A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens (Mechanik, Werkstoffe und Konstruktion im Bauwesen) [1st ed. 2022] 3658367873, 9783658367879

Table of contents :
Abstract
Zusammenfassung
Contents
Glossaries
Abbreviations
Sub- and Superscripts
Symbols
1 Introduction
1.1 Motivation
1.2 State of the Art
Strength of Glass
Statistical Treatment
Computational Models
Head Injury Probability
1.3 Structure
Journal Articles
Book Contributions
Conference Proceedings
2 Theoretical Background
2.1 Glass
2.1.1 Definition of Glass
2.1.2 Properties and Application
2.1.3 Production
2.2 Mechanical Behavior of a Crack
2.2.1 Stress Concentration
2.2.1.1 Stress at Circular Holes
2.2.1.2 Stress at Elliptic Holes
2.2.1.3 Energy-Based Fracture Criterion
2.2.1.4 Stress Field at the Crack Tip
2.2.2 Stress Intensity Factor
2.2.3 Geometry Factor
2.3 Statistical Treatment of Glass Strength
2.3.1 Probability Distribution Functions
2.3.1.1 Two-Parameter Weibull Distribution
2.3.1.2 Three-Parameter Weibull Distribution
2.3.1.3 Left-Truncated Weibull Distribution
2.3.2 Parameter Estimation Methods
2.3.2.1 Least-Square Method
2.3.2.2 Weibull Plot
2.3.2.3 Maximum Likelihood
2.3.2.4 Modified Maximum Likelihood
2.4 Acoustic Emission Localization
3 A Stochastic Fracture Model for Glass
3.1 Introduction
3.2 Subcritical Crack Propagation
3.3 Stochastic fracture Model
3.3.1 Initialization Procedure
3.3.1.1 Estimating Initial Flaw Sizes
3.3.1.2 Geometry Factor Shift
3.3.2 Fracture Calculation
3.3.3 Post-Fracture Behavior
3.4 Model Validation
3.4.1 Mesh Dependency
3.4.2 Surface Size
3.4.3 Stress Rate
4 Mechanical Parameter Quantification
4.1 Introduction
4.2 Fundamental Mechanical Parameters
4.2.1 Young’s modulus
4.2.2 Poisson’s Ratio
4.3 Subcritical Crack Growth Parameters
4.3.1 Experimental Part
4.3.1.1 Specimen Preparation
4.3.1.2 Dynamic Fatigue Experiments
4.3.2 Evaluation
4.3.2.1 Parameter Estimation
4.3.2.2 Evaluation of Environmental Influences
4.3.3 Discussion
4.4 Fracture Toughness
4.5 Stress Field Factor
4.6 Crack Growth Threshold
4.7 Geometry Factor
4.8 Summary
5 Stochastic Strength of an Automotive Windscreen
5.1 Introduction
5.2 Lower Bound for Glass Strength
5.3 Differentiation of Strength Populations
5.4 Experimental Part
5.4.1 Specimen Preparation
5.4.2 Surface Strength
5.4.3 Edge Strength
5.5 Residual Stress
5.6 Statistical Evaluation
5.7 Fractographic Verification
5.8 Summary
6 Displacement-Controlled Windscreen Tests
6.1 Introduction
6.2 Experimental Part
6.2.1 Experimental Setup
6.2.2 Experimental Results
6.3 Numerical Part
6.3.1 Finite Element Model
6.3.1.1 Head Impactor
6.3.1.2 Windscreen
6.3.1.3 Test Frame
6.3.2 Interlayer Stiffness Evaluation
6.4 Comparison of the Results
6.4.1 Impactor Displacement
6.4.2 Fracture Origin
6.5 Summary
7 Free-Flying Head Impact
7.1 Introduction
7.2 Head Injury Criterion
7.3 Head Impact Replacement Test
7.3.1 Experimental Part
7.3.2 Numerical Part
7.3.3 Comparison of Results
7.4 Head Injury Probability Distribution
7.4.1 Stochastic Head Impact Simulation
7.4.2 Head Injury Distribution
8 Summary and Future Research Topics
8.1 Summary
8.2 Future Research Topics
References
Appendix A Subcritical Crack Growth Parameters
A.1 Experimentally Determined Parameters
A.2 Literature Values
Appendix B Windscreen Strength Test Results
B.1 Coaxial Ring-on-Ring Tests
B.2 Four-Point-Bending Tests
Appendix C Geometric Correction Factor Shift
Appendix D Newman and Raju Equation

Citation preview

Mechanik, Werkstoffe und Konstruktion im Bauwesen | Band 63

Christopher Brokmann

A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens

Mechanik, Werkstoffe und Konstruktion im Bauwesen Band 63 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 Institut für Mechanik und Materialforschung der TH Mittelhessen in Gießen bündeln die Forschungs- 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.

