heat transfer coefficient between surface and the hole area is assumed (Figure 3.17).

3.5.2 FE-Model and Element Definition The goal of FE-simulations of quenching is to obtain residual stresses of soda-lime-silica glass plates. The radiation effect is neglected. The methodology of the FE-simulation of the thermal tempering process is described by means of an "infinite" round plate with a hole positioned in the center of the plate. The FE-model is produced as a solid model. Due to symmetry, only a piece of the plate under an angle of 5◦ and half the thickness is modelled. The boundary conditions are defined according to the symmetry utilization. In order to have an area for the residual stresses that is not influenced by the hole and the edge (far-field) it is sufficient to define a radius of 200 mm from the center of the hole. A sketch of the FE-model with the definition of the boundary conditions is shown in Figure 3.18 (a). The coordinate system of the FE-

(a)

(b)

Figure 3.18 (a) Sketch of the "infinite" round plate model with a hole in the center and the definition of boundary conditions (b) Meshed model

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3 Numerical Simulation of the Thermal Tempering Process

model is oriented so that r is the radial direction, θ is the tangential direction and z is the thickness direction. The discretization of the thickness is done according to Table 3.1 with 10 elements through half the thickness applied with a bias factor of 4. The meshed FEmodel is shown in Figure 3.18 (b). The transient finite element simulation of the residual stresses is carried out in two calculation steps. First, the temperature history during the tempering process is determined in a transient temperature calculation using the 3-D 20-Node Thermal Solid element SOLID90 (available in FE-program Ansys 18.1), which is a thermal element with temperature as the only degree of freedom. The geometry, node locations and the coordinate system for the element SOLID90 with the available convection surfaces are shown in Figure 3.19 (a). In order to calculate the temperature curve over the time of quenching for each node of the meshed FE-model, the corresponding surfaces of the SOLID90 elements located at the po-

(a)

(b) Figure 3.19 (a) 3-D 20-Node Thermal Solid, from (A NSYS, 2009); convection surfaces: 1 J-I-L-K, 2 I-J-N-M, 3 J-K-O-N, 4 K-L-P-O, 5 L-I-M-P, 6 M-N-O-P (b) Convection surfaces for a 10 mm thick plate with a hole diameter of 50 mm; values of the heat transfer coefficients h [W/m2 K] applied for a scaling factor of the heat transfer coefficients of β = 1.57

3.5 3D FE-Modelling of the Thermal Tempering Process

51

sition of the convection surfaces are provided with the respective values of the heat transfer coefficient, see Figure 3.19 (b). The element orientation must be taken into account. In the second calculation step the temperature change over time is put in terms of load steps on a structural mechanical model. The material properties listed in Table A.1 in the appendix as well as the parameters of the Prony series (Table A.2) to simulate the viscous and structural relaxation of glass during the thermal tempering process is considered in the second step. The consideration of the thermo-mechanical material behaviour of glass in order to calculate stresses requires a change of element type. Therefore, the thermal element SOLID90 is converted into 3-D 20-Node Viscoelastic Solid element VISCO89 with displacements in X−,Y − and Z−direction as degrees of freedom. The geometry, node locations and the coordinate system for the element remain unchanged. The results of the residual stress calculation in radial and tangential direction of the hole area for a 10 mm thick infinite round glass plate with a hole (φ 50 mm) in the center after the cooling process is shown in Figure 3.20. The calculations are carried out in the finite element program Ansys 18.1 (A NSYS, 2017). The scaling factor of the heat transfer coefficient is β = 1.57. The corresponding heat transfer coefficients are set on the convection surfaces according to Figure 3.17. The residual surface compressive stress far away from the hole and the edge (far-field) in radial and in tangential direction is approx. 100 MPa. The initial temperature is set to T0 = 650 ◦C (923.15 K) and the ambient temperature to T∞ = 20 ◦C (293.15 K). The stress in the area of the holes that is decisive for the glass strength is the tangential compressive stress, since this ensures that the micro-cracks that arise as a result of the production and handling are in a permanent state of compression and thus the resistance to external loads in this area increases. In Figure 3.20 (b), the tangential stress in hole area is presented.

52

3 Numerical Simulation of the Thermal Tempering Process

(a)

(b) Figure 3.20 FE-simulation of residual stress in (a) Radial direction σr [Pa], and (b) Tangential direction σθ [Pa] in the hole area; 10 mm thick plate with a scaling factor for the heat transfer coefficients of β = 1.57, T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

3.5 3D FE-Modelling of the Thermal Tempering Process

53

3.5.3 Definition of Time Steps Just as in the simulation of 1D forced convection, time stepping also has a significant influence on the accuracy of residual stresses in the 3D FE-modelling of the thermal tempering process. In order to save computing time the total time for the FE-simulation of the quenching process, which is the total calculation time, is divided into two cooling steps in terms of time steps. In the first cooling time step t1 the viscous glass heated above the glass transition is cooled in small time increments Δt1 to a temperature below an assumed glass transition temperature Tg = 550 ◦C. A fine time stepping in the beginning of the cooling is necessary to simulate the rapid cooling of the glass with the initial temperature T0 . The temperature time curve steeply declines at the beginning of the cooling process and afterwards levels off at the ambient temperature T∞ , see Figure 3.8 (a). Therefore, at the beginning of the cooling process, a fine time stepping is required until the model, which was heated to approx. 100 ◦C above Tg with an initial temperature T0 , reaches a temperature below the glass transition temperature at the surface as well as in the mid-plane. For the selection of the time increments in the first cooling time step t1 a convergence study has been carried out in P OURMOGHADDAM et al. (2018a) for glass thicknesses 6 mm, 10 mm and 15 mm using a 2D-axisymmetric model with the 2-D 8-Node Thermal Solid element PLANE77 for the temperature calculations, which was changed into 2-D 8-Node Viscoelastic Solid element VISCO88 for the stress calculations. The results are shown in Figure 3.21. The time increments determined in P OURMOGHADDAM et al. (2018a) correspond to the values from the 1D forced convection simulations in Table 3.1(a).

Figure 3.21 Time increment convergency for the first cooling time step t1 from P OURMOGHADDAM et al. (2018a); glass thicknesses 6 mm, 10 mm and 15 mm; Deviation from a model using the time increment Δt = 0.005 s; T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

54

3 Numerical Simulation of the Thermal Tempering Process

Figure 3.22 Parabolic function (Eq. 2.3) and the FE-simulation of the residual stress in far-field area; glass thickness 10 mm; h = 152.06 W/m2 K; T0 = 650 ◦C (923.15 K) and T∞ = 20 ◦C (293.15 K)

After the whole glass plate is cooled down below the transition temperature and solidifies, in the second cooling time step t2 larger time increments Δt2 can be set for cooling the glass plate to the ambient temperature T∞ . However, experience has shown that the second cooling step t2 below the glass transition temperature cannot be performed in one step to simulate the parabolic stress distribution along the plate thickness (P OURMOGHADDAM et al., 2018a). The difference between the FE-simulation and the parabolic function according to Eq. 2.3 of the residual stresses in the far-field area for the plate thickness of 10 mm is shown in Figure 3.22. In order to simulate the stress development at the beginning of the process (Time ≤ 2 s) a time increment convergence study has been carried out in P OURMOGHADDAM et al. (2018a) for glass plates with the thickness of 6 mm, 10 mm and 15 mm. The results agree with those of the 1D simulation (Figure 3.12 (b)). In order to achieve high accuracy and save computing time, FE-simulations of the process start can be carried out in three steps. The time increment Δt1 for the first two seconds is chosen according to Table 3.1(b). In the second step the time increment is changed to Δt2 = 0.05 s (2 s ≤ Time ≤ 5 s) and carried on with Δt3 = 0.1 s to the end of the calculation (Time = 10 s).

3.5 3D FE-Modelling of the Thermal Tempering Process

55

3.5.4 Convergence Study of Mesh Density In order to save calculation time and to maximize the accuracy of the results of the residual stress calculations, the convergence of element size respectively the mesh density in the observation area must be investigated in addition to time stepping. A convergence study for the number of elements along the thickness is carried out in Section 3.4.4 and in N IELSEN (2009). The convergence analyses have shown that 10 elements in a non-equidistant grid through half the thickness with a bias factor of 4 approximately provide an accuracy of 30 unbiased elements (equidistant grid). This saves computation time and ensures accuracy for the simulation of the parabolic stress distribution along the thickness. The convergence study of the mesh density around an observation point is carried out on a 3D infinite plate with the thickness of 10 mm (P OURMOGHADDAM et al., 2018a). The fine-meshed area around the observation point, which is the focus of the convergence analysis, is called the observation area. So the FE-model is divided into a fine-meshed observation area with elements of equal edge length and an area with a coarse mesh, see Figure 3.23. The mesh consists of 3-D 20-Node hexa hedron elements (see Section 3.5.2). The edge length of the elements at the surface in the observation area is chosen equal to the element thickness at the surface. The bias factor along the thickness is set to 4. The

Figure 3.23 Sketch of the infinite plate with a fine-meshed observation area with nE elements of equal size around the observation point

56

3 Numerical Simulation of the Thermal Tempering Process

Figure 3.24 Convergency of the mesh density ρE [1/mm2 ] around the observation point at the surface of the plate (thickness of 10 mm); Deviation from a model using ρE = 2 1/mm2 in the observation area

symmetry of the model is used for the calculations, see Figure 3.23. The mesh density ρE within the observation area is expressed as: ρE =

nE Aobs

(3.51)

Where nE is the number of elements within the observation area with the surface of Aobs around the observation point. For the convergence study, nE is changed while the resulting mesh density ρE is determined. The solutions are compared to a mesh density of ρE = 2 1/mm2 . In Figure 3.24, the results of the convergence study of the mesh density around the observation point for the residual stresses at the surface and in the mid-plane of an infinite plate is presented.

3.6 Summary The numerical simulation of 1D forced convection in order to calculate residual stresses after and during the thermal tempering process of soda-lime silica glass in far-field area was performed. The mathematically described transient temperature field developing on the basis of Newton’s law of cooling and the resulting stress field along the glass thickness due to the thermo-mechanical material behaviour of glass was implemented numerically. The MATLAB source code required for the simulation can be found in Appendix A.2. The calculation results of the residual stresses for different heat transfer coefficients were validated by experimental test data and showed good agreement. In summary, the one-

3.6 Summary

57

dimensional model of the thermal tempering process provides sufficiently accurate results and is well suited for residual stress calculations in the far-field area of glass plates. Convergence studies were carried out on the model parameters time increment Δt and grid node number Nn for the equidistant and non-equidistant grid discretization along the glass thickness. It was shown that the calculation results depend above all on the choice of the model parameters mentioned. The thickness also plays an important role. Due to the high cooling rates required for thin glass plates in order to achieve the same residual stress as for thicker plates, smaller time increments must be selected. For the calculation of the maximum surface tensile stress, even smaller time increments have to be selected due to the large temperature change over the time at the beginning of the cooling process (steep temperature-time curve) and due to the change in material behaviour in the glass transition range. From the results of the convergence analyses, minimum values of the model parameters for different glass thicknesses were proposed with which a sufficiently high accuracy of the stress results can be achieved. Furthermore, a method for 3D FE-modelling of tempered glass plates with a perturbation in the form of drill holes or cut-outs was presented. The significant difference is the different treatment of the heat transfer coefficient due to different cooling rate for the different convection surfaces: inner surface of the hole, chamfer and edge. In Appendix A.3, residual compressive stresses and the required heat transfer coefficients h based on the simulation of 1D forced convection are given for different glass thicknesses. For the identification of the heat transfer coefficients of different convection surfaces based on experimental data (B ERNARD et al., 2009) a scaling factor β was defined in order to assume a simple relation of the heat transfer coefficient between surface and hole area. The described FE-model can be used to calculate residual stresses for glass plates with holes or cut-outs. The results of the numerically simulated residual stresses for tempered glass plates with the variation of geometrical parameters for typical geometries of holes and cut-outs have been presented in P OURMOGHADDAM et al. (2018a). The influence of the hole, edge and corner distances on the minimum residual compressive stresses at holes after the tempering process has been demonstrated in P OURMOGHADDAM et al. (2016). In contrast to the thermal tempering process, during additive manufacturing (AM) the glass must be cooled slowly and in a controlled manner (annealing process) from a processing temperature well above the glass transition temperature (approx. 1030 ◦ C for soda-lime-silica glass and approx. 1260 ◦ C for borosilicate glass). The aim is to minimize a possible build-up of stress due to the resulting temperature gradients during the cooling process. The stresses occurring during the cooling process can lead to cracking in the process and produce undesired residual stresses in the finished product. The knowledge of glass strength in the range up to the glass transition temperature can thus also be used for the further development and optimization of innovative manufacturing techniques such as

58

3 Numerical Simulation of the Thermal Tempering Process

the additive manufacturing with glass (K LEIN et al., 2015; KOTZ et al., 2017). Particularly, in numerical simulations of the cooling process of glass using Fused Deposit Modelling (FDM) techniques, glass strength at elevated temperatures up to the glass transition temperature plays a decisive role as it is necessary for the design due to the stresses that arise during the process. An improvement of the process technology for additive manufacturing (AM) with glass can only be achieved by precise knowledge of the glass strength in this temperature range (P OURMOGHADDAM et al., 2018c).

4 Experimental Investigations into the Fragmentation of Tempered Glass 4.1 General Glass is one of the most popular building materials today. However, the tensile strength is governed by small flaws that significantly reduce the actual engineering strength of annealed float glass, see e.g. S CHULA (2015), H ILCKEN (2015) or S CHNEIDER et al. (2016). Thermally tempered glass shows a higher resistance to external loads due to its residual stress state and, in case of failure, is quite safe with regard to cutting and stitching due to the small, blurred fragments. For this reason, thermally tempered glass is also referred to as safety glass. The residual stress state is obtained by the tempering process (Chapter 3) and is approximately parabolic distributed along the thickness with the compressive stress on both surfaces and an internal tensile stress in the mid-plane (Figure 2.6). By imposing a compressive residual stress at the surface, the surface flaws will be in a permanent state of compression which has to be exceeded by externally imposed stress before failure can occur (S CHNEIDER, 2001; N IELSEN et al., 2010a; P OURMOGHADDAM et al., 2016; P OURMOGHADDAM et al., 2018a). The amount of the residual surface compressive stress largely depends on the cooling rate and therefore on the heat transfer coefficient between the glass and the cooling medium (G ARDON, 1965; A RONEN et al., 2017) (cf. Chapter 3). However, thermally tempered glass will fragmentize completely into many pieces, if the equilibrated residual stress state within the glass plate is disturbed sufficiently and if the elastic strain energy in the glass is large enough (Figure 4.1). The so-called Rupert’s drop with bulbous head and thin tail can withstand high impact or pressure applied to the head, but explodes immediately into small particles when the tail is broken (S ILVERMAN et al., 2012). The dynamic fragmentation process develops fractal through repeated branching of propagating cracks. The fragmentation of tempered glass sets standards for the quality of the thermal tempering process and the degree of safety. The European Standard EN 12150-1 (2015) defines the minimum number of fragments required for soda-lime-silica safety glass based on fragmentation test results. As an example, for glass thicknesses of 4 mm to 12 mm the counted number of fragments in an observation field of 50 mm × 50 mm should not be less © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6_4

60

4 Experimental Investigations into the Fragmentation of Tempered Glass

(a)

(b)

Figure 4.1 (a) Fragmentized tempered glass (b) Tempered glass fragments

than 40 pcs. Several studies on the fragmentation behaviour in tempered glasses showing the relation between the residual stress, glass thickness and fragment density have been reported (ACLOQUE, 1956a; A KEYOSHI et al., 1965; BARSOM, 1968; G ULATI, 1997; S HUTOV et al., 1998; M OGNATO et al., 2017; P OURMOGHADDAM et al., 2018d). The interrelation between the glass thickness and the fragmentation behaviour in tempered thin glasses with glass thicknesses 2.1 mm to 3.2 mm has been studied by L EE et al. (2012). The first investigations of the characteristic fracture behaviour of tempered glass were carried out by Acloque in the late 1950s and early 1960s, see e.g. ACLOQUE (1956b), ACLOQUE (1962), and ACLOQUE (1963). Thermally tempered glass was fragmented by the impact of a sharp steel tip in order to penetrate the surface layers with residual compressive stresses, minimizing the external energy added to the system. Furthermore, pictures of the fragmentation process were captured at a frame rate of 1 μs. Kerkhof also performed high-speed recording of crack branching in glass panes under different failure stresses at frame rates of up to 2.5 μs (K ERKHOF, 1976). The Cranz-Schardin method with rotating mirrors (C RANZ et al., 1929), introduced in 1929, was used for the highspeed recordings. The in-plane shape of the crack front was shown in ACLOQUE (1956b) by fragments with clearly visible and regular line structures occurring on the fracture surface, the so-called Wallner-lines (WALLNER, 1939). The velocity of the crack propagation in soda-lime glass has been determined experimentally by several authors in the past to be approximately 1500 m/s, see e.g. S CHARDIN et al. (1937), ACLOQUE (1956b), D ÖLL (1975), C HAUDHRI et al. (1986), TAKAHASHI (1999), and N IELSEN et al. (2009). The velocity was also derived theoretically by energy considerations and verified experimentally by K ERKHOF (1963). High-speed observations of the catastrophic fracture propagation in thermally tempered soda-lime-silica glass were also provided by N IELSEN et al. (2009). The characteristic fragments were generated by a whirl-like crack propagation. Pictures

4.1 General

61

of the in-plane shape of the fracture front, captured during fragmentation showed that the fracture front is more complex than what was derived from Wallner-lines by Acloque. In this chapter experimental investigations on the fragmentation behaviour and the fracture pattern morphology of heat treated glass plates with different thicknesses and different residual stress levels respectively elastic strain energies are presented. For this purpose, the theoretical background of the energy conditions in an elastic body is described in Section 4.2. The fracture tests are carried out in order to determine the degree of tempering required to ensure the safe character of glass fragmentation, the fragment density resulting from fracture and the average fragment size. The results of the fracture tests are also used for the development and calibration of a model for the prediction of the 2D macro-scale fragmentation of glass (Chapter 5). A simplified model and an experimental validation of the residual stress levels is provided in Section 4.3 for the determination of the engine power of the tempering oven in order to product differently heat treated specimens used in the fracture tests. The fracture tests and the experimental results are presented in Section 4.4. Fractographic investigations of the fracture surface are presented in Section 4.5 in order to outline the influence of the residual stress state on the fractographic features of the fracture surface. In Section 4.6, the fragments from the fracture tests are examined by Computer Tomography (CT). Here, the fragment and fracture surface area are determined. To describe the process of fracture in tempered glass, a few terms are defined: The word crack refers to the local development of a single crack before it branches. Fragmentation is used for describing the overall fracture process, including multiple crack- and crack branching processes. The results provided in Section 4.3 and the results of the fracture tests in Section 4.4 have been pre-published in P OURMOGHADDAM et al. (2018d) and P OURMOGHADDAM et al. (2019).

62

4 Experimental Investigations into the Fragmentation of Tempered Glass

4.2 Strain Energy Conditions Fragmentation is the direct consequence of the energy that is stored inside the material. In tempered glass, an elastic strain energy U is generated within the glass plate due to the parabolic stress state that governs the fragmentation behaviour and the subsequent fragment size (BARSOM, 1968; G ULATI, 1997; S HUTOV et al., 1998; WARREN, 2001; TANDON et al., 2005; N IELSEN, 2017; P OURMOGHADDAM et al., 2018e). Thus, the fragment size depends on the amount of the stored energy. Small fragments are caused by a high stored strain energy due to the high residual stress state originating from the extremely rapid cooling. And lower residual stress states result in larger fragments due to lower stored strain energy. Thus, not only the stress but also the thickness of the glass plate plays a role in determining the strain energy. When external forces or residual stresses deform an elastic body, work is performed, as their points of application are displaced. This work is stored in the body as elastic strain energy. The total elastic strain energy UT stored in a deformed linear elastic, isotropic body is obtained by integrating the energy over the volume of the body: UT =

1 2

 V

σi j εi j dV

(4.1)

where σi j is the stress tensor and εi j is the strain tensor. The residual stress in thermally tempered glass is distributed parabolically along the glass thickness t (Figure 2.6). This parabolic stress distribution σ (z), presented in Eq. (2.3), can be written in terms of the surface stress σs as: 1 2z σ (z) = σs (1 − 3ζ 2 ), ζ = 2 t

(4.2)

using symbols defined in Figure 2.6. The parabolic stress distribution is in equilibrium and symmetric about the mid-plane. Because of the parabolic distribution it can be assumed that the surface compressive stress is twice the mid-plane tensile stress in magnitude (2σm = −σs ). The zero stress contour is located at a depth of approximately 20% of the thickness t. The stress state σ is assumed to be planar. Using cylinder coordinates r for the radius and θ for the polar angle the constitutive relationship for plane stress is defined by: ⎤ ⎡ 1 σr ⎣ σθ ⎦ = E ⎣ν 1 − ν2 τrθ 0 ⎡

ν 1 0

⎤⎡ ⎤ 0 εr 0 ⎦ ⎣ εθ ⎦ 1−ν γrθ 2

(4.3)

with the Young’s modulus E and the Poisson’s ratio ν. The equilibrated stress state is also assumed to be hydrostatic for field stresses. A planar hydrostatic stress state means that no

4.2 Strain Energy Conditions

63

shear stresses occur (τrθ = 0) and the normal stresses are always principal stresses, which are equal in the plate plane (σr = σθ = σ (z)) and zero in the direction of the thickness (σz = 0). This assumption is reasonable for thin glass plates and as long as locations far from edges (far-field) are considered. The total elastic strain energy UT can be written as: UT =

1 2

 V

(σr εr + σθ εθ ) dV =

1−ν E

 V

σ 2 (z) dV

(4.4)

with εr = εθ =

1−ν σ (z) E

(4.5)

Inserting the residual stress field from Eq. (4.2) and integrating over the cylinder we find: UT =

1−ν E

= πR2

z=0.5t r=R   θ=2π

σ 2 (z) rdθ dr dz

z=−0.5t r=0 θ =0

(4.6)

(1 − ν) 2 tσs 5E

which is the initially stored elastic strain energy in a cylindrical body of the radius R, thickness t and the residual surface stress of σs in case of a parabolic stress distribution along the thickness (see Eq. 4.2). The stored energy in Eq. (4.6) can be written in terms of the strain energy per unit surface area of any given base shape of a body by dividing with the base area for the cylinder (BARSOM, 1968; G ULATI, 1997; WARREN, 2001; N IELSEN, 2017; P OURMOGHADDAM et al., 2018e): U=

(1 − ν) 2 tσs 5E

(4.7)

The energy can also be written in terms of the energy density UD : UD =

1 (1 − ν) 2 4 (1 − ν) 2 σs = σm 5 E 5 E

(4.8)

which is the amount of elastic strain energy stored in the system per unit volume and thus only depends on the residual stress and the material properties. In Figure 4.2, four examples of fracture patterns showing the fragment size for different heat treated tempered glass plates with the same thickness of 12 mm are shown. The amount of the elastic strain energy U is calculated using Eq. (4.7) considering the mid-plane tensile stress with 2σm = −σs . The influence of the elastic strain energy on the fragmentation behaviour respectively the fragment size is evident. Under the assump-

64

4 Experimental Investigations into the Fragmentation of Tempered Glass

(a)

σm = 57.9 MPa; U = 354.7 J/m2

(c)

σm = 31.4 MPa; U = 104.1 J/m2

(b)

(d)

σm = 38.4 MPa; U = 156.2 J/m2

σm = 27.2 MPa; U = 78.1 J/m2

Figure 4.2 Fragmentation for different residual stress states at a plate thickness of t = 12 mm; indication of the biaxial tensile residual stress in the mid-plane σm [MPa] and the resulting elastic strain energy U [J/m2 ]

tion of a cylindrical fragment, while at a high energy of U = 354.7 J/m2 the size of the fragments (approximately equal to the diameter) is smaller than 10 mm, at the same thickness a smaller energy of U = 78.1 J/m2 results in a multiple of the fragment size. In Figure 4.3 (a), the relation between the elastic strain energy per unit surface area and the residual tensile stress in the mid-plane of a glass plate dependent on its thickness is shown. The strain energy per unit surface area due to the residual stress increases proportionally to the thickness of the glass plate. Since the fragmentation is governed by the residual stress respectively the released strain energy, for two glass plates with the same residual stress state the one with the higher thickness will fragmentize in smaller fragments than the thinner glass plate, see Figure 4.2. However, the energy density is independent of the thickness. In the case of the elastic strain energy density UD , the curves for different glass thicknesses coincide to a single curve, see Figure 4.3 (b). Accordingly, the elastic strain

4.3 Heat Treatment of Glass Specimens

(a)

65

(b)

Figure 4.3 (a) Elastic strain energy U [J/m2 ] per unit surface area vs. Mid-plane tensile stress σm [MPa] for different glass thicknesses (1.8 mm to 19.0 mm) (b) Elastic strain energy density UD [J/m3 ] vs. Mid-plane tensile stress σm [MPa]; note that all curves are coinciding

energy density depends only on the stress distribution function (constant, linear, parabolic, etc.), i.e. on the loading condition and the linear elastic material properties.

4.3 Heat Treatment of Glass Specimens In order to produce glass specimens with systematically varying residual stresses for a subsequent fragmentation analysis (Section 4.4) float glass specimens have to be differently heat treated due to the variation of the cooling rate during the tempering process. This requires the variation of the engine power of the tempering oven. In this section, a simplified model is presented for the determination of the engine power as a function of the residual stress using quench parameters by forced convection. The necessary empirical equations of integral heat and mass transfer coefficients are introduced, which were suggested by M ARTIN (1977). For a residual stress dependent production of thermally tempered glass, float glass plates are thermally tempered due to heat treatment of the glass with different heat transfer coefficients. In the method presented, quench parameters for determining the engine power required to reach the target residual stresses are taken into account. The plausibility of the model is checked on the basis of experimental data. The results have been pre-published in P OURMOGHADDAM et al. (2019) The residual stresses are strongly process-related and vary under different boundary conditions such as the cooling rate, the nozzle arrangement, the nozzle diameter, the distance

66

4 Experimental Investigations into the Fragmentation of Tempered Glass

between the nozzles and the glass surface as well as the roller distances. The initial temperature as well as the cooling rate in particular have a significant influence on the residual stress development during the tempering process (Chapter 3), see e.g. NARAYANASWAMY et al. (1969) and A RONEN et al. (2017). The heat transfer coefficient, which describes the cooling property of a surface and governs the cooling process is difficult to determine experimentally. Therefore, analytical or complex numerical determination of the heat transfer coefficient is required for the numerical simulation of the tempering process. This section presents equations compiled by M ARTIN (1977) from experimental data to determine the heat transfer coefficient with knowledge of the process limits as described above. This can also be used for the determination of the desired residual stresses in glass plates from the tempering process. It is also important for the manufacturer to be able to calculate the residual stress before the process for an effective utilization of the tempering oven and also for an accurate residual stress result. In order to study the correlation between the residual stress, particle count, particle weight and particle size, glass plates with three different thicknesses of 4 mm, 8 mm and 12 mm were tempered with heat treatment conditions for a predetermined residual stress range. For the different heat treatment of the specimens the engine power of the tempering oven was estimated and varied by recalculating the air velocity and the air pressure from the iteratively determined heat transfer coefficients needed for yielding the target residual stress. After the tempering process, the residual stress in the glass specimens were measured, using a scattered light polariscope (SCALP).

4.3.1 Specimen Preparation for the Thermal Tempering For the fracture tests three series of tempered glass specimens of size 360 mm × 1100 mm with three different thicknesses of 4 mm, 8 mm and 12 mm were produced (see Table 4.1). There were 24 specimens in each series divided into eight groups of three specimens for each run of the tempering process. Hence, there were eight runs of the tempering process per series for achieving eight different residual stresses in each series. In order to choose the target residual stresses for the heat treatment of the series, the elastic strain energy level for the start of the micro-crack branching of glass according to F INEBERG (2006) was considered. However, this level of energy represents the start of local micro-crack branching by one rapidly moving tensile crack. The question was how this level would influence the fracture pattern of a tempered glass. Therefore, in the first step the target residual stresses were chosen above as well as below the strain energy level of 35 J/m2 . The aim was to achieve residual stresses that lead to a range of fine to coarse fragmentation of the specimens. As it is shown in Figure 4.4, eight target residual mid-

4.3 Heat Treatment of Glass Specimens

67

Table 4.1 Series for the tempering process

Series

Specimens per series

Specimens per run

Runs per series

Dimension [mm]

Thickness [mm]

ScA ScB ScC

24 24 24

3 3 3

8 8 8

360 × 1100 360 × 1100 360 × 1100

4 8 12

Figure 4.4 Eight target residual mid-plane tensile stresses for the heat treatment of the series with three different thicknesses 4 mm, 8 mm and 12 mm considering the level of strain energy for the start of micro-crack branching according to F INEBERG (2006)

plane tensile stresses were chosen from 10 MPa to 60 MPa for the 8 mm and 12 mm thick specimens and from 10 MPa to 45 MPa for the 4 mm thick specimens. For different heat treatment of the series, the engine power of the thermal tempering oven was varied. Thus, the engine power required for the corresponding residual stress was determined considering the procedure sketched in Figure 4.5. The engine power for the cooling section of the thermal tempering oven was recalculated by the determination of the required heat transfer coefficient h, which in turn leads to a certain cooling air velocity w and subsequently a required air pressure P. The engine power was then calculated considering the experience values for the relation between the adjusted engine power and the resulting air pressure P. The specimens were tempered using the tempering oven of the company Semcoglass Holding GmbH. In Figure 4.6, the tempering oven (a), the cooling device (b) and the nozzle arrangement in the cooling section (c) are shown.

68

4 Experimental Investigations into the Fragmentation of Tempered Glass

Figure 4.5 Procedure for the determination of the engine power of the tempering oven

(a)

(b)

(c)

Figure 4.6 (a) Tempering oven of the company Semcoglas Holding GmbH, (b) Cooling device upper side, (c) Rollers and nozzle arrangement in the cooling section

4.3 Heat Treatment of Glass Specimens

69

4.3.2 Cooling Air Velocity The heat transfer coefficient describes the cooling properties of a surface and thus is a significant parameter of the quenching process. However, the heat transfer coefficient cannot be set in the system parameters of the tempering oven. Only the power of the engine, which leads to a certain cooling air velocity of the nozzles, is adjustable for quenching process. In order to obtain different residual stress levels in the tempered glass plate series, it is therefore necessary to calculate various cooling air velocities. This starts with the heat transfer coefficient required for the target residual stress and continues with the cooling air velocity to the engine power, see Figure 4.5. Hence, before the cooling air velocity of the nozzles can be determined, the heat transfer coefficient required for yielding the target residual stress must first be identified. A correlation between the heat transfer coefficient in the far-field area of an infinite glass plate and the resulting numerical residual stress was simulated and described in Chapter 3 using the numerical model for the 1D forced convection. The values for the heat transfer coefficient to the respective residual stresses are taken from the tables in Appendix A.3. This section describes a method for determining the cooling air velocity using the known heat transfer coefficient. The various cooling air velocities w are calculated using the empirical equations of integral heat and mass transfer coefficients suggested by M ARTIN (1977). For the calculations the dimensionless Nusselt number Nu, Reynolds number Re and Prandtl number Pr are required, which are material-dependent and can be written as: Nu =

hD k

wD υ υ Pr = a Re =

(4.9) (4.10) (4.11)

where D is the inner diameter of a nozzle, h the heat transfer coefficient and w is the cooling air velocity. The material values of the cooling medium, air, are the thermal conductivity k, the kinematic viscosity υ and the thermal diffusivity a at an arithmetically averaged fluid temperature Tm = (TN + TS )/2 between the temperature at nozzles outlet TN and the temperature at the surface of the glass plate TS . Accordingly, the heat transfer coefficient h is non-dimensionalized with the Nusselt number Nu as the comparison between the conduction and convection heat transfer rates. Reynolds number Re gives informations about the type of fluid (lamina, turbulent, etc.) and Prandtl number Pr assesses the relation between momentum transport and thermal transport capacity of a fluid. A sketch of the impinging jet flow which the empirical equations for the empirical determination of the heat and mass transfer coefficient are based on is shown in Figure 4.7.

70

4 Experimental Investigations into the Fragmentation of Tempered Glass

The arrangement and type of the nozzles, as shown in Figure 4.8, influence the calculation (S CHABEL et al., 2013). Typical arrangements in the area of thermal tempering process are array of round nozzles (ARN) and array of slotted nozzles (ASN) (cf. Figure 4.8). The cooling section of the tempering oven, which was used for the thermal tempering of the specimens, had sheet metal boxes connected in parallel in the operating direction with triangular array of holes as air nozzles, see Figure 4.6 (c). For further calculations, therefore a triangular arranged array of round nozzles ARNΔ is assumed. The empirical equations for the integral heat and mass transfer coefficients for ARNΔ , as shown in Figure 4.8 (c), in terms of the practical application is suggested by M ARTIN (1977) as: NuARN = G · Re2/3 · Pr0.42

(4.12)

   −0.05 10 · h∗ · d ∗ 6 d ∗ · (1 − 2.2d ∗ ) · 1+ G= 1 + 0.2 · (h∗ − 6) · d ∗ 6

(4.13)

with the validity range for G: 0.004 ≤

(d ∗2 = f ) ≤ ∗

0.04,

2≤

(h = H/D) ≤

12,

2000 ≤

Re ≤

100 000

In the case of triangular arranged array of round nozzles (ARNΔ ) the relative nozzle area f can be written as: π D2 f= √ · 2 2 3 LT  D d ∗ = f = 0.9523 LT

(4.14) (4.15)

where LT is the distance between the nozzles and H is the distance between the nozzles and the surface of the glass plate, see Figure 4.7. The cooling air velocity was recalculated using the Eqs. (4.9) - (4.15). By inserting the Eqs. (4.9), (4.10) and (4.11) in Eq. (4.12) the heat transfer coefficient h is determined:   wD 2/3  υ 0.42 Gk · · (4.16) h= D υ a

4.3 Heat Treatment of Glass Specimens

71

Figure 4.7 Sketch of the impinging jet flow, inner diameter of a nozzle D and distance H between the nozzles and the surface of the glass plate

(a)

(b)

(c)

(d)

Figure 4.8 Nozzle configuration: (a) Single round nozzle (SRN), (b) Single slotted nozzle (SSN), (c) Array of round nozzles (ARN), (d) Array of slotted nozzles (ASN) according to VDI Heating Atlas (S CHABEL et al., 2013)

72

4 Experimental Investigations into the Fragmentation of Tempered Glass

(a)

(b)

Figure 4.9 Cooling air velocity w [m/s] vs. (a) Heat transfer coefficient h [W/m2 K] and (b) Target residual surface compressive stress σs [MPa]; solid curves are trend lines including the corresponding function and the coefficient of determination R2 ; geometry values: H = 50 mm, D = 5 mm and LT = 5 mm (ARNΔ ); material values of dry air at p = 0.101325 MPa: k = 48.334×10−3 W/mK, υ = 57.217×10−6 m2 /s and a = 80.606 × 10−6 m2 /s (L EMMON et al., 2010); TN = 25 ◦C; TS = 650 ◦C

In Figure 4.9 (a), the cooling air velocity w of the nozzles is shown as a function of the heat transfer coefficient required for the target residual stresses. The value for the cooling air velocity w was varied iteratively with Eq. (4.16) until the required heat transfer coefficient h for the target residual stress was reached. The cooling air velocities in correlation of the target residual surface compressive stresses are shown in Figure 4.9 (b). The calculations were carried out with the nozzle values LT = 5 mm, H = 50 mm and D = 5 mm for the cooling section of the thermal tempering oven for an arithmetically averaged temperature Tm from the temperatures TN = 25 ◦C and TS = 650 ◦C. The thermodynamic material properties of the cooling medium air k, υ and a were taken from the literature (L EMMON et al., 2010) for dry air at an atmospheric pressure of p = 0.101325 MPa.