Weitere Bände in der Reihe http://www.springer.com/series/13824

Christopher Brokmann

A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens

Christopher Brokmann ME/IMM Technische Hochschule Mittelhessen Gießen, Germany Vom Promotionszentrum für Ingenieurwissenschaften des Forschungscampus Mittelhessen zur Erlangung des akademischen Grades des Doktors der Ingenieurwissenschaften (Dr.-Ing.) genehmigte Dissertation von Christopher Brokmann M.Sc. aus Hannover 1. Gutachten: Prof. Dr. Stefan Kolling (Technische Hochschule Mittelhessen, Gießen) 2. Gutachten: Prof. Dr. Sangam Chatterjee (Justus Liebig Universität, Gießen) Tag der Einreichung: 30.08.2021 Tag der mündlichen Prüfung: 16.12.2021 Gießen 2021

ISSN 2512-3238 ISSN 2512-3246 (electronic) Mechanik, Werkstoffe und Konstruktion im Bauwesen ISBN 978-3-658-36787-9 ISBN 978-3-658-36788-6 (eBook) https://doi.org/10.1007/978-3-658-36788-6 Springer Vieweg © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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

Abstract The strength of glass is not a material-specific value. During production and handling, microscopic cracks are induced. The length, shape and occurrence of these flaws are subjected to statistical scatter. The fracture behavior of glass, which is dominated by these flaws, is therefore difficult to predict numerically. An important case for the stochastic fracture behavior of glass is the head impact of a pedestrian on an automobile windscreen. The time of fracture of the windscreen has a significant influence on the risk of injury and should therefore be considered in the development process of windscreens. In this study, a model for the stochastic fracture behavior of glass is developed. The model, which is based on physical and fracture mechanical properties, enables the stochastic fracture behavior of components to be predicted. The applied stress rate, the size of the glass ply as well as the discretization within the finite element simulation are investigated. The strength of a windscreen was determined as the basis for the simulations performed. For this purpose, the windscreen and each individual surface have been divided into printed and un-printed areas. Samples obtained by water jet cutting were tested for their fracture strength by means of a coaxial ring-on-ring test. The strength of the edge was also determined by means of four-point bending tests. Eight different strength distributions could be determined. Experiments with free-flying impactors and impactors guided by an electric cylinder serve as the experimental basis for the simulations. The experiments were carried out at different impact velocities. The comparison of experimental and numerical results was based on the time of fracture and the location of the fracture origin. The stochastic failure model was able to reproduce the experimentally determined fracture behavior under certain deviations. Subsequently, a realistic head impact of a pedestrian on an automotive windscreen was simulated. For the first time, the scatter of the injury probability could be quantified. The head injury probability distribution obtained can be used as a basis for the design of future crash tests.