4.3.3 Engine Power of the Tempering Oven The glass plates were tempered in groups of three using the thermal tempering oven of the company Semcoglas Holding GmbH. Once the cooling air velocity w for different target residual stress states is determined, the air pressure P can be calculated using Eq. (4.17) for the total pressure with the air velocity pressure and the hydrostatic air pressure: 1 P = ρw2 + ρgh j 2

(4.17)

4.3 Heat Treatment of Glass Specimens

73

Figure 4.10 Engine power EP [%] in percentage of the total power vs. air pressure P [Pa]; points show experience values of the tempering oven and solid curve show the trend line including the corresponding function and the coefficient of determination R2

where h j is the distance between the pressure chamber and the air jets, ρ the air density and g is the gravity. For the determination of the air pressure, an average atmospheric pressure of 0.101325 MPa and the temperature of 25 ◦C was assumed, resulting in an air density of ρ = 1.184 kg/m3 (L EMMON et al., 2010). The distance between the pressure chamber and the air jets h j was measured in this case with approximately 0.65 m. After the determination of the required air pressure for each target residual stress considering the cooling air velocity and the heat transfer coefficient, the air pressure was matched to the empirical function of the thermal tempering oven in terms of the correlation between the engine power EP as the percentage of the total power of the thermal tempering oven and the resulting air pressure in the height of the air jets. The empirical function of the thermal tempering oven is shown in Figure 4.10. The points show experience values of the tempering oven. Based on the empirical function of the tempering oven, the required values of the engine power EP for achieving the target residual stresses was determined. The target residual surface stress is shown in correlation with the air pressure P in Figure 4.11 (a) and with the required engine power EP in Figure 4.11 (b). The eight target residual mid-plane tensile stresses (Figure 4.4) and the required process parameters of the tempering oven, i.e. cooling air velocity, air pressure and subsequently the required engine power are summarized in Table 4.2 for the three thicknesses of 4 mm, 8 mm and 12 mm.

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4 Experimental Investigations into the Fragmentation of Tempered Glass

(a)

(b)

Figure 4.11 (a) Air pressure P [Pa] vs. Target residual surface compressive stress σs [MPa], (b) Engine power EP [%] vs. Target residual surface compressive stress σs [MPa]; solid curves are trend lines including the corresponding function and the coefficient of determination R2

Table 4.2 Target residual surface stresses σs [MPa] and the corresponding system parameters: cooling air velocities w [m/s], air pressure P [Pa] and the required engine power EP [%] as the percentage of the total power, h j = 0.65 m, ρ = 1.184 kg/m3 (atmospheric pressure and 25 ◦C)

t = 4 mm σs w [MPa] [m/s] -20 -30 -40 -50 -60 -70 -80 -90

9.0 16.0 25.0 35.0 49.0 61.0 78.0 103.0

P [Pa] 55.5 159.1 377.5 732.7 1428.9 2210.4 3609.3 6288.1

t = 8 mm EP σs w [%] [MPa] [m/s] 6.6 12.1 19.7 28.8 42.1 54.0 71.3 97.8

-20 -30 -40 -50 -60 -80 -100 -120

5.1 8.9 13.3 19.2 26.1 43.7 69.8 102.4

P [Pa] 22.9 54.4 112.3 225.8 410.8 1138.1 2891.8 6215.1

t = 12 mm EP σs w [%] [MPa] [m/s] 4.0 6.6 9.9 14.7 20.7 37.0 62.9 97.2

-20 -30 -40 -50 -60 -80 -100 -120

P [Pa]

EP [%]

3.8 16.1 3.3 6.2 30.3 4.7 9.2 57.7 6.8 12.7 103.0 9.4 17.0 178.6 12.9 27.4 452.0 21.9 41.2 1012.4 34.6 59.0 2068.3 52.0

4.3 Heat Treatment of Glass Specimens

75

4.3.4 Residual Stress Measurements and Approach Assessment After thermal tempering, it must be checked whether the target residual stresses have been reached. In order to check the actual stress state in the specimens the residual stresses were measured after the tempering process using a scattered light polariscope (SCALP) developed by GlasStress Ltd. The residual stresses in vertical and horizontal direction in thirteen measurement points were measured at both surfaces of the specimens, see Figure 4.12. The anisotropy of the residual compressive surface stresses in the area of the measurement points was quite low with a coefficient of variation around 3%. In Figure 4.13, the optical path difference under polarized light is shown for three different thicknesses. It was observed that the anisotropy and the inaccuracies in yielding the target residual stresses increased for the 12 mm thick plates. In Figure 4.14 (a), the target residual stresses in comparison to the average of the measured mid-plane tensile stresses in the measurement points after the tempering process is shown. Due to the lack of accuracy of the thermal tempering oven for the low engine power range, it was not possible to reach the lowest target residual stress states, especially for the 12 mm thick plates, see Figure 4.14 (a). However, in view of the objective of producing glass specimens with different residual stress levels for a meaningful stress or strain energy dependent fragmentation analysis, the heat treatment of the specimen series was both necessary and successful. In Figure 4.14 (b), the correlation between the measured residual mid-plane stress and the engine power of the tempering oven is shown. The values for the engine power indicated on the horizontal axis are the required fan power of the motor in relation to the maximum power. With the help of SCALP measurements it could be shown that this approach is promising. In about 70% of the glass panes, a smaller deviation than 5% to the target residual stress was observed.

Figure 4.12 Positions and directions of the residual stress measurements

76

(a) t = 3.8 mm

4 Experimental Investigations into the Fragmentation of Tempered Glass

(b) t = 7.9 mm

(c) t = 12.0 mm

Figure 4.13 Optical path difference under polarized light, (a) t = 3.8 mm, σs = −91.3 MPa (rel. deviation of 1.28%) (b) t = 7.9 mm, σs = −91.1 MPa (rel. deviation of 1.15%), (c) t = 12.0 mm, σs = −97.4 MPa (rel. deviation of 2.01 %)

(a)

(b)

Figure 4.14 (a) Elastic strain energy U [J/m2 ] vs. Mid-plane tensile stress σm [MPa]; coloured markers show the average residual mid-plane stress from thirteen measurement points; hollow circles show the target residual mid-plane stress from Figure 4.4, (b) Mid-plane tensile stress σm [MPa] vs. Engine power EP [%] of the tempering oven; coloured markers show the measured residual stress after the tempering process, solid curves are trend lines including the corresponding function and the coefficient of determination R2

4.4 Fracture Tests and Fragmentation Analysis

77

It is important for the manufacturer to be able to determine and set the engine power according to the residual stress. This increases or optimizes the effectiveness of the tempering oven and production. It was observed that the accuracy of residual stresses decreased with thicker glass plates and at lower cooling rates. Inaccuracies of the residual stress results for thicker glass plates (e.g. 12 mm) at lower engine powers than 20% were expected. Due to the glass cooling property, based on the viscoelastic material behavior at temperatures above glass transition temperature and the temperature dependency of the glass structure, in thicker plates very high residual stresses occur at very low cooling rates. The inaccuracy due to the lower cooling rates may be due to the inaccuracy of the fan performance. The determined engine power for achieving low residual stresses could not be set exactly. However, despite the inaccuracies at low residual stresses, the process parameters for quenching glass sheets can be determined from these simplified calculations at the typical industrial cooling rates and engine power range of 30% to 100%.

4.4 Fracture Tests and Fragmentation Analysis The fracture tests and the results of the investigation into the fragment size of tempered glass plates with various residual stress levels are presented in order to correlate the fragment size or fragment density in an observation field with the residual stress or elastic strain energy resulting from the thermal tempering process. This section presents the relation between the fragment density and the permanent residual stress in fragmented tempered glasses of various thicknesses. The specimen preparation was discussed in Section 4.3. Fracture tests were carried out on tempered glass plates and the number of fragments in certain observation fields were counted. The term Fragment density is used for the number of fragments within an observation field of size 50 mm × 50 mm and is abbreviated as N50 . The average fragment density in the observation fields was set in correlation with the average measured residual stress or the elastic strain energy U of each specimen. Furthermore, the average particle weight of 130 particles per specimen chosen by random was determined. The relation between the average particle weight and the measured residual stress is given. The volume and the base surface as well as the radius of the particles are calculated assuming cylindrical fragments with approximately unchanged thicknesses. The relation between the residual stress and the particle base surface of regular polygonal shapes n = 3 to n = 8 edges in addition to the cylindrical fragment (n → ∞) is also determined. The glass used for the fracture tests was commercial soda-lime-silica glass with three different thicknesses 4 mm, 8 mm and 12 mm. The results in this work are a basis for the establishment of a theoretical model to predict 2D macro-scale fracture patterns from elastic strain energy in tempered glass which is the subject of Chapter 5. The results of the investigations in this section have been pre-published in P OURMOGHADDAM et al. (2018d).

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4 Experimental Investigations into the Fragmentation of Tempered Glass

4.4.1 Fragmentation and Fracture Patterns The heat treated series were fractured by impact according to EN 12150-1 (2015). In EN 12150-1 (2015), the boundary of the fragmentation analysis, the position of impact and the observation field of size 50 mm × 50 mm for counting fragments is regulated, see Figure 4.15. The fracture tests were carried out using a sharp steel center punch and a hammer. The glass plates were fixed at each edge and then fractured by impact with the impact point located on the center axis of the glass plate and 13 mm away from the edge. When elastic strain energy in the glass is large enough and the tip of the center punch penetrates into the tensile stress zone, the glass plate will fragmentize, since the equilibrated residual stress state within the glass plate is disturbed sufficiently. However, the influence of the elastic strain energy on the fracture structure and the fragment size is investigated. In order to keep the influence of the supplied energy uniform and low throughout all series, it was qualitatively ensured that the impact velocity of the center punch remains constant for all samples. For the recording of the fracture pattern, the so-called CulletScanner of the company SoftSolution GmbH was used (Figure 4.16). The CulletScanner desk is equipped with a scanner head, which can be driven over the fractured glass plate and scan the fracture pattern. The scanned fracture patterns were then processed with a program for the analysis of fracture patterns of the same company (C ULLET S CANNER, 2017). The program was used to count fragments in selected observation fields. Some examples of the scanned fracture patterns of the heat treated glass specimens with three different glass thicknesses are shown in Figures 4.17, 4.18 and 4.19. The residual mid-plane tensile stresses given in Figures 4.17, 4.18 and 4.19 are the average values of the thirteen measurement points according to Figure 4.12. The elastic strain energy U was calculated using Eq. 4.7 for the average residual mid-plane tensile stresses considering the parabolic stress distribution

Figure 4.15 Fracture test and fragmentation analysis according to EN 12150-1 (2015); the position of impact; the area excluded from analysis is indicated by the dashed line; observation field of size 50 mm × 50 mm

4.4 Fracture Tests and Fragmentation Analysis

(a)

79

(b)

Figure 4.16 (a) Fracture test using center punch and hammer, (b) CulletScanner of the company Soft Solution GmbH used for scanning the fracture pattern

along the thickness (2σm = −σs ) and the corresponding thicknesses. For the calculation of the elastic strain energy the actual thickness of the specimens were measured. It is obvious that with the same thickness, the fracture structure becomes finer with increasing residual stress level. This applies to all thicknesses. However, it should be noted that the elastic strain energy is the governing mechanical parameter for crack generation. The elastic strain energy stored in the glass specimens due to the residual stress state is released in the case of fracture mainly in the form of crack surface generation (N IELSEN, 2017; P OURMOGHADDAM et al., 2018e). The greater the energy released, the more new crack surfaces are generated. In Figures 4.17, 4.18 and 4.19, the amount of generated crack surfaces increases from fracture patterns (a) to (e), i.e. with increasing elastic strain energy. In the impact tests, the additional energy supply by striking with the hammer must be taken into account. Therefore, depending on the impact hardness, primary cracks appear around the impact point. However, this effect decreases with increasing stored energy, as the ratio of the impact energy to the stored energy decreases. This can be observed, for example, in a comparison of the two fracture patterns in Figures 4.17 (e) and 4.18 (e). Some of the primary cracks initiated at the point of impact will propagate. The proportion of propagating primary cracks also depends on the stored energy. For example, in the fracture pattern in Figure 4.17 (a) at a thickness of 3.9 mm and an elastic strain energy of U = 8.3 J/m2 , only one crack from the primary cracks spreads further, while the other cracks stop propagating at a certain length around the impact point. If enough stored energy is available to release, the cracks will spread to the edge. On the way there, they can also be deflected by the slight change in the material composition or the stress state, see e.g. Figure 4.17 (c). The maximum crack propagation velocity in soda lime glass is approx. 1500 m/s, see e.g. S CHARDIN et al. (1937), ACLOQUE (1956b), D ÖLL (1975), C HAUDHRI et al. (1986), TAKAHASHI (1999), and N IELSEN et al. (2009). If the

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4 Experimental Investigations into the Fragmentation of Tempered Glass

propagating primary crack has reached the maximum fracture velocity before reaching the edge and there is still sufficient energy to release for the crack propagation, new crack surfaces are generated by crack branching. Fragmentation, which is the result of multiple crack branching of the primary cracks, occurs when sufficient stored or supplied energy is available for release. Therefore, the higher the stored energy, the higher the number of crack branching and thus the finer the fracture structure. The scans in Figures 4.20 and 4.21 show fracture patterns of glass heat treated specimens with approximately the same residual stress level but with different glass thicknesses. Due to the different glass thicknesses, different elastic strain energies result per unit surface area. Hence, the same residual stress state results in different fracture patterns respectively different fragment densities. Thus, the residual mid-plane stress of approximately 29 MPa creates a relatively fine fracture structure for a 12 mm thick glass plate (Figure 4.20 (c)), whereas for a 4 mm thick glass plate it just creates a few through cracks (Figure 4.20 (a)). The same can be observed in Figure 4.21, where a higher residual mid-plane stress of approximately 45 MPa also gives different fracture structures depending on thickness. After scanning the fracture structures, the fracture patterns were analysed using the software C ULLET S CANNER (2017), which can count the fragments in selected observation fields.

4.4 Fracture Tests and Fragmentation Analysis

81

(a) t = 3.9 mm; EP = 12%; σm = 15.5 MPa; U = 8.3 J/m2

(b) t = 3.8 mm; EP = 29%; σm = 24.8 MPa; U = 20.6 J/m2

(c) t = 3.8 mm; EP = 54%; σm = 33.7 MPa; U = 38.0 J/m2

(d) t = 3.8 mm; EP = 71%; σm = 40.5 MPa; U = 54.8 J/m2

(e) t = 3.8 mm; EP = 98%; σm = 47.3 MPa; U = 74.8 J/m2 Figure 4.17 Scans of the fracture patterns; engine power EP [%]; avg. measured residual mid-plane stresses σm [MPa]; elastic strain energy U [J/m2 ]; plate thickness t ≈ 4 mm

82

4 Experimental Investigations into the Fragmentation of Tempered Glass

(a) t = 7.8 mm; EP = 10%; σm = 21.2 MPa; U = 30.8 J/m2

(b) t = 7.7 mm; EP = 15%; σm = 24.8 MPa; U = 41.7 J/m2

(c) t = 7.9 mm; EP = 21%; σm = 28.2 MPa; U = 55.3 J/m2

(d) t = 7.7 mm; EP = 37%; σm = 36.4 MPa; U = 89.8 J/m2

(e) t = 7.9 mm; EP = 97%; σm = 53.7 MPa; U = 200.5 J/m2 Figure 4.18 Scans of the fracture patterns; engine power EP [%]; avg. measured residual mid-plane stresses σm [MPa]; elastic strain energy U [J/m2 ]; plate thickness t ≈ 8 mm

4.4 Fracture Tests and Fragmentation Analysis

83

(a) t = 12 mm; EP = 7%; σm = 26.9 MPa; U = 76.4 J/m2

(b) t = 12 mm; EP = 10%; σm = 30.0 MPa; U = 95.0 J/m2

(c) t = 12 mm; EP = 22%; σm = 38.1 MPa; U = 153.3 J/m2

(d) t = 12 mm; EP = 35%; σm = 48.1 MPa; U = 244.3 J/m2

(e) t = 12 mm; EP = 52%; σm = 58.3 MPa; U = 358.9 J/m2 Figure 4.19 Scans of the fracture patterns; engine power EP [%]; avg. measured residual mid-plane stresses σm [MPa]; elastic strain energy U [J/m2 ]; plate thickness t = 12 mm

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4 Experimental Investigations into the Fragmentation of Tempered Glass

(a) t = 3.8 mm; EP = 42%; σm = 29.2 MPa; U = 28.5 J/m2

(b) t = 7.8 mm; EP = 21%; σm = 29.3 MPa; U = 58.9 J/m2

(c) t = 12 mm; EP = 7%; σm = 27.2 MPa; U = 78.1 J/m2 Figure 4.20 Scans of the fracture patterns; engine power EP [%]; avg. measured residual mid-plane stresses σm ≈ 29 MPa; elastic strain energy U [J/m2 ]; different plate thickness

4.4 Fracture Tests and Fragmentation Analysis

85

(a) t = 3.8 mm; EP = 98%; σm = 45.4 MPa; U = 68.9 J/m2

(b) t = 7.9 mm; EP = 63%; σm = 45.4 MPa; U = 143.3 J/m2

(c) t = 12 mm; EP = 35%; σm = 48.0 MPa; U = 243.3 J/m2 Figure 4.21 Scans of the fracture patterns; engine power EP [%]; avg. measured residual mid-plane stresses σm ≈ 45 MPa; elastic strain energy U [J/m2 ]; different plate thickness

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4 Experimental Investigations into the Fragmentation of Tempered Glass

4.4.2 Elastic Wave Propagation The impact generates elastic waves which spread through the solid from the point of impact to the mutual edge of the glass plate and are reflected there. Space waves emanate from the impact point as the activator source and the impressed energy is radiated into the solid as a wave movement. Figure 4.22 shows the propagation of the elastic waves based on the internal energy generated by the impact. The elastic wave propagation is simulated using the FE-program LS-DYNA (LSTC, 2017). The impact is simulated by the fall of a steel ball onto a rigid base glass plate with an isotropic linear elastic material behaviour. Solid elements are used for the simulation of the glass plate. The impact position is 13 mm from the edge according to EN 12150-1 (2015). The status of the elastic wave at the respective propagation times is shown in the Figures 4.22 (a) to (h). At any point in time, the red bordered wave ranges are the areas with the highest internal energy density. The elastic wave reaches the opposite side after approx. 1.12 × 10−4 s. With a propagation width of 347 mm from the impact position to the mutual edge this results in a elastic wave propagation velocity of approx. 3100 m/s. For verification of the numerically calculated wave propagation velocity, the so-called Rayleigh wave (R-wave) propagation velocity is determined. R-waves propagate at the surface and parallel to it. They are one of the three waveforms that occur in wave theory in an elastic body. The other two waveforms are pressure or primary waves (P-waves) and secondary or shear waves (S-waves). R-waves contain P- and S-waves in their form, the latter proportionally stronger. For linear elastic materials with positive Poisson’s ratio (ν > 0), the Rayleigh wave propagation velocity can be approximated as (F REUND, 1998; P ETERSEN et al., 2017): 0.862 + 1.14ν cR ≈ cS 1+ν

(4.18)

where cR is the R-wave propagation velocity and cS the S-wave propagation velocity. The propagation velocity of the shear wave depends on the material density and the shear modulus G = E/2(1 + ν) and can be written as:  cS =

E 2(1 + ν)ρ

(4.19)

For the material glass the relation of R- to S-wave propagation velocity in Eq. (4.18) is cR /cS ≈ 0.91. Thus, the Rayleigh wave propagation velocity for material glass results in cR = 3070 m/s, while the magnitude of the fracture front velocity is v f r ≈ 1500 m/s according to experimental data reported by e.g. S HARON et al. (1999) and N IELSEN et al. (2009). This results in the relations v f r ≈ 0.49cR ≈ 0.45cS . This means that the fracture front follows the elastic wave front and the fracture propagation can be influenced by the reflection of the elastic wave at the mutual edge in addition to the impact. In Figure 4.22 (g), for example, it can be seen that the elastic wave at the time 1.3 × 10−4 s is

4.4 Fracture Tests and Fragmentation Analysis

(a) Time = 10−5 s

(b)

87

Time = 2.7 × 10−5 s

(c) Time = 4.1 × 10−5 s

(d) Time = 5.7 × 10−5 s

(e) Time = 8.1 × 10−5 s

(f)

(g)

Time = 1.3 × 10−4 s

Figure 4.22

Time = 10−4 s

(h) Time = 2.5 × 10−4 s

FE-simualtion of the elastic wave propagation due to impact

reflected at the edge and is on its way back towards the impact position. At this point, the fracture front has only covered a distance of about 0.195 m, i.e. about half the width of the specimen, and will hit the returning elastic wave. It can also be observed that the elastic waves propagate at an angle of about 45◦ and therefore there are areas to the right and left of the impact point that are practically unaffected by the impact. However, the area below the impact position is unsuitable for a fracture analysis due to the high proportion of elastic wave traffic.

4.4.3 Fragmentation Analysis Boundary According to EN 12150-1 (2015) the fragments within the dashed line, as shown in Figure 4.15, should be taken into account for the evaluation of the fragment density. However, in the area below the impact point over the whole width to the mutual edge of the specimen the fracture behaviour, i.e. crack branching and finally the fragment density, is influenced by the low distance of the impact point and the mutual edge. The elastic wave propagation

88

4 Experimental Investigations into the Fragmentation of Tempered Glass

due to impact is shown in Figure 4.22. The cracks propagate in direction of the mutual edge and branch more often along the width of the specimen than along its length to release energy. And since the fracture front propagation and elastic wave front propagation are time-shifted, the energy release for the fracture development is most likely additionally influenced by the reflected elastic wave. In specimens with a higher energy state and the same impact the crack propagation and branching from the location of the impact is much more intensive so that the described effect is less important in determination of the fragment density. In this work, the area in which the fragmentation is influenced by the low distance of the impact point and the mutual edge of the specimen is called the impact influence zone. Therefore, only the fragments in the areas under an assumed angle of 45◦ left and right from the impact point were considered for the investigations. The assumed impact influence zone is shown in Figure 4.23 (a). For the fragmentation analysis the boundary of analysis is defined based on the impact influence zone, see Figure 4.23 (b).

(a)

(b) Figure 4.23 (a) Impact influence zone under 45◦ left and right from the impact point, (b) Boundary of the fragmentation analysis; the area which is excluded from the fragmentation analysis is indicated by the dashed line; observation field of size 50 mm × 50 mm

4.4 Fracture Tests and Fragmentation Analysis

89

4.4.4 Fragment Density For the fragmentation analysis of the fractured tempered glass specimens the fragment density was determined for each specimen in eight observation fields with the size of 50 mm × 50 mm positioned around the residual stress measurement points (Figure 4.12) 1, 4, 5, 6, 8, 9, 10 and 13 within the defined boundary of the fragmentation analysis. The observation fields are shown in Figure 4.24. The reason for the eight observation fields was to have more fragments per specimen for quantitative results of the investigations. Since the main objective was to analyse fragmentation with just the influence of the residual stress state on the crack propagation and crack branching, the fragments in the chosen observation fields were assumed to be caused free from the impact energy as well as from the distance of the impact point and the edge of the specimen. The fragment density N50 was evaluated for each fractured specimen in terms of the average fragment number of the eight observation fields of size 50 mm × 50 mm. To better understand the fracture test results for the fragment density, the definition of fragmentation is required. The fracture structure in which the fracture test resulted in at least one crack branching respectively one fragment within one of the eight observation fields is defined as "fragmentation". Only the specimens for which this fragmentation definition applies were evaluated. In Figure 4.25, two examples for "fragmentation" and "no fragmentation" are shown for a glass thickness of approximately 8 mm. The fracture pattern in Figure 4.25 (a) shows crack formation in impact influence zone due to the increase of energy by the impact, but no fragmentation within the selected boundary of the fragmentation analysis. In contrast, Figure 4.25 (b) shows crack formation within the fragmentation analysis boundary. However, the fragmentation in impact influence zone is also greater here due to the increase in energy. Fragments at the edges of the observation field were counted as half a fragment. The assessment of whether fragments were within or only partially within the observation field was made from case to case. For an overview of the test results with respect to the stress dependency of the fragment

Figure 4.24 Eight observation fields with size of 50 mm×50 mm around the residual stress measurement points 1, 4, 5, 6, 8, 9, 10 and 13 under an angle of 45◦ from the impact point

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4 Experimental Investigations into the Fragmentation of Tempered Glass

(a) t = 7.8 mm; U = 30.8 J/m2

(b) t = 7.8 mm; U = 58.9 J/m2 Figure 4.25

Definition of fragmentation with (a) No fragmentation and (b) Fragmentation

density, the average number of fragments within the eight observation fields is correlated with the measured mid-plane tensile residual stress in the glass specimens. In Figure 4.26, the correlation between the evaluated average fragment density N50 in the observation fields with the size of 50 mm × 50 mm according to Figure 4.24 and the mid-plane tensile residual stress of the specimens is presented. The solid curves show trend lines of the experimental data with the corresponding function and coefficient of determination R2 . As expected, the fragment density increases with increasing residual stress. With the 4 mm thick glass specimens, only 6 out of 24 specimens are fragmented according to the above definition. However, Figure 4.26 shows that to achieve the same number of fragments due to fragmentation, thinner glass plates must have a higher residual stress level than thicker plates. For example, to obtain a fragment number of 10 per observation field of 50 mm × 50 mm, the mid-plane residual tensile stress in a 8 mm thick glass would have to be about 30 MPa, while a 4 mm thick glass would require approximately 40 MPa. In Figure 4.27 (a), the fragment density is shown in correlation with the elastic strain energy per unit surface area of the fragmented specimens. The elastic strain energy was calculated using Eq. (4.7) taking into account −σs = 2σm . Also here the trend lines are added with the corresponding function and coefficient of determination R2 . The trend

4.4 Fracture Tests and Fragmentation Analysis

91

Figure 4.26 Fragment density N50 [−] vs. Mid-plane tensile stress σm [MPa]; the markers show the experimental data for the plate thicknesses t = 4 mm, t = 8 mm and t = 12 mm; solid curves are trend lines including the corresponding function and coefficient of determination R2

(a)

(b)

Figure 4.27 (a) Elastic strain energy U [J/m2 ] vs. Fragment density N50 [−]; solid curves are trend lines; the dashed line represents the start of micro-crack branching (35 J/m2 ) according to F INEBERG (2006), (b) Elastic strain energy density UD [J/m3 ] vs. Fragment density N50 [−]; trend line converges to 5712.7 J/m3 for N50 = 1

92

4 Experimental Investigations into the Fragmentation of Tempered Glass

lines of the three studied thicknesses are higher than the limit of micro-crack branching of 35 J/m2 according to F INEBERG (2006), which is plausible with respect to Fineberg’s limit as the start of the local micro-branching of a propagating crack. This means that the value of 35 J/m2 is the energy release when crack instability sets in and the fracture surface progressively becomes rougher at the moment of crack branching (F INEBERG et al., 1991; F INEBERG et al., 1992; S HARON et al., 1996). For fragmentation purposes, however, greater energy is required to create crack branching not only on the fracture surface level, but also with respect to a fracture structure in the glass plate. This is discussed in detail in Chapter 6. A better understanding is obtained if the thickness dependency is first eliminated from the experimental results by applying the elastic strain energy density UD per unit volume over the fragment density N50 . This correlation is shown in Figure 4.27 (b). The values of UD can be calculated using Eq. 4.8 or just by dividing the values of the elastic strain energy per unit surface by the respective thickness. Thus, the three trend lines for three different thicknesses from Figure 4.27 (a) coincide to one trend line with a polynomial of the second degree as the corresponding function. Figure 4.27 (b) shows that there is a certain deviation for fragment density values less than 10, which correlates with the small number of fragmented glass plates with a thickness of 4 mm. Nevertheless, the trend line is representative of all thicknesses with the help of which a minimum limit necessary for fragmentation can be determined. The trend line of the elastic strain energy density values in Figure 4.27 (b) converges to UD = 5712.7 J/m3 for an average number of the fragment density N50 = 1, which, according to the experimental results of the fracture tests, is recognized as the minimum required elastic strain energy density for crack building. This does not mean that a fragment must necessarily occur in an observation field or the fracture must lead to crumbed fracture, but that at this energy density it will in any case lead to the formation of more than one through crack so that a fracture structure is obtained. However, the minimum limit of the elastic strain energy per unit surface area for crack building depends on the glass thickness and therefore differs with different thicknesses. Now the correlation between the elastic strain energy per unit surface area U and the fragment density N50 for different glass thicknesses can be established with the help of the energy density values from Figure 4.27 (b) multiplied by thickness U = UD · t, which is shown in Figure 4.28. Thus, the curves are shifted vertically in the diagram depending on the glass thickness. The correlation curves in Figure 4.28 can also be determined in terms of the residual stress and the thickness. For this purpose, the experimentally determined values for the elastic strain energy U (Figure 4.28) are set equal to the part of the energy equation that provides the stress and thickness dependency with 5E/4(1 − ν) · U = σm2 · t. This correlation is shown in Figure 4.29. In this way, the fragment density can be determined as a function of thickness and residual stress.

4.4 Fracture Tests and Fragmentation Analysis

93

Figure 4.28 Elastaic strain energy U [J/m2 ] vs. Fragment density N50 [−] for different glass thicknesses from t = 1.8 mm to t = 19 mm

Figure 4.29 Fragment density N50 [−] vs. σm2 · t [MPa2 ·m] for different glass thicknesses from t = 1.8 mm to t = 19 mm

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4 Experimental Investigations into the Fragmentation of Tempered Glass

According to EN 12150-1 (2015) for the classification of glass as thermally tempered single-pane safety soda-lime-silica glass the counted number of fragments in the observation field should not be less than 40 pcs for the plate thicknesses from 4 mm to 12 mm. However, the residual stress state and the corresponding elastic strain energy vary for different thicknesses. A list of fragment numbers N50 in correlation with the σm2 · t from Figure 4.29 is given in Table A.8 (Appendix A.4). If the fragment density is set in correlation with the mid-plane tensile stress, all curves would coincide and are independent of thickness, since the original curve is motivated by the strain energy density UD . The number of fragments can be estimated as a function of the residual stress. A thickness-independent estimation formula as a rule of thumb for estimating the number of fragments is proposed here: N50 = 0.04 MPa−2 · σm2

σm ≥ 30 MPa

(4.20)

In Eq. 4.20, the fragment density does not depend on the thickness. This formula can be used for estimating the range of fragment density as a function of residual stress. However, at low residual stress values (σm < 30 MPa) the estimation deviates from the determined correlations. As can be seen in Figure 4.28, the same elastic strain energy provides for a higher fragment density in thinner plates than in thicker plates. In other words, in thinner plates a lower strain energy state provides for the same fragment density as in thicker plates. The reason is that the elastic strain energy is linearly proportional to the thickness but square proportional to the residual mid-plane tensile stress. In order to achieve the same elastic strain energy in thinner plates as in thicker plates, a corresponding higher residual midplane tensile stress in thinner plates is required. The residual mid-plane tensile stress can be written in terms of the elastic strain energy as:  5 E U (4.21) σm = 4 1−ν t As an example, the required residual mid-plane tensile stresses to achieve an elastic strain energy of 50 J/m2 calculated with Eq. 4.21 are shown in Figure 4.30. Thus, for the same target elastic strain energy in glass plates with different thicknesses, the required residual mid-plane tensile stresses are 37.7 MPa (t = 4 mm), 26.7 MPa (t = 8 mm) and 21.8 MPa (t = 12 mm). With decreasing plate thickness, the required residual stress state increases with the same elastic strain energy.

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95

Figure 4.30 Mid-plane tensile stress σm [MPa] vs. Elastic strain energy U [J/m2 ]; example: U = 50 J/m2 and corresponding mid-plane tensile stresses for the plate thicknesses t = 4 mm (37.7 MPa), t = 8 mm (26.7 MPa) and t = 12 mm (21.8 MPa)

4.4.5 Particle Weight and Volume A further examination of the fragments was the determination of the weight in order to relate it to the residual stress. This also allows a statement to be made about the stressdependent fragmentation behaviour of glasses. In addition, the particle weight can be used to determine the volume of the fragments and thus the fragmented volume as a function of the stress. In addition to the number of fragments, the weight and the volume are further criteria for the stress-dependent analysis of the fracture behaviour of glasses. Under the assumption of a cylindrical fragment, the volume determined here is used to determine the diameter of a fragment as a function of residual stress or elastic strain energy. In order to determine the weight and volume of the glass fragments as a function of residual stress, fragments were collected during the fracture tests. The average particle weight was determined from more than 130 particles per specimen chosen at random in the observation fields. The fragments were weighed using a precision scale with a weighing range of 0.02 g to 220 g. The particle weight can be used in determining the degree of tempering. On the basis of the knowledge about the residual mid-plane tensile stress and the elastic strain energy release rate combined with the assumption that a fragment of fractured tempered glass has a hexagonal cross section with sides equal to the length of the average mean fracture path, BARSOM (1968) suggests the mid-plane tensile stress σm

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to the power of four multiplied with the mass of a glass fragment M normalized with the thickness t to be constant:   M lb5 4 σm · (4.22) = 7.73 · 1012 9 t in Eq. (4.22) converted to the fragment mass M can be written as:   t lb M= · 7.73 × 1012 Psi4 4 σm in

(4.23)

with Pound-force per square inch Psi = inlb2 = 6.89 × 10−3 MPa, the British unit of mass pound (1 lb = 453.59 g) and the British unit of length inch (1 in = 25.4 mm) the particle mass M can be written as:   g t (4.24) · 3.11 × 105 MPa4 M= σm4 mm The correlation of mid-plane tensile stress and average particle weight in thermally tempered glass plates is shown in the diagrams of Figure 4.31. Each point in the three diagrams represents the mean value of at least 130 fragments per specimen. In Figure 4.31 (a), the particle weight is compared to the predicted particle weight according to BARSOM (1968) evaluated by means of Eq. (4.24). Therefore, the residual mid-plane tensile stresses of the respective specimens were used for the evaluation. The two trend lines are almost parallel and coincide with each other with little deviation. The standard deviation of the particle weight increases with lower tension or larger fragments. The correlation between the particle weight and the residual stress can be applied separated by thickness, see Figure 4.31 (b). Eq. (4.24) is evaluated for different thicknesses in Figure 4.31 (c) and compared with the experimental results. The values match very well except for the 4 mm thick plates, where a small deviation from Barsom’s prediction is observed. This could be due to the comparatively small number of broken glass specimens at low residual stresses. The fragments generated by the fracture may have different base shapes with different edge numbers n as shown in Figure 4.32. For the determination of the base area, the volume of the fragment can be used. Knowing the density of glass (ρ = 2500 kg/m3 ) the volume of the fragments can be recalculated from the weight with V = M/ρ. Figure 4.33 (a) shows the relation between the fragment volume and the residual stress. As expected, the trend line of the particle volume has the same gradient as the particle weight. Based on the influence of the residual stress state on the particle volume there is a corresponding influence of the residual stress state in a tempered glass on the size of the particles. For an assumed cylindrical base shape of the fragments, the radius r of the base area in relation to residual stress is shown in Figure 4.33 (b). A correlation between the particle base area for fragments with regular

4.4 Fracture Tests and Fragmentation Analysis

(a)

97

(b)

(c) Figure 4.31 Particle weight [g] vs. Mid-plane tensile stress σm [MPa] (a) Experimental results of all fractured specimens in comparison with the particle weights evaluated according to B ARSOM (1968), (b) Experimental results of the fractured specimens separated by thickness, (c) Experimental results in comparison with Eq. (4.24) evaluated for different thicknesses; each marker represents the average of at least 130 particles per specimen

Figure 4.32 Estimated particle base shapes, from P OURMOGHADDAM et al. (2018d)

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(b)

Figure 4.33 (a) Particle volume [mm3 ] vs. Mid-plane tensile stress σm [MPa] with volume V = M/ρ and density ρ = 2500 kg/m3 , (b) Radius of the basic area of a fragment with assumed cylindrical shape in [mm] vs. Mid-plane tensile stress σm [MPa]

polygonal shapes with n = 3 to n = 8 sides/number of edges in addition to the cylindrical fragment (n → ∞) is also determined and shown in Figure 4.34. The curves in the diagram were determined with the assumption that all fragments in the observation field have the same base shape respectively the same number of edges n. Each curve in Figure 4.34 represents a fragmented glass plate with the same edge number n. For the calculation of the particle base area a surface radius r of a cylindrical fragment was recalculated from the particle volume (Figure 4.33 (b)). The radius r was used to calculate the base area for different edge numbers n. In reality, the fragments in the observation field have different base shapes with different edge numbers. However, the influence of the residual stress on the base area of the fragments was clarified. A table containing the results for particle weight, volume and base area as an average of at least 130 fragments per specimen in relation to the measured residual stress is given in Appendix A.5. The fragment size correlates with the strain energy, which, as mentioned before, is the measure of fragmentation. Assuming a cylindrical fragment (n → ∞), Figure 4.35 shows the dependency of the diameter of the base area for at least 130 fragments per specimen on the corresponding strain energy density UD . The dashed line shows the limit of fragmentation at UD = 5712.7 J/m3 according to the results of the fracture tests.