v

Zusammenfassung Die Festigkeit von Glas ist kein material-spezifischer Wert. Während der Produktion und dem Handling werden mikroskopische Risse induziert, welche in Länge, Form und Häufigkeit einer statistischen Verteilung unterliegen. Das daraus resultierende stochastische Bruchverhalten von Glas ist aus diesem Grund nur erschwert numerisch prognostizierbar. Dies wird besonders deutlich bei dem Kopfaufprall eines Fußgängers auf eine automobile Windschutzscheibe. Der Bruchzeitpunkt der Windschutzscheibe hat einen signifikanten Einfluss auf das Verletzungsrisiko und sollte daher bei dem Entwicklungsprozess von Windschutzscheiben berücksichtigt werden. Im Rahmen dieser Arbeit wird ein Modell für das stochastische Bruchverhalten von Glas vorgestellt. Das auf physikalische und bruchmechanische Eigenschaften basierende Modell ermöglicht das stochastische Bruchverhalten von Bauteilen vorherzusagen. Die Berücksichtigung von Einflussgrößen wie die applizierte Spannungsrate, die Größe der Glasscheibe sowie der Diskretisierung innerhalb der Finite-Elemente Simulation wurden untersucht. Als Grundlage für die erfolgten Simulationen wurden die Festigkeiten einer Windschutzscheibe bestimmt. Die Windschutzscheibe ist dafür in bedruckt und unbedruckt, sowie jede einzelne Oberfläche unterteilt worden. Mittels Wasserstrahlschneiden gewonnene Proben wurde mittels Doppelring-Biegeversuch auf ihre Bruchfestigkeit untersucht. Die Festigkeit der Kante wurde ebenfalls mittels vierPunkt-Biegeversuche ermittelt. Es konnten acht verschiedene Festigkeitsverteilungen bestimmt werden. Als experimentelle Basis für die Simulationen dienten Versuche mit freifliegenden sowie mittels Elektrozylinder geführten Impaktor. Die Versuche wurden bei unterschiedlichen Geschwindigkeiten und Lagerungen der Windschutzscheibe durchgeführt. Der Vergleich experimenteller und numerischer Ergebnisse wurde anhand des Bruchzeitpunktes und des Ortes des initialen Versagens vorgenommen. Das stochastische Versagensmodell konnte unter gewissen Abweichungen das experimentell ermittelte Bruchverhalten reproduzieren. Im Anschluss wurde ein realer Kopfaufprall eines Fußgängers auf die Windschutzscheibe simuliert. Erstmals konnte die Streuung der Verletzungswahrscheinlichkeit quantifiziert werden und kann als Basis für die Auslegung zukünftiger Crash-Tests verwendet werden.

vii

Contents Glossaries

xiii

1 Introduction 1.1 Motivation ............................................................................ 1.2 State of the Art ..................................................................... 1.3 Structure .............................................................................. 1.4 Achievements of This Study .....................................................

1 1 2 5 6

2 Theoretical Background 2.1 Glass ................................................................................... 2.1.1 Definition of Glass ........................................................ 2.1.2 Properties and Application ............................................. 2.1.3 Production .................................................................. 2.2 Mechanical Behavior of a Crack ................................................ 2.2.1 Stress Concentration ..................................................... 2.2.2 Stress Intensity Factor ................................................... 2.2.3 Geometry Factor .......................................................... 2.3 Statistical Treatment of Glass Strength ...................................... 2.3.1 Probability Distribution Functions ................................... 2.3.2 Parameter Estimation Methods ....................................... 2.4 Acoustic Emission Localization .................................................

9 9 9 9 11 12 12 16 17 18 19 21 25

3 A Stochastic Fracture Model for Glass 3.1 Introduction.......................................................................... 3.2 Subcritical Crack Propagation .................................................. 3.3 Stochastic fracture Model ........................................................ 3.3.1 Initialization Procedure ................................................. 3.3.2 Fracture Calculation ..................................................... 3.3.3 Post-Fracture Behavior .................................................. 3.4 Model Validation ................................................................... 3.4.1 Mesh Dependency......................................................... 3.4.2 Surface Size ................................................................. 3.4.3 Stress Rate..................................................................

29 29 30 34 34 38 39 42 43 44 46

4 Mechanical Parameter Quantification 49 4.1 Introduction.......................................................................... 49 ix

x

Contents

4.2

Fundamental Mechanical Parameters.......................................... 4.2.1 Young’s modulus .......................................................... 4.2.2 Poisson’s Ratio ............................................................ Subcritical Crack Growth Parameters......................................... 4.3.1 Experimental Part ........................................................ 4.3.2 Evaluation .................................................................. 4.3.3 Discussion ................................................................... Fracture Toughness ................................................................ Stress Field Factor ................................................................. Crack Growth Threshold ......................................................... Geometry Factor .................................................................... Summary..............................................................................

50 50 51 52 52 54 57 60 60 61 62 64

5 Stochastic Strength of an Automotive Windscreen 5.1 Introduction.......................................................................... 5.2 Lower Bound for Glass Strength................................................ 5.3 Differentiation of Strength Populations ....................................... 5.4 Experimental Part.................................................................. 5.4.1 Specimen Preparation.................................................... 5.4.2 Surface Strength........................................................... 5.4.3 Edge Strength.............................................................. 5.5 Residual Stress ...................................................................... 5.6 Statistical Evaluation .............................................................. 5.7 Fractographic Verification ........................................................ 5.8 Summary..............................................................................

67 67 68 69 71 71 72 74 79 80 81 84

6 Displacement-Controlled Windscreen Tests 6.1 Introduction.......................................................................... 6.2 Experimental Part.................................................................. 6.2.1 Experimental Setup ...................................................... 6.2.2 Experimental Results .................................................... 6.3 Numerical Part ...................................................................... 6.3.1 Finite Element Model .................................................... 6.3.2 Interlayer Stiffness Evaluation ......................................... 6.4 Comparison of the Results ....................................................... 6.4.1 Impactor Displacement .................................................. 6.4.2 Fracture Origin ............................................................ 6.5 Summary..............................................................................

87 87 87 87 88 91 91 92 96 96 96 102

7 Free-Flying Head Impact 7.1 Introduction.......................................................................... 7.2 Head Injury Criterion ............................................................. 7.3 Head Impact Replacement Test ................................................. 7.3.1 Experimental Part ........................................................

103 103 103 104 104

4.3

4.4 4.5 4.6 4.7 4.8

Contents

7.4

xi

7.3.2 7.3.3 Head 7.4.1 7.4.2

Numerical Part ............................................................ Comparison of Results ................................................... Injury Probability Distribution.......................................... Stochastic Head Impact Simulation .................................. Head Injury Distribution................................................

106 108 110 110 111

8 Summary and Future Research Topics 115 8.1 Summary.............................................................................. 115 8.2 Future Research Topics ........................................................... 116 References

119

A Subcritical Crack Growth Parameters 135 A.1 Experimentally Determined Parameters ...................................... 135 A.2 Literature Values ................................................................... 137 B Windscreen Strength Test Results 139 B.1 Coaxial Ring-on-Ring Tests ...................................................... 139 B.2 Four-Point-Bending Tests ........................................................ 141 C Geometric Correction Factor Shift

143

D Newman and Raju Equation

147

Glossaries Abbreviations 2PW 3PW

two-parameter Weibull three-parameter Weibull

AIS

abbreviated injury scale

CDF CEN CRRT CST

cumulative distribution function European Committee for Standardization coaxial ring-on-ring test constant surface test

FE FEM

finite element finite element method

HIC

head injury criterion

LS LTW LUCY

least-square left-truncated Weibull location uncertainty value

MLE MMLE

maximum likelihood estimator modified maximum likelihood estimator

NCAP

new car assessment program

PDF PMMA PVB PVC

probability density function poly(methyl methacrylate) poly(vinyl butyral) poly(vinyl chloride)

TDOA

time difference of arrival

xiii

xiv

Glossaries

Sub- and Superscripts min max 0 ∞ bond s f c I,II,III t b

minimum maximum initial, residual infinite atomic bond surface fracture critical crack opening modes tension bending

n fail hyp in cos el th alt maj res p

Symbols 1 a A b β c C d E η η˙ F F fY G γ Gc h Γ J

unity tensor crack depth, acceleration surface, fracture mirror constant specimen width shape parameter half crack width right Cauchy Green tensor distance Young’s modulus, Energy strain strain rate force deformation gradient geometry shift factor shear modulus location parameter, surface energy Griffith’s critical energy release rate specimen height incomplemete gamma function relative volume

sample size failure hyperbola initial crack-opening stress element threshold alternative major resultant principal

Glossaries

K k L l λ L υ n ~ ∇ P p r R2 ρ S S σ σ σ˙ τ U V v W wt χ Y

xv

bulk modulus, stress intensity stiffness Likelihood function specimen length principal stretch log likelihood function Poisson’s ratio total number, crack growth exponent nabla operator cummulative probability occurence probability radius coefficient of determinition density square of residuals 2nd Piola-Kirchhoff stress tensor stress stress tensor stress rate truncation point strain energy volume speed strain energy density weight percent stress field factor geometry factor