4.4 Fracture Tests and Fragmentation Analysis

99

Figure 4.34 Particle base area [mm2 ] vs. Mid-plane tensile stress σm [MPa] for fragment shapes with different numbers of edges n = 3 to n = 8 in addition to a cylindrical fragment (n → ∞), from P OURMOGHADDAM et al. (2018d)

Figure 4.35 Elastic strain energy density UD [J/m3 ] vs. Diameter of the base area [mm] for a cylindrical fragment (n → ∞); black circles with the trend line show the average of more than 130 particles per specimen; dashed line shows the limit of fragmentation at UD = 5712.7 J/m3 according to fracture test results

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4.5 Fractographic Examinations of the Fracture Surface The characteristic fracture pattern and fractographic features of the fracture surface of fragments in tempered glass are the result of its unique parabolic residual stress state. As previously described in Section 2.4, the residual stress state in thermally tempered glass plates is highly compressive at both surfaces, sharply decreases to zero at about 20% of the plate thickness, and changes to tensile stress in the interior. If the external loading is large enough to overcome the compressive stress on the surface of the plate so that the surface flaws might extend beyond the compressive layer or the tensile stress region is penetrated with a pointed object so that the stress state is disturbed, catastrophic crack propagation is initiated. The result is fractured surfaces with unique features. In this section, the fragments from the fracture tests (Section 4.4) are examined in more detail with regard to their fracture surfaces and the influence of residual stress state on the fractographic features of the fracture surface.

4.5.1 The Fracture Surface in Tempered Glass In order to understand and analyse the fracture surface in fragmented tempered glass, the formation and evolution of the fracture surface must first be clarified. The fracture surface caused by the rapid fracture is the result of the fast crack growth which can be divided into two stages: in the initial stage, the crack shape is nearly circular and the crack accelerates rapidly towards the direction of crack propagation. As the crack tip approaches its terminal velocity (≈ 1500 m/s), the crack’s acceleration decreases drastically. In the second stage, the crack front is extended along the plane of symmetry and the crack propagates at a velocity almost equal to half the Rayleigh wave speed of the material (YOFFE, 1951) (cf. Section 4.4.2). Near the terminal velocity, the crack front shape does not change significantly (ACLOQUE, 1958), which means that the appearance of the fracture surface does not vary significantly along the fracture propagation direction. The in-plane shape of the fracture front has been captured in N IELSEN et al. (2009) using high-speed digital cameras. Figure 4.36 shows the in-plane shape of the fracture front in the advanced stage of the fracture process of a drilled tempered soda-lime-silica glass plate with the thickness of 19 mm. It can be clearly seen that the fracture front near the surfaces of the glass plate is delayed compared to the front in the interior respectively in the tensile zone. This creates a complex shape of the fracture front. The post fracture investigations of the fragments obtained in the fracture tests show that the fracture surface generated are rather characteristic due to this type of crack propagation, which is a result of the unique loading condition originating from the residual stress state. Figure 4.37 shows the crack reached terminal velocity in the interior where the tensile stresses were greatest. The red arrow indicates the direction of crack propagation and the onset of mist

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Figure 4.36 Image of the in-plane shape of the fracture front obtained from high-speed video tests on 19 mm thick tempered glass plates (N IELSEN et al., 2009)

Figure 4.37 Mist and velocity hackle in tempered glass for two different fragments; figure shows the crack reached terminal velocity in the interior; the red arrow shows the direction of crack propagation and the onset of mist

and velocity hackle region, which by definition are markings formed on the surface of an accelerating crack close to its terminal velocity, initially visible as a misty appearance and with increasing velocity revealing a fibrous texture and discrete elongated steps aligned in the direction of crack propagation (F RECHETTE, 1990; H ULL, 1999; Q UINN, 2016). Each fragment has usually the pattern shown in Figure 4.37 since cracks slow at a branch then accelerate to terminal velocity again. Thus, the mist and velocity hackle region evolves during the generation of the fracture surface. Figure 4.38 (a) shows an instance of the fracture surface at the crack terminal velocity. The direction of crack propagation is shown. As can be seen in Figures 4.38 (a), (b), (c) and (d), there are three different zones that are identified as characteristic of the fracture surface at the crack terminal velocity in tempered glasses. The first rough zone is the mist and velocity hackle layer with the thickness th as described before and shown in Figure 4.38 (b). The second zone is characterized by being almost perfectly smooth showing some characteristic lines. The arc shaped lines occurring

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Figure 4.38 Fractographic features of the fracture surface from fracture tests on tempered glass with the outlined direction of propagation, (a) Fracture surface at the crack terminal velocity for a 12 mm thick tempered glass specimen. The ’mist band’ at the center of the glass, the ’mist and velocity hackle’ layer of width th and the compression layer of width tc , (b) Detailed view of the ’mist and velocity hackle’, (c) Detailed view of the ’Wallner lines’ and the lower compression layer, (d) Detailed view of the ’Twist hackles’ in the lower compression layer

in the second zone are secondary Wallner lines, see Figure 4.38 (c). The secondary Wallner lines are generated by an elastic pulse released by a discontinuity in the progress of the crack front, typically one of the rough details which arise as the crack approaches its terminal velocity (F RECHETTE, 1990; H ULL, 1999; Q UINN, 2016). The third zone is the compression layer that occurs near the outer surfaces (cf. Figure 4.38 (d)) and corresponds to the compressive zone in the theoretical residual stress distribution. Crack propagation is retarded at the outer surfaces due to the residual compressive stress. Final break-through to the outer surfaces creates vertical twist hackles, also referred to as river deltas (Q UINN, 2016), near the outer surfaces of the glass, see Figure 4.38 (d). Here the crack propagates

4.5 Fractographic Examinations of the Fracture Surface

103

in the compression layer almost perpendicular to the outer surface and the crack surface is separated as indicated by twist hackles. Thus, the twist hackles on the compression zone and the Wallner lines on the smooth zone reveal the direction of crack propagation in each zone. The mist and velocity hackle on the rough zone indicate a complex fracture process. The topography shows that there must be radical change between these three zones. These three zone types appear in any kind of glass fracture, however, their appearance and arrangement in the fracture surface is changed based on the loading situation.

4.5.2 Analysis of the Fracture Surface In order to characterize the fracture surface with regard to the residual stress, the fractured surface was examined. As described in Section 4.5.1, three different zones of the fracture surface are characterized as the rough zone with the mist and velocity hackle, the smooth zone with the Wallner lines and the compression zone with the twist hackles. These are identical for all investigated fragments, independent of the glass thickness. The three zones are layers with a certain thickness. Fractographic analysis of the fracture surface was performed to investigate the influence of residual stress on the depth of the compression layer (tc ) and the mist and velocity hackle layer (th ). For the fracture surface analysis 12 fragments of each specimen was examined. Since all three specimens in the series have approximately the same residual stress level, this means 36 fragments for each residual stress level were investigated. The depth of the compression layer is usually fixed at about 42% (21% on each side) of the glass thickness, which would lead to the assumption that the parabolic residual stress state is engraved over the thickness during quenching. This assumption could apply to fully tempered glass respectively to normal tempering conditions. However, the results of the fracture surface analysis in this thesis showed a correlation between the measured depth of the compression layer and the residual stress. In Figure 4.39 (a), the correlation between the relative compression layer tc /t and the residual mid-plane tensile stress σm is shown. The results of one compression layer are shown here. It can clearly be seen that the thickness of the compression layer increases with increasing residual stress. The ratio of the compression layer thickness to the total glass thickness corresponds to the above mentioned value of approximately tc /t = 0.21 (21% of the glass thickness) at mid-plane tensile stresses above 50 MPa. This is the maximum value achieved in the investigations. However, at lower mid-plane tensile stresses the value drops to a minimum of approximately tc /t = 0.10, which is about half of the maximum value achieved. The results show that the compression layer is always located in the compression zone of the residual stress curve (tc /t ≤ 0.21), although not always, i.e. not at lower residual stress levels, as deep as the compression zone of the residual stress curve. The relative compression layer thickness varies for different glass thicknesses at the same mid-plane tensile stress. As mentioned before, the reason for this lies in the energetic consideration and means that there must

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(a)

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(b)

Figure 4.39 (a) Relative compression layer tc /t [−] vs. Mid-plane tensile stress σm [MPa] for three different glass thicknesses, (b) Elastic strain energy U [J/m2 ] vs. Relative compression layer tc /t [−]

also be a dependency on the stored energy. The same residual stress leads to higher elastic strain energy with thicker glass panes (see Eq. 4.7). The correlation between the elastic strain energy U and the relative compression layer tc /t is shown in Figure 4.39 (b). This diagram clearly shows that the larger the stored elastic strain energy, the larger the depth of the compression layer. From the point of view of the author of this thesis, the reason for this is the way in which the crack propagates as described in Section 4.5.1. As described before the crack propagation is retarded at the outer surfaces and the final break-through to the outer surfaces occurs when the fracture front (cf. Figure 4.36) close to the outer surfaces separates the crack surface as indicated by twist hackles, which are almost perpendicular to the outer surfaces. As already shown in the last sections of this chapter, the smaller the energy, the larger the fragments are. This means that the fracture surface is also larger respectively the crack propagation path before crack branching is longer. Thus, in a glass plate with lower stored elastic strain energy, less energy is released to separate the crack surface via the twist hackles. However, in a glass plate with higher stored elastic strain energy, more energy is released to separate the crack surface and thus a thicker compression layer is generated. In Figure 4.40, the correlation between the measured depth of the mist and velocity hackle layer and the residual stress is shown. The mist and velocity hackle layer with the thickness th remains almost unchanged and independent of the residual stress level. The average value of the relative mist and velocity hackle layer of all examined specimens was th /t = 0.25 ± 0.01 with a coefficient of variation CV = 0.04. Thus, the thickness of the mist and velocity hackle layer at terminal velocity is 1/4 of the glass thickness and independent of the residual stress as well as the glass thickness. In the case of dynamic

4.5 Fractographic Examinations of the Fracture Surface

105

Figure 4.40 Relative mist and velocity hackle layer th /t [−] at terminal velocity vs. Mid-plane tensile stress σm [MPa]; the average value of all examined specimens is th /t = 0.25 ± 0.01 (CV = 0.04)

fracture and when the terminal velocity is reached, the fracture surface of the fragments in the hackle area behaves in the same way. Although the length of the mist and velocity hackle layer varies with different residual stresses, as this area builds up and evolves during crack formation (cf. Figure 4.37) until terminal velocity is reached, however, the ratio th /t does not change thereafter.

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4.6 Computer Tomographic (CT) Examination and Determination of the Fragment and Fracture Surface Area Computer-assisted tomographic (CT) examinations of the fragments obtained in the fracture tests were performed. CT scans are used to measure the actual fracture surface and the fragment surface area due to the fracture tests as physical quantities. In addition to the correlations between the residual stress or elastic strain energy and the fragment density respectively the number of fragments within an observation field, which was described in Section 4.4.4, the correlation between the residual stress or the associated elastic strain energy and the physical quantities, i.e. the measured fracture surface and fragment surface is also determined in this section by means of CT scans. The extracted files from the CT scan are represented as 3D models by the software package myVGL for the visualization of voxel data projects (M Y VGL, 2010) with interface to CAD programs. For the measurements, the scans of the fragments are processed as 3D models in the CAD program R HINOCEROS (2017).

4.6.1 CT System and 3D Reconstruction CT system is a versatile high-resolution X-ray system capable of 2D X-ray inspection (NDT), 3D computed tomography (Mirkro-CT and Nano-CT) and 3D metrology without destroying objects. In this thesis the computer tomograph Phoenix v/tome/x s 240d (Figure 4.41) from GE Sensing & Inspection Technologies with nano- and microfocus tube (with analogue equipments for the radiography of components) of the Department of Ma-

Figure 4.41 Computer tomograph Phoenix v/tome/x s 240d (Image source: Department of Materials Science Darmstadt MPA/IfW)

4.6 Computer Tomographic (CT) Examination and Determination of the Fragment and 107 Fracture Surface Area

Figure 4.42 (a) Principle of the X-ray measurement and (b) Difference between the micro- and nanofocus system (Image source: Department of Materials Science Darmstadt MPA/IfW)

terials Science (MPA/IfW Darmstadt) of the Technical University of Darmstadt is used for scanning the fragments. 4.6.1.1 Principle of the X-ray measurement Figure 4.42 (a) shows the basic principle of the X-ray measurement. A magnet lens focuses the electron beam on the target in the form of a focal spot diameter. Depending on the diameter of the focal spot, a micro- or nanofocus X-ray tube can be used (cf. Figure 4.42 (b)). The nanofocus system with a focal spot diameter of 200 nm was used here to achieve better resolution for smaller components such as fragments. By the target (from tungsten platelets), the acceleration of the electrons is reduced, which leads to X-rays. These X-rays are then reflected onto the sample. This creates an image of the sample on the detector, which is registered as the reconstruction (scan) of the sample. The 3D reconstruction of the sample is outlined in Section 4.6.1.2. The position of the sample can be changed horizontally (x,z) and vertically (y) between tubes and detector. Rotation is also possible during blasting. If required, the detector can extend its measuring range in the x-direction by motion devices and produce a virtual magnification of the imaging surface. The resolution is influenced by three factors: (1) Voxel size V, i.e. volumetric information of the object, which is called pixel in 2D. It is also about data points in a graphical grid, (2) Focal spot diameter limits the resolution, depends on the X-ray tube. The smaller the object, the better the quality of the scanned file,

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(a)

(b)

Figure 4.43 (a) sample chamber of the CT system with two different X-ray tube systems of microand nanofocus, the detector and the specimen holder, (b) Glued fragment on a glass rod

(3) The closer the object is to the X-ray tube, the greater the geometric magnification. 4.6.1.2 3D Reconstruction of the Fragment A 3D model is obtained from 2D series images, which are taken by automatic rotation step by step from 1◦ to 360◦ (cf. Figure 4.42). These projections contain the information about the sample position and the recorded sample density of the respective pixels. The number of projections depends on the sample size. Because with a 360◦ rotation per angle degree "more pixels" are overpainted. When the X-rays hit the glass, they are attenuated by the transmission of glass and reach the detector in an mitigated form. According to this, X-ray images show the attenuation of the X-rays, which pass through different areas of the object and as a result can produce different gray values. The deeper the object is, the higher the energy required by the photons to penetrate the matter. This leads to a measurement artifact. By using a filter, the soft or medium soft and long-wave rays can be sorted out. The fragments from the fracture tests (Section 4.4) were used for computer tomography. Fragments of the three series ScA, ScB and ScC (cf. Table 4.1) were chosen randomly for the investigations. To determine the actual fracture surface, care was taken to ensure that the fragment did not exhibit significant depth cracks. The fragment was first cleaned with ethanol so that no other material remains stuck to it. Each fragment was marked for further investigations. As can be seen in Figure 4.43 (a), the fragment is stored in the sample chamber of the CT system. To store the fragment, it was glued to a glass rod with hot glue (see figure 4.43 (b)) to create a boundary between the scanned fragment and the glass rod due to the higher transmission of the glue, which later simplified the processing of the scanned file. The extracted files from the CT scan were displayed in 3D by the software package M Y VGL (2010). The CT scanned fragment was reconstructed using the volume rendering tool of the program called Opacity manipulation area. This tool allows the specification of the opacity of the object’s voxels as a function of its gray value by means of

4.6 Computer Tomographic (CT) Examination and Determination of the Fragment and 109 Fracture Surface Area

Figure 4.44 3D reconstruction of the fragment due to volume rendering process

Figure 4.45 Example of the 3D reconstructed fragment

an opacity transfer function. This can be used to render the volume of the sample, see Figure 4.44. The software has different rendering settings with different algorithms. Here the Isosurface renderer was used as rendering algorithm. This algorithm produces high quality photorealistic images at interactive speeds almost independently of the size of the data set. The most common use of the isosurface renderer is to render the object at the iso-level defined by the surface determination, i.e. showing the actual determined surface of the object. Figure 4.45 shows an example of a 3D reconstructed fragment based on the CT scan. The voxels remaining after the rendering process that do not belong to the sample are removed in a further step. At this point it is mentioned that for an examination of the inner material, e.g. to detect cracks in the material, different section planes can be considered. However, since only the fragment surface area and the fracture surface for fragments without cracks in the material are subject of the investigation, the inner material is not further investigated and discussed.

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(a)

(b)

(c)

Figure 4.46 (a) Polygonized model of a fragment, (b) Deletion of the polygons on the straight surfaces on the top and bottom side, (c) Remaining fracture surface

4.6.2 Model Processing for Measurement The 3D reconstructed model must be further processed for the measurement of the fragment and the fracture surface area. The measurement of the fragment surface area and the actual fracture surface area of the fragment was done in the program package R HINOCEROS (2017). The 3D reconstructed model was first poligonized or meshed by a multitude of small planar triangles (facets or polygons), cf. Figure 4.46 (a). The mesh has been refined so that there are no discontinuities at the transitions from the top and bottom to the fracture surface or at the irregularities of the fracture surface itself. For this purpose, the mesh width of the polygon mesh was selected so that the average area of a triangle is about max. 0.005% of the total area. This value can decrease to approximately 0.0005% for the largest investigated fragment with a surface area of approximately 2660 mm2 . The sum of the areas of all triangles results in the surface area of the fragment. To determine the fracture surface, it was cut out of the model, cf. Figure 4.46 (b) and (c). The polygons (triangles) which were on a straight surface, i.e. on the upper and lower side, were deleted so that only the polygons which were on the actual fracture surface remained. This procedure ensures minimal damage to the fracture surface, since some polygons are also deleted in the transition areas or remain on both surfaces. However, due to the highly selected mesh fineness, this effect is ignored and a maximum error in the fracture area of ±0.005% of the total area is accepted. The sum of the areas of all triangles of the cut out polygons results in the fracture surface area of the fragment. In order to determine the ratio of the actual fracture surface to a smooth fracture surface ρˆ = A f r,act /A f r,sm , respectively the fracture surface roughness, the model of the fragment was somewhat manipulated. For this purpose, the boundary lines of the upper and lower surfaces at two

4.6 Computer Tomographic (CT) Examination and Determination of the Fragment and 111 Fracture Surface Area

(a)

(b)

Figure 4.47 Creating a smooth fracture surface (a) Connecting the boundary lines of the top and bottom surfaces, (b) Sweeping the line; surface is smooth in thickness direction

superimposed points were connected by a line (Figure 4.47 (a)). Then a smooth surface was created by sweeping the line along the two boundary lines (Figure 4.47 (b)). Thus the resulting area, which is irregular along the boundary lines of the upper and lower surfaces, is smooth in direction of the thickness and can be easily determined.

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4.6.3 Results of the Fragment and Fracture Surface Area For the investigations, five fragments were randomly selected from each broken glass plate. The series for the fracture tests were subdivided according to residual stress level. For each residual stress level three glass plates were thermally tempered (cf. Section 4.3). Accordingly, 15 fragments per residual stress level were scanned. In the diagrams of this section (Figure 4.51 to Figure 4.53), each point represents the average value of five examined fragments. All results are additionally presented in Appendix A.6 in tabular form, indicating the standard deviation and the coefficient of variation. The fragment surface area AS , the actual fracture surface area A f r,act and the smooth fracture surface area A f r,sm are shown for some examples of CT scans of fragments with three different thicknesses in Figures 4.48, 4.49 and 4.50. The shape of the fragments and the fracture surface are clearly visible. The smooth surface is created to compare the actual fracture surface and the assumption of a smooth fracture surface. This can be used to determine a fracture surface roughness ˆ f r,sm ). The ρˆ to estimate the actual fracture surface for a known base shape (A f r,act = ρA relationship between the residual stress and the fracture surface roughness ρˆ is shown in Figure 4.51 (a) for different thicknesses. The diagram in Figure 4.51 (a) and the CT scans in Figures 4.48, 4.49 and 4.50 show that the fracture surface becomes more and more irregular or rougher with larger glass thicknesses and that the fracture surface roughness ρˆ is correspondingly larger than with thinner glass plates. This conclusion can also be reached by observing fragments with the same elastic strain energy density UD but different thicknesses. For example, the fragments in Figure 4.48 (c) (t = 3.8 mm) and Figure 4.49 (b) (t = 7.9 mm) both have approximately the same elastic strain energy density of about UD = 18100 J/m3 due to the same residual mid-plane stress of about σm = 45 MPa. However, despite the same energy density they have clearly different fracture surfaces. Additionally, the fracture surface roughness ρˆ = 1.147 is larger for the thicker fragment in Figure 4.49 (b). The reason for this lies in the fracture formation. For a thin plate, a smaller amount of elastic strain energy per base area must be released to form the fragment than for a thicker plate for the same elastic strain energy density. In the above example, the elastic strain energy per base area is about U = 69 J/m2 for the thin glass plate (t = 3.8 mm) and about U = 143 J/m2 for the thicker glass plate (t = 7.9 mm). Since almost the entire energy is largely converted into fracture surface formation, a slightly larger actual fracture surface is generated in relation to the smooth fracture surface of thicker glass plates. The maximum value of the fracture surface roughness of the fragments examined in this thesis is ρˆ = 1.220 for a glass plate with the thickness of t = 12 mm and the minimum value is ρˆ = 1.026 for a glass plate with the thickness of t = 3.8 mm. Figure 4.51 (b) shows the relationship between the elastic strain energy per unit surface area U and the fracture ˆ It can be clearly seen that the roughness of the fracture surface area surface roughness ρ. increases with increasing elastic strain energy per unit surface area. This means that at higher elastic strain energies more fracture surface is required than is available over the smooth thickness to generate a fragment. This is also the reason that at the same residual

4.6 Computer Tomographic (CT) Examination and Determination of the Fragment and 113 Fracture Surface Area

(a)

(b)

(c) Figure 4.48 CT scan examples of fragments with a thickness of t = 3.8 mm, (a) σm = 45.55 MPa (UD = 18255.18 J/m3 ); ρˆ = 1.036, (b) σm = 47.60 MPa (UD = 19941.91 J/m3 ); ρˆ = 1.032, (c) σm = 45.35 MPa (UD = 18095.21 J/m3 ); ρˆ = 1.026

stress the thicker fragments have a higher fracture surface roughness (Figure 4.51 (a)). The correlation between the residual stress and the fragment and fracture surface area determined from the CT scans is shown in Figures 4.52 (a) and (b) for three glass plate thicknesses. The elastic strain energy density UD resulting from the residual stress thus has an influence on the fragment and fracture surface area. The correlation between the elastic strain energy density UD and the fragment and fracture surface area determined from the CT scans is shown in Figures 4.53 (a) and (b). It can be clearly seen that the physical quantities AS and A f r,act increase with decreasing elastic strain energy density. Thus, in addition to the fragment density N50 (Section 4.4.4), the quantities AS and A f r,act were determined from the fracture tests, which help to characterize the fracture behaviour.

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(a)

(b)

(c) Figure 4.49 CT scan examples of fragments with a thickness of t = 7.9 mm, (a) σm = 53.69 MPa (UD = 25365.57 J/m3 ); ρˆ = 1.167, (b) σm = 45.40 MPa (UD = 18138.21 J/m3 ); ρˆ = 1.131, (c) σm = 36.61 MPa (UD = 11793.08 J/m3 ); ρˆ = 1.075

4.6 Computer Tomographic (CT) Examination and Determination of the Fragment and 115 Fracture Surface Area

(a)

(b)

(c) Figure 4.50 CT scan examples of fragments with a thickness of t = 12 mm, (a) σm = 55.44 MPa (UD = 27046.12 J/m3 ); ρˆ = 1.210, (b) σm = 38.35 MPa (UD = 12944.95 J/m3 ); ρˆ = 1.160, (c) σm = 27.10 MPa (UD = 6464.64 J/m3 ); ρˆ = 1.102

116

(a)

4 Experimental Investigations into the Fragmentation of Tempered Glass

(b)

Figure 4.51 (a) Fracture surface roughness ρˆ [−] vs. Mid-plane tensile stress σm [MPa] for three different glass plate thicknesses t = 4 mm, t = 8 mm and t = 12 mm, (b) Elastic strain energy per unit surface area U [J/m2 ] vs. Fracture surface roughness ρˆ [−]

(a)

(b)

Figure 4.52 (a) Fragment surface area AS [mm2 ] vs. Mid-plane tensile stress σm [MPa], (b) Actual fracture surface area A f r,act [mm2 ] vs. Mid-plane tensile stress σm [MPa] for three different glass plate thicknesses t = 4 mm, t = 8 mm and t = 12 mm

4.7 Summary

(a)

117

(b)

Figure 4.53 Elastic strain energy density UD [J/m3 ] vs. (a) Fragment surface area AS [mm2 ], (b) Actual fracture surface area A f r,act [mm2 ]

4.7 Summary In this chapter comprehensive investigations and fragmentation analysis based on fracture tests on thermally tempered glass plates with different residual stress levels were performed. The fragmentation as a direct consequence of the energy stored inside the material and the fracture behaviour was characterized. The fracture tests were carried out on three series of tempered glass specimens of size 360 mm × 1100 mm with three different thicknesses of 4 mm, 8 mm and 12 mm. The specimens were divided into three series according to thickness. Each series contained 24 specimens, divided into eight groups of three specimens for each run of the tempering process. For the production of glass specimens with systematically different residual stresses, float glass plates were heat treated differently due to the variation of the cooling rate during the tempering process. In order to choose the target residual stress for the heat treatment of the series, the elastic strain energy of 35 J/m2 for the start of the micro-crack branching of glass according to F INEBERG (2006) was considered as an orientation level. For different heat treatment of the series, the engine power for the cooling section of the thermal tempering oven was calculated and varied due to the experience values for the relation between the adjusted engine power and the resulting air pressure. For this purpose, the required heat transfer coefficient was determined, which in turn leads to a certain cooling air velocity and subsequently to a required air pressure to achieve the target residual stress. A simplified model was presented for the determination of the engine power as a function of the residual stress using quench parameters by forced convection based on the empirical equations of integral heat and mass transfer coefficients suggested by M ARTIN (1977).

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The fracture tests were carried out by impact according to EN 12150-1 (2015). It was shown that the elastic strain energy which is generated within the glass plate due to the parabolic stress state governs the fragmentation behaviour with regard to the fragment density and subsequently the fragment size. For the fragmentation analysis of the fractured tempered glass specimens the average number of fragments N50 was determined for each specimen within eight observation fields of size 50 mm × 50 mm and set in correlation with the average measured residual stress or the elastic strain energy U per unit surface area of each specimen. In addition, the glass thickness dependence was eliminated from the experimental results by applying the elastic strain energy density UD per unit volume over the fragment density N50 . Thus correlation curves for the three thicknesses coincide to one curve with a polynomial of the second degree as the corresponding function. This correlation is representative of all thicknesses used to determine a minimum limit required for fragmentation. The determined correlation curve of the elastic strain energy density values converges to UD = 5712.7 J/m3 for an average number of the fragment density N50 = 1, which is recognized as the minimum required elastic strain energy density for crack building according to the experimental results of the fracture tests. This does not mean that a fragment must necessarily occur in an observation field or the fracture must lead to crumbed fracture, but that at this energy density it will in any case lead to the formation of more than one through crack so that a fracture structure is obtained. For the practical application of the experimental results, the relationship between the elastic strain energy density and the fragment density was expressed in terms of residual stresses and as a function of thickness. For this the energy density values were multiplied by thickness (U = UD · t) and the elastic strain energy U per unit surface area was set equal to the part of the energy equation that provides the residual stress and thickness dependence with 5E/4(1 − ν) · U = σm2 · t. Hence, a correlation between the fragment density and σm2 · t was established. In addition to the number of fragments, further investigations on fragment weight and volume were carried out as further parameters of the stress-dependent analysis of the fracture behaviour. In order to determine correlation between the weight and volume of the glass fragments and the residual stress, the average particle weight was determined from more than 130 particles per specimen chosen randomly in the observation fields. The results were compared with those of BARSOM (1968) and matched very well. The fragments from the fracture tests were examined with regard to their fracture surfaces and the influence of residual stress state on the fractographic features of the fracture surface such as the compression layer, mist and velocity hackle. It was shown that the thickness of the compression layer is correlated to the residual stress level, however, the thickness of the mist and velocity hackle layer at terminal velocity of the crack remains unchanged and independent of the residual stress at 14 t. Furthermore, computer-assisted tomographic (CT) examinations of the fragments were performed in order to measure the actual fracture surface A f r,act and the fragment surface area AS as physical quantities. In

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119

addition to the correlations between the residual stress and the fragments density, the relationship between the residual stress or the associated elastic strain energy and the physical quantities, i.e. the measured fragment and fracture surface area was also determined by means of CT scans. The CT scan files were presented as 3D models. As a further characteristic of the fracture surface and for a better understanding of the fracture behaviour, the fracture surface roughness ρˆ was also determined and correlated with the residual stress or the elastic strain energy. The fracture surface roughness increased from ρˆ = 1.03 for the elastic strain energy of U ≈ 60 J/m2 to ρˆ = 1.22 for U ≈ 310 J/m2 . This means that at higher elastic strain energies more fracture surface is required than is available over the smooth thickness to generate a fragment. It was therefore observed that with the same residual stress, the thicker fragments exhibit a higher fracture surface roughness.

5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass 5.1 General The fracture behaviour of glass concerns scientists all over the world since decades. As glass is considered a brittle material, the strength of glass is typically described by the Linear Elastic Fracture Mechanics (LEFM). First pioneering work within fracture mechanics of glass was conducted by Griffith in 1920 (G RIFFITH, 1921). Since then numerous authors have contributed to analysing the fracture structure of glass. In this chapter, the fracture mechanical and mathematical foundation, deduction and application of a Machine Learning inspired simulation approach for the prediction of fracture patterns of thermally tempered glass via stochastic tessellation is shown. The main goal of the presented method, called ’BREAK’, is Bayesian reconstruction and prediction and thus a fast simulation of the glass fracture patterns. The theoretical method, based on the energy criterion of Griffith and merged with specific fracture pattern statistics obtained from the fracture tests, is used to predict the 2D macro-scale fragmentation of glass. In order to predict the fragmentation of glass the 2D Voronoi tessellation of distributed points based on Spatial Point Processes is used. The mathematical background on spatial point patterns is given in detail in M ARTINEZ et al. (2015), BADDELEY et al. (2016), and W IE GAND et al. (2014) while (random) Voronoi tessellations are concerned in O KABE et al. (1992) and OYANA et al. (2015). Within the context of this thesis just the main concepts are repeated in order to allow the understanding of the MATLAB implementation and results, which are presented in the following. Certain parts of this chapter were obtained and elaborated within an cooperation of the author of this thesis with Michael Kraus (K RAUS, 2019), furthermore some parts of this chapter have already been published (P OURMOGHADDAM et al., 2018f; P OURMOGHADDAM et al., 2018b; P OURMOGHADDAM et al., 2018e; K RAUS et al., 2019b; K RAUS et al., 2019a). The fragment density in an observation field, the fragment shape and thus the entire fracture pattern depends on the magnitude of the strain energy resulted by the imposed residual stress state. This was shown in Chapter 4 for differently heat treated glass plates with different residual stress levels. Also several studies on the fragmentation behaviour of tempered glasses have proven relationships between the residual stress state, the glass © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6_5

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Figure 5.1 Schematic framework of the method BREAK, showing the connections of experimental observations to the elements of spatial point patterns and linear elastic fracture mechanics

5.1 General

123

thickness and the fragment density, see e.g. ACLOQUE (1956a), A KEYOSHI et al. (1965), BARSOM (1968), G ULATI (1997), S HUTOV et al. (1998), L EE et al. (2012), M OGNATO et al. (2017), and P OURMOGHADDAM et al. (2018d). It has therefore been proven that the fragment size increases with decreasing residual stress level respectively the strain energy. Further experimental and numerical investigations of fragment’s deformation and elastic strain energy were carried out by N IELSEN et al. (2017). The change in strain was determined by comparing the surface shape of a fragment before and after fracture. Several models for relating the fragment size to the residual stress state have been suggested in the literature, see e.g. ACLOQUE (1956a), BARSOM (1968), G ULATI (1997), S HUTOV et al. (1998), WARREN (2001), and TANDON et al. (2005). Some of these works have proposed models for the fragments size based on an energy approach. Most of the works try to establish an analytical model for the fragment size considering the release of the so-called tensile strain energy defined as the part of the strain energy resulting from the mid-plane tensile stress alone. Within this thesis, a machine learning motivated stochastic point process model for the prediction of the fragmentation of tempered glass is presented. Parts of this chapter were recently pre-published, (P OURMOGHADDAM et al., 2018f; P OUR MOGHADDAM et al., 2018b; P OURMOGHADDAM et al., 2018e; K RAUS et al., 2019b; K RAUS et al., 2019a), where a theoretical model for the prediction of the fragmentation of tempered glass is presented. The schematic connections of the theories and experiments involved are given in Figure 5.1. The model is based on the combination of an energy criterion (G RIFFITH, 1921) of LEFM with respect to the elastic strain energy state before and after the fragmentation and the statistical analysis of the fracture pattern of tempered glass in order to determine characteristics of the fracture pattern (e.g fragment size, fracture intensity, etc.) within an observation field. The basic idea behind the modelling approach is that the final fracture pattern is a Voronoi tessellation induced by a stochastic point process whose parameters can be derived from fracture mechanical considerations and statistical evaluation of images of several fractured glass specimens. Thus, the fracture pattern is simulated by calibrating a stochastic point process, taking into account the strain energy stored in the glass plate and the empirical reality of a fracture pattern, and consecutive Voronoi tessellation of the region of interest. This approach is in contrast to traditional simulation approaches of glass fracture patterns, as these works handle the fracture of tempered glass numerically explicit or implicit within a Finite Element Analysis (FEA). Several computational methods strive to model unrestricted fracture growth in a continuum using an explicit representation of the fracture surfaces, e.g. extended finite element method (XFEM) (M OËS et al., 1999; S UKUMAR et al., 2015) and embedded discontinuities (J IRASEK, 2000). The evolution of cracks within the FEA can also be described by means of e.g. cohesive zone elements (G EUBELLE et al., 1998; M AITI et al., 2004; M AITI et al., 2005; B ISHOP et al., 2016; VOCIALTA et al., 2017), meshless methods (B E LYTSCHKO et al., 1995; N GUYEN et al., 2008; A ZEVEDO et al., 2015; B ERMBACH , 2017),

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

Figure 5.2 Schematic of the RB-method; red dots are ’snapshots’ φh (θn ), 1 ≤ n ≤ N on the parametric manifold Mh , the RB space VN = span{ζ1 , ..., ζN } = span{φh (θ1 ), ..., φh (θN )} here indicated by the hyperplane and the RB solutions φh (θN ) ∈ VN , θN ∈ P, represented by the red curve for the case of P = 1 parameter (Q UARTERONI et al., 2015)

element deletion (P ELFRENE et al., 2016a; P ELFRENE et al., 2016b; B ERMBACH, 2017; A LTER, 2018), phase-field methods (K ARMA et al., 2004; A MBATI et al., 2016; K IENDL et al., 2016; S TEINKE et al., 2018) and peridynamic models in the class of particle methods (J OHNSON et al., 2002; F OSTER et al., 2011; R EN et al., 2017; B UTT et al., 2018). A semi-static fracture mechanics based model to describe the evolution of a single crack in tempered and strengthened glass plates is implemented in D UGNANI et al. (2014). A FEA of the fracture process suffers from e.g. computing time or numerical instability. In addition, the resulting fracture pattern may not reflect statistical properties of real fracture patterns such as e.g. the distribution of fracture particle areas or diameters etc. The proposed BREAK method avoids these problems and avoids them by directly simulation of the fracture pattern after the fracture process rather than the fracture process itself. However, the method is restricted to the boundary conditions of the limited training sample. In mathematical theory apart from the machine learning techniques, this concept is known as Reduced Basis Method (RB-Method) (Q UARTERONI et al., 2015; H ESTHAVEN et al., 2015), which is graphically given in Figure 5.2. The idea as given in Figure 5.2 is, that an unknown model of truth (black line) causes observations (red dots) within experimental possibilities (blue plane). As experiments are limited in resolution etc. an approximation of the unknown model of truth is deduced from the experimental observations (red dots). More specifically for the context of this thesis, in Figure 5.2 the red points (= ’snapshots’) are the scanned glass fracture patterns (see Chapter 4), which are used to form a reduced basis system for modelling the unknown truth causing the real fracture pattern (black line). While the original concept of the Reduced Basis Methods from ’snapshots’ stems from the theory of parametrized partial differential equations (PPDE), where causalities of inputs and outputs of the PPDE are given by the PPDE, within this section, a stochastic point process is used instead of the PPDE.

5.2 Geometrical Evolution of a 2D Voronoi Cell

125

5.2 Geometrical Evolution of a 2D Voronoi Cell Among the many possibilities of tessellations, the Voronoi tessellation, which was first introduced in 1908 by VORONOI (1908), might be the most popular and useful especially in the field of brittle fracture. It divides a region into convex polygons, or cells, that fill space without overlap. Especially for the simulation of the fracture of brittle ceramic materials, Voronoi tessellation seems to be quite suitable (B ISHOP, 2009; B ISHOP et al., 2016). The shape of cells of Voronoi tessellations clearly reflects the manner of configuration of points. Therefore, the point processing method has a significant influence on the shape of the Voronoi cells and on the geometry. In order to predict the fracture pattern of broken glass, it is necessary to define a problem-specific point process method. A point process is a random mechanism that results in a point pattern. Different spatial point process methods and the mathematical background will be explained in detail later in Section 5.5. The 2D Voronoi tessellations are generated due to the Delaunay triangulation of the planar distributed nuclei. The cell borders are obtained by the connection of the centres of the circumcircles and thus are perpendicular bisectors for each pair of adjacent points. The geometric evolution of a 2D Voronoi cell for randomly distributed points is shown in Figure 5.3. The cell region around point P3 is the Voronoi cell V3 and has a defined border that depends on the location of the neighbouring points. However, this does not apply to points P1, P2, P4, P5 and P6. Here the cells are infinitely large to the outside, since no further points provide for the delimitation of the cells. Each point within the Voronoi cell V3 has the smallest direct distance to point P3.

Figure 5.3 Geometrical evolution of a 2D Voronoi cell, (a) Randomly distributed points, (b) Delaunay triangulation and circumcircles, (c) Voronoi cell region of the point P3, from P OURMOGHADDAM et al. (2018f)

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5.3 Strain Energy Conversion As it was discussed in the Eqs. (4.1) to (4.8) in Section 4.2, strain energy expresses the relationship between stress, strain or deformation, displacement, material properties and external effects in the form of internal and external forces. The fracture pattern, i.e. the fracture structure, the fragment size and thus the fragment density are the direct consequence of the amount of the initial elastic strain energy U0 which is available before the fragmentation sets in and can be written as: U0 = UResidual stress +UExternal loading

(5.1)

where it should be noted that in this work, it is assumed that the glass is only subjected to residual stress from the thermal tempering and U0 = UResidual stress can be calculated using Eq. (4.7) as initial elastic strain energy per unit surface area. Since the glass fracture structure consists of vertical cracks, i.e. the fracture surface runs along the glass thickness and is almost perpendicular to glass surfaces, the strain energy per unit surface area can be taken into account. When the glass is fragmentized the internal energy converts into different forms of energies (R EICH et al., 2013; N IELSEN, 2017), e.g. for creating new fracture surfaces (cracking and crack branching), kinetics and sound energy. Hence, the remaining elastic strain energy in a fragment U1 after the fragmentation sets in must be expressed as: U1 = U0 − (U f r,sur f ace +Ukinetic +Usound +Uother )

(5.2)

where Uother represents other kinds of energy such as heat. Within the context of this thesis, just the relative remaining elastic strain energy UR,Rem as the ratio between the remaining elastic strain energy and the initial elastic strain energy is considered. The relative remaining elastic strain energy can be written as: UR,Rem =

U1 U0

(5.3)

All other forms of energy are not considered here. The remaining stress state and the resulting remaining elastic strain energy in a single fragment has been calculated numerically by N IELSEN (2017). Further experimental and numerical investigations of fragment’s deformation and elastic strain energy were carried out by N IELSEN et al. (2017). The change in strain was determined by comparing the surface shape of a fragment before and after fracture.

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5.4 Methodology and Fracture Mechanics Approach In order to predict the 2D macro-scale fragmentation of glass, model parameters with regard to the fragment size and the intensity of the fracture structure are estimated with the help of the energy criterion in fracture mechanics (G RIFFITH, 1921), taking into account stochastic fracture pattern analyses. Before introducing the concepts of random point processes and spatial statistics as well as the stochastic modelling framework, the basic modelling approach and the incorporation of fracture mechanics is emphasized.

5.4.1 Basic Modelling Approach The basic idea for the theoretical prediction and simulation of the fracture patterns of tempered glass is to distribute seed points with a certain distance δ to each other based on the locally acting stress respectively the stored elastic strain energy, and with a fracture intensity of λ obtained by evaluation of fracture patterns in the plane. The final fracture pattern is created by Voronoi tessellation over the generated point cloud. As it will be introduced in Section 5.5, the seed points for the Voronoi tessellation can be distributed using different distribution methods. Taking into account the fragment size and the number of fragments within an observation field ND , two model parameters for the prediction of the 2D macro-scale fracture pattern can be elaborated: the point intensity λ ≈ 1/ND as the fracture intensity parameter and a minimum inter-point distance δ as the fragment size parameter. Thus the parameters of the prediction model are: θ = {δ , λ }

(5.4)

This work focus on the elastic strain energy resulting from the residual stress state from the thermal tempering process. In order to predict the fragmentation and for the recognition and the generation of the fracture pattern, the fragment size parameter δ and the fracture intensity parameter λ are determined. As will be shown later, the fragment size parameter δ is equal to the hard-core distance of the Matérn Hard-core and Strauss spatial point process and thus can be understood as a local minimum inter-point distance. For the determination of the fragment size parameter δ the linear elastic fracture mechanics based on the energy criterion introduced by G RIFFITH (1921) is applied to incorporate the residual stress due to the thermal tempering as well as further kinds of stresses such as stress resulting from external loads. A part of the energy is released by new surfaces generated from cracking and branching of progressive cracks. In high-speed images obtained by N IELSEN et al. (2009), it was observed that so-called ’whirl-fragments’ were generated by a whirl-like crack propagation (cf. Figure 5.4). It was also observed that progressing cracks branched at an angle of 60◦ and formed a hexagonal fragment. A hexag-

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(a)

(b)

(c)

(d)

Figure 5.4 High-speed images by N IELSEN et al. (2009) showing the formation of a whirl-fragment due to 60◦ bifurcation

(a)

(b)

(c)

Figure 5.5 (a) Hexagonal close packing distribution of points with an ’energy circle’ of radius r0 as the area of influence of each point; the point distance is δ , (b) Delaunay triangulation of HCP, (c) Honeycomb as the result of the Voronoi tessellation of the seed point according to HCP

onal fracture structure is assumed to be the ’perfect/ideal’ fracture pattern or ’perfectly’ broken glass plate. Predicting such a ’perfect’ fracture structure is easy, provided δ is known. All points in the plane have the same distance to each other and can be distributed by Hexagonal Close Packing (HCP) of points (cf. Figure 5.5 (a)). The Voronoi tessellation is conducted via the Delaunay triangulation of the HCP-distributed points (Figure 5.5 (b)) and results in the cell structure of a honeycomb (Figure 5.5(c)). Thus the number of fragments in an observation field can be predicted as well. However, since the fracture pattern will not be perfect in reality (cf. Figure 5.6), due to stochastic nature of the material properties as well as the stress distribution from the production process, another parameter λ is introduced into the model. This parameter can be obtained from experimental fracture pattern data and describes the intensity of the fragments and thus the empirical reality of

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129

(a)

(b)

(c) Figure 5.6 Fracture patterns of glass specimens of size 1100 mm × 360 mm and the thickness of t = 12 mm for different residual stresses: (a) σm = 26.9 MPa (UD = 6367.8 J/m3 ), (b) σm = 30.0 MPa (UD = 7920.0 J/m3 ) and (c) σm = 38.1 MPa (UD = 12774.2 J/m3 )

the fracture pattern. Hence, λ is closely related to the number of fragments in an observation field ND . The Fracture Intensity Parameter λ , which is a characteristic value for the number of fragments in an observation field, can be determined using the experimental data of the fracture tests as described in Section 4.4. In Figure 5.6, three samples of fracture patterns are shown for the same glass thickness but different residual stresses respectively elastic strain energy density UD . As it was outlined and shown in Section 4.4, fracture intensity can be determined by placing 50 mm × 50 mm observation fields on the fracture patterns and determining the average number of fragments in correlation to the energy density for each specimen. As will be discussed and shown in Section 5.7, the estimation of the two model parameters is not restricted to 50 mm × 50 mm observation fields but can be done over the whole fracture pattern. The evaluation by the 50 mm × 50 mm observation fields is motivated by EN 12150-1 (2015).

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5.4.2 Fragment Size Parameter δ The determination of the fragment size parameter δ is based on the energy conditions described in Section 4.2. As it is shown in Figure 5.5 (a), the fragment size parameter δ = 2r0 is the distance between two neighboured points. In order to avoid the necessity to introduce the fracture intensity parameter λ and to outline the fragment size parameter δ purely physically, it is assumed in the following that the points in the observation plane are distributed according to the HCP. Three assumptions are necessary for the determination of the fragment size parameter respectively for the minimum distance of two seed points. Assumption 1: The first assumption is that the glass plate will break into cylindrical fragments. It is assumed that the sphere of influence for the stored elastic strain energy U of each point representing a fragment is a circle, which is here called "Energy circle" respectively a cylinder in 3D with a radius r0 before fragmentation. This assumption means that the fracture surface of a fragment is the lateral surface of a cylinder. As can be seen in Figure 5.5 (a), the energy circles touch but do not overlap. Consequently, all energy in the observed field is distributed in these energy circles. Thus, each radius r0 depends on the elastic strain energy in the influence area of the respective point. The free gaps between the energy circles are zero energy areas and have no influence on the further calculation. This is legitimate because the energy circles are subjected to the total strain energy in the observed field. Assumption 2: In the case of fracture the stored elastic strain energy will release and there will be a relaxation of energy in the fragment. However, it is assumed that in the case of fracture, the elastic strain energy is not completely relaxed, but only by a relaxation factor η: η=

r1 r0 − r1 = 1− r0 r0

(5.5)

This means, that there will be a remaining elastic strain energy U1 in the fragment after the fragmentation. This was shown numerically in N IELSEN (2017) by FE-simulations on a fragment before and after the fracture sets in. Hence, the relative remaining elastic strain energy UR,Rem can also be described as: UR,Rem =

r2 U1 = 12 = (1 − η)2 U0 r0

(5.6)

For the energy sphere described under Assumption 1, this means that the energy circle shrinks to a smaller circle with a radius r1 after fragmentation (see Figure 5.7). The relative released/relaxed elastic strain energy can be written as:

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131

Figure 5.7 Energy circle before fragmentation with radius r0 and after fragmentation with radius r1 ; Aη is the surface of the energy relaxation zone, from P OURMOGHADDAM et al. (2018e)

UR,Rel = 1 −UR,Rem = η(2 − η)

(5.7)

The area in which the energy relaxes is called the energy relaxation zone. The area of the energy relaxation zone Aη can be easily calculated as follows: Aη = πr0 2 η(2 − η) = πr0 2 ·UR,Rel

(5.8)

In addition, the elastic strain energy Uη in the relaxation zone can be calculated as follows:

Uη =

σˆ (1 − ν) tAη E

(5.9)

wherein σˆ describes the stress state as a factor of the integrated stress function through the thickness. The integration pre-factor changes depending on the stress curve σ (z) over the thickness and is therefore dependent on the load situation. In the case of a parabolic residual stress distribution due to thermal tempering, σˆ can be determined as σˆ m = 45 σm2 by calculating with the residual tensile mid-plane stress or σˆ s = 15 σs2 for the residual compressive surface stress. Uη = U0 −U1 is therefore the elastic strain energy released during fragmentation. Since the elastic strain energy in Eq. (5.9) is related to the energy relaxation zone, Uη is therefore called the ’relaxed energy’ respectively the elastic strain energy released during fragmentation. This is the energy that causes the creation of new fracture surfaces due to cracking and branching.

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Assumption 3: The energy −dΠ released on an infinitesimal crack propagation dA is called energy release rate: G=−

dΠ dA

(5.10)

The energetic relation used by G RIFFITH (1921) for the first time as a fracture condition is: dΠ dΓ + =0 dA dA

(5.11)

This means that during the fracture process the sum of the potential dΠ = dΠa + dΠi of the external and internal forces and the fracture surface energy Γ is zero. The fracture condition in Eq. (5.11) is also called Griffith’s fracture criterion. In the case of crack formation, the work of the forces must be formed on both opposite crack surfaces. Therefore, the fracture surface energy applies: dΓ = 2γdA

(5.12)

Considering the energy release rate G according to Eq. (5.10), Eq. (5.11) can be written with the fracture surface energy from Eq. (5.12) and the critical energy release rate Gc = 2γ also in terms of the energy release rate G: G = Gc

(5.13)

In other words, when the crack propagation begins and continues, the energy released must be equal to the energy required for the fracture process. This leads to the assumption that in the event of fracture, a portion of the stored elastic strain energy, here Uη is released through the generation of new fracture surfaces. Other forms of energy such as acoustic or heat are neglected. Therefore, the third required assumption is that all released elastic strain energy Uη is converted to the fracture surface energy Γ. This results in the following condition: !

Γ = Uη = 2γA f r

(5.14)

where A f r = 2πr1t ρˆ is the fracture surface respectively the lateral surface by assuming a cylindrical fragment of height or thickness t. ρˆ is the roughness of the fracture surface, since the fracture surface is not smooth in reality but irregularly shaped (cf. Section 4.6). As shown and discussed in Section 4.6, the roughness of the actual fracture surface increases with increasing elastic strain energy to a value of ρˆ = 1, 220 for U = 310 J/m2 . However, for an assumption of a cylindrical fragment with a smooth lateral surface, the

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fracture surface roughness is ρˆ = 1. The specific fracture surface energy γ can be expressed in terms of the critical energy release rate depending on the stress state as: ⎧ 2 K ⎪ Plane stress ⎪ 2EIc , ⎨ Gc (5.15) = γ= ⎪ 2 ⎪ ⎩ KIc2 (1−ν 2 ) , Plane strain 2E wherein KIc is the critical stress intensity factor. Inserting the Eq. (5.9) in Eq. (5.14) and resolving it to r0 , we obtain:  UR,Rem 4γ ρˆ 4γ ρˆ (1 − η) = · · (5.16) r0 = UD,0 η(2 − η) UD,0 UR,Rel with UD,0 =

(1 − ν) σˆ E

(5.17)

According to the first assumption, the glass plate will break into cylindrical fragments. In Figure 5.8 the relative remaining elastic strain energy UR,Rem in correlation with the area of the base shape of the fragment A f r,base normalized by t 2 is shown in comparison with the FE results in N IELSEN (2017). As can be observed in Figure 5.8, the analytical results

Figure 5.8 Relative remaining elastic strain energy UR,Rem [−] vs. Fragment size given by A f r,base [mm2 ] normalized by t 2 for fractured soda-lime-silica glass, analytical result (black line) in comparison to the FE results (dashed line) by N IELSEN (2017), from (P OURMOGHADDAM et al., 2018e)

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Figure 5.9 Fragment density N50 [−] in an observation field of size 50 mm × 50 mm vs. Residual surface compressive stress |σs | [MPa] for different relaxation factors η = 0.05 to η = 0.9, from (P OURMOGHADDAM et al., 2018e)

of the fragment size from a value of approximately A f r,base /t 2 ≥ 1.5 corresponds to the FE results. The difference with very small fracture surfaces results from the change of the stress state from plane hydrostatic to multi-axial in the fracture state in the FE simulation, which was carried out on a fragment volume. In the FE-simulation stresses in the thickness direction occurred in the fracture state, whereas the analytical solution is based on a continuous plane hydrostatic stress state. Now the distance between points was elaborated to δ = 2r0 , which directly affects the fragment size and so the fragmentation density ND in dependence on the residual stress respectively the elastic strain energy U. The method significantly depends on the relaxation factor η (see Figure 5.9), which can take values in the interval [0; 1]. The curves in Figure 5.9 were created by assuming cylindrical fragments with N50 = 502 /(πr02 ) and different relaxation factors η. A relaxation factor of η = 0 would mean that there is no fracture and η = 1 that the energy is relaxed completely. Thus the fragment size or the minimum distance between two points for a hard-core spatial point process (Section 5.5) is mainly dependent on the energy relaxed in the fracture state. As described before and shown in Figure 5.9, the fragment density is affected by the relaxation factor η. The greater η, the higher the percentage of the stored elastic strain energy that is released during the fragmentation. A larger energy produces more crack surfaces and thus also a finer fracture pattern or in other words a larger fragment number within an observation field. For example, if an observation field of size 50 mm × 50 mm is given, for a residual surface compressive stress of 100 MPa one can calculate a fragment density of N50 = 14 for

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a relaxation factor of η = 0.05 and N50 = 132543 for a relaxation factor of η = 0.9. The Fragment Size Parameter δ was determined using the experimental data of fracture tests. In Section 4.4 specimens with different thicknesses were thermally tempered with different degree of tempering so that specimens with different residual stresses were obtained for the fracture tests. After the fracture tests the average fragment weight was determined from more than 130 fragments per specimen. Knowing the density of glass (2500 kg/m3 ) and the glass thickness t the area of the base shape of the fragment (fragment volume divided by thickness) can be recalculated from the weight. Using the energy density UD for the determination of the correlation, the curves of base area and thus the Fragment Size Parameter δ , which is the diameter of the base area, for different thicknesses coincide to one line (cf. Figure 4.35). In Figure 5.10, the elastic strain energy density UD calculated from the different residual stresses of the specimens is applied over the Fragment Size Parameter δ . Each of the black circles represents the average of more than 130 fragments per specimen. For the calculation of δ from the experimental data a fragment with the number of edges n → ∞ (cylindrical fragment) is assumed which contains other edge numbers and the Fragment Size Parameter δ is determined over the radius of the fragment. The red line represents the fragment size calculated analytically using Eq. (5.16). For the analytical calculations the measured residual stress values of the specimens used in Section 4.4, a Young’s modulus of E = 70000 MPa, a Poisson’s ratio of ν = 0.23, a fracture surface roughness of ρˆ = 1 and the plane stress state have been taken into account. As it can be observed in Figure. 5.10 the analytical curve fits for a relaxation factor of η = 0.123. A relaxation factor of η = 0.123 results in the relative remaining elastic strain energy of UR,Rem = 0.77. Thus, approximately 77% of the initial elastic strain energy U0 remains in the fragment and correspondingly 23% of the initial energy is converted into fracture surface energy and therefore is used during the fragmentation for the generation of new fracture surfaces by crack generation and branching. This applies to thin glass plates with the assumption of cylindrical fragments with smooth fracture surfaces. In Figure 5.11, the correlation between the elastic strain energy density and the fragment density is shown for the initial energy density UD,0 , remaining energy density UD,1 and the released energy density UD,η . Thus a direct correlation between the energy in pre-fracture, fracture and post-fracture state is shown.

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Figure 5.10 Elastic strain energy density UD [J/m3 ] vs. Fragment Size Parameter δ [mm], black circles with the black trend line show the average of more than 130 fragments per specimen (P OURMOGHADDAM et al., 2018d); the red line represents the calculated fragment size for η = 0.123; dashed line shows the limit of fragmentation at UD = 5712.7 J/m3 according to fracture test results

Figure 5.11 Elastic strain energy density UD [J/m3 ] vs. Fragment density N50 [−] for the initial, remaining and released energy density, η = 0.123

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5.4.3 Fracture Intensity Parameter λ The fracture intensity is another important model parameter for the theoretical prediction of the fracture pattern of tempered glass. The fracture intensity is a characteristic value for the fragmentation behaviour and contains information on the fracture structure and the fragment density in a given area. For the determination of the Fracture Intensity Parameter λ the two cases deterministic and stochastic fragmentation process have to be distinguished. For the deterministic case a HCP-distribution of the seed points of the Voronoi tessellation can be assumed. Thus, in the deterministic fragmentation process, the resulting cell structure is always a honeycomb. In contrast, the stochastic fragmentation process results in a polygonal cell structure induced by the Voronoi tessellation over a spatial point process. In the following, the two cases of fracture intensity are explained in more detail. 5.4.3.1 Deterministic Fragmentation Process From a deterministic Fracture Mechanics point of view, the material can be considered perfectly homogeneous, so that the spatial location of fragment centres (i.e. seed points) within a studied object is equally likely. As a starting point for an estimation of the locations of the centres of the glass fragments this assumption serves well and is called ’Deterministic Fragmentation Process’. field a hexagonal For the computation of the fragment density ND in an observation √ fracture structure (HCP) with a fragment side length a = 2r0 / 3 is now assumed (Figure 5.5 (c)). In a square observation field with the side length D the fragment density ND,HCP can be described as: ND,HCP =

D2 √ 2r02 3

(5.18)

Assuming that the point process is homogeneous with the intensity λ (which is constant over the whole domain), this further implies that the expected number of points falling into an observation region with the side length D is proportional to its area: ND = λ · D2

(5.19)

In order to obtain the hexagonal Voronoi cell structure (honeycomb), the seed points have to be distributed according to Hexagonal Close Packing (HCP). Combining Eq. (5.18) with Eq. (5.19), an approximation for the expected fracture intensity λ˜ HCP of a honeycomb, motivated from deterministic fracture mechanics, can be expressed in terms of the Fragment Size Parameter δ = 2r0 as: 2 1 λ˜ HCP = √ · 2 3 δ

(5.20)

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(a)

(b)

Figure 5.12 Fracture intensity λ˜ HCP [1/mm2 ] of a honeycomb vs. (a) Fragment size parameter δ [mm], (b) Energy relaxation factor η [−], from P OURMOGHADDAM et al. (2018e)

In Figure 5.12 (a), a double logarithmic interrelation between fracture intensity of the honeycomb and fragment size is shown. The fracture intensity decreases with a coarser fracture structure. λ˜ HCP can further be rewritten in terms of the energy relaxation factor η as: 1 η 2 (2 − η)2 · λ˜ HCP = √ 2 (1 − η)2 2 3ˆr

(5.21)

with rˆ =

4E ρˆ γ σˆ (1 − ν)

(5.22)

In Figure 5.12 (b), a double logarithmic interrelation between fracture intensity of a honeycomb and the energy relaxation factor is shown. The fracture intensity increases with higher energy relaxation. In Figure 5.13 (a)-(c) the Voronoi tessellation of HCP-distributed seed points with the point distance δ is shown for a plate with the residual mid-plane tensile stress of 50 MPa. The respective fragmentation density is shown in an observation field of size 50 mm × 50 mm. The relaxation factor η increases from η = 0.04 in Figure 5.13 (a) to η = 0.2 in Figure 5.13 (c). It is shown that for the same residual stress of 50 MPa the point distance decreases with higher values for η.

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(a)

(b)

139

(c)

Figure 5.13 Fragmentation in an observation field of size 50 mm × 50 mm vs. Energy relaxation factor η with regard to the Voronoi tessellation of HCP-distributed points with the distance δ , the residual mid-plane tensile stress σm = 50 MPa, (a) η = 0.04; δ = 17.89 mm; N50 = 9; λ˜ HCP = 0.0036 1/mm2 , (b) η = 0.1; δ = 6.92 mm; N50 = 60; λ˜ HCP = 0.0240 1/mm2 , (c) η = 0.2; δ = 3.25 mm; N50 = 274; λ˜ HC = 0.1096 1/mm2 , from P OURMOGHADDAM et al. (2018e)

(a)

(b)

(c)

Figure 5.14 Fragmentation in an observation field of size 50 mm × 50 mm vs. Energy relaxation factor η with regard to the Voronoi tessellation of uniform distributed points with the minimum distance δmin , the residual mid-plane tensile stress σm = 50 MPa, (a) η = 0.04; δmin = 17.89 mm; N50 = 7; λ˜ HC = 0.0028 1/mm2 , (b) η = 0.1; δmin = 6.92 mm; N50 = 44; λ˜ HC = 0.0176 1/mm2 , (c) η = 0.2; δmin = 3.25 mm; N50 = 183; λ˜ HC = 0.0732 1/mm2 , from P OURMOGHADDAM et al. (2018e)

5.4.3.2 Stochastic Fragmentation Process If fracture tests are conducted on glass panes, it can be observed, that the centre locations (seed points) of fragments do vary stochastically within an object under investigation due to different reasons such as inhomogeneous thermal pre-stressing of a glass pane, inhomogeneous load application etc. If the assumptions of a hexagonal fracture structure (HCP) are eased to the assumption of a polygonal fracture structure induced by the Voronoi Tessellation over a spacial point process, the resulting fragmentation patterns now depend on two parameters δ and λ (Figure 5.14). Analogously to Section 5.4.3.1, the same investigation can be carried out where this time using random seed point locations in an observation field of size 50 mm × 50 mm with a uniform distribution for the seed point locations instead of HCP condition. This kind of randomly distribution leads to a varying inter-point distance, where the distance determined on the basis of the relaxation factor η is given as the minimum distance for the point process. In Figure 5.14 (a)-(c), the Voronoi tessella-

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Figure 5.15 Elastic strain energy density UD [J/m3 ] vs. Fracture Intensity Parameter λ [1/mm2 ] determined from fracture tests in Section 4.4

tion of uniformly distributed points with the minimum point distance δmin is shown for a plate with the residual mid-plane tensile stress of 50 MPa. In comparison to the fragment density for the case of HCP-distributed points, the number of fragments is underestimated for uniformly distributed points. Until η = 0.04 the difference is not significant yet. However, the difference is more pronounced for the higher relaxation factor of η = 0.2. This is because on the one hand, Eqs. (5.20) to (5.22) are derived as approximations for the underlying stochastic point process as prior expectations motivated from the deterministic fracture process and on the other hand, in the deduction of Eqs. (5.20) to (5.22) a minimum energy requirement was not introduced for the randomly distributed points. 5.4.3.3 Experimental Determination of the Fracture Intensity Parameter λ The Fracture Intensity Parameter λ has been determined experimentally from fracture tests on thermally tempered glass specimens. In Section 4.4, the results of the fracture tests on differently heat treated glass specimens of size 1100 mm × 360 mm and three different thicknesses of 4 mm, 8 mm and 12 mm with different residual stress levels were discussed. The average fragment density N50 of eight square observation fields with a side length of D = 50 mm was determined and set into correlation with the strain energy density UD (see Figure 4.27 (b)). Now assuming a constant intensity in each observation field the average Fracture Intensity Parameter λ can be calculated for each specimen using Eq. (5.19). Using the energy density UD for the determination of the correlation, the curves of the fragment density N50 and thus the fracture intensity λ for the different thicknesses coincide to one line. In Figure 5.15, the correlation between the elastic strain energy density UD , calculated from the measured residual stresses of each specimen, and the

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Figure 5.16 Elastic strain energy density UD [J/m3 ] vs. Fracture Intensity Parameter λ [1/mm2 ] (experimental: black circles) and Fragment Size Parameter δ [mm] (experimental: black dots); red line represents the Fragment Size Parameter determined analytically with the energy relaxation factor of η = 0.123

Fracture Intensity Parameter λ is shown. The two model parameters δ and λ are applied over the elastic strain energy density UD in Figure 5.16.

5.4.4 Comparison of the Prediction Method with Literature In Figure 5.17, the predicted fragment density N50 within a square observation field of size 50 mm × 50 mm in terms of a deterministic fragmentation due to a hexagonal fracture structure (HCP) based on the Eq. (5.18) for the energy relaxation factor of η = 0.123 is shown in comparison with the fracture test results in A KEYOSHI et al. (1965) and the fragment density prediction according to BARSOM (1968). Considering Barsom’s particle mass equation (4.24), it can be rewritten as a function of the fragment base are A f r,base , since M = ρV = ρtA f r,base : 

1 σm = 105.61 MPa4 mm2 A

1/4 (5.23)

This clearly shows that the expression is independent of the thickness t. In order to compare the results with the fragment density results of A KEYOSHI et al. (1965), Eq. (5.23) 2 to obtain can also be rewritten considering a fragment base area of A f r,base = 2500mm N50

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Figure 5.17 Fragment density N50 [−] vs. Mid-plane tensile stress σm [MPa]; black dots show fracture test results in this thesis; fragment density prediction in terms of a hexagonal fracture structure (HCP) with η = 0.123 (black line) in comparison with the prediction curve according to B ARSOM (1968) (dashed red line) and the curves according to A KEYOSHI et al. (1965)

the average number of fragments in a square observation field of size 50 mm × 50 mm (fragment density N50 ): N50 =



4 σm 14.96MPa

(5.24)

The fragment density as a function of the residual mid-plane tensile stress is shown in Figure 5.17. It can be clearly seen that the prediction of fragment density in this thesis is very well in line with the prediction according to BARSOM (1968). Especially since the Barsom’s fragment density curve can be exactly reproduced with an energy relaxation factor of η = 0.14. Using Eq. 5.7, this means that applying the analytical model developed by BARSOM (1968), about 26% of the initial energy U0 is used during fragmentation. According to the prediction model in this thesis, the fraction of the initial elastic strain energy that is converted into fracture surface respectively is dissipated in generating new fracture surfaces by crack generation and branching is about 23% (see Section 5.4.2) which is in good agreement with 26% predicted according to BARSOM (1968). This applies to thin glass plates under the assumption of cylindrical fragments with smooth fracture surfaces. Assuming squared fragments, G ULATI (1997) estimates the fraction of used elastic strain energy to be 43%. However, using the frangibility model indicated by G ULATI (1997) and applying the analytical model developed by BARSOM (1968) yields a fraction of the elastic strain energy of 35%. The overestimation of the dissipated energy in G ULATI (1997) is due to the fact that the

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prediction model is elaborated with the elastic strain energy in the tensile region of the glass plate. A thickness fraction of 0.578t was used as the integration limit. Thus, Gulati obtained an integration pre-factor of σˆ m = 0.309σm2 . In this thesis, however, the stored elastic strain energy is calculated over the entire glass thickness. Therefore, the integration pre-factor is σˆ m = 45 σm2 . Thus, the stored elastic strain energy in G ULATI (1997) is only 38.6% of the value used for the elaboration of the prediction model in this thesis. This means that the initial elastic strain energy density in Eq. (5.16) has to be changed to 0.386UD,0 in order to compare the results with G ULATI (1997). This increases the value of r0 and subsequently leads to a higher energy relaxation factor η due to the Eq. (5.5). In this respect, the fracture tests can be fitted with the higher energy relaxation factor of η = 0.26. Accordingly, the fraction of the dissipated elastic strain energy used for the generation of new fracture surfaces increases from 23% to approximately 45% which is in good agreement with 43% predicted by G ULATI (1997). In the view of the author of this thesis, however, this is an overestimation, since the energy density stored in the glass plate is not only released over the tensile region but over the entire thickness of the plate. In contrast to the fragment density curves given in A KEYOSHI et al. (1965), the fragmentation in this thesis as well as in BARSOM (1968) and G ULATI (1997) are solely governed by the residual stress and are independent of the glass thickness. In Chapter 4 of this thesis, the correlation between fragmentation and the elastic strain energy (density) was shown. It can be observed in Figure 5.17 that the fragment density curve presented in this thesis as well as the reproduced curve from BARSOM (1968) are almost tangential to the Akeyoshicurves which drop steeply depending on the corresponding thickness. The results show that the consideration of a stored elastic strain energy approach using a plane stress state as performed in G ULATI (1997) and BARSOM (1968) and this thesis leads to a thickness independence of the prediction model. Up to now, only one small section in M OLNÁR et al. (2016) raised the idea of using a Voronoi tessellation to investigate the glass fracture pattern. In M OLNÁR et al. (2016), however, it is neither elaborated nor clearly stated, which point process was used for the presented Voronoi tessellation. Furthermore in that paper nothing was said about the connections of the parameters from linear elastic fracture mechanics to those of an underlying stochastic point process. Considerations of the author on the two results presented in M OL NÁR et al. (2016) led to the conclusion, that a homogeneous Poisson point process without further restrictions was used to generate the given fracture pattern. However, the equation presented in this paper is unphysical, since negative energies result for certain pre-stresses. As a conclusion, the work presented in M OLNÁR et al. (2016) is rather a superficial attempt to combine the Voronoi tessellation idea with the fracture pattern description, without going into the stochastic point process part and the search for the connections between point pattern statistics and fracture mechanics. The same basic idea of connecting point processes with fracture mechanics to predict glass fracture patterns for different pre-stress

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levels in glass panes was gathered in a recent collaboration of Michael Kraus (K RAUS, 2019) and the author of this thesis independent of M OLNÁR et al. (2016). Furthermore the level of research and elaboration of the connections in that direction as presented in P OUR MOGHADDAM et al. (2018f), P OURMOGHADDAM et al. (2018e), K RAUS et al. (2019b), and K RAUS et al. (2019a) and this thesis is far beyond what is shown in M OLNÁR et al. (2016).

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5.5 Mathematical Foundations of Point Processes and Point Pattern Analysis In addition to the fracture mechanics, the framework of the method BREAK is also motivated from spatial statistics, which deals with statistical methods that explicitly consider the spatial arrangement of data (W IEGAND et al., 2014; M ARTINEZ et al., 2015; BADDE LEY et al., 2016). The spatial data can be viewed as measurements that are observed at discrete locations in a two- or three-dimensional region and might be spatially correlated. Typically, spatial statistics methods can be divided into one of three categories based on the type of spatial data to be analysed. These types of data are called: point patterns, geostatistical data, and lattice data. Spatial point patterns are data made up of the location of point events. It is of interest whether the relative data positions represent a significant pattern or not. For example, if seismologists have data about the distribution of earthquakes in a region, pattern analysis allows them making predictions about future earthquakes. For Geostatistical data (also called spatially continuous data) a measurement is attached to the location of the observed event. As an example, if geologists take ore samples at locations in a region, these data allow them to estimate the extent of the mineral deposit over the entire region. The third type of spatial data is called Lattice data. These data are often associated with regularly or irregularly spaced, where the objective of analysis of lattice data is to model spatial patterns in the attributes associated with the fixed areas. For example, a political party uses data representing the geographical voting patterns in a previous election to determine a campaign schedule for their candidate. For the deduction of the method BREAK the spatial point patterns category is used, hence in the context of this thesis further details are given only on this category in the latter of this chapter. For more details on statistical methods for geostatistical or lattice data it is referred to W IEGAND et al. (2014), M ARTINEZ et al. (2015), and BADDELEY et al. (2016). Since different notations are common in the literature, the notation from BADDELEY et al. (2016) is adopted within this thesis.