1 Introduction 1.1 Motivation Traffic accidents claim an estimated 1.35 million lives worldwide every year. Approximately 26 % of these victims are pedestrians and cyclists [1]. This corresponds to 970 deaths per day. Around 80 % of all serious injuries to a pedestrian in an accident occur in the head region [2, 3]. Depending on the size of the pedestrian and the vehicle, the head hits the windscreen with a certain probability. Despite these enormous numbers, the stochastic fracture behavior of glass is not considered in crash tests and crash simulations regarding automotive windscreens. To reproduce the pedestrian head impact on an automotive windscreen under laboratory conditions, specially developed head impactors are fired at the hood or windscreen under defined boundary conditions. Within the laboratory tests, the resulting acceleration of the impactor is measured as a function of time. By integrating the resulting acceleration, the head injury criterion (HIC) is determined according to [4]. While in the case of head impacts on hoods, the HIC can now be controlled very well by various measures, such as optimized design or active hoods, and can be predicted in crash simulation, head impacts on windscreens represent a major problem in the vehicle development process. When the windscreen is hit by an impactor, the glass plies fracture. Due to the stochastic distribution of microscopic flaws on the glass surface, the strength of glass is subjected to a high statistical scatter which leads to the fracture of the windscreen. It is not possible to map this scatter solely on the basis of laboratory tests due to the time and expense involved. An important step in the process of modern vehicle development is the construction of virtual prototypes, which are validated by numerical simulations. Ideally, real crashes should merely confirm the final simulation results. In the future, this should also be possible for the head impact on windscreens. Thus, there is an urgent need for simulation models that can calculate the head injury probability by taking into account the stochastic fracture behavior of glass. This investigation aims to develop a model for the stochastic fracture behavior of glass within finite element (FE) simulations. The model is intended to be generally applicable for the reproduction and prediction of the stochastic fracture behavior of glass. The main application is to consider the stochastic scatter for crash simulations of head impacts on automotive windscreens. The aim is to predict the probability of the HIC value and thus the severity of the injury. This © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_1

1

2

1 Introduction

would enable the investigation and optimization of the geometry of windscreens and their influence on the injury severity in the case of a head impact within the development stage of vehicles.

1.2 State of the Art The current challenge for numerical prediction and reproduction of the fracture behavior of glass is the stochastic nature of glass strength. The strength of glass is subject to many influences. In the following, current investigations and unsolved problems for the numerical simulation of the stochastic fracture behavior of glass are presented. Although a general consideration of glass is intended, the primary focus of this study lies on automotive windscreens.

Strength of Glass The difficulty in predicting the fracture behavior of glass is that the strength of glass is not a material-specific value as it is for example for steel or most plastics. The strength of glass is characterized by microscopic flaws [5]. These flaws are mainly caused by the handling, production and damage during the lifetime of the glass [6]. If an external stress is applied to these flaws, they can grow subcritically until the fracture of the whole glass component [7]. Due to the random molecular structure of glass, there are no slip planes that would allow plastic deformation [8]. Exceptions are extreme load cases like the Vickers indentation test, where densification underneath the indentation zone occurs [9]. Detecting and measuring cracks in glass for strength determination is possible by means of light scattering methods [10, 11] or by deep learning computation methods [12]. Despite promising results, the cost-efficient application within an industrial environment is not yet possible. Direct testing of glass strength is defined in standards for the surface strength [13] or the edge strength [14] of flat glass. Despite an existing standardization of the test methods, a discussion on the further development of these methods is necessary [5, 15]. The testing of flat glasses leads directly to the local fracture stresses at the fracture origin, while for more complex structures usually only dimensional data such as bursting pressures for container glasses are determined [16]. Simple changes within the production process of glass components can lead to a change in strength. Currently it is not possible to make predictions on the changed strength of glass components by standardized simple flat glass tests. Findings from structural and container glass with regard to edge strength and surface strength cannot be transferred to automotive windscreens due to the curvature and the different manufacturing processes. A detailed experimental investigation on the topics of edge strength, surface strength, the influence of

1.2 State of the Art

3

screen printings and residual stress with regard to automotive windscreens is not known, neither at the level of the small specimen nor at the level of the whole windscreen [17]. While influences on the edge strength of glass are known [18–22], their influence on windscreen fracture during a pedestrian head impact has not been studied yet.

Statistical Treatment The statistical treatment of glass fracture is a topic of ongoing research. The different origins of flaws within glass lead to different strength populations. Problems arise with the assignment of probabilities to experimental values [23]. The probabilities are determined by means of probability estimators. Within the last decades, many probability estimators were introduced for a wide range of applications [24–30]. An overview of these estimators is given by [31]. The most utilized estimator with regard to glass strength is the Weibull probability estimator [32]. While the Weibull probability estimator is widely used [33] and mathematically proposed [34], this statement is not accepted by all investigations due to legitimate doubts[35, 36]. Among the numerous statistical distributions available, the Weibull, log-Normal and other extreme value distributions are commonly used to describe the strength of glass [37]. By far the most common is the two-parameter Weibull (2PW) distribution [38–40], which is based on Weibull’s weakest-link theory [32]. However, it is shown that experimental data of float glass systematically deviate from the Weibull distribution [41]. This seems obvious due to the dependency of strength on the geometry and depth of flaws on the glass surface. The initial flaws are caused by a wide variety of influences and thus do not lead to a single glass strength population. Because existing cracks grow when external stress is applied, stress rate, stress state and environmental conditions also exert an influence on the strength population [42–44]. One disadvantage of the most common 2PW distribution is the problem of capturing experimental data at the lower bound for glass. It is assumed that a lower bound for the strength of glass exists [45]. This assumption may be correct due to modern manufacturing processes and quality control mechanisms which prevent cracks above a certain size. This assumption is supported by the investigation of sandblasted glass [46]. No further reduction in the strength of glass samples was observed after a certain duration of sandblasting. [45] recommends a left-truncated Weibull (LTW) distribution for the lower bound of glass. A problem in determining the left-truncation point for glass is the usually small sample size of strength data [47]. Other investigations showed that the lower bound for glass should follow a power law from a micro-mechanical point of view, which is given by the LTW distribution [48, 49].