5.5.1 Spatial Point Process and Spatial Point Pattern A spatial point process X is a random mechanism whose outcome is a point pattern. If the process is repeated under identical conditions, the observed point pattern would be different each time. Figure 5.18 shows ten different possible results or realizations of the same random point process. A spatial point pattern x is a set x = {x1 , ..., xn }

(5.25)

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

Figure 5.18

Ten different possible outcomes (’realizations’) of the same point process

of points xi in a study region B, which is especially in the context of this thesis in the twodimensional space (B ⊆ R2 ). Each point location xi is a vector containing the coordinates of the i-th event:   xi1 (5.26) xi = xi2 The term event can refer to any spatial phenomenon that occurs at a point location. The number n(x) of points in the pattern is not fixed in advance and may be any finite nonnegative number including zero. A location in the region B, where an observation of the phenomenon was taken, is called point. If, in addition to the coordinate locations of the events, further quantities of interest associated with the events are indicated, this type of spatial data is referred to as a marked point pattern. Within this chapter, however, it is assumed that the data represent a mapped point pattern, i.e. all relevant events in the study region B were measured, while B can have any form. According to M ARTINEZ et al. (2015), the analysis of point patterns is sensitive to the definition of B as edge effects might arise in the analysis of point patterns. Thus the analysis has to be conducted for different guard areas and/or different study regions. Since spatial point process is a random process, the behaviour can be viewed in terms of first-order and second-order properties. These are related to the expected value (i.e. the mean) and the covariance (M ARTINEZ et al., 2015; BADDELEY et al., 2016). The first-order property is described by the intensity λ , which is defined as the mean number of events per unit area, where the second order property reflects the spatial dependence in the process. The probability density of a point pattern x with a number of points n, point process parameters θ and distribution of the point locations fnθ (x) is given by: fn (x, θ ) = P[N(B) = n|θ ]n! fnθ (x)

(5.27)

5.5 Mathematical Foundations of Point Processes and Point Pattern Analysis

(a)

(b)

147

(c)

Figure 5.19 Examples of the point distribution outcomes of the same number of points on the same plane with three different methods: (a) Homogeneous Poisson process / Complete Spatial Randomness (CSR), (b) Regular / Hard-core process and (c) Cluster process, from P OURMOGHADDAM et al. (2018e)

Different models for the spatial point processes are known in the literature, an overview can be found in W IEGAND et al. (2014) and BADDELEY et al. (2016). In general, there are three main categories of spatial point processes: (a) Homogeneous Poisson point process / Complete Spatial Randomness (CSR), (b) Regular / hard-core processes and (c) Cluster processes. The three types of point processes are graphically illustrated in Figure 5.19 In order to predict and simulate the 2D macro-scale fragmentation of tempered glass, only three special types of spatial point processes are considered: (1) Homogeneous Poisson process (HPP) (2) Matérn Hard-core process (MHCP) (3) Strauss process (SP) In the further, important features and the mathematical background of these spatial point processes is introduced and explained in more detail. 5.5.1.1 Homogeneous Poisson Process (HPP) / Complete Spatial Randomness (CSR) A point process is called ’completely random’ as illustrated in Figure 5.19 (a) if it is characterised by two key properties: •

homogeneity: the points have no preference for any spatial location and the intensity does not vary over the study region

•

independence: information about the outcome in one region of space has no influence on the outcome in other regions

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

•

the number n(X ∩ B) of random points falling in a test region B has a Poisson distribution

Thus, there are no interactions between the distributed points, i.e. a given number n(X ∩ B) of random points in the point process X are uniformly (equal likelihood) and independently distributed over the study region B. A point process with these properties is called homogeneous Poisson point process (HPP) or complete spatial randomness (CSR) and its probability density of a point pattern x is given by fn (x, λ ) = λ n(x) exp((1 − λ )|B|)

(5.28)

where n(x) is the number of points in x. The right-hand side is the probability density of observing the pattern x if the true intensity is λ . This probability density depends only on the number of points, because all spatial locations are equally likely under CSR. The CSR is important in many ways as it is a realistic model of some physical phenomena such as radioactivity etc. and it serves as a benchmark against other patterns (i.e. the HPP is the null hypothesis in statistical tests) (BADDELEY et al., 2016). In this sense Homogeneity means that the expected number of points falling in a region B should be proportional to its area |B| on average: En(X ∩ B) = λ |B|

(5.29)

where λ is constant and is in effect the average number of random points per unit area. λ is known as the intensity of the point process. For the HPP, the random number of points n(X ∩ B) has a Poisson distribution with mean μ = λ |B|. The HPP has in total one parameter, which is the intensity λ . The simulation of a HPP is easy under the given properties. Given a region B where the realization is to be generated and specifying an intensity value λ , at first the total number of points is generated as a random number N according to a Poisson distribution with mean μ = λ |B|, then these N points are placed independently in B with a uniform distribution for the x1 and x2 coordinate. 5.5.1.2 Matérn Hard-Core Processes (MHCP) (Matérn) Hard-core point processes (MHCP) or simple inhibition processes (M ATÉRN, 1960) as illustrated in Figure 5.19 (b) are characterized by any two points maintain a minimal distance rHC (≡ δHC ) but do not incorporate further departures from CSR. For MHCP the observation window B is covered by n non overlapping disks with diameter rHC . The MHCP has in total two parameters, which are the intensity λ and the hard-core distance rHC . The simulation of MHCP patterns is possible in several ways. One approach is to remove all pairs of points that have a separation distance shorter than rHC from an existing

5.5 Mathematical Foundations of Point Processes and Point Pattern Analysis

(a)

(b)

149

(c)

Figure 5.20 Formation of Matérn’s Model I. (a) Points are generated by HPP, (b) Any point lying within distance rHC is deleted (×), (c) Resulting thinned point pattern, from B ADDELEY et al. (2016)

(a)

(b)

(c)

Figure 5.21 Formation of Matérn’s Model II. (a) Points are generated by HPP marked by independent arrival times, (b) Any point lying within distance rHC of an earlier point is deleted (×), (c) Resulting thinned point pattern, from B ADDELEY et al. (2016)

CSR pattern (Matérn Model I), see Figure 5.20. Another method is to use a sequential algorithm that generates a series of CSR points, but the only series of points accepted are those that are further away than the distance rHC from any point generated earlier in the sequence (Matérn Model II), see Figure 5.21. Within this thesis the first simulation method ((Matérn Model I) is used. 5.5.1.3 Gibbs and Strauss Process MHCP are often too restrictive or simple to characterize inhibition processes to generate observed point patterns (W IEGAND et al., 2014). Interactions between individual points may not result in total inhibition of other points within the zone of influence rHC . Instead, interactions between points within the zone of influence rHC may be less likely and not impossible. A Gibbs point process is the model of choice for a point pattern with inhibitions

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between points. It is the only kind of spatial point process that involves explicit interaction between the points (BADDELEY et al., 2016). With a fixed number of points n within a given observation window B the Gibbs point process is defined by a location density function fn (x). This is a measure of the likelihood of a particular spatial configuration of points under a given Gibbs process defined by the so-called pair-potential function Ψ(r), which is a function of the distance r = ||xi − x j || between two points xi and x j . Interactions are typically assumed to be symmetric, hence the location density function does not depend on the order of points. For a Gibbs point process with a fixed number of points n and pairwise interactions, the location density function can be given by   1 1 i= j exp − ∑ y(||xi − x j ||) (5.30) fn (x) = Zn 2 i, j where Zn is a normalization constant ensuring fn being a probability density function, and y(||xi − x j ||) is the pair-potential function. Eq. (5.30) can be rewritten by using a quantity called pairwise interaction function h(r) = exp(−Ψ(r)). The pairwise interaction function yields a value of 1 if the two points do not interact, since in this case Ψ(r) = 0. However, pairs of points located at distance r exhibit hyperdispersion if h(r) < 1 and clustering if h(r) > 1 (I LLIAN et al., 2008). The location density function (5.30) can be expressed in terms of the pairwise interaction function h(r) as: n−1

fn (x) = ∏ i=1

n

1 h(||xi − x j ||) Z j=i+1 n

∏

(5.31)

Point patterns x that yield high values of the location density function fn (x) are more likely to occur than those with lower values, as dedicated by the Gibbs process, which is defined by the pairwise interaction function h(r). A Gibbs hard-core process with a fixed number of points has pairwise interaction function:  0 r ≤ rHC (5.32) h(r) = 1 r > rHC This means, that a MHCP is a special case of Gibbs processes, but usually a MHCP is preferred of the Gibbs hard-core process due to reasons of simplicity of simulation as mentioned later in this subsection. An extension of the Gibbs hard-core process (with fixed number of points) is the so-called Strauss process (SP) (S TRAUSS, 1975) with a fixed number of points. In this case, the

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151

Figure 5.22 Simulated realizations of Strauss processes in the unit region with fixed intensity and interaction distance rHC but different values of γ ranging from 0 to 1, from B ADDELEY et al. (2016)

pairwise interaction function for a point xi inside rHC yields a value 0 ≤ γ ≤ 1. The SP possesses the pairwise interaction function:  γ = exp(−β ) r ≤ rHC (5.33) h(r) = 1 r > rHC β is the corresponding value of the pair-potential function for points with a distance of r ≤ rHC . For the case of γ = 0 in Eq. (5.33) a MHCP and for γ = 1 a HPP is obtained, see Figure 5.22. Combining Eqs. (5.30) and (5.33) yields:   1 β i= j exp − ∑ I(||xi − x j || ≤ rHC ) fn (x) = Zn 2 i, j (5.34) 1 s(x,rHC ) 1 exp(−β s(x, rHC )) = γ = Zn Zn where the quantity s(x, rHC ) counts the number of pairs of points that are located closer than distance rHC . The SP has in total three parameters, which are the intensity λ , the hard-core distance rHC and the interaction parameter γ. The simulation of a Gibbs process and thus Strauss Process requires indirect methods as they are defined by means of a high-dimensional probability density function fn (x), which returns the likelihood of a given point configuration. As the likelihood is related to the assumed interaction structure of pairs of points governed by the pairwise interaction function h(r) the Gibbs process is simulated by trial and error, starting with a suitable initial pattern. Points are then removed at random and are replaced only by points that are likely to occur, given the remaining point configuration and the probability density function fn (x). Thus, the sampling of a Gibbs point process is in general not a simple task (as e.g. the normalizing constant Zn may be unknown as in the SP case). Therefore, specific Markov Chain Monte Carlo (MCMC) methods have to be applied as they do not require normalized probabilities for sampling (M ØLLER, 1999; BADDELEY et al., 2016).

152

(a) Intensity (first-order)

5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

(b)

Second-order

(c)

Third-order

Figure 5.23 Product densities. (a) First-order product density: probability of a point being located within a randomly placed disk of infinitesimal area dx, (b) Second-order product density: probability of two points being located in two randomly placed disks separated by distance r, (c) Third-order product density: probability of three points being located in three randomly placed disks characterized by two distances r1−2 and r2−3 and one angle α, from W IEGAND et al. (2014)

5.5.2 Exploratory Data Analysis Within an exploratory context, point process models are used to provide a detailed description of the spatial structure of a point pattern. In this case, a point process model is presented as a parametric model that fits the data to provide a concise description of the spatial structure (W IEGAND et al., 2014). Before stochastic modelling of a point pattern dataset after having obtained spatial point patterns, an exploration of the statistical properties is conducted. According to W IEGAND et al. (2014) and BADDELEY et al. (2016), the typical steps involved in a point-pattern analysis are: (1) determination of the point pattern data type and decision on whether the pattern is homogeneous or not (2) selection of appropriate summary statistics (3) selection of appropriate null models and point process models (4) comparison of observed data and null models In this section, density estimation techniques to estimate the intensity or first-order property as well as methods to explore the distributions of nearest neighbour distances to estimate second-order properties are introduced. A graphical illustration of estimating different orders of statistics of a spatial point pattern is given in Figure 5.23.

5.5 Mathematical Foundations of Point Processes and Point Pattern Analysis

153

5.5.2.1 First-Order Statistics: Intensity The exploratory analysis begins with an assessment of spatial homogeneity by estimation of the intensity λ , which is a first-order statistic. This can be done using tools such as the quadrant counting test, kernel estimation or LISA (Local Indicators of Spatial Association) methods (BADDELEY et al., 2006; BADDELEY et al., 2016). An estimate of the intensity is obtained by dividing the study region B into a regular grid and then counting the number of events falling into each square. These counts are each divided by the area of the square. Homogeneity is assumed when the variation of this fraction is small over the entire B region. Alternatively, a kernel method can be used to obtain a smoother estimate of intensity than the quadrant method. Let x denote a point in the study region B and represent the event locations. Then an estimate of the intensity using the kernel method is given by   x − xi 1 n 1 K (5.35) λˆ h (x) = ∑ h2 δh (x) i=1 h where K is the kernel function (bivariate probability density) and h is the bandwidth or smoothing parameter. In Eq. 5.35 the edge correction factor is    x−u 1 K du (5.36) δh (x) = 2 h Bh which represents the volume under the scaled kernel centred on x inside the study region B. By the kernel width h the smoothness of the estimate can be controlled. A recommended choice for the bandwidth is h = 0.68n−0.2 , when B is the unit square (M ARTINEZ et al., 2015). A schematic representation of the kernel estimation method is shown in Figure 5.24. 5.5.2.2 Second-Order Statistics: Spatial Dependence Second-order statistics are based on the spatial relationship of pairs of points. In contrast to the first-order statistics of point processes, where there is only one relevant summary statistic, that is, the intensity function, there are several second-order statistics. After investigating the intensity λ and thus first-order statistics of the point pattern, the next step is to explore the second-order properties respectively the spatial dependence of the point pattern. The following exploratory methods investigate the second-order and spacing properties by studying the distances between events in the study region B. Spacing: Nearest-Neighbour Function G and Empty-Space Function F Valuable information about the spatial arrangement of points is conveyed by the cumulative distribution function of the nearest-neighbour and the empty-space distances. The nearest-neighbour distance is the distance from a data point to the nearest other data point

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

Figure 5.24 A schematic representation of the kernel estimation method applied to study region B, from OYANA et al. (2015)

(event-event distance rg ), where as the empty-space distance is the distance from a fixed location to the nearest data point (location-event distance r f ). Note that nearest neighbour distances provide information at small physical scales, which is a reasonable approach if there is variation in the intensity over the region B. If B is a disc of radius r, the |B| = πr2 and for the CSR model with the intensity λ , the empty-space cumulative distribution function is given by F(r f ) = 1 − exp (−λ πr2f )

(5.37)

with 0 ≤ r f . Eq. (5.37) is the theoretical empty-space function of the Poisson process (HPP). The nearest-neighbour cumulative distribution function is analogous to the emptyspace function G(rg ) = 1 − exp (−λ πrg2 )

(5.38)

with 0 ≤ rg . For a completely random pattern (CSR), the distribution of the nearestneighbour distance is the same as that of the empty-space distance G(rg ) ≡ F(r f ). For a general point process, F and G will be different functions. The second-order properties of a spatial point patterns can now be investigated by comparing the observed empirical cumulative distribution function of r f or rg against the respective CSR functions. The empirical cumulative distribution function (CDF) for the location-event distances r f is given by ˆ f ) = #(r f ,i ≤ r f ) F(r m

(5.39)

5.5 Mathematical Foundations of Point Processes and Point Pattern Analysis

(a)

(b)

155

(c)

Figure 5.25 Empirical empty-space distance distribution function Fˆ (solid lines) for each of the three categories of point patterns in Figure 5.19 and the theoretical F-function for a HPP (dashed lines), (a) HPP pattern, (b) Regular pattern (c) Clustered pattern, from B ADDELEY et al. (2016)

(a)

(b)

(c)

Figure 5.26 Empirical nearest-neighbour distance distribution function Gˆ (solid lines) for each of the three categories of point patterns in Figure 5.19 (a) HPP pattern, (b) Regular pattern (c) Clustered pattern; and the theoretical G-function for a HPP (dashed lines), from B ADDELEY et al. (2016)

Similarly, the empirical cumulative distribution function (CDF) for the event-event distances rg is ˆ g ) = #(rg,i ≤ rg ) G(r n

(5.40)

where m respectively n is the number of points randomly sampled from the study region. An example the representative F- and G-function for the three categories of spatial point processes as given in Figure 5.19 is illustrated in Figure 5.25 and Figure 5.26. ˆ g ) and F(r ˆ f ) provides possible evidence of inter-event interactions. For A plot of G(r the case of clustering in the point pattern, lot of short distance neighbours are to be exˆ g ) would increase steeply for smaller values of rg and flatten out as pected, i.e. that G(r the distances get larger. On the other hand, if the point pattern under investigation shows ˆ g ) would be flat at small regularity, there should be more long-distance neighbours and G(r distances and increase steeply at larger distances rg or r f . For the examination of the plot ˆ f ), the opposite interpretation holds (e.g. if there is an excess of long distances of F(r

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

ˆ g) ˆ f ), then that is evidence for clustering). A third possible plot is that of G(r values in F(r ˆ against F(r f ). If the relationship follows a straight line, then this is evidence that there is ˆ g ) to exceed F(r ˆ f ), with no spatial interaction. If there is clustering, one can expect G(r the opposite situation occurring if the point pattern exhibits regularity. So far, the handling of edge effects (influence of events near the boundary of the region B) has been neglected. Different strategies for that are presented and discussed in BAD DELEY et al. (2016), W IEGAND et al. (2014), M ARTINEZ et al. (2015), and L UNN et al. (2012). One possible method is to force a guard area inside the perimeter of B, where the nearest-neighbour distances for points or events is not computed, but these events are used in computing nearest-neighbours for points or events inside the rest of B. Third- and higher-order summary statistics are not of interest within this thesis. Further information on that can be found in BADDELEY et al. (2016). Correlation: K- and L- Function A standard statistical tool for measuring dependence is correlation or more generally covariance, which both are classified as a second moment quantity. A very popular technique for analysing spatial correlation is the K-function proposed by R IPLEY (1977). The emˆ g ) and F(r ˆ f ) use distances to the nearest neighbour, which means, that pirical CDF G(r they consider the spatial point pattern over the smallest scales. To gain insight about the point pattern at several scales, the K-function is used as this quantity is related to the second-order properties of an isotropic process, where first-order homogeneity is valid. The K-function is defined as K(r) = λ −1 E[#extra events within distance d of an arbitrary event]

(5.41)

where λ is the intensity over a region and E[·] is the expectation operator. The empirical K-function can be estimated through: Ir (di j ) |B| ˆ K(r) = 2 ∑∑ n i= j rg,i j

(5.42)

where n is the number of events, di j = ||xi − x j || is the distance between the i-th and jth events in a point pattern x and Ir is an indicator function that takes on the value of one if di j ≤ r and zero otherwise. The rg,i j in Eq. (5.42) is a correction factor for the edge effects. If a circle is centred at event i and passes through event j, then rg,i j is the proportion of the circumference of the circle that is in region B. The estimated Kfunction can be compared to what would be expect if the process that generated the data is completely spatially random. For a CSR spatial point process, the theoretical K-function is K(r) = πr2 . Specifically, the observed point process exhibits regularity for a given value ˆ > πr2 . An of r if the estimated K-function is less than πr2 , alternatively for clustering K(r) example for a representative K-function for the three categories of spatial point processes

5.5 Mathematical Foundations of Point Processes and Point Pattern Analysis

(a)

(b)

157

(c)

Figure 5.27 Empirical K-function (solid lines) for each of the three categories of point patterns in Figure 5.19 (a) HPP pattern, (b) Regular pattern (c) Clustered pattern; and the theoretical K-function for a HPP (dashed lines), from B ADDELEY et al. (2016)

(a)

(b)

(c)

Figure 5.28 Empirical L-function (solid lines) for each of the three categories of point patterns in Figure 5.19 (a) HPP pattern, (b) Regular pattern (c) Clustered pattern; and the theoretical L-function for a HPP (dashed lines), from B ADDELEY et al. (2016)

as given in Figure 5.19 illustrates the aforementioned, Figure 5.27. transformation of K-function is the L-function  K(r) L(r) = π

A commonly used

(5.43)

which transforms the theoretical Poisson K-function πr2 to a straight line L(r) = r and thus making visual assessment of the L graph easier. An estimator for the L-function is:  ˆ K(r) ˆ (5.44) L(r) = π An example for a representative L-function for the three categories of spatial point processes as given in Figure 5.19 is shown in Figure 5.28.

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

Summarizing this subsection, the K- and L-functions are useful guidelines for building point process models as they classify the correlations at several length scales. The two functions will be used in the latter of this chapter to support the model building and statistical evaluation of fracture pattern images of thermally tempered glass plates.

5.6 Geometrical Properties of Random 2D Voronoi Tessellations The statistical distributions of the geometrical properties of Voronoi tessellations can in general be only obtained through simulation as analytically tractable expectation operations are in general not available due to the probability density of most spatial point patterns. In literature, few numerical studies on the Voronoi tessellation over HPP, MHCP or SP have been provided. For the HPP these kind of study was conducted by e.g. Z HU et al. (2001) and TANEMURA (2003), for the MHCP by e.g. O KABE et al. (1992), D EREUDRE et al. (2011), and I BRAHIM et al. (2013) and for the SP by e.g. G HOSH et al. (1997) and TAYLOR et al. (2012). In this section, two-dimensional Voronoi tessellations of planar distributed points are used for the simulation of the fragmentation of tempered glass. Therefore, points are numerically generated in a 2D plane using the three different point process types. The Voronoi tessellation of each point pattern is then generated and the resulting cells are subject to statistical examination wit respect to geometrical characteristics such as the cell area, cell perimeter and number of edges per cell as they are connected to certain characteristics of fracture mechanics.

5.6.1 Random Tessellations over Spatial Point Patterns and Statistics of their Geometrical Properties During the analysis of a spatial point pattern it can be very useful to apply methods that do not form part of the point process statistics. With this, new geometrical structures, referred to as ’secondary structures’ (BADDELEY et al., 2016; F EIGELSON et al., 2012; O HSER et al., 2009; I LLIAN et al., 2008), are constructed based on the points in the pattern, and statistical methods suitable for the specific type of geometric structure are applied. Secondary structures include: random sets, random fields, tessellations and networks or graphs. Within this thesis, just tessellations are of interest, as the single fracture pieces are interpreted within the BREAK context as the cells / tiles formed by the Voronoi tessellation, which itself is induced by the spatial point pattern. The geometrical evolution of a 2D Voronoi cell was described in Section 5.2. According to O HSER et al. (2009) and I LLIAN et al. (2008) a tessellation / mosaic divides the plane B into non-overlapping polygon. As described in Section 5.2, the most important tessellation model is the Voronoi tessellation, which is a regular tessellation of planes and

5.6 Geometrical Properties of Random 2D Voronoi Tessellations

159

higher-dimensional spaces. The Voronoi tessellation is constructed with respect to a point process X in Rd with point pattern (generators) x = {x1 , ..., xn }. The Voronoi cell VC is defined as the set of those points, having smaller distance to point xi than any other point x j , j ∈ In = {1, ..., n}:  VC (xi ) = z ∈ Rd : ||z − xi || p ≤ ||z − x j || p ∀

j = i, j ∈ In

(5.45)

The Voronoi tessellation of Rd with respect to x is the set of all Voronoi cells generated by the points in x: VT (x) = {VC (xi ) : xi ∈ x}

(5.46)

By using the Euclidean distance (p = 2) in Eq. (5.45), Voronoi cells are guaranteed to be convex polygons. In a broader sense, Voronoi tessellations are the mathematical tool to capture the idea of dividing space into zones of influence with a set of generators. Within the context of this thesis, the 2D Voronoi tessellations are generated by means of the Delaunay triangulation of the planar distributed nuclei (cf. Section 5.2), see Figure 5.29. The cell borders are obtained connecting the centres of the circumcircles and are thus perpendicular bisectors for each pair of adjacent points. In Figure 5.29 a sample of Voronoi tessellation of a HPP as well as a MHCP (point process with regularity) is given. Some of the statistical properties of the spatial point patterns are reflected in the properties of the morphological structure of the corresponding Voronoi tessellation. Thus, the Voronoi tessellation plays an important role in the morphological investigation of the glass fracture pattern structure. Artificial results have to be considered for a Voronoi tessellation over a finite inspection region B. Here, cells lying at the edge become infinitely large (Figure 5.3) and therefore should not be considered in the statistical evaluations. For this purpose an evaluation boarder is defined in which the Voronoi cells are finite. This is shown as an example in Figure 5.30. In literature, specific terms for Voronoi tessellations over specific point processes such as the Poisson Voronoi Tessellation, Hard-Core Voronoi Tessellation etc. are known (O KABE et al., 1992; O HSER et al., 2009; BADDELEY et al., 2016). For the Poisson-Voronoi tessellation (PVT), the only degree of freedom is the intensity λ of the generating Poisson point process. Thus for the PVT, analytical expressions for different characteristics such as the cell area AC , vertices number nV or cell perimeter PC are summarized in Table 5.1. Tessellations of stochastic point processes that differ from PVT usually do not possess analytically viable cell characteristics and thus have been estimated by large simulation studies. This was done within the context of this thesis and pre-published in P OURMOGHADDAM et al. (2018f) for the MHCP and SP for different levels of regularity, which is highlighted in the following.

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

(a) Figure 5.29 MOGHADDAM

(b)

Sample of Voronoi tessellation of generated points (a) HPP and (b) MHCP, from P OUR et al. (2018f)

Figure 5.30 2D Voronoi tessellation sample with infinitely large border cells. The dotted red line indicates the evaluation border for the statistical investigations, from P OURMOGHADDAM et al. (2018f)

Table 5.1 Statistic of PVT for a HPP with intensity λ , from I LLIAN et al. (2008)

Characteristic Cell area AC Cell perimeter PC Number of vertices nV

Mean

Variance

λ −1

0.2802λ −2 0.9455λ −1 1.7808

√ 4 λ /λ 6

5.6 Geometrical Properties of Random 2D Voronoi Tessellations

161

5.6.2 Point Pattern Generation Two-dimensional Voronoi tessellations of point patterns are used to simulate the fragmentation of tempered glass. Within the context of this thesis and the prepublication (P OUR MOGHADDAM et al., 2018f), MATLAB program package (MATLAB, 2017) was used for the simulation study. First, ND nuclei are generated within an observation field of side length D according to the point probability distribution of the respective process. ND can be interpreted as the number of fragments within this observation field. In this study, HPP, MHCP and SP are investigated, where their respective point densities are given by Eqs. (5.28) to (5.31). As already outlined in Section 5.5.1, the HPP is a one-parameter model with intensity λ as the only degree of freedom, the MHCP possesses two parameters (intensity λ and minimum hard-core distance δHC ) and the SP has three parameters (intensity λ , hard-core distance δHC and acceptance probability γ). For the minimum hardcore distance δHC ≡ rHC applies. For the HPP with intensity λ the total number of points in an observation field of size D2 can be computed with ND = λ · D2 according to Eq. (5.19). As outlined in Section 5.4.3, the most regular Voronoi tessellation over a domain is obtained for points distributed due to Hexagonal Close Packing (HCP), which is generated by equidistantly distributed points under an angle of 60◦ with distance δHCP . For the HCP hexagonal cells (six sides and vertex angles of 120◦ ) are found, cf. Figure 5.31. √ Assuming a HCP fracture structure with a fragment side length of a = δHCP / 3 in a quadratic observation field with the side length D, the fragment number respectively the

Figure 5.31 Regular honeycomb tessellation resulting from hexagonal close packed points in an observation filed. The regular hard-core distance between any two adjacent points is δHCP , from P OUR MOGHADDAM et al. (2018f)

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

number of cells ND,HCP from Eq. (5.18) can also be written in terms of the point distance δHCP : ND,HCP =

2D2 √

2 δHCP

3

(5.47)

From Eqs. (5.19) and (5.47), the uniform equidistance δHCP between any two adjacent nuclei can be computed by:   2D2 2 1 √ = √ · (5.48) δHCP = ND,HCP 3 3 λ In order to compare the regularity of a Voronoi tessellation over a stochastic spatial point process to the HCP, the uniformity parameter α is introduced as: α=

δHC δHCP

(5.49)

For the boundary values of the uniformity parameter, if α = 1 ⇔ δHCP = δHC the Voronoi tessellation is a HCP (honeycomb) and if α = 0 ⇔ δHC = 0 the Voronoi tessellation is a HPP, i.e. completely random. For the uniformity parameter values 0 ≤ α ≤ 1 the MHCP and the SP are related to the HCP. Given the uniformity parameter α and the intensity λ of the point process, the hard-core distance δHC for the realization of the MHCP and SP point processes can be obtained by using Eqs. (5.48) and (5.49):  2 1 (5.50) δHC = α · √ · 3 λ For practical simulation, in order to construct a random Voronoi tessellation according to HPP, MHCP or SP with ND cells in the observation field with the area D2 , the hard-core distance δHC has to be less than the HCP equidistance δHCP , otherwise, it is impossible to obtain ND cells. The process parameters for the simulation study are summarized in Table 5.2. For the statistical evaluation of the resulting Voronoi cell characteristics, each simulation is performed at 104 realizations (i.e. generations of Voronoi tessellations on the observation field). However, considering the number of cells within the evaluation border, the actual number of cells considered for the statistical evaluations varies according to the intensity λ , the uniformity parameter α and the acceptance probability γ from 105 to 2 × 106 .

5.6 Geometrical Properties of Random 2D Voronoi Tessellations

Table 5.2

163

Process parameters for the generation of the nuclei, from P OURMOGHADDAM et al. (2018f)

Point process

HPP MHCP SP

λ [1/mm2 ]

Uniformity parameter α [−]

Acceptance probability γ [−]

0.0025; 0.005; 0.01; 0.02 0.0025; 0.005; 0.01; 0.02 0.0025; 0.02

0 0.1;0.2;...;0.7 0.1;0.2;...;0.7

0.001; 0.2

Intensity

5.6.3 Results of the Voronoi Cell Structures Since this section is an excerpt from the prepublication P OURMOGHADDAM et al. (2018f), only the main findings are repeated in the following. In Figure 5.32 to Figure 5.34, the 2D Voronoi tessellations for HPP, MHCP and SP are presented. The intensity λ is the decisive parameter of the point distribution method HPP, which governs the number of cells per area. The greater the intensity, the higher the number of the cells ND in the observation field of the length D, cf. Figure 5.32. However, regarding the cell shape for the HPP, it is observed in Figure 5.32 that the shape of the cells is rather spiky and irregular (not HCP-like), even for a number of ND = 200. Considering the MHCP with the hard-core point distance δHC calculated by Eq. (5.50) for a given intensity λ and uniformity parameter α, the number of Voronoi cells can be influenced. In Figure 5.33 it is now recognized that the cell shape is governed by the uniformity parameter α but still looks spiky for a much less number of cells compared to the HPP, cf. Figure 5.33. For a low uniformity parameter of α = 0.1, the Voronoi cell structure of the MHCP (Figure 5.33 (a)) is very similar to that of the HPP (Figure 5.32 (a)) for the same intensity λ = 0.005 1/mm2 . However, for a high uniformity parameter of α = 0.8, the Voronoi cell structure tends, as expected, towards a honeycomb pattern with hexagonal cells, cf. Figure 5.33 (b). The hard-core process can be further specified to a SP by introducing an acceptance probability γ for point pairs with a smaller distance than δHC , cf. Section 5.5.1. The influence of the acceptance probability γ for the SP point process is shown in Figure 5.34 for a high intensity λ = 0.02 1/mm2 and uniformity parameter α = 0.8. This figure was chosen to assess the influence of the difference in acceptance probability γ at a high degree of regularity according to the parameters α and λ to be close to the HCP. In Figure 5.34 (a), it can be seen, that for a low value of acceptance probability γ = 0.001, the resulting Voronoi tessellation appears quite regular and qualitatively near to ’real’ fracture patterns. For higher values of γ more spiky cells are observable in the Voronoi cell structure in Figure 5.34 (b). It is concluded that a SP with a high uniformity parameter α and a low acceptance probability γ can be assumed a priori for the modelling of glass fracture patterns.

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(a)

(b)

Figure 5.32 2D Voronoi tessellation generated based on the HPP in an observation field of length D for two intensities of (a) λ = 0.005 1/mm2 (ND = 50); (b) λ = 0.02 1/mm2 (ND = 200), from P OUR MOGHADDAM et al. (2018f)

(a)

(b)

Figure 5.33 2D Voronoi tessellation generated based on the MHCP in an observation field of length D for λ = 0.005 1/mm2 (ND = 50) and two uniformity parameters of (a) α = 0.1 and (b) α = 0.8, from P OURMOGHADDAM et al. (2018f)

(a)

(b)

Figure 5.34 2D Voronoi tessellation generated based on the SP in an observation field of length D for λ = 0.02 1/mm2 (ND = 200); α = 0.8 and two acceptance probabilities of (a) γ = 0.001 and (b)γ = 0.2, from P OURMOGHADDAM et al. (2018f)

5.6 Geometrical Properties of Random 2D Voronoi Tessellations

165

5.6.4 Results of the Statistical Distributions of Voronoi Cell Characteristics After simulating the realizations of the Voronoi tessellation over different spatial point processes as described before, the statistical distributions of the geometrical properties of the Voronoi cell structure in terms of the cell area AC (in [mm2 ]), cell perimeter PC (in [mm]) and the cell edge number have been determined using the numerical software MATLAB (2017). Within this section the HPP, MHCP and SP distribution functions of the cell area are shown for comparison and with the focus on the SP the distribution functions of the cell perimeter and the cell edge number are shown in Figure 5.35 to Figure 5.39. The HPP and MHCP distribution functions of the cell perimeter and the cell edge number are shown in Appendix A.7. In these diagrams, the probability density functions (PDF) as well as the cumulative distribution functions (CDF) of the investigated 2D Voronoi cells are shown for the different point processes HPP, MHCP and SP. In the diagrams, the ordinate axis is the PDF respectively the CDF and the abscissa axis is the inspected characteristic parameter of the Voronoi cells. For the cell area and the cell perimeter the obtained values were evaluated after taking the logarithm to gather a better impression across different scales. For HPP, the CDF curves of the cell area and the cell perimeter are shifted horizontally to the left for increasing intensity λ . This means that both cell area and cell perimeter decrease (Figure 5.35). For MHCP, the CDF curves of the cell area and the cell perimeter are steeper for larger uniformity parameter α = 0.7 compared to α = 0.1, cf. Figure 5.36. This means, that the greater the uniformity parameter α, the more Voronoi cells possess almost the same content and perimeter. In Figure 5.36 (b), for λ = 0.02 and α = 0.7 (black dotted curve) almost 80% of the obtained Voronoi cells possess an log-area content of about 1.6 log(mm2 ). Considering the fact that high values of the uniformity parameter

(a)

(b)

Figure 5.35 HPP: (a) Probability Density Function (PDF) and (b) Cumulative Distribution Function (CDF) of the logarithmic cell area log AC [log(mm2 )] for intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OURMOGHADDAM et al. (2018f)

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5 Prediction of 2D Macro-Scale Fragmentation of Tempered Glass

(a)

(b)

Figure 5.36 MHCP: (a) Probability Density Function (PDF) and (b) Cumulative Distribution Function (CDF) of the logarithmic cell area log AC [log(mm2 )] for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OURMOGHADDAM et al. (2018f)

α signify behaviour close to HCP, this result could be expected a priori. For SP, the two CDF curves for the log cell area content lie close together for identical α whereas for greater acceptance probability the CDF is more flat. This is in accordance with the previous findings on the MHCP, as the more points within the hard-core distance are allowed, the more variation in Voronoi cell area and perimeter is to be expected as the Voronoi cells are less regular compared to the HCP, cf. Figure 5.37 and Figure 5.38. It is interesting that the SP CDF curves of the log cell area content and the log cell perimeter for a small acceptance probability of γ = 0.001 are quite close to the CDFs of the respective quantities of a

(a)

(b)

Figure 5.37 SP: (a) Probability Density Function (PDF) and (b) Cumulative Distribution Function (CDF) of the logarithmic cell area log AC [log(mm2 )] for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.02} [1/mm2 ] and acceptance probability γ ∈ {0.001; 0.2}, from P OURMOGHAD DAM et al. (2018f)

5.6 Geometrical Properties of Random 2D Voronoi Tessellations

(a)

167

(b)

Figure 5.38 SP: (a) Probability Density Function (PDF) and (b) Cumulative Distribution Function (CDF) of the logarithmic cell perimeter log PC [log(mm)] for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.02} [1/mm2 ] and acceptance probability γ ∈ {0.001; 0.2}, from P OUR MOGHADDAM et al. (2018f)

(a)

(b)

Figure 5.39 SP: (a) Probability Density Function (PDF) and (b) Cumulative Distribution Function (CDF) of the cell edge number for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.02} [1/mm2 ] and acceptance probability γ ∈ {0.001; 0.2}, from P OURMOGHADDAM et al. (2018f)

MHCP. It is interpreted, that this is due to the close relation of the SP to the MHCP, as the low probability of occurrence of pairs of points closer than a distance smaller than δHC . For all stochastic point process models under investigation it was found, that about 50% of the examined cells have an edge number of six, which is shown for the SP in Figure 5.39. In summary, the presented simulation study on Voronoi tessellations over three stochastic spatial point patterns allowed to deduce theoretical findings on several geometrical characteristics such as the area content, diameter and perimeter of the cells, which can be connected to parameters and variables from fracture mechanics. It was observed, that by

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introduction of a uniformity parameter α the shape of a typical Voronoi cell from a HPP, MHCP and SP can be characterized in comparison to the HCP. With the uniformity parameter α it is furthermore possible to relate an observed Voronoi tessellation to the HCP. The simulation study showed, that a SP with high uniformity parameter α qualitatively comes nearest to observed glass fracture patterns. The obtained simulation results furthermore give advise on the order of magnitude of the parameters of the stochastic point processes for the process identification for fractured glass, as it will be introduced in the next section. Besides the fracture mechanical relevance a general mathematical value can be addressed to the obtained results, as the posterior distributions for the geometrical properties can be used in other applications without loss of generality.

5.7 Methodology of Fracture Pattern Recognition and Generation (Method BREAK ) After laying the foundations of deterministic fracture mechanics in Section 5.4 and Voronoi tessellation of stochastic point processes in Sections 5.5 and 5.6, the basic modelling idea of Section 5.4.1 is now connected to a method called ’Bayesian Reconstruction and Prediction of Glass Fracture Patterns (BREAK)’. The idea of the method BREAK is that the glass fracture pattern can be estimated and simulated by the 2D Voronoi tessellation of seed points generated by a planar stochastic point process, which is motivated by the energy concept from fracture mechanics based on the elastic strain energy conditions as outlined in Section 4.2. Methods of Bayesian spatial point statistics are fed with elastic strain energy conditions and experimental data maintain statistical parameters of the fracture pattern that lead to a simulation and prediction of the fracture pattern. The overall schematic framework of the method with the incorporated theories is depicted in Figure 5.1, the concrete concept is given in Figure 5.40. It is intended to calibrate one fracture pattern prediction model for one specific residual stress state in a thermally tempered glass plate. The results within this section have been pre-published in K RAUS et al. (2019b). In order to determine the statistical values of the fracture structure such as fragment edge number, the fragment perimeter, fragment base area, etc., at first the fracture image has to be recorded and morphologically processed. The fracture pattern image is then evaluated for the statistics of the identified fracture particles. Then a spatial point process model (HPP, MHCP, SP) is chosen. The two process parameters fragment size δ (respectively the hard-core distance between point pairs) and fracture intensity λ are calibrated according to the information from the fracture images. Subsequently, the calibrated candidate point process model is evaluated against a fracture test image that was not part of the calibration data set. If the validation test is positive, the further statistically representative fracture patterns can be simulated by the calibrated model.

5.7 Methodology of Fracture Pattern Recognition and Generation (Method BREAK )

169

Figure 5.40 Flowchart of the method ’BREAK ’

5.7.1 Fracture Image Recording and Morphological Fracture Image Processing As described in Section 4.4.3 and shown in Figure 4.23, fragmentation was affected by impact in an area below the impact position (impact influence zone). Therefore, only the fragments in the areas under an assumed angle of 45◦ left and right from the impact point were considered for the investigations. In this last part of this chapter, the observation fields, as shown in Figure 4.24, are referred to as Ω due to the brevity of the notation. Figure 5.41 shows glass fragments, computed centroids of each fragment and the resulting Voronoi tessellation in observation field of size 50 mm × 50 mm as an example for three different residual stress levels. For the further obtaining of the Voronoi structure from the fracture images, it is assumed, that the centroid of each identified pixel region of a glass fragment is a good estimator of the x and y coordinates of the seed of the respective Voronoi cell. This may from a strict geometrical point of view not be fully true, as the glass fragment boundaries are no straight lines but rather curved. In the context of this thesis, however, this approximation of the seed of the Voronoi cell for the glass fragment seems rather sufficient. This assumption is further supported by a qualitative comparison of the resulting Voronoi tessellation derived by the centroids and the actual fracture pattern in Figure 5.41, where it can be observed, that the number of centroids coincide with the number of glass fragments (thus the estimation of λ is unbiased). The estimate of the inter point distance δ is biased by the centroid assumption, which is, however, considered as unavoidable and negligible due to the construction of the Voronoi cell.

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 5.41 Glass fragments, computed centroid (red cross) and Voronoi tessellation in observation field of size 50 mm × 50 mm on three different locations on the glass plate with the residual mid-plane tensile stress of (a) - (c) σm = 27.24 MPa, (d) - (f) σm = 31.53 MPa and (g) - (i) σm = 38.06 MPa; t = 12 mm

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171

An alternative modelling approach would require setting up the Voronoi tessellation of the pixel image as a least-squares approximation approach, which would be possible for the given pictures of the fracture patterns. However, this is not intended in the further of the context of this thesis, as the main focus of this assessment is on the properties of the inferred fracture patterns for the stochastic point process. However, the mentioned point on the fitting of the Voronoi tessellation will be recaptured in the outlook of this chapter, cf. Section 5.8, as more points can be mentioned for future enhancement of BREAK. In the next step, the model parameters intensity λ , hard-core radius r0 as well as the acceptance probability γ are estimated from the fracture patterns within MATLAB in order to allow a calibration of the spatial statistical models.

5.7.2 Spatial Statistics Model Calibration and Evaluation of Candidate Models Within the context of this thesis, the calibration of a stochastic point process model according to the method BREAK is shown for a specific magnitude of residual mid-plane tensile stress level of σm = 31.53 MPa (UD = 8748 J/m3 ) as the work flow for calibration of further models at other residual stress levels is the same. The calibration was performed on specimens experimentally investigated in the fracture tests of Section 4.4 with three specimens provided for each residual stress level. In Appendix A.8 to this thesis, the Voronoi tessellation together with the seed points for all three glass test specimens are given as a total view. The calibration of the spatial statistics model HPP, MHCP and SP can be conducted using the fracture image statistics and especially due to the computed centroid information. Within this thesis, the parameters of the point processes are inferred by inspection of the first- and second-order statistics as introduced in Section 5.5.2 by a MATLAB script, which guarantees an automated and thus repeatable and fast evaluation of the fracture patterns. As already stated, three samples are provided for each residual stress level. In the following, results for just the first test specimen are given graphically, the respective figures and diagrams for all other test specimens are given in Appendix A.8. The estimation of the process parameters, however, is given for each specimen and the whole set of available specimens. At first, the intensity λ is estimated from each fracture picture. The intensity is estimated by a kernel density estimator over the whole evaluation region Ω without edge correction. The edge correction is later considered when evaluating and interpreting the results. An example for the estimation of the spatial point process intensity and thus fragment density is depicted in Figure 5.42 (a) for specimen ’a’ (note, that in this computation no edge correction was applied as the choice of a correction method itself is non-trivial (BADDELEY et al., 2016; M ØLLER, 1994). In Figure 5.42 (b), the estimation of the intensity λ of the underlying point process as well as the induced Voronoi tessellation for the fracture pattern of one of the tested glass panes is shown as 3D view and a top view.

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(a)

(b) Figure 5.42 Estimation of the process intensity λˆ for specimen ’a’ (t = 12 mm;) with the residual midplane tensile stress σm = 31.53 MPa (UD = 8748 J/m3 ), (a) top view and (b) 3D view, from K RAUS et al. (2019b)

As the intensity estimation results may depend on the specific choice of the kernel width, the estimation was repeated for three distinct kernel widths of h ∈ [25; 50; 100] [mm]. The results are given in Figure 5.43 for the first specimen ’a’ of the three specimens of the ScC series for the tempering process (see Table 4.1) with the residual mid-plane tensile stress σm = 31.53 MPa (UD = 8748 J/m3 ). From Figure 5.43, it is seen, that the mean intensity is quite stable at a level of μ1 λˆ = 0.0137 under different kernel widths, the spread however is slightly influenced. The significant peak right to the zero value is interpreted to be caused by not applying an edge correction. Inspection of Figure 5.42 supports this deduction, as the intensity λ near the evaluation borders of Ω are ’pressed’ down toward zero. The results of repeating this estimation of the process intensity λ for the remaining specimen are given in Table 5.3. The estimation of the intensity λ via kernel method as a first-order statistics method presented in Section 5.5.2.1 is an alternative to the method presented in Section 5.4.3 where the fracture intensity parameter λ has been determined experimentally from fracture tests (Section 4.4) on thermally tempered glass specimens (P OURMOGHADDAM et al., 2018d; P OURMOGHADDAM et al., 2018e). The average fragment density ND was determined by counting the fragments of eight square observation fields with a side length D = 50 mm for each observation field (as shown in Figure 4.24). Under assumption of a constant intensity in each observation field, the average fracture intensity parameter λ was

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Figure 5.43 Comparison of the influence of different kernel widths h on the estimator for the process intensity λˆ for specimen ’a’ (t = 12 mm) with the residual mid-plane tensile stress σm = 31.53 MPa (UD = 8748 J/m3 )

there computed for each specimen using Eq. (5.19). In Figure 5.15 or in Figure 5.16, the correlation between elastic strain energy density UD (computed from the measured residual stresses of each specimen) and the fracture intensity parameter λ was presented. It was shown that the accuracy of the experimental results of the fracture intensity decreases with lower energy density respectively for larger fragments. According to Figure 5.16 a constant fracture intensity of λ ≈ 0, 01 1/mm2 is achieved in the observation field for the specimen ’a’ used in this section with the elastic strain energy density UD = 8748 J/m3 . Thus, the results for intensity λ as estimated in this section via kernel fit quite well to the estimation method presented in Section 5.4.3 respectively in the prepublication P OUR MOGHADDAM et al. (2018e). In a further step, the hard-core radius r1 as well as the acceptance probability γ are estimated from second-order statistics by means of the empirical nearest-neighbour and empty-space distribution functions Gˆ and Fˆ as well as the Ripley’s K-function and the L-function. note that the nomenclature ’r1 ’ in the following does not specify that the assigned value for r1 is the true hard-core distance of a MHCP or a SP. The reason for this nomenclature will become clear from further deductions and interpretations of the rest of this section. The obtained functions are compared to their corresponding theoretical (and analytically available) functions for a CSR process. In the following only the results for specimen ’a’ are presented, the other results are given in Appendix A.8. From inspection of the empirical nearest-neighbour and empty-space distribution functions Gˆ and Fˆ given in Figure 5.44 and the theory as given in Section 5.5, it is concluded, that both functions support a regular process (MHCP or SP). For the Gˆ function, regularity is supposed, if there are more long distance neighbours and thus the the Gˆ function is flat at small distances and climbs steeply at larger distances compared to a CSR process. The opposite argumentation must hold for the Fˆ function to support regularity, which is the case for Figure 5.44 (b). From the Gˆ and Fˆ functions the hard-core distance can be estimated to

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(a)

(b)

Figure 5.44 Empirical nearest-neighbour distribution functions Gˆ and empirical empty-space distribution function Fˆ vs. their respective theoretical CSR graph, evaluated for specimen ’a’; t = 12 mm; σm = 31.53 MPa (UD = 8748 J/m3 )

(a)

(b)

Figure 5.45 Empirical Ripley’s K-function Kˆ vs. the respective theoretical CSR graph and L-function ˆ evaluated for specimen ’a’; t = 12 mm; σm = 31.53 MPa (UD = 8748 J/m3 ) L,

be at r1 = 2 mm. In Figure 5.45, the empirical Ripley’s K-function and the respective L-function is shown for the specimen ’a’. In the K-function the hard-core distance still seems to be 2 mm, but in the L-function a second kink is visible at 5 mm. The interpretation at this point is that this kink can be seen in the SR process sense as the L-function indicates clustering for peaks of positive values and regularity for negative values for the corresponding scale r. In the negative value range at the beginning of the r-axis close to zero, the value of 5 mm is now considered as the estimator for the hard-core distance rHC , while all smaller distances contribute to the acceptance probability γ of a SP. The radius of 2 mm (named r1 ) is interpreted as an artificial result of the F−, G− and K-functions, since they do not further consider a SP, i.e. these functions estimate the hard-core distance based on the minimum distance obtained from the point pattern, without taking into ac-

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175

Figure 5.46 Empirical L function Lˆ at a greater distance scale, evaluated for specimen ’a’; t = 12 mm; σm = 31.53 MPa (UD = 8748 J/m3 )

count that a spatial point process might allow a certain number of points to fall closer than the hard-core distance. An interesting point is that the L-function despite inhibition/regularity also indicates clustering (positive values) on the length scale between r ∈ [30; 100] mm, cf. Figure 5.46. A closer inspection of Figure 5.42 (a) is in accordance with that finding. Over this length scales, there is indeed clustering behaviour obtained at the outer boarders of the glass specimen (indicated by a red colour representing a higher intensity λ in that region). At this point it is concluded, that on the one hand, the residual stress might be influencing (distance of rollers of the tempering oven is at about 150 mm). On the other hand, the inducement of the fracture due to hit with a hammer and a prick punch a subsequent elastic wave is travelling through the glass pane (Section 4.4.2). In addition, further elastic waves are released from each breaking or branching fragment at the fracture front during the fracture process. These elastic waves are furthermore reflected at the edges of the specimen, which can strongly influence this fracture pattern. Future research could try to establish a correlation of the wavelength of these elastic waves to the finding of a clustering at a length scale of 100 mm. So far, two process parameters for a regular point process have been determined. Having obtained the intensity and the hard-core distance is sufficient to calibrate a MHCP. In addition, the acceptance probability γ has to be inferred for the calibration of the SP. Unfortunately, no closed form solution for the calibration of all three parameters are available (BADDELEY et al., 2016), thus within this section, it is assumed that the hard-core distance r0 as well as the acceptance probability γ can be estimated from the L-function Lˆ to a very good approximation for the reason mentioned before. In the following, the acceptance probability γ is estimated due to its original definition of s(x, rHC ) in Eq. (5.34), which counts the number of pairs points that are located closer than the hard-core distance. The estimated γ from the data of all three specimen is given in Table 5.3. The uniformity parameter respectively the regularity of the point pattern is found to be α = 0.55 by applying Eq. (5.49). At this point it is repeated, that within this

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Table 5.3 Estimated parameters of the spatial point pattern from the fracture pictures of three specimens, from K RAUS et al. (2019b)

Series ScC Specimen ’a’ Specimen ’b’ Specimen ’c’ Mean St. Dev.

Figure 5.47

λ [1/mm2 ]

r1 [mm]

rHC [mm]

γ [−]

rHCP [mm]

α [−]

0.0137 0.0138 0.0143 0.0139 0.0003

2.00 1.48 1.50 1.66 0.295

5.00 5.00 5.00 5.00 0.00

3.878E-05 3.603E-05 5.03E-05 4,169E-05 7.547E-06

9.18 9.15 8.99 9.10 0.104

0.54 0.55 0.56 0.55 0.006

PDF and CDF of inter point distances r for specimen ’a’

section, only the results for specimen ’a’ are presented for reasons of brevity, however, these results are representative for the findings of the other specimens. The details on the results of the other specimens are given in Appendix A.8. Finally, the statistics of the inter point distances r (PDF of the distances between all points to the remaining points) is inspected. For specimen ’a’ it is given in Figure 5.47. Interestingly, the PDF in Figure 5.47 shows two peaks at about 150 mm and 700 mm. Taking into consideration again Figure 5.42 (a) this aspect hints into the direction, that the specimen as well as the elastic wave length might have influenced the whole fracture pattern.

5.7.3 Stochastic Fracture Pattern Simulation Having obtained the process parameters λ , rHC , γ, the simulation of fracture patterns according to a spatial point process is possible. Within this section, only the SP is considered for simulation. This is based on the findings presented in Section 5.6.4, where it was elaborated, that the SP is the most suitable process for describing fracture patterns of thermally tempered glass.

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177

Figure 5.48 Realization of a fracture pattern of a thermally tempered glass plate (t = 12 mm) for a residual mid-plane tensile stress of σm = 31.53 MPa (UD = 8748 J/m2 ) simulated with a SP, from K RAUS et al. (2019b)

For the final comparison of the quality of the estimated SP based on the presented findings, a simulation of just three realizations of a SP with the mean parameters according to Table 5.3 is statistically evaluated and compared to the statistics of the fracture pattern images. Such a small number of simulations is selected in order not to represent the number of simulations too high in comparison to the number of calibration specimens. On the other hand, the simulations rely on the mean values of the parameters as given in Table 5.3 a greater simulation number would give a better insight on variability of the prediction results. The comparison within this thesis is on the basis of the fragment area only, other statistics could for sure be compared as well. In Figure 5.48, one realization of the simulation of the fracture pattern of a thermally tempered glass according to the SP is given. Inspecting the realization of the simulated fracture pattern, several things can be recognised. The distribution of fragment sizes qualitatively looks as one would expect a fracture pattern of thermally tempered glass to be. There is variation in the size of the fragments and there is order on several length scales, e.g. some clusters with ’bigger’ fragments at x = 1000 mm and y = 300 mm or x = 550 mm and y = 200 mm. The conclusion that can be drawn from Figure 5.49 is that the mean cell area is very well covered by the SP simulation results, but the distribution does not fully coincide. However, this should not be interpreted too much as a disadvantage of the SP simulation, instead the fracture inducing process as discussed above for the "real" glass specimen must be taken into account. It has already been mentioned in a previous conclusion that the fracture inducing process can cause reflected elastic waves to propagate through the glass during fragmentation. The fracture pattern can be affected by these elastic waves. An interesting future research outcome for this purpose would be if other ways of crack initiation would lead to differently distributed PDFs and CDFs for the base shape area of a fragment and if the SP cell area statistics would fit better into the variance. To this end, it is finally concluded that the SP is generally validated for the use of the simulation of fracture patterns of thermally tempered glass and for the present case in

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Figure 5.49 Comparison of the PDF of the cell areas log AC [log(mm2 )] of the SP simulations and the fracture patterns, from K RAUS et al. (2019b)

particular. Future research however has to improve the better capturing of the spread of the obtained statistics, some points on that are given in the next subsection as outlook.

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179

5.8 Summary and Outlook In this chapter, a method for prediction of 2D macro-scale fragmentation of thermally tempered glass based on fracture mechanics considerations and statistical analysis of the fracture structure was deduced and applied. The main idea of the method is that the glass fracture pattern can be estimated by the Voronoi tessellation of seed points generated by a stochastic point process, which is motivated by the energy concept from fracture mechanics. In a basic approach two process parameters λ (Fracture Intenity Parameter) and δ (Fragment Size Parameter) were defined in order to predict the fracture structure of thermally tempered glass. A methodology based on the energy criterion in fracture mechanics considering the elastic strain energy conversion during the fragmentation was elaborated and applied for the prediction of the fragment size as well as the number of fragments within certain observation fields. On the basis of three assumptions respectively preconditions •

cylindrical fragments,

•

relaxation of the stored energy (energy relaxation factor η) to a remaining elastic strain energy in the fragment in the post-fracture state,

•

energetic criterion: released elastic strain energy is converted to the fracture surface energy

the base area of a fragment was analytically determined as a function of the initial elastic strain energy density UD,0 and an energy relaxation factor η. The relaxation energy factor η = 0.123 was identified by fitting the analytical model to the results of the fracture tests in Section 4.4. Using the analytical model, the elastic strain energy remaining in the fragment was calculated to 77% of the initial elastic strain energy. Correspondingly 23% of the initial elastic strain energy converts into fracture surface energy and therefore is used during the fragmentation for the generation of new fracture surfaces by crack generation and branching. A direct correlation between the energy in pre-fracture, fracture and post-fracture state was shown. On the basis of this outcome, an energy release rate required for crack branching can be determined. The phenomenon of crack branching and the associated energy release rate is discussed in more detail in Chapter 6. In addition, the number of fragments as a function of the elastic strain energy respectively a fracture intensity in a deterministic context of deterministic fracture mechanics was calculated under the assumption of a honeycomb as fracture structure and compared with data from the literature. Furthermore, a method called BREAK (Bayesian Reconstruction and Prediction of Glass Fracture Patterns) was deduced and applied. Here Bayesian spatial point statistics are fed with strain energy conditions and experimental data maintain statistical parameters of the fracture pattern that lead to a simulation and prediction of the fracture pattern. It was

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(a)

(b)

(c)

Figure 5.50 Fragmentation pattern in an observation field of size 200 mm × 200 mm for different distance norms for the Voronoi tessellation, (a) 1-norm (b)π-norm (c) Inf-norm

shown that the parameters of the method can be calibrated to the actual fracture patterns of thermally tempered glass and that a simulation of statistically representative fracture patterns for this level of residual stress is possible. For the thermally tempered glass used within this section, the process parameters for the stochastic point processes were obtained not just as point estimates but with their empirical distribution. Thus further insight (especially the sources of uncertainty) of the modelling approach and the model behaviour is possible. It was found, that the glass with a residual stress level of σm = 31.53 MPa results in a fracture pattern of uniformity α = 0.55. In addition, the simulation of representative fracture patterns by the calibrated SP could successfully be validated against the statistics of the fracture patterns. In the present deduction of BREAK, varying residual stress is implicitly modelled in a greater uncertainty of the model parameters δ and λ . Future research could work on separating the variability in the residual stress from the uncertainty of the fracture process as these sources of variability may not be caused by the same physical mechanism. One approach could be to use random fields for modelling the variability of the residual stress over the glass plate. To the authors knowledge, scanning machines are available on the market, which allow the determination of direction and magnitude of the residual stress within a glass specimen, which delivers data for the calibration of such random fields. A calibrated random field could then be used further as sensible prior distribution in the calibration process of the point process model for the glass fracture pattern prediction. Another point for improvement of the proposed method BREAK is that up to now, the Euclidean distance (2-norm) for generation the Voronoi tessellation was used within the method. First simulations with other than the 2-norm look promising for the aim to capture non-straight fracture particle boarders. In Figure 5.50, Voronoi cells with identical number of seed points but three distinct norms (1-, π- and Inf-norm) are given in order to allow a graphical comparison and assessment. The deduction and estimation of the Voronoi distance norm from fracture pattern pictures is an interesting task, as it will provide further

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181

insight into the properties of the stochastic fracture modelling. Furthermore the more precise estimation of the boundaries of the glass fracture particles is very important to assess quantitatively the crack surface area. Inspection of Figure 5.50 makes qualitatively clear, that different norms induce different Voronoi cell perimeters and area contents, which are central quantities of the fracture mechanical considerations as presented within the introduction of this section. Thus the more specific and accurate the glass fracture pictures can be assessed and processed, the more precise the uncertainty quantification of the process parameters as well as the fracture pattern prediction becomes. This also gives raise to the point that future work could concentrate on a reformulation the calibration of this method within a least-squares setting, which works directly on the pixel image and allows deeper Bayesian handling of the problem. Future research has to investigate the limits of prediction of this method for decreasing levels of residual stress as fracture patterns are less dense compared to the discussed case within this thesis. It is assumed that the method BREAK has to be enhanced for that case e.g. by handling stochastic line processes instead of point processes in order to be able to capture that fragmentation behaviour correctly. The same holds, if this method is used for the simulation of the fracture pattern of thin glass. Finally, the MATLAB implementation of BREAK allows to export the simulated fracture pattern as mesh for further processing in a FEA context. With this feature, the author is convinced, that the method BREAK will support finding a numerical set-up for the computation of fractured laminated glass structures with incorporation of all the thermomechanical considerations from the polymeric interlayer.

6 Investigations into the Phenomenon of Crack Branching

6.1 General

A particularly peculiar phenomenon in fracture mechanics is bifurcation or crack branching, which occurs when a crack runs rapidly under high stress intensity. Bifurcation also occurs with a crack under simple tensile stress, which initially propagates perpendicularly to the tensile stress that has been applied to the crack. Various criteria have been proposed for specifying branching conditions in brittle materials. Early experiments to determine the role of crack velocity in branching were performed by K IES et al. (1950), S CHARDIN (1959), and ACLOQUE (1962). It was found that the crack velocity on an advancing crack changed little or not at all after branching. These results and further experiments led C LARK et al. (1966) and C ONGLETON et al. (1967) to conclude that crack branching is primarily controlled by a critical value of the elastic strain energy release rate or stress intensity factor, rather than a crack velocity criterion. D ÖLL (1975) referred the concept of a critical energy release rate to finding a constant crack velocity before and after branching. In his study on three glass specimens, the energy release rate at branching was about twice the energy release rate measured at the point where the maximum (constant) crack velocity was achieved. According to this criterion, branching occurs when both crack branches can continue at maximum speed. In this chapter, the fracture process and the branching of a advancing crack are described in more detail. The state of crack propagation is characterized by the phenomena that occur from the beginning of the crack propagation to crack branching. The energy release rate required for the crack branching is determined based on the outcomes in Section 5.4 with respect to the elastic strain energy released during fragmentation process and compared with some results of the branching energy release rate reported in the literature. Further results of investigations on tensile tests on float glass specimens pre-damaged on one side are presented. © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6_6

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6 Investigations into the Phenomenon of Crack Branching

6.2 Fracture Process and Associated Energy Release Rate The phenomenon of crack branching that occurs in the high velocity region of crack propagation is subject to two conditions (D ÖLL, 1975; K ERKHOF, 2011): (1) The crack attains the maximum fracture velocity v f r,max in the material (2) The energy release rate exceeds some critical value Gb The question which arises is, given a crack travelling with v f r = v f r,max in a material, what determines the value of the energy release rate G at which branching occurs? To answer this question, the term "crack branching" must first be interpreted and defined. This thesis distinguishes between the micro-branching that occurs on the fracture surface and the macro-branching (or simply branching) that describes the bifurcation of an advancing crack. Taking this definition into account, the fracture process is described in more detail below.

6.2.1 Characterization of Crack Propagation In order to characterize the state of a propagating crack, the fracture process respectively the crack propagation is divided into three different phases. The schematic of these three phases of the crack propagation with respect to the principal dependence of the fracture velocity v f r on energy release rate G is shown in Figure 6.1 (D ÖLL, 1975; K ERKHOF, 2011). 6.2.1.1 Phase A: Start of Crack Propagation The first state of a crack (crack generation) is when at the tip of an initial crack the energy release rate exceeds a critical value due to sufficient stress intensity, the crack velocity increases steadily and the crack begins to become unstable so that crack propagation begins. Thus, Phase A of the crack propagation describes the fracture process if G ≤ Gc applies. In pure mode I, according to Eq. (5.15) there is an unambiguous relationship between the energy release rate G and the stress intensity factor KI . For the plane stress state, the critical energy release rate Gc can be written in terms of the critical stress intensity factor KIc as: Gc =

2 KIc E

(6.1)

KIc is also often referred to as fracture toughness and Gc as crack resistance. The expression G = Gc in Eq. (5.13) can be interpreted as follows: the fracture process occurs when the energy released during crack propagation corresponds to the energy required

6.2 Fracture Process and Associated Energy Release Rate

185

Figure 6.1 Schematic of the three phases of crack propagation with respect to the principal dependence of the fracture velocity v f r on energy release rate G, course of the curve from K ERKHOF (2011)

√ (G ROSS et al., 2011). With a critical stress intensity factor of 0.75 MPa m for sodalime-silica glass (L AWN, 1993) the critical energy release rate can be easily calculated to Gc = 8.04 J/m2 . This is the energy per m2 of fracture surface which is required to start the crack propagation from the fracture origin respectively the initial crack. This is the first phase of the crack propagation before crack branching sets in. 6.2.1.2 Phase B: Micro-Branching In Phase B (Gc < G ≤ Gb,micro ), the fracture velocity of the unstable crack increases and correspondingly more energy is released via the fracture surface. The first unevenness, roughness or peaks form on the fracture mirror and the fracture surface begins to become rough and matt. This phenomenon is also referred to as micro-branching. Both the roughness on the fracture surface and the formation of micro-cracks take place in this phase of the crack propagation. In F INEBERG (2006) tensile tests were carried out on sodalime-silica glass samples which had been pre-damaged on one side as seed cracks. The test results with respect to the fracture surface energy which is required for the microbranching is shown in Figure 6.2. It can be clearly observed that the energy release rate at micro-branching is Gb,micro = 35 J/m2 which is not a function of the crack velocity. This shows that the energy release rate investigated by Fineberg is to be positioned in Phase B of the crack evolution and must not be described as crack branching but only as microcrack branching. In the evolution of the fracture surface from its origin to crack branching,

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6 Investigations into the Phenomenon of Crack Branching

Figure 6.2 Fracture surface energy Γ [J/m2 ] vs. Fracture velocity v f r [m/s]; test results for microbranching from F INEBERG (2006)

Figure 6.3 Schematic fracture surface evolution during the three phases of the fracture process from fracture origin to crack branching

this is the energy required for the fracture surface to slowly become rough and matt. The schematic evolution of the fracture surface during the three phases of the fracture process from fracture origin to crack branching is shown in Figure 6.3. With regard to the fracture surface evolution, Phase B of the crack propagation describes the transition area from mirror to mist region respectively the mirror/mist boundary. In other words, the area in which

6.2 Fracture Process and Associated Energy Release Rate

187

roughness is formed. However, in order to achieve crack branching, more energy must be released. 6.2.1.3 Phase C: Crack Branching In Phase C (Gb,micro < G ≤ Gb ) the crack propagation reaches the last step on the way to branching. The fracture velocity reaches its maximum which in case of soda-lime-silica glass is v f r,max ≈ 1500 m/s and remains constant while still more energy releases over the fracture surface. The fracture surface evolves and completes with the formation of the Hackle region (Figure 6.3) and after exceeding a critical value of the energy release rate Gb the crack branches with the generation of at least two new fracture surfaces. Hence, in this thesis Gb is called branching energy release rate and is responsible for the generation of new fracture surfaces. Gb is a multiple of Gb , micro and Gc . Based on the achievements in Section 5.4 with respect to the elastic strain energy released during the fragmentation process, the branching energy release rate Gb can be determined. For this purpose, the fracture surface energy Γ can be calculated from the elastic strain energy density share UD,η , which is released during fragmentation, taking into account a cylindrical fragment with the diameter δ of the base area: Γ = UD,η · δ π

(6.2)

Considering the requirement in Eq. (5.14) that the released energy from fragmentation process is directly converted into the generation of new fracture surfaces, the released energy is set equal to the released energy required for branching: Γ = Gb

(6.3)

Furthermore, the branching energy release rate Gb for the plane stress state can be expressed via the stress intensity kb which is required for the crack branching: Gb =

Kb2 E

(6.4)

Kb is the stress intensity at crack branching and is referred to as branching stress intensity. This results in a direct dependence of the released elastic strain energy UD,η during fragmentation and the branching stress intensity Kb : ! Kb = UD,η · E · δ π (6.5) From the results of the fracture tests in Section 4.4 it emerged that the initial elastic strain energy density required for one fragment is UD,0 = 5712.7 J/m3 (see Figure 4.27 (b)). Additionally, in Section 5.4 was shown that about 23% of the initial elastic strain energy

188

6 Investigations into the Phenomenon of Crack Branching

density UD,0 is converted into fracture surface energy and is therefore used during the fragmentation for the generation of new fracture surfaces. Accordingly, the released elastic strain energy density is UD,η = 0.23UD,0 = 1313.9 J/m3 . Furthermore, in Section 5.4 a dependence of the elastic strain energy density on the diameter of the fragments base area or on the fragment size parameter δ was analytically determined and fitted to experimental data (see Figure 5.10). For UD,0 = 5712.7 J/m3 the diameter of the fragment base area is δ = 21.37 mm. Based on these informations the branching stress intensity √ is calculated as Kb = 2.485 MPa m. And the branching energy release rate in Phase C is Gb = 88.22 J/m2 . This means that the energy required for crack branching is about 2.5 times greater than the energy required for micro-branching according to F INEBERG (2006) (Gb ≈ 2.5Gb,micro ). This can be confirmed by D ÖLL (1975), who found out that the energy release rate at branching was approximately twice the energy release rate that was measured at the point where the maximum (constant) crack velocity was achieved. The branching energy release rate is about 11 times greater than the energy release rate required for the start of crack propagation (Gb ≈ 11Gc ). A comparison of the results of the branching stress intensity Kb and the branching energy release rate Gb determined within the context of this thesis with values from the literature is given in Table 6.3. The values of the energy release rate G at different crack propagation phases are summarized in Table 6.4.

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass The crack propagation in thin float glass sheets under uni-axial tensile stress is investigated by conducting tensile tests on one-sided pre-damaged specimens. A schematic representation of the system studied is shown in Figure 6.4. The vertical direction is defined as the z-direction. A small seed notch is introduced at the edge of the sample midway between the vertical boundaries of a thin float glass plate. To initiate fracture, the glass sheet is then stretched by applying a uniform displacement at its vertical boundaries until the crack starts to propagate in the x-direction. The dashed line in Figure 6.4 shows the crack’s direction of propagation starting from an imposed initial crack. Stress was applied in the z-direction to the undersides of rectangular strips bonded onto the upper and lower edges of the plate. In order to determine the crack velocity at branching respectively the velocity of the crack front v f r , high-speed techniques were applied in cooperation with assoc. Prof. Jens Henrik Nielsen from the Department of Civil Engineering of the Technical University of Denmark (DTU). As described before, in brittle materials such as glass characteristic patterns "mirror, mist, hackle" (M ECHOLSKY et al., 1984; Q UINN, 2016) appear along the fracture surface. The mirror or macroscopically smooth surface appears as fracture initiates. As the crack

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass

189

Figure 6.4 Schematic representation of the method of loading used in the tensile tests: stress was applied in the z-direction (arrows). The dashed line shows the crack’s direction of propagation

progresses, this featureless region evolves into the "mist" region, where first roughness occur, which appears misty or diffuse to the eye. As the crack progresses still further, the surface becomes progressively rougher until, in the rough "hackle" regime, surface structure is in evidence on many scales. After the hackle region has formed completely, the crack branches according to Section 6.2. In this section, the crack branching behaviour and associated energy release rate as well as the crack velocity is investigated based on the results of the tensile tests on one-sided pre-damaged float glass sheets.

6.3.1 Preparation of the Specimens The specimens were pre-damaged on one side for the tensile tests. A seed notch with a semi-circular tip is introduced at the edge of the specimen midway between the vertical boundaries of a thin float glass sheet. The notch from which fracture should initiate is produced by cutting the glass sheet using a sheet wire saw (Figure 6.5). In order to vary the fracture stress, the length of the notch was varied. For the cutting process the specimen was fixed and the rotating wire (low speed of 0.4 m/s) was set to cutting position. The wire was then manually and slowly inserted into the specimen until the desired notch length was reached and the wire was carefully pulled out of the specimen to complete the cutting process. An example of the notch formed after the cutting process is shown in Figure 6.5 (c). The round cross-section wire with a diameter of 0.17 mm provides a notch width and a round semi-circular notch head of the same diameter with a deviation of ±0.003 mm

190

6 Investigations into the Phenomenon of Crack Branching

(a)

(b)

(c)

Figure 6.5 (a) Image of the sheet wire saw that was used to insert the notch into the specimens, (b) Detail view of the fixed specimen, (c) Example of a notch formed after the cutting process

for all specimens. However, the notch length aN was introduced manually and therefore the desired length could not be reached exactly. The notch lengths were therefore subsequently measured with a light microscope. The obliqueness of the introduced notch was also measured to verify the orthogonality of the notch to the edge. The notch parameters and some examples of notches with different notch lengths are shown in Figure 6.6. The cutting process can result in chipping along the notch, which could possibly influence the fracture process. The chipping can be seen in the notch examples in Figure 6.6 (b) - (f). However, these defects are considered negligible for the fracture and bifurcation process. Only the oblique angle of the notch ϕN can influence the direction of crack propagation.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6.6 (a) Notch parameters: notch length aN , oblique angle ϕN and diameter of the notch head φN , (b) - (f) Examples of notches with different notch lengths

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass

(a)

191

(b)

Figure 6.7 Example of a prepared specimen with steel strips bonded onto the upper and lower edge and a notch introduced at one edge midway between the vertical boundaries (a) Top view, (b) Perspective view

The specimens are divided into 5 series (a1 ... a5 ) according to notch length: a1 ≤ 0.5 mm; a1 < a2 ≤ 1 mm; a2 < a3 ≤ 1.5 mm; a3 < a4 ≤ 2 mm and a4 < a5 ≤ 3 mm. For load initiation, rectangular steel strips were bonded onto the upper and lower edges of the glass sheet with epoxy-resin adhesive to form steps at the top and bottom boundaries of the specimen that were parallel to within 0.02 mm. These strips uniformly transferred the applied stress to the specimen. An example of a prepared specimen is shown in Figure 6.7. The series and the information on the specimens, such as the measurement of the notch parameters are summed up in Table A.11 of Appendix A.9.