4

1 Introduction

Computational Models Computational models for the fracture behavior of glass already exist, wand they exhibit a wide variety of approaches and methods. Many models can reproduce the fracture behavior of glass, but without taking into account the statistical scatter of the strength. Therefore, no predictions can be made about future load cases. The FE models of [50–52], models using cohesive zone formulations [53–57], discrete FE models [58, 59] and the extended finite element method (FEM) [60, 61] are mentioned here. For the purpose of using glass in the construction sector, the models [62–64] should be mentioned. Long calculation times, lack of physical interpretation of input parameters and their inability to predict the stochastic fracture make it challenging to use the mentioned models within a wide industrial range [65]. Numerical models taking into account the stochastic fracture behavior of glass rare. One model that is worth mentioning here is endowed with an approach modeling the critical flaw [66] or with a given stress distribution [63] in the field of civil engineering. [67] calculates fracture based on a statistical distribution of cracks. Another stochastic model is [68], which reproduces the stochastic scatter within a limited range of stress rates from standardized coaxial ring-on-ring tests.

Head Injury Probability Investigations concerning the stochastic nature of the head injury are e.g. [69], who performed 26 head impact tests on a Seat Leon windscreens and obtained a HIC between 680 and 1752. A HIC value of HIC = 1000 is equivalent to a probability of 18 % for a severe injury, a probability of 55 % for a serious injury and a probability of 90 % for moderate head injury to an average adult [70]. A historical review of the HIC is given in [71]. A detailed description of the head injury criterion is found in [52, 72]. Head impact values from Mercedes C-Class windscreens obtained by seven experiments were determined in [17]. Three of the tests with impact on the exterior side resulted in a head injury criterion between 296 and 539, while the remaining four tests with impact on the interior side resulted in a head injury criterion between 295 and 575. All seven experiments were performed with central impact on the windscreen. [73] showed three identical head impact experiments with the head injury criterion ranging from 410 to 1084. Designing windscreens regarding the stochastic fracture behavior is recommended in [74]. Apart from laminated safety glass, head impact simulations on an automotive side window made out of poly(methyl methacrylate) (PMMA) obtained head injury criterion values between 21 and 3300 [75]. Some varieties of PMMA have a similar stochastic fracture behavior compared to glass.

1.3 Structure

5

1.3 Structure A universally applicable model for predicting and reproducing the stochastic fracture behavior of glass is not available. Within this study, such a model shall be created and examined for its possibilities and disadvantages. Chapter 2 presents a brief discussion of fundamental mechanical and statistical principles which are necessary for this thesis. A new stochastic fracture model for glass is developed in Chapter 3. The model is based on physical observations with regard to subcritical crack growth in glass. A validation based on various possible influences is performed afterwards. The influences of the size of FE as well as of the applied stress rate and stress state are considered. A study of parameters describing the mechanical behavior of glass is presented in Chapter 4. It was found that the mechanical parameters for glass in the pertinent research literature are subject to some uncertainties. This may be caused by the influence of the chemical composition of the glass. These uncertainties can have a considerable influence on the mechanical behavior. Furthermore, there are many different approaches to determining mechanical parameters, which may also cause some deviations. The stochastic strength of a windscreen is investigated in Chapter 5. The estimation of the windscreen strength samples is divided into two parts: the surface strength and the edge strength. The surface strength is determined by coaxial ring-on-ring tests. Samples for this purpose were extracted from windscreens by means of water jet cutting. The existing curvature of the windscreens and thus also the obtained specimen is investigated. Using the mechanical parameters