6.3.2 Experimental System and Methods Experiments are conducted on soda-lime-silica glass specimens of size 100 mm × 100 mm made by the manufacturer Schott AG. To enable comparison with two-dimensional elastic theory, experiments were performed on thin (1.0 mm ± 0.05 mm-thick) sheets of float glass. Float glass was chosen as thermally relaxed glass to investigate the crack behaviour under uni-axial tensile stress with a constant stress distribution along the thickness. The test bench and testing device is shown in Figure 6.8. The specimen boundaries were strained by means of a computer-controlled tensile testing device. The device consisted of two specially made translating grips made to fit against the undersides of the strips bonded onto the top and bottom edges of the specimen (Figure 6.8). The grips were constrained to travel in the z-direction (vertical) and were moved apart by means of an Instron Universal Testing Machine (Instron 8502). First, a tensile force of 200 N with a displacement rate of 5 mm/min is aimed at to ensure that the specimen is drawn without slippage. The rate of displacement can be chosen so as to have at most 0.1 μm displacement step (in direction of load) occurs within the typical duration time of fracture (here

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6 Investigations into the Phenomenon of Crack Branching

Figure 6.8 Test bench with testing device and alliggnment of cameras and light source: 1) Universal Testin Maching (Instron 8502), 2) Translating grippers with specimen, 3) Load cell, 4) High-speed camera (Phantom v2512) and 5) Light source

about 100 mm/1500 m/s = 66.67 μs), i.e. 90 mm/min. Therefore, the Universal Testing Machine for the tensile tests is set path-controlled to run 1 mm at 90 mm/min. The applied force is measured by means of a load cell with a capacity of 5 kN, which was attached between the translating grippers and the universal testing machine (Figure 6.8). To capture the crack path, the specimen was illuminated obliquely with two light sources. This motivated the reflection of the light at both crack surfaces. In order to capture the crack propagation two high-speed cameras (Phantom v2512) with different set frame rates were used. The cameras were positioned so that the specimen is taken obliquely from below (about 20◦ ). Thus the reflection of the light at crack surface was recorded. The direction of the light and the direction of the cameras for the tensile tests is shown in Figure 6.8. A 5 mm grid serves as a scale for the high-speed images and is positioned below the notch for each specimen.

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass

193

The high-speed cameras have a ring memory that is used for permanent recording. This simplifies the triggering of the system. At the time of the fracture, manual triggering was used. The camera stores a sufficiently large range of frames until the time of triggering, when the fracture process was felt manually and of course took place after the beginning of the fracture process. A high frame rate is of great importance for the assessment and evaluation of extremely fast crack formation and propagation. Since the resolution decreases with the increase of the frame rate, different settings were tested for the evaluation. The applied frame rates, shutter times and picture resolutions are stated in Table 6.1. Table 6.1 Set-up data for cameras

Camera

fps [kHz]

Shutter [μs]

Resolution [Pixels]

A1a) A2 B

1000 780 380

0.261 0.261 0.6

128 × 16 128 × 64 256 × 128

a)

Setting data of the camera A for the specimen series a1 and a2

6.3.3 Results of Crack Branching Behaviour and Fracture Velocity Figure 6.9 shows the crack propagation for five different fracture stresses σ f at four propagation times. The fracture front spreads out circularly around the fracture origin, which in the tensile tests is the introduced notch. It can clearly be seen that the number of bifurcations increases with increasing fracture stress. In the following, the branching behaviour with regard to the supplied energy due to tensile stress is outlined. In addition, the crack velocity is determined with the help of the high-speed images. 6.3.3.1 Crack Branching Behaviour and Associated Energy Release Rate The crack branching process is motivated by the history of crack propagation and is not self-evident. As described in Section 6.2, after the energy release rate at the maximum fracture velocity reaches a certain value (Table 6.4), the progressive crack begins to branch. After this primary fracture bifurcation, the further branching behaviour depends on the stress or resulting energy density stored in the specimen. The fragmentation of thermally tempered glass plates was outlined in Section 4.4. In the investigations of Section 4.4, a correlation between the number of fragments and the initial energy was determined on thermally tempered glass with different residual stress level. Also in the tensile tests of this section different stress levels are achieved due to the pre-damage. Hence, with regard to elastic strain energy density, a correlation between crack branching and energy level can

194

6 Investigations into the Phenomenon of Crack Branching

(a)

(b)

(c)

(d)

(e) Figure 6.9 Examples of the recorded crack branching at propagation times 0; 26.32; 52.63; 69.74 [μs] with the fracture stresses (a) σ f = 13.85 MPa, (b) σ f = 23.55 MPa, (c) σ f = 29.70 MPa, (d) σ f = 34.31 MPa, (e) σ f = 41.25 MPa; fps: 380 kHz, Grid: 5 mm × 5 mm

be established. Considering Eqs. (4.4) and (4.5) the elastic strain energy density at fracture can be calculated in terms of the fracture stress: UD, f =

(1 − ν) σˆf E

(6.6)

with the factor of the integrated stress function σˆf = σ 2f due to a constant tensile stress distribution over the thickness. The crack branching behaviour can be evaluated with regard to the elastic strain energy density. After the first branching, the number of crack branching increases with the energy density UD, f at fracture. In order to investigate the crack branching behaviour, a correlation between the energy density and the number of fracture bifurcations is elaborated. The number of fracture bifurcations is determined on the basis of the branching nodes nB using the fracture patterns as shown in Figure 6.10. In addition, the crack length at the first branching rB and the corresponding branching angle

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass

195

ϕB were determined for each specimen and set into correlation with the corresponding elastic strain energy density. In Figure 6.11, the relationship between the elastic strain energy density at fracture and the three branching parameters is shown. Figure 6.11 (a) shows the increasing number of branching nodes nB with increasing elastic strain energy density UD, f which results from the fracture stress σ f according to Eq. 6.6. In contrast, the crack length at the first branching rB decreases with increasing energy density, suggesting that the higher the energy density, the earlier the branching begins, see Figure 6.11 (b). Figure 6.11 (c) shows the angle ϕB of the first crack branching, which increases with increasing elastic strain energy density, however, converges to 60◦ at energy values above approx. 6000 J/m3 . According to the results of the tensile tests, branching is generated at an elastic strain energy density of UD, f (nB = 1) ≈ 1086 J/m3 . This is the elastic strain energy density which is required for the formation of the first crack branching (nB = 1) and can be calculated using the function of the trend line in Figure 6.11 (a). Provided that this energy is completely converted to fracture surface generation (UD, f (nB = 1) = UD,η ), a branching energy release rate Gb can be calculated according to Eqs. (6.4) and (6.5). The only unknown parameter is the crack length at the first branching node rB = δ which can be calculated using the function of the trend line in Figure 6.11 (b) for UD, f (nB = 1) = 1086 J/m3 . This results in a crack length of rB (1086 J/m3 )= 25.73 mm if there is only one crack branching in the glass sheet. Inserting in Eq. (6.5) results in a branching stress inten√ sity of Kb = 2.479 MPa m, which fits very well with the value from Section 6.2 (Kb = √ 2.485 MPa m) and also with the values from the literature, which are summarized in Table 6.3. According to Eq. (6.4) the energy release rate at crack branching can be determined to Gb = 87.79 J/m2 . Interestingly, this value, which is obtained by the results of the tensile tests on float glass sheets, is compatible with the value of the branching energy release rate obtained from the fracture tests on thermally tempered glass plates at an energy relaxation rate of 23% of the initial energy (Gb = 88.22 J/m2 ), see Table 6.3.

196

6 Investigations into the Phenomenon of Crack Branching

Figure 6.10 Branching parameters: branching nodes nB (circles) exemplary for σ f = 23.55 MPa; rB is the crack length at the first branching and ϕB the angle of the first branching

(a)

(b)

(c) Figure 6.11 Elastic strain energy density UD, f [J/m3 ] in correlation with (a) Number of branching nodes nB , (b) Crack length at the first branching rB normalized by the width of the specimen (100 mm) and (c) Angle of the first branching ϕB [◦ ]

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass

197

Figure 6.12 Determination of the fracture velocity exemplary for a specimen with a fracture stress of σ f = 33.49 MPa, dashed line is the fracture front; fps = 380 kHz; Δ f rames = 5; r f r is the distance from the specimen edge to the fracture front

6.3.3.2 Fracture Velocity The fracture velocity is determined on the basis of the high-speed recordings. A frame with an initial fracture front distance r f r,0 , i.e. distance from the specimen edge to the fracture front, is taken as the reference state and the velocity of the fracture front v f r is determined area by area over the width of the specimen. The mean value of the range velocities is representative for the specimen. For each area, consisting of up to 5 frames (Δ f rames ≤ 5), the increase of the fracture front Δr f r is determined and the fracture velocity is calculated as follows: vfr =

f ps · Δr f r Δ f rames

(6.7)

For the fracture velocity calculations a frame rate of fps = 380 kHz is applied according to Table 6.1. Figure 6.12 shows an example of how the fracture velocity is determined. The velocity of the fracture propagation front was determined for 17 specimens. In Figure 6.13, the fracture velocity at different fracture stresses is shown. The average velocity of 20 frames in four steps of each specimen (Δ f rames = 5) was evaluated at a frame rate of 380 kHz. The average fracture velocity of all specimens is determined to v¯ f r = 1471.81 m/s. As shown in Figure 6.13, there is no correlation between the fracture velocity and the fracture stress. Furthermore, the velocity of a crack path is determined in order to show the variability of the velocity during the fracture process. For this purpose a crack path was selected at low (σ f = 13.85 MPa) and high (σ f = 33.49 MPa) fracture stress with respect to the various number of crack branching (see Figure 6.14) and

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6 Investigations into the Phenomenon of Crack Branching

Figure 6.13 Fracture velocity v f r [m/s] vs. Fracture stress σ f [MPa]; average of all specimens v¯ f r = 1471.81 m/s (±13.64 m/s); Δ f rames = 5; frame rate: 380 kHz

the velocity of crack propagation was determined along the path. In order to determine the fracture velocity along the crack path and to gather the change in velocity between the branches, the crack growth was evaluated for each individual frame (Δ f rames = 1) at a frame rate of 380 kHz. Figure 6.15 shows the change in fracture velocity during crack propagation for the crack paths according to Figure 6.14. The time axis is given in microseconds (μs). It can be clearly seen that the fracture velocity does not remain constant in the branch area, but varies greatly during the crack propagation process due to crack branching. The peaks in the diagrams in Figure 6.15 show that the fracture velocity increases up to the branching and suddenly decreases after the crack branches, before increasing again immediately before the generation of the next crack branching. The reason for the velocity peak in the branching area is the instability of the crack motion. The increase in the energy release rate reaches a limit at which the crack can no longer progress stably and leads to the formation of at least two new cracks or at least four new fracture surfaces. This process also influences the crack velocity. If there is enough time between two branches, the crack stabilizes and the velocity remains constant, see Figure 6.15 (a). The velocity of the fracture front was found to be constant between the branches and after the last branching. However, the actual fracture velocity behaviour is similar to an oscillation that starts with a large amplitude and flattens towards the end of the fracture process. The crack velocity at the first branching reaches values about 1600 m/s, while at the last branching the velocity is approximately 1550 m/s. This could be due to the fact that the energy released for fracture surface formation is somewhat lower at each branching due to conversion into other forms of energy such as sound energy or kinetics. However, the fracture velocity converges to the mean value over the crack path at the end of the fracture.

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass

199

(a)

(b) Figure 6.14 Selected crack paths with the number of branching for the determination of the variability of fracture velocity during fracture process (a) σ f = 13.85 MPa (b) σ f = 33.49 MPa

(a)

(b)

Figure 6.15 Fracture velocity v f r [[m/s] vs. Time t [μs] during crack propagation for the crack paths according to Figure 6.14 with identification of the crack branching (a) σ f = 13.85 MPa; mean fracture velocity v¯ f r = 1461.62 m/s; three branches (b) σ f = 33.49 MPa; mean fracture velocity v¯ f r = 1477.37 m/s; four branches, Δ f rames = 1; frame rate: 380 kHz

6.3.4 Branching Stress Intensity Factor Based on Fracture Surface Examinations To determine the energy release rate at branching Gb , the fracture surface in the area before the first branching is examined with a light microscope. However, the stress intensity at branching Kb would first have to be determined. Many authors have been able to de-

200

6 Investigations into the Phenomenon of Crack Branching

termine a stress intensity factor at branching by examining the fracture surface, see e.g. J OHNSON et al. (1966), A NTHONY et al. (1970), M ECHOLSKY et al. (1974), K IRCH NER et al. (1976), K IRCHNER et al. (1979), M ECHOLSKY et al. (1984), R ICE (1984), and M ECHOLSKY (1995). These authors suggested that the mechanism for branching was available at the mist/hackle boundary, and that sufficient energy must be available to produce an additional fracture surface at each end of an advancing crack. Thus, the branching stress intensity for the material soda-lime-silica glass was determined on the basis of fracture surface examinations by means of double cantilever beam or flexure tests with respect to the distance from the fracture origin to the mist/hackle boundary. The energy-balance criterion resulted in the following equation: Ah σf = √ πrh

(6.8)

where σ f is the fracture stress, rh the distance from the fracture origin to the mist/hackle boundary, and Ah a proportionality constant also referred to as mirror constant. BANSAL (1977) and K IRCHNER et al. (1979) suggested that the constant Ah in Eq. (6.8) could be replaced by a branching stress intensity factor and a shape and size parameter with Kb = AhY . This substitution continues to assume that the branching mechanism is associated with hackle formation and results in the following stress intensity criterion for crack branching: Kb σf = √ Y πrh

(6.9)

where Kb is the critical stress intensity factor for branching as described before and Y is a parameter that is dependent on flaw shape, specimen size, and the stress distribution. According to G ROSS et al. (2011) the shape parameter for a tensile loaded one-sided notched specimen can be calculated as:  Y=

r πr 2W πrh 0.752 + 2.02 Wh + 0.37(1 − sin 2Wh )3 · tan h πrh 2W cos πr 2W

(6.10)

where W is the width of the specimen in the direction of the notch. Both the energy and stress intensity criteria have been reviewed by R ICE (1984), with regard to fractographic features. Thus, the branching stress intensity Kb can be written in terms of the fracture stress σ f and the distance from the fracture origin to the mist/hackle boundary rh as: √ Kb = Y σ f πrh

(6.11)

After the determination of the branching stress intensity, the energy release rate at branching can be calculated using Eq. (6.4).

6.3 Tensile Test on One-Sided Pre-Damaged Float Glass

201

(a)

(b) Figure 6.16 Fracture surface of a specimen fractured in the tensile tests (a) Overview of the fracture surface (b) Detail view of the mirror/mist/hackle region

Figure 6.16 shows an example of the fracture surface of a specimen fractured in the tensile tests in this thesis, showing the fracture surface from the notch to the first branching. The inclined formation of the mist and hackle region is due to the low imperfection in the geometry of the manually inserted notch, which, however, does not influence the macroscopic branching behaviour of the specimen. In order to determine the branching stress intensity Kb according to Eq. (6.11) the distance from the fracture origin, the end of the notch, to the mist/hackle boundary rh is determined due to microscope measurements, see Figure 6.16 (a). The decision on what to consider as mist/hackle boundary was made on the basis of unevenness in the mist/hackle region characterized with the naked eye at 200× scale size. In other words, the first peak visible to the naked eye in the mist/hackle region was considered a mist/hackle boundary. As a rule, these peaks have a height of 50 μm to 80 μm. In case of indecision, the mean value of the distance from the notch end of 5 sighted peaks of 50 μm to 80 μm was considered as the mist/hackle boundary. Figure 6.17 shows the view of the fracture surface of a branched crack. It can be clearly seen that after branching and when a new fracture surface is generated, the process of fracture surface formation (mirror/mist/hackle) repeats itself. The branching stress intensity Kb determined from the fracture surface examinations is summarized in Table 6.2. The associated branching energy release rate Gb is calculated according to Eq. (6.4) and the shape parameter is evaluated according to Eq. (6.10). Thus, based on the fracture surface

202

6 Investigations into the Phenomenon of Crack Branching

Figure 6.17 View of the fracture surface of branched crack with repetition of the fracture surface evolution

examinations of 12 specimens, which were fractured by tensile tests, an average stress in√ tensity at branching of K¯ b = 2.494 MPa m (±0.041) and an associated average branching energy release rate of G¯ b = 88.85 J/m2 (±2.95) is obtained. Interestingly, this fits both to the considerations on Gb in Section 6.2.1, which came from the fracture tests on thermally tempered glass plates (acc. to Section 4.4) and a theoretical assumed energy relaxation of the initial elastic strain energy density by 23% during fragmentation (acc. to Section 5.4), and to the considerations in Section 6.3.3, which were based on the elastic strain energy density required for one branching node. The comparison with some values of the branching stress intensity and the associated energy release rate from the literature is presented in Section 6.4.

6.4 Summary of Results and Comparison with Literature

203

√ Table 6.2 Branching stress intensity Kb [MPa m] and associated branching energy release rate 2 Gb [J/m ] vs. fracture stress σ f [MPa] based on fracture surface examinations

σf [MPa]

W [mm]

rh [mm]

Y [−]

Kb√ [MPa m]

Gb [J/m2 ]

11.56 12.88 12.94 17.88 18.33 26.78 28.02 29.70 33.49 34.31 38.37 49.20

100.18 99.86 99.90 100.06 99.84 100.03 99.77 100.04 99.92 100.14 99.86 100.12

9.98 9.13 8.75 4.85 4.56 2.13 1.97 1.75 1.38 1.29 1.05 0.62

1.1952 1.1857 1.1815 1.1462 1.1442 1.1300 1.1292 1.1282 1.1267 1.1263 1.1254 1.1239 Average Stdev. VAR

2.447 2.586 2.535 2.530 2.510 2.475 2.489 2.484 2.484 2.460 2.480 2.441 2.494 0.041 0.016

85.55 95.51 91.81 91.44 89.99 87.53 88.53 88.16 88.17 86.47 87.89 85.09 88.85 2.95 0.033

6.4 Summary of Results and Comparison with Literature In this chapter, the phenomenon of crack branching during the fracture process was investigated. The fracture process was characterized by dividing the crack propagation into three phases of Phase A (Start of crack propagation), Phase B (Micro-branching) and Phase C (Crack branching). In each phase of the crack propagation the associated energy release rate was outlined. Especially in Phase C, a required energy release rate at crack branching was determined with respect to elastic strain energy released during fragmentation. For this purpose, the results of the fracture tests on thermally tempered glass plates in Section 4.4 and a theoretically estimated energy relaxation of the initial elastic strain energy density by 23% (acc. to Section 5.4) during fragmentation was used. Tensile tests on one-sided pre-damaged float glass sheets were carried out in order to investigate the crack branching behaviour. The measured notch parameters, i.e. notch head diameter φN , notch length aN and notch oblique angle ϕN as well as the results of the tensile tests, i.e. fracture load Ff , fracture stress σ f and the elastic strain energy density at fracture UD, f and the evaluation parameters of the branching behaviour, i.e. branching nodes nB , crack length at the first branching node rB and the angle of the first branching ϕB are summarized in Appendix A.9. Correlations between the elastic strain energy density

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6 Investigations into the Phenomenon of Crack Branching

at fracture and the branching nodes, crack length at branching and the branching angle were elaborated. The branching energy release rate was also determined using the tensile tests with respect to the elastic strain energy density at fracture. The required energy release rate was determined for one crack branching. The same energy release rate was determined due to fracture surface examinations. For this purpose, the branching stress intensity factor with respect to the distance from the fracture origin to the mist/hackle boundary was determined. Thus, in this thesis, the energy required for crack branching, or in other words the energy released during crack branching, was determined at about 88 J/m2 on basis of experimental results and theoretical considerations. Table 6.3 summarizes the results. In the following a comparison to some values from the literature is drawn. For the comparison of the branching energy release rate Gb , the findings of some authors who have experimentally investigated the branching stress intensity factor Kb are used here, see e.g. J OHNSON et al. (1966), A NTHONY et al. (1970), M ECHOLSKY et al. (1974), K IRCHNER et al. (1976), K IRCHNER et al. (1979), M ECHOLSKY et al. (1984), R ICE (1984), and M ECHOLSKY (1995). These authors determined the branching stress intensity for the material soda-lime-silica glass on the basis of fracture surface investigations using double cantilever beam or flexure tests with respect to the distance from the fracture origin to the mist/hackle boundary as outlined in Section 6.3.4. In contrast, Fineberg, as already outlined, measured the first peaks or roughness that form in the mirror area of the fracture surface and interpreted them as branching. For this purpose, he has carried out tensile tests on specimens that have been pre-damaged on one side. Hence, in the context of this thesis, the branching energy release rate according to F INEBERG (2006) is defined the micro-branching limit rather than the macro-branching and is positioned between the crack resistance Gc and the branching energy release rate Gb with respect to the crack propagation phases. The results for the branching stress intensity and the corresponding branching energy release rate are summed up in Table 6.3. The values of the branching stress intensity from the literature have been converted to the branching energy release rate Gb using Eq. (6.4). As can be seen in Table 6.3, the branching energy release rate of Gb = 88.22 J/m2 that has been determined in this thesis using the results of the fracture tests on thermally tempered glass plates (Section 4.4) as well as Gb = 87.79 J/m2 , which was determined based on the elastic strain energy density considerations from the tensile tests on one-sided predamaged float glass sheets (Section 6.3) and Gb = 88.85 J/m2 according to fracture surface examinations from tensile tests are both well compatible with the results obtained from the literature. Taking into account the literature values, Table 6.4 lists the values for the energy release rate at different phases of the fracture process as described in Section 6.2.1. The values for Gc in Table 6.4 are obtained by converting the fracture toughness from Table 2.3 to energy release rate using Eq. (6.4). Furthermore, the fracture velocity, i.e. the velocity of the fracture front, was determined

6.4 Summary of Results and Comparison with Literature

205

Table 6.3 Comparison of the results for branching stress intensity Kb and branching energy release rate Gb with some results from the literature

Author [−] J OHNSON et al. (1966) M ECHOLSKY et al. (1974) K IRCHNER et al. (1976) R ICE (1984) Present thesis: fracture testsa) Present thesis: tensile testsb) Present thesis: tensile testsc) a) b) c)

Kb√ [MPa m]

Gb [J/m2 ]

2.243 2.479 2.579 2.467 2.485 2.479 2.494

71.87 87.79 95.02 86.94 88.22 87.79 88.85

According to elastic strain energy considerations from fracture tests on thermally tempered glass plates (Section 6.2.1) According to elastic strain energy considerations from tensile tests (Section 6.3.3) According to fracture surface examinations from tensile tests (Section 6.3.4)

Table 6.4 Values of the energy release rate G at different phases of the fracture process

Crack propagation phase

Fracture surface evolution

A B C

Mirror Mirror/Mist Mist/Hackle

Start of crack propagation (Gc ) Start of micro-crack branching (Gb,micro ) Start of crack branching (Gb )

a)

Values obtained by converting the fracture toughness from Table 2.3

b)

according to F INEBERG (2006) Summary of the literature sources from Table 6.3

c)

G [J/m2 ] 7.41 - 9.61a) 35b) 71.87 - 95.02c)

using high-speed recordings of the crack propagation during the tensile tests. The fracture process was recorded with different frame rates. However, it turned out that the images are not usable at high frame rates of 780 kHz and 1000 kHz due to low resolution and the invisibility of the cracks. New experiments would have to be done with high frame rates and the light setting would have to be adjusted so that the crack is visible at the low resolution. The average velocity of the fracture propagation front was determined for 17 specimens, using more than 20 pictures of each specimen. An average velocity of the fracture front was found to be v¯ f r = 1471.81 m/s (±13.64 m/s). A correlation between the fracture velocity and the fracture stress was not found. The fracture velocity was determined in correlation of time along two selected crack paths with different fracture stresses in order to examine the change in velocity at crack branching. The fracture velocity curve has peaks in the branching area and increases to 1600 m/s at the first branching.

7 Summary and Outlook This thesis offered comprehensive analysis of the fracture behaviour and significant characteristics of the fracture pattern morphology of fragmented thermally tempered sodalime-silica glass. Investigations on influencing parameters and variables of the fracture process as well as the fracture mechanical properties and the physical quantities of the fracture pattern morphology were provided. Fragmentation analyses to determine correlations between the residual stress and the morphological properties of the fracture pattern of soda-lime-silica glass were carried out due to fracture tests on differently heat treated glass specimens of different thickness. A simplified model was presented for the determination of the engine power of the tempering oven at different heat transfer coefficients and as a function of the residual stress using quench parameters by forced convection. For the purpose of fragmentation analysis, i.a., the relationship between the fragment density as the number of fragments within a certain observation field, particle weight and volume, fragment and fracture surface area were set into correlation with the permanent residual stress respectively the resulting stored elastic strain energy. The fracture tests were carried out by impact according to EN 12150-1 (2015). The fragment and fracture surface area of randomly collected fragments were measured by means of Computer Tomography (CT) scans. The fracture surface roughness was also determined and correlated with the residual stress and the associated elastic strain energy. The experimental investigation of the fracture pattern morphology of thermally tempered glass and the determination of the correlation between the residual stress and the fragment density or fracture surface area can be extended by investigating further thicknesses. Above all, the minimum residual stress level in the specimens should be increased for further investigations, as the investigations in this thesis have shown that the orientation limit of 35 J/m2 (according to F INEBERG (2006)) as the micro-branching limit is not sufficient for full fragmentation in thinner specimens. Therefore, an orientation at approximately 90 J/m2 is recommended for the heat treatment of the specimens as the limit of macroscopic crack branching, as determined in this thesis on the basis of crack branching investigations. As a further perspective for the investigation of catastrophic fragmentation in thermally tempered glass, high-speed recordings of fracture tests at different residual stress levels can be performed to document the residual stress dependence of crack propagation and crack branching during the fracture process. For example, two high-speed cameras directed next to each other perpendicular to the glass plane can be used to record crack © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6_7

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7 Summary and Outlook

propagation and crack branching at two adjacent locations. With two cameras the focus is to be doubled, so that the probability of a crack recording between two branches is increased. The images are evaluated with regard to crack acceleration and crack terminal velocity between two branches at different residual stress levels. Furthermore, fracture tests can be carried out considering investigation of the influence of impact points and bearing conditions as well as the documentation of the wave propagation of the impact. In order to assess the fracture behaviour under different load and bearing conditions, different impact positions and different bearing arrangements can be implemented. The stress state in the plate can be changed by a defined compressive stress on the glass edges, which can be applied by several hydraulic presses. The actual stress state in the glass plate changed due to storage should be determined photoelastically by optical stress measurement using a SCALP. The impact is achieved by means of an impulse hammer with integrated force transducer in order to measure the impact force with regard to a numerical simulation of the fracture process. Shock sensors can be used to document the load or wave propagation as a result of the impact. These should be mounted at defined distances from the impact position to the corner and to the edge one behind the other on the glass plate. The data about the actual wave propagation in the glass body can be provided for the analysis of the influence of the wave propagation on the fragmentation and used for the validation of possible simulations of the fracture process. A machine learning inspired approach for the prediction of 2D macro-scale fragmentation of thermally tempered glass based on fracture mechanics considerations and statistical analysis of the fracture pattern morphology was deduced and applied. The approach is based on the combination of the LEFM energy criterion related to the initial elastic strain energy before fragmentation and the remaining elastic strain energy in the fragments after fragmentation and statistical analysis of the fracture pattern of tempered glass to determine the features of the fracture pattern morphology within observation fields. A method called ’Bayesian Reconstruction and Prediction of Glass Fracture Patterns (BREAK)’ was presented. Here, the fracture pattern of thermally tempered glass is predicted and simulated due to Bayesian spatial point statistics fed with elastic strain energy conditions and experimental data maintain statistical parameters of the fracture pattern. The fracture pattern simulation was performed by Voronoi tessellation of spatial point patterns. It was shown that the parameters of the method can be calibrated to the actual fracture patterns of thermally tempered glass and that a simulation of statistically representative fracture patterns for this level of residual stress is possible. Further research in this area is the evaluation of the approach with regard to its suitability for predicting structural fragmentation. In particular, the realistic representation of the influence of local residual stresses on fragment density and fragment morphology are decisive criteria for the assessment of the model. Furthermore, an improvement of the approach is to be tested and, if necessary, implemented by means of knowledge gained in

209

the model. The improvement of the basic understanding of decisive fracture mechanisms in tempered glass structures as well as the development of efficient analysis and prognosis models can be pursued. However, the implementation of the presented prediction method allows to export the simulated fracture pattern as mesh for further processing in a FEA context in order to find a numerical set-up for the computation of fractured laminated glass structures with incorporation of all the thermo-mechanical considerations from the polymeric interlayer. Furthermore, the fracture behaviour of glass was investigated with regard to the phenomenon of crack branching. As an essential aspect, the evolution states of a propagating crack in the fracture process, i.e. start of crack propagation, start of micro-crack branching and the beginning of the crack branching were characterized related to the associated energy release rate. The stress intensity factor and the associated energy release rate at crack branching was determined in three different ways, all leading to the same result. First, by theoretical estimate of the initial elastic strain energy density relaxation during fragmentation using Griffith’s energy criterion in LEFM and based on the fracture test results. Then, by elastic strain energy considerations from tensile tests on one-sided pre-damaged 1 mmthick float glass sheets. And finally by fracture surface examinations of the specimens from the mentioned tensile tests. Furthermore, the fracture front velocity and the crack velocity at branching was determined by high-speed recording of the crack propagation during the tensile tests. Further high-speed tests on thicker glass panes could be of interest to investigate the origin and process of fracture and the development of the fracture front along the thickness. Overall, this thesis provides a comprehensive insight into the fracture behaviour and fracture pattern morphology of thermally tempered soda-lime-silica glass with numerous outcomes and recommendations for further investigations and applications in the field of glass fracture. For example, as setting parameters for the fracture behaviour or for comparison and validation, the achieved results can be used, i.a., for the application and implementation of numerical models for the simulation of any kind of glass fracture.

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© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6

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Appendix A.1 Properties of Soda-Lime-Silica Glass In this section of the appendix, the material properties as well as the viscoelastic and structural relaxation characteristics of soda-lime-silica glass is summarized.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Pour-Moghaddam, On the Fracture Behaviour and the Fracture Pattern Morphology of Tempered Soda-Lime Glass, Mechanik, Werkstoffe und Konstruktion im Bauwesen 54, https://doi.org/10.1007/978-3-658-28206-6

224

APPENDIX

Table A.1 Material properties of soda-lime-silica glass (C ARRE et al., 1999; D UFFRENE, 1994; G UILLEMET et al., 1992)

Property

Symbol

Value

Unit

ρ E ν αg

2500 70 0.23 9 × 10−6

kg/m3 GPa − 1/K

αl

25 × 10−6

1/K

k

0.975 + 8.58 × 10−4 893 + 0.4T − 1.8 × 10−7 /T 2 1433 + 6.5 × 10−3 T 55 000 0.5 20

W/(mK), T in ◦ C J/(kg K), T in K J/(kg K), T in K K − ◦C

Density Young’s modulus Poisson’s ratio Thermal expansion coefficient for solid glass Thermal expansion coefficient for liquid glass Thermal conductivity Specific heat of solid glass Specific heat of liquid glass Ratio H/R Constant x (0 < x < 1) Ambient temperature

c p,g c p,l

T∞

Table A.2 Viscoelastic and structural relaxation characteristics; weights and relaxation times (C ARRE et al., 1999; D UFFRENE, 1994; G UILLEMET et al., 1992)

Shear relaxation i 1 2 3 4 5 6 1

w1i

τ1i

0.05523 0.08205 0.1215 0.2286 0.2860 0.2265

6.658 × 10−5 1.197 × 10−3 1.514 × 10−2 0.1672 0.7497 3.2920

Bulk relaxation

Structural relaxation

w2i

τ2i

Ci

λi

0.02221 0.02239 0.02870 0.2137 0.3942 0.3189

5.009 × 10−5

0.05523 0.08205 0.1215 0.2286 0.2860 0.2265

5.965 × 10−4 1.077 × 10−2 0.1362 1.505 6.747 29.63

9.945 × 10−4 2.022 × 10−3 1.925 × 10−2 0.1199 2.0330

Values given at the reference temperature Tre f = 864 K.

A.2 1D Forced Convection: MATLAB Source Code

225

A.2 1D Forced Convection: MATLAB Source Code The source code required to calculate the residual stress field is specified. The functions are listed in the order in which they are executed.

226

APPENDIX

A.2 1D Forced Convection: MATLAB Source Code

227

228

APPENDIX

A.2 1D Forced Convection: MATLAB Source Code

229

230

APPENDIX

A.3 1D Forced Convection: Heat Transfer Coefficient

231

A.3 1D Forced Convection: Heat Transfer Coefficient Scaling factor β for the identification of the heat transfer coefficient h for different convection surfaces: Edge: hEdge = 72β ; Surface: hSur f = 96β ; Chamfer: hCham = 125β ; Hole: hHole = 69β . In Table A.4 to A.7 scaling factor β = h/96 applies. T0 = 650 ◦C (923.15 K) T∞ = 20 ◦C (293.15 K)

232

APPENDIX

Table A.3 Heat transfer coefficient h [W/m2 K] vs. Residual surface compressive stress σs [MPa]; thickness 1.8 mm and 3.0 mm

t = 1.8 mm σs [MPa] -5.04 -10.82 -15.78 -20.86 -24.42 -31.29 -36.52 -40.30 -45.72 -52.52 -55.22 -59.37 -64.26 -69.73 -75.61 -81.02 -84.58 -88.61 -93.59 -101.63 -106.64 -111.27 -115.07 -121.13 -125.05 -129.34 -135.06 -141.77 -147.36 -152.23 -156.08 -160.24 -164.96 -170.35 -176.38 -183.45 -188.05 -193.03 -197.63 -202.16 1

h

[W/m2 K] 36 76 110 145 170 220 260 290 335 395 420 460 510 570 640 710 760 820 900 1000 1090 1180 1260 1400 1500 1620 1800 2050 2300 2560 2800 3100 3500 4050 4800 5900 6800 8000 9300 10800

t = 3.0 mm β [−]

Δt [s]

σs [MPa]

h [W/m2 K]

β [−]

Δt [s]

0.375 0.792 1.146 1.510 1.771 2.292 2.708 3.021 3.490 4.115 4.375 4.792 5.313 5.938 6.667 7.396 7.917 8.542 9.375 10.417 11.354 12.292 13.125 14.583 15.625 16.875 18.750 21.354 23.958 26.667 29.167 32.292 36.458 42.188 50.000 61.458 70.833 83.333 96.875 112.500

0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005

-4.83 -9.31 -14.93 -20.33 -25.04 -30.10 -34.88 -40.24 -45.21 -49.83 -55.48 -60.63 -65.35 -70.70 -75.53 -80.73 -85.36 -91.39 -95.98 -99.89 -104.83 -109.44 -114.01 -118.66 -125.49 -129.99 -134.45 -139.32 -144.63 -151.12 -155.62 -160.58 -164.60 -168.51 -172.45 -178.52 -184.67 -189.06 -193.27 -196.65

21 40 64 88 110 135 160 190 220 250 290 330 370 420 470 530 590 680 760 800 900 1010 1140 1300 1600 1850 2150 2550 3100 4000 4800 5900 7000 8300 10000 14000 20000 28000 50000 150000

0.219 0.417 0.667 0.917 1.146 1.406 1.667 1.979 2.292 2.604 3.021 3.438 3.854 4.375 4.896 5.521 6.146 7.083 7.917 8.333 9.375 10.521 11.875 13.542 16.667 19.271 22.396 26.563 32.292 41.667 50.000 61.458 72.917 86.458 104.167 145.833 208.333 291.667 520.833 1562.500

0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005

Intermediate values can be interpolated.

A.3 1D Forced Convection: Heat Transfer Coefficient

233

Table A.4 Heat transfer coefficient h [W/m2 K] vs. Residual surface compressive stress σs [MPa]; thickness 4.0 mm and 6.0 mm

t = 4.0 mm σs [MPa] -5.49 -9.88 -15.55 -21.74 -25.32 -29.65 -35.14 -40.33 -45.24 -49.86 -54.85 -60.83 -66.27 -70.45 -74.34 -78.65 -83.86 -90.77 -95.79 -100.16 -105.15 -109.67 -113.78 -118.23 -123.48 -129.11 -134.35 -138.44 -143.34 -148.31 -152.71 -157.43 -164.22 -169.11 -171.38 -175.56 -181.85 -185.91 -189.53 -193.74 1

h

[W/m2 K] 18 32 50 70 82 97 117 137 157 177 200 230 260 285 310 340 380 440 485 528 580 630 685 750 845 950 1080 1172 1310 1450 1610 1820 2240 2630 2840 3300 4200 5000 5900 7300

t = 6.0 mm β [−]

Δt [s]

σs [MPa]

h [W/m2 K]

β [−]

Δt [s]

0.188 0.333 0.521 0.729 0.854 1.010 1.219 1.427 1.635 1.844 2.083 2.396 2.708 2.969 3.229 3.542 3.958 4.583 5.052 5.500 6.042 6.563 7.135 7.813 8.802 9.896 11.250 12.208 13.646 15.104 16.771 18.958 23.333 27.396 29.583 34.375 43.750 52.083 61.458 76.042

0.050 0.050 0.050 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.005

-6.58 -9.15 -14.81 -19.54 -25.64 -31.11 -35.53 -41.06 -45.16 -50.28 -55.19 -59.89 -65.40 -70.30 -74.93 -79.89 -84.83 -89.95 -94.73 -99.07 -104.53 -110.63 -116.24 -121.42 -125.43 -129.20 -133.43 -138.60 -145.54 -150.65 -155.27 -159.45 -163.61 -168.02 -172.25 -181.62 -187.46 -190.82 -195.29 -199.85

15 20 32 42 55 67 77 90 100 113 126 139 155 170 185 202 220 240 260 275 300 330 360 390 415 440 470 510 570 620 670 720 775 840 910 1100 1250 1350 1500 1680

0.156 0.208 0.333 0.438 0.573 0.698 0.802 0.938 1.042 1.177 1.313 1.448 1.615 1.771 1.927 2.104 2.292 2.500 2.708 2.865 3.125 3.438 3.750 4.063 4.323 4.583 4.896 5.313 5.938 6.458 6.979 7.500 8.073 8.750 9.479 11.458 13.021 14.063 15.625 17.500

0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020

Intermediate values can be interpolated.

234

APPENDIX

Table A.5 Heat transfer coefficient h [W/m2 K] vs. Residual surface compressive stress σs [MPa]; thickness 8.0 mm and 10.0 mm

t = 8.0 mm σs [MPa] -4.76 -9.07 -14.07 -20.39 -25.42 -30.38 -35.25 -39.99 -44.60 -50.16 -55.50 -60.64 -65.58 -70.33 -74.90 -80.15 -84.32 -89.51 -95.16 -100.48 -104.76 -109.60 -114.18 -118.52 -123.97 -130.29 -134.98 -140.45 -145.51 -150.21 -154.59 -159.45 -165.38 -170.73 -175.60 -180.04 -184.11 -188.73 -194.46 -200.14 1

h

[W/m2 K] 8 15 23 33 41 49 57 65 73 83 93 103 113 123 133 145 155 168 183 198 210 225 240 255 275 300 320 345 370 395 420 450 490 530 570 610 650 700 770 850

t = 10.0 mm β [−]

Δt [s]

σs [MPa]

h [W/m2 K]

β [−]

Δt [s]

0.083 0.156 0.240 0.344 0.427 0.510 0.594 0.677 0.760 0.865 0.969 1.073 1.177 1.281 1.385 1.510 1.615 1.750 1.906 2.063 2.188 2.344 2.500 2.656 2.865 3.125 3.333 3.594 3.854 4.115 4.375 4.688 5.104 5.521 5.938 6.354 6.771 7.292 8.021 8.854

0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

-5.89 -11.32 -16.80 -20.74 -25.47 -30.16 -34.78 -40.80 -45.93 -50.90 -55.72 -60.38 -64.90 -68.28 -75.30 -79.92 -84.92 -89.72 -94.33 -99.04 -103.70 -108.15 -113.22 -118.39 -124.02 -129.28 -134.58 -140.10 -144.69 -149.55 -155.07 -159.72 -165.09 -170.04 -174.60 -179.66 -185.75 -192.48 -197.26 -200.06

8 15 22 27 33 39 45 53 60 67 74 81 88 95 105 113 122 131 140 150 160 170 182 195 210 225 241 259 275 293 315 335 360 385 410 440 480 530 570 595

0.083 0.156 0.229 0.281 0.344 0.406 0.469 0.552 0.625 0.698 0.771 0.844 0.917 0.990 1.094 1.177 1.271 1.365 1.458 1.563 1.667 1.771 1.896 2.031 2.188 2.344 2.510 2.698 2.865 3.052 3.281 3.490 3.750 4.010 4.271 4.583 5.000 5.521 5.938 6.198

0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200

Intermediate values can be interpolated.

A.3 1D Forced Convection: Heat Transfer Coefficient

235

Table A.6 Heat transfer coefficient h [W/m2 K] vs. Residual surface compressive stress σs [MPa]; thickness 12.0 mm and 15.0 mm

t = 12.0 mm σs [MPa] -7.11 -11.26 -14.52 -20.16 -24.86 -30.52 -37.00 -39.73 -45.09 -49.45 -56.21 -61.90 -65.06 -68.92 -76.35 -80.62 -86.12 -90.74 -97.69 -98.81 -104.16 -109.25 -114.12 -118.77 -124.66 -131.55 -137.15 -141.22 -145.12 -149.60 -154.85 -160.43 -165.94 -171.63 -176.90 -181.80 -186.37 -190.65 -196.59 -201.98 1

h

[W/m2 K] 8 12.5 16 22 27 33 40 43 49 54 62 69 73 78 88 94 102 109 120 122 131 140 149 158 170 185 198 208 218 230 245 262 280 300 320 340 360 380 410 440

t = 15.0 mm β [−]

Δt [s]

σs [MPa]

h [W/m2 K]

β [−]

Δt [s]

0.083 0.130 0.167 0.229 0.281 0.344 0.417 0.448 0.510 0.563 0.646 0.719 0.760 0.813 0.917 0.979 1.063 1.135 1.250 1.271 1.365 1.458 1.552 1.646 1.771 1.927 2.063 2.167 2.271 2.396 2.552 2.729 2.917 3.125 3.333 3.542 3.750 3.958 4.271 4.583

0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200

-5.25 -12.32 -16.98 -20.49 -25.20 -29.91 -34.57 -40.31 -44.82 -51.42 -54.62 -59.86 -66.93 -70.84 -75.60 -80.22 -84.69 -89.89 -94.90 -99.71 -105.85 -110.28 -114.54 -121.31 -126.45 -131.35 -135.46 -139.43 -143.77 -149.91 -156.16 -161.57 -167.04 -171.07 -174.86 -178.49 -183.67 -191.55 -194.46 -198.65

4.8 11 15 18 22 26 30 35 39 45 48 53 60 64 69 74 79 85 91 97 105 111 117 127 135 143 150 157 165 177 190 202 215 225 235 245 260 285 295 310

0.050 0.115 0.156 0.188 0.229 0.271 0.313 0.365 0.406 0.469 0.500 0.552 0.625 0.667 0.719 0.771 0.823 0.885 0.948 1.010 1.094 1.156 1.219 1.323 1.406 1.490 1.563 1.635 1.719 1.844 1.979 2.104 2.240 2.344 2.448 2.552 2.708 2.969 3.073 3.229

0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500

Intermediate values can be interpolated.

236

APPENDIX

Table A.7 Heat transfer coefficient h [W/m2 K] vs. Residual surface compressive stress σs [MPa]; thickness 19.0 mm

t = 19.0 mm σs [MPa]

h [W/m2 K]

β [−]

Δt [s]

-5.51 -9.07 -14.09 -21.49 -26.70 -30.40 -34.82 -40.64 -45.65 -50.56 -55.36 -60.04 -65.26 -70.33 -75.26 -80.04 -84.70 -90.32 -94.68 -98.90 -104.04 -108.99 -115.62 -120.16 -124.54 -129.61 -132.88 -140.66 -145.10 -149.36 -154.81 -159.98 -166.12 -170.78 -175.21 -180.46 -185.46 -190.17 -194.67 -199.76

4 6.5 10 15 18.5 21 24 28 31.5 35 38.5 42 46 50 54 58 62 67 71 75 80 85 92 97 102 108 112 122 128 134 142 150 160 168 176 186 196 206 216 228

0.042 0.068 0.104 0.156 0.193 0.219 0.250 0.292 0.328 0.365 0.401 0.438 0.479 0.521 0.563 0.604 0.646 0.698 0.740 0.781 0.833 0.885 0.958 1.010 1.063 1.125 1.167 1.271 1.333 1.396 1.479 1.563 1.667 1.750 1.833 1.938 2.042 2.146 2.250 2.375

0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.100 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

1

Intermediate values can be interpolated.

A.4 Fracture Tests: Results of the Fragment Density

237

A.4 Fracture Tests: Results of the Fragment Density In this section of the appendix, results of the fracture tests are summarized with respect to the fragment density N50 as a function of the residual stress and the glass thickness.

238

APPENDIX

Table A.8 Fragment density N50 [−] vs. σm2 · t [MPa2 · m] from Figure 4.29; results based on fracture tests on differently heat treated glass specimens of thickness t = 4 mm, t = 8 mm and t = 12 mm

t [mm] 1.8

3

4

1

8 σm2 · t

N50 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200

6

1.26 1.37 1.49 1.61 1.74 1.86 1.99 2.12 2.26 2.39 2.53 2.68 2.82 2.97 3.12 3.27 3.42 3.58 3.74 3.90 4.07 4.24 4.41 4.58 4.76 4.93 5.11 5.30 5.48 5.67 5.86 6.06 6.25 6.45 6.66 6.86 7.07 7.28 7.49 7.70

2.10 2.29 2.49 2.69 2.90 3.11 3.32 3.54 3.76 3.99 4.22 4.46 4.70 4.94 5.19 5.45 5.70 5.97 6.23 6.50 6.78 7.06 7.34 7.63 7.93 8.22 8.52 8.83 9.14 9.45 9.77 10.10 10.42 10.76 11.09 11.43 11.78 12.13 12.48 12.84

2.80 3.06 3.32 3.59 3.86 4.14 4.43 4.72 5.02 5.32 5.63 5.94 6.27 6.59 6.92 7.26 7.61 7.96 8.31 8.67 9.04 9.41 9.79 10.18 10.57 10.96 11.37 11.77 12.19 12.61 13.03 13.46 13.90 14.34 14.79 15.24 15.70 16.17 16.64 17.12

Intermediate values can be interpolated.

4.20 4.58 4.98 5.38 5.79 6.21 6.64 7.08 7.53 7.98 8.44 8.92 9.40 9.89 10.39 10.89 11.41 11.93 12.47 13.01 13.56 14.12 14.69 15.27 15.85 16.44 17.05 17.66 18.28 18.91 19.55 20.19 20.85 21.51 22.18 22.87 23.56 24.25 24.96 25.68

10

12

15

19

8.39 9.17 9.95 10.76 11.58 12.42 13.28 14.16 15.05 15.96 16.89 17.83 18.80 19.78 20.77 21.79 22.82 23.87 24.94 26.02 27.12 28.24 29.38 30.53 31.70 32.89 34.10 35.32 36.56 37.82 39.09 40.39 41.70 43.02 44.37 45.73 47.11 48.51 49.92 51.35

10.49 11.46 12.44 13.45 14.48 15.53 16.60 17.70 18.81 19.95 21.11 22.29 23.50 24.72 25.97 27.23 28.52 29.84 31.17 32.52 33.90 35.30 36.72 38.16 39.63 41.11 42.62 44.15 45.70 47.27 48.87 50.48 52.12 53.78 55.46 57.16 58.89 60.63 62.40 64.19

13.29 14.51 15.76 17.04 18.34 19.67 21.03 22.42 23.83 25.27 26.74 28.24 29.76 31.31 32.89 34.50 36.13 37.79 39.48 41.20 42.94 44.71 46.51 48.34 50.19 52.08 53.98 55.92 57.89 59.88 61.90 63.94 66.02 68.12 70.25 72.41 74.59 76.80 79.04 81.31

[MPa2 · m] 5.60 6.11 6.64 7.17 7.72 8.28 8.85 9.44 10.03 10.64 11.26 11.89 12.53 13.18 13.85 14.53 15.21 15.91 16.62 17.35 18.08 18.83 19.58 20.35 21.13 21.93 22.73 23.55 24.37 25.21 26.06 26.92 27.80 28.68 29.58 30.49 31.41 32.34 33.28 34.24

7.00 7.64 8.30 8.97 9.65 10.35 11.07 11.80 12.54 13.30 14.07 14.86 15.66 16.48 17.31 18.16 19.02 19.89 20.78 21.68 22.60 23.53 24.48 25.44 26.42 27.41 28.41 29.43 30.47 31.51 32.58 33.65 34.75 35.85 36.97 38.11 39.26 40.42 41.60 42.79

A.4 Fracture Tests: Results of the Fragment Density

239

...continuation of Table A.8

t [mm] 1.8

3

4

1

8

10

12

15

19

52.80 54.27 55.75 57.25 58.77 60.31 61.86 63.43 65.02 66.63 68.25 69.89 71.55 73.22 74.92 76.63 78.36 80.10 81.86 83.64 85.44 87.25 89.08 90.93 92.80 94.68 96.59 98.50 100.44 102.39 104.36 106.35 108.36 110.38 112.42 114.48 116.55 118.65 120.76 122.88

66.00 67.84 69.69 71.57 73.47 75.39 77.33 79.29 81.28 83.29 85.31 87.36 89.44 91.53 93.65 95.78 97.94 100.12 102.33 104.55 106.80 109.07 111.36 113.67 116.00 118.36 120.73 123.13 125.55 127.99 130.46 132.94 135.45 137.98 140.53 143.10 145.69 148.31 150.95 153.61

83.60 85.93 88.28 90.65 93.06 95.49 97.95 100.44 102.95 105.49 108.06 110.66 113.29 115.94 118.62 121.33 124.06 126.82 129.61 132.43 135.28 138.15 141.05 143.98 146.93 149.92 152.93 155.96 159.03 162.12 165.24 168.39 171.57 174.77 178.00 181.26 184.54 187.86 191.20 194.57

σm2 · t [MPa2 · m]

N50 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400

6

7.92 8.14 8.36 8.59 8.82 9.05 9.28 9.52 9.75 9.99 10.24 10.48 10.73 10.98 11.24 11.49 11.75 12.01 12.28 12.55 12.82 13.09 13.36 13.64 13.92 14.20 14.49 14.78 15.07 15.36 15.65 15.95 16.25 16.56 16.86 17.17 17.48 17.80 18.11 18.43

13.20 13.57 13.94 14.31 14.69 15.08 15.47 15.86 16.26 16.66 17.06 17.47 17.89 18.31 18.73 19.16 19.59 20.02 20.47 20.91 21.36 21.81 22.27 22.73 23.20 23.67 24.15 24.63 25.11 25.60 26.09 26.59 27.09 27.60 28.11 28.62 29.14 29.66 30.19 30.72

17.60 18.09 18.58 19.08 19.59 20.10 20.62 21.14 21.67 22.21 22.75 23.30 23.85 24.41 24.97 25.54 26.12 26.70 27.29 27.88 28.48 29.08 29.69 30.31 30.93 31.56 32.20 32.83 33.48 34.13 34.79 35.45 36.12 36.79 37.47 38.16 38.85 39.55 40.25 40.96

Intermediate values can be interpolated.

26.40 27.13 27.88 28.63 29.39 30.15 30.93 31.72 32.51 33.31 34.13 34.95 35.77 36.61 37.46 38.31 39.18 40.05 40.93 41.82 42.72 43.63 44.54 45.47 46.40 47.34 48.29 49.25 50.22 51.20 52.18 53.18 54.18 55.19 56.21 57.24 58.28 59.32 60.38 61.44

35.20 36.18 37.17 38.17 39.18 40.21 41.24 42.29 43.35 44.42 45.50 46.59 47.70 48.82 49.95 51.09 52.24 53.40 54.57 55.76 56.96 58.17 59.39 60.62 61.87 63.12 64.39 65.67 66.96 68.26 69.58 70.90 72.24 73.59 74.95 76.32 77.70 79.10 80.50 81.92

44.00 45.22 46.46 47.71 48.98 50.26 51.55 52.86 54.19 55.52 56.88 58.24 59.62 61.02 62.43 63.86 65.30 66.75 68.22 69.70 71.20 72.71 74.24 75.78 77.33 78.90 80.49 82.09 83.70 85.33 86.97 88.63 90.30 91.98 93.68 95.40 97.13 98.87 100.63 102.40

240

APPENDIX

A.5 Fracture Tests: Results of Particle Weight, Volume and Base Area In this section of the appendix, the fracture test results with respect to the particle weight, volume and base area as a function of the residual stress is summarized. The values represent the average from at least 130 particles (fragments) per specimen.

A.5 Fracture Tests: Results of Particle Weight, Volume and Base Area

241

Table A.9 Measured residual stress σi [MPa] vs. Measured particle weight M [g], volume V [mm3 ] and base area Abase = V /t [mm2 ]; values represent the average from at least 130 particles per specimen

Specimen

t [mm]

σs [MPa]

σm [MPa]

M [g]

V [mm3 ]

Abase [mm2 ]

ScA-8-a Stdev. CV [%] ScA-8-b Stdev. CV [%] ScA-8-c Stdev. CV [%]

3.8

-91.3 1.2 1.4 -92.3 1.4 1.6 -93.1 1.4 1.5

45.5 1.2 2.6 45.3 1.3 2.8 47.6 1.7 3.5

0.457 0.5 58.4 0.461 0.5 55.3 0.344 0.3 38.2

182.9

48.1

184.3

48.5

137.4

36.2

-73.1 1.0 1.4 -72.9 1.2 1.6 -74.9 2.1 2.8 -90.4 0.6 0.7 -91.1 1.0 1.2 -93.1 2.0 2.1 -107.0 1.3 1.2 -106.3 1.3 1.3 -109.0 1.8 1.6

36.6 0.7 1.9 36.3 0.6 1.6 37.2 0.9 2.4 45.4 0.9 1.9 45.5 0.8 1.8 46.7 0.7 1.5 53.5 0.8 1.5 53.7 0.9 1.7 55.2 0.6 1.2

1.459 0.4 29.3 1.619 0.5 31.9 1.387 0.4 29.4 0.613 0.2 28.7 0.586 0.1 22.9 0.549 0.2 27.8 0.325 0.1 28.0 0.307 0.1 30.0 0.310 0.1 27.3

583.6

73.9

647.5

84.1

554.8

71.1

245.1

31.0

234.6

29.7

219.7

27.8

130.2

16.7

122.7

15.5

124.1

15.7

ScB-6-a Stdev. CV [%] ScB-6-b Stdev. CV [%] ScB-6-c Stdev. CV [%] ScB-7-a Stdev. CV [%] ScB-7-b Stdev. CV [%] ScB-7-c Stdev. CV [%] ScB-8-a Stdev. CV [%] ScB-8-b Stdev. CV [%] ScB-8-c Stdev. CV [%]

3.8

3.8

7.9

7.7

7.8

7.9

7.9

7.9

7.8

7.9

7.9

242

APPENDIX

...continuation of Table A.9

Specimen

t [mm]

σs [MPa]

σm [MPa]

M [g]

V [mm3 ]

Abase [mm2 ]

ScC-3-a Stdev. CV [%] ScC-3-b Stdev. CV [%] ScC-3-c Stdev. CV [%] ScC-4-a Stdev. CV [%] ScC-4-b Stdev. CV [%] ScC-4-c Stdev. CV [%] ScC-5-a Stdev. CV [%] ScC-5-b Stdev. CV [%] ScC-5-c Stdev. CV [%] ScC-6-a Stdev. CV [%] ScC-6-b Stdev. CV [%] ScC-6-c Stdev. CV [%]

12

-55.2 0.8 1.5 -54.3 0.8 1.4 -54.2 1.3 2.3 -59.4 0.9 1.6 -59.6 0.8 1.3 -58.8 1.2 2.0 -63.6 1.7 2.6 -63.1 1.6 2.5 -63.5 1.6 2.6 -77.1 1.6 2.1 -77.7 2.4 3.1 -77.4 1.8 2.3

27.1 0.9 3.5 26.9 0.6 2.3 26.9 0.9 3.3 30.0 1.1 3.6 29.2 1.2 4.1 30.1 1.0 3.4 31.5 1.1 3.5 31.5 1.4 4.6 31.7 0.8 2.5 37.6 0.6 1.6 38.4 1.2 3.2 38.9 1.4 3.6

9.488 4.7 49.3 10.798 5.0 46.0 11.122 5.5 49.7 5.474 2.2 39.6 6.022 2.6 43.3 5.729 2.4 42.7 3.806 1.3 34.3 4.252 1.4 33.9 3.424 1.1 32.7 1.980 0.5 24.1 1.899 0.5 28.3 1.913 0.5 26.8

3795.2

316.3

4319.1

359.9

4448.8

370.7

2189.5

182.5

2408.6

200.7

2291.5

191.0

1522.5

125.8

1700.8

141.7

1369.4

114.1

792.1

65.5

759.6

63.3

765.4

63.8

12

12

12

12

12

12.1

12

12

12.1

12

12

A.6 Fracture Tests: CT Results of Fragment and Fracture Surface Area

243

...continuation of Table A.9

Specimen

t [mm]

σs [MPa]

σm [MPa]

M [g]

V [mm3 ]

Abase [mm2 ]

ScC-7-a Stdev. CV [%] ScC-7-b Stdev. CV [%] ScC-7-c Stdev. CV [%] ScC-8-a Stdev. CV [%] ScC-8-b Stdev. CV [%] ScC-8-c Stdev. CV [%]

12

-93.0 2.6 2.8 -93.4 2.1 2.3 -94.5 1.7 1.8 -110.2 1.9 1.7 -110.9 2.0 1.8 -110.9 4.0 3.6

46.7 1.6 3.5 45.4 1.2 2.7 47.7 1.0 2.0 55.4 1.8 3.2 55.7 1.1 1.9 55.5 1.8 3.3

1.172 0.3 25.1 1.220 0.3 24.4 1.115 0.3 24.7 0.743 0.2 26.9 0.702 0.2 31.1 0.684 0.2 28.5

468.9

39.1

488.2

40.7

446.1

37.2

297.4

24.6

280.9

23.4

273.6

22.8

12

12

12.1

12

12

A.6 Fracture Tests: CT Results of Fragment and Fracture Surface Area The results of the fragment surface area AS , actual fracture surface area A f r,act and the fracture surface roughness ρˆ determined from CT scans of fragments are summarized. For each specimen, the average value of 5 examined fragments is given.

244

APPENDIX

Table A.10 Summary of the results of the fragment and fracture surface area of the fragments by CT scan examinations

Specimen

t [mm]

σm [MPa]

AS [mm2 ]

A f r,act [mm2 ] ρˆ [−]

ScA-8-a Stdev. CV %] ScA-8-b Stdev. CV %] ScA-8-c Stdev. CV %]

3.8

45.6

3.8

45.4

3.8

47.6

302.3 53.7 17.8 395.7 111.2 28.1 286.3 34.6 12.1

150.1 24.9 16.6 169.1 38.1 22.5 128.0 30.4 23.7

1.036 0.008 0.8 1.026 0.006 0.6 1.032 0.015 1.4

ScB-6-a Stdev. CV %] ScB-6-b Stdev. CV %] ScB-6-c Stdev. CV %] ScB-7-a Stdev. CV %] ScB-7-b Stdev. CV %] ScB-7-c Stdev. CV %] ScB-8-a Stdev. CV %] ScB-8-b Stdev. CV %] ScB-8-c Stdev. CV %]

7.9

36.6

7.7

36.3

7.8

37.2

7.9

45.4

7.9

45.5

7.9

46.7

7.8

53.5

7.9

53.7

7.9

55.2

413.5 113.7 27.5 480.1 165.8 34.5 419.2 127.8 30.5 281.5 42.6 15.1 239.5 22.9 9.5 289.2 53.5 18.5 194.0 10.4 0.1 197.1 13.8 0.1 148.8 11.2 0.1

281.2 63.9 22.7 315.7 93.2 29.5 283.6 75.4 26.6 208.1 25.9 12.5 183.0 12.6 6.9 212.0 28.2 13.3 155.0 7.5 0.1 156.7 11.0 0.1 122.6 6.6 0.1

1.075 0.028 2.6 1.066 0.010 1.0 1.078 0.018 1.6 1.131 0.018 1.6 1.139 0.040 3.5 1.127 0.033 3.0 1.161 0.036 3.1 1.167 0.015 1.3 1.156 0.047 4.1

A.6 Fracture Tests: CT Results of Fragment and Fracture Surface Area

245

...continuation of Table A.10

Specimen

t [mm]

σm [MPa]

AS [mm2 ]

A f r,act [mm2 ] ρˆ [−]

ScC-3-a Stdev. CV %] ScC-3-b Stdev. CV %] ScC-3-c Stdev. CV %] ScC-4-a Stdev. CV %] ScC-4-b Stdev. CV %] ScC-4-c Stdev. CV %] ScC-5-a Stdev. CV %] ScC-5-b Stdev. CV %] ScC-5-c Stdev. CV %] ScC-6-a Stdev. CV %] ScC-6-b Stdev. CV %] ScC-6-c Stdev. CV %] ScC-7-a Stdev. CV %] ScC-7-b Stdev. CV %]

12

27.1

12

26.9

12

26.9

12

30.0

12

29.2

12

30.1

12.1

31.5

12

31.5

12

31.7

12.1

37.6

12

38.4

12

38.9

12

46.7

12

45.4

1582.9 170.5 8.6 1875.1 1110.3 59.2 1538.8 417.3 27.1 1222.7 300.6 24.6 1648.3 443.4 26.9 1496.2 64.2 4.3 800.0 125.0 13.1 1114.0 88.9 8.0 869.8 179.8 20.7 583.2 87.1 14.9 505.2 93.8 18.6 503.9 85.0 16.9 335.4 53.4 11.9 320.1 19.7 4.8

911.3 410.3 45.0 1039.5 455.3 43.8 910.1 203.7 22.4 775.1 130.8 16.9 973.2 197.4 20.3 929.9 73.8 7.9 659.5 77.4 9.7 737.5 49.8 6.8 612.3 99.5 16.3 456.7 59.5 13.0 401.0 65.7 16.4 390.8 53.9 13.8 269.0 41.9 11.4 245.9 15.5 4.5

1.049 0.103 9.3 1.059 0.009 0.8 1.042 0.011 1.1 1.088 0.028 2.6 1.071 0.018 1.7 1.076 0.026 2.4 1.087 0.034 3.1 1.093 0.020 1.8 1.097 0.031 2.8 1.130 0.043 3.7 1.123 0.077 6.5 1.125 0.023 2.0 1.189 0.044 3.7 1.190 0.056 4.5

246

APPENDIX

...continuation of Table A.10

Specimen

t [mm]

σm [MPa]

AS [mm2 ]

A f r,act [mm2 ] ρˆ [−]

ScC-7-c Stdev. CV %] ScC-8-a Stdev. CV %] ScC-8-b Stdev. CV %] ScC-8-c Stdev. CV %]

12

47.7

12.1

55.4

12

55.7

12

55.5

315.9 28.0 6.4 200.1 31.6 10.2 195.7 39.7 13.2 205.3 36.6 10.8

256.4 17.3 4.9 170.6 25.1 9.3 158.7 35.1 13.6 185.9 32.0 11.2

1.180 0.068 5.7 1.210 0.090 7.0 1.220 0.060 4.9 1.213 0.062 5.3

A.7 Statistical Distributions of Voronoi Cell Characteristics In this section of the appendix the results of the investigations on the statistical distributions of Voronoi cell characteristics (cell area AC , cell perimeter PC and the number of cell edges) are summarized for random Voronoi tessellations due to point process models HPP, MHCP and SP. In the figures the probability density function (PDF) and the cumulative distribution function (CDF) are shown.

A.7 Statistical Distributions of Voronoi Cell Characteristics

(a)

247

(b)

Figure A.1 HPP: (a) PDF and (b) CDF of the logarithmic cell area log AC [log(mm2 )] for intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OURMOGHADDAM et al. (2018f)

(a)

(b)

Figure A.2 HPP: (a) PDF and (b) CDF of the logarithmic cell perimeter log PC [log(mm)] for intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OURMOGHADDAM et al. (2018f)

(a)

(b)

Figure A.3 HPP: (a) PDF and (b) CDF of the cell edge number for intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OURMOGHADDAM et al. (2018f)

248

APPENDIX

(a)

(b)

Figure A.4 MHCP: (a) PDF and (b) CDF of the logarithmic cell area log AC [log(mm2 )] for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OUR MOGHADDAM et al. (2018f)

(a)

(b)

Figure A.5 MHCP: (a) PDF and (b) CDF of the logarithmic cell Perimeter log PC [log(mm)] for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OURMOGHADDAM et al. (2018f)

(a)

(b)

Figure A.6 MHCP: (a) PDF and (b) CDF of the cell edge number for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.005; 0.01; 0.02} [1/mm2 ], from P OURMOGHADDAM et al. (2018f)

A.7 Statistical Distributions of Voronoi Cell Characteristics

(a)

249

(b)

Figure A.7 SP: (a) PDF and (b) CDF of the logarithmic cell area log AC [log(mm2 )] for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.02} [1/mm2 ] and acceptance probability γ ∈ {0.001; 0.2}, from P OURMOGHADDAM et al. (2018f)

(a)

(b)

Figure A.8 SP: (a) PDF and (b) CDF of the logarithmic cell perimeter log PC [log(mm)] for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.02} [1/mm2 ] and acceptance probability γ ∈ {0.001; 0.2}, from P OURMOGHADDAM et al. (2018f)

(a)

(b)

Figure A.9 SP: (a) PDF and (b) CDF of the cell edge number for a uniformity parameter: α = 0.1 and α = 0.7 at intensities λ ∈ {0.0025; 0.02} [1/mm2 ] and acceptance probability γ ∈ {0.001; 0.2}, from P OURMOGHADDAM et al. (2018f)

250

APPENDIX

A.8 Stochastic Fracture Pattern Simulation of Thermally Tempered Glass via Method BREAK In this section of the appendix the results of the investigations on three thermally tempered glass specimens (t = 12 mm) with a residual stress level of ’a’ σm = 31.53 MPa (UD = 8748 J/m3 ), ’b’ σm = 31.12 MPa (UD = 8522 J/m3 ) and ’c’ σm = 31.40 MPa (UD = 8676 J/m3 ) are summarized.

A.8 Stochastic Fracture Pattern Simulation of Thermally Tempered Glass via Method 251 BREAK

(a)

(b) Figure A.10 Estimation of the process intensity λˆ for specimen ’a’ (t = 12 mm;) with the residual mid-plane tensile stress σm = 31.53 MPa (UD = 8748 J/m3 ), (a) top view and (b) 3D view

Figure A.11 Estimation of the process intensity λˆ for specimen ’b’ (t = 12 mm;) (top view) with the residual mid-plane tensile stress σm = 31.12 MPa (UD = 8522 J/m3 )

Figure A.12 Estimation of the process intensity λˆ for specimen ’c’ (t = 12 mm;) (top view) with the residual mid-plane tensile stress σm = 31.40 MPa (UD = 8676 J/m3 )

252

APPENDIX

(a)

(b)

Figure A.13 Empirical nearest-neighbour distribution functions Gˆ and empirical empty-space distribution function Fˆ vs. their respective theoretical CSR graph, evaluated for specimen ’a’

(a)

(b)

Figure A.14 Empirical nearest-neighbour distribution functions Gˆ and empirical empty-space distribution function Fˆ vs. their respective theoretical CSR graph, evaluated for specimen ’b’

(a)

(b)

Figure A.15 Empirical nearest-neighbour distribution functions Gˆ and empirical empty-space distribution function Fˆ vs. their respective theoretical CSR graph, evaluated for specimen ’c’

A.8 Stochastic Fracture Pattern Simulation of Thermally Tempered Glass via Method 253 BREAK

(a)

(b)

Figure A.16 Empirical Ripley’s K-function Kˆ vs. the respective theoretical CSR graph and L-function ˆ evaluated for specimen ’a’ L,

(a)

(b)

Figure A.17 Empirical Ripley’s K-function Kˆ vs. the respective theoretical CSR graph and L-function ˆ evaluated for specimen ’b’ L,

(a)

(b)

Figure A.18 Empirical Ripley’s K-function Kˆ vs. the respective theoretical CSR graph and L-function ˆ evaluated for specimen ’c’ L,

254

Figure A.19 PDF and CDF of inter point distances r for specimen ’a’

Figure A.20 PDF and CDF of inter point distances r for specimen ’b’

Figure A.21 PDF and CDF of inter point distances r for specimen ’c’

APPENDIX

A.8 Stochastic Fracture Pattern Simulation of Thermally Tempered Glass via Method 255 BREAK

(a)

(b)

(c) Figure A.22 Fracture pattern, computed centroid (red cross) and Voronoi tesselation for the specimens (a) ’a’ σm = 31.53 MPa (UD = 8748 J/m3 ) , (b) ’b’ σm = 31.12 MPa (UD = 8522 J/m3 ) and (c) ’c’ σm = 31.40 MPa (UD = 8676 J/m3 )

256

APPENDIX

A.9 Tensile Tests: Parameters and Results The measured notch parameters, i.e. notch head diameter φN , notch length aN and notch oblique angle ϕN as well as the results of the tensile tests on one-sided pre-damaged specimens, i.e. fracture load Ff , fracture stress σ f and the elastic strain energy density at fracture UD, f and the evaluation parameters of the branching behaviour, i.e. branching nodes nB , crack length at the first branching node rB and the angle of the first branching ϕB are summarized.

W [mm]

100.24 99.86 100.04 100.03 100.12 100.06 100.04 99.92 100.08 99.85 99.9 99.88 99.86 100.1 100.14 99.86 100.18 99.9 99.84 99.83 99.77 99.94 100.01 99.85 99.97 0.13 0.001

T-a1-03 T-a1-04 T-a1-05 T-a1-06 T-a1-07 T-a2-01 T-a2-02 T-a2-03 T-a2-05 T-a2-06 T-a2-07 T-a2-08 T-a2-09 T-a2-10 T-a2-12 T-a3-01 T-a3-02 T-a3-03 T-a3-04 T-a3-06 T-a4-01 T-a4-02 T-a5-01 T-a5-02 Average Stdev.: CV:

99.83 100.14 99.75 100.14 99.85 100.15 100 100.01 99.98 100.07 100.14 99.92 99.96 100.1 100.01 100.23 100.07 99.91 99.96 100.11 100.14 100.14 100.23 99.9 100.04 0.12 0.001

H [mm] 1.01 1.04 1.03 1.03 1.02 1.04 1.04 1.05 1.00 1.01 1.01 1.00 1.00 1.04 1.01 1.03 1.05 1.01 1.00 1.04 1.04 1.05 1.03 1.02 1.03 0.02 0.017

t [mm] 0.1702 0.1715 0.1704 0.1710 0.1700 0.1705 0.1701 0.1708 0.1702 0.1701 0.1701 0.1703 0.1701 0.1701 0.1704 0.1708 0.1701 0.1705 0.1703 0.1700 0.1720 0.1700 0.1700 0.1700 0.1704 0.0005 0.003

φN [mm] 0.4912 0.5031 0.5833 0.2634 0.5199 1.0112 0.7289 0.8606 1.0696 0.9537 0.8517 0.8884 0.8319 0.9191 0.7438 1.4600 1.3865 1.1016 1.1239 1.3707 1.8490 1.8920 2.8710 3.0120

aN [mm] 4.62 4.29 4.04 0 0 1.63 1.61 3.22 -1.79 2.69 3.42 3.35 1.72 2.4 0.72 1.23 1.94 1.72 -1.99 1.73 -1.602 1.03 -1.23 -1.11

ϕN [◦ ] 1374.7 3099.2 1604.7 2759.1 5024.8 1860.9 3089.6 3513.5 2356.7 2433.3 3446.4 3427.3 3832 4294.3 3470.4 1324.4 1216.3 1305.8 1829.8 2131.6 2907.6 1453.8 934.06 435.89

Ff [N] 13.58 29.84 15.57 26.78 49.20 17.88 29.70 33.49 23.55 24.13 34.16 34.31 38.37 41.25 34.31 12.88 11.56 12.94 18.33 20.53 28.02 13.85 9.07 4.28

σf [MPa] 2028.07 9795.85 2667.83 7888.47 26631.05 3517.64 9700.26 12336.42 6099.67 6403.91 12833.65 12952.09 16197.97 18717.27 12950.72 1823.78 1470.73 1842.35 3694.80 4636.77 8637.65 2111.27 904.44 201.49

UD, f [J/m2 ] 3 11 2 12 40 5 16 18 8 7 17 20 20 25 20 2 1 2 4 6 10 4 0 0

nB [−]

Summarized parameters and results of the tensile tests on one-sided pre-damaged float glass specimens

Specimen [−]

Table A.11

20.21 6.35 19.36 8.35 4.36 12.58 7.13 6.63 10.1 9 6 6.1 6 6 6.61 22.13 22.77 14.6 15.64 9.56 8.78 15.35 -

rB [mm] 34.3 57.35 41.32 53.8 59.5 51.6 60.5 59.34 58.9 59.1 58.5 61 60.7 57.6 60.6 26.7 38 42.2 52.8 51.1 59.6 48 -

ϕB [◦ ]

A.9 Tensile Tests: Parameters and Results 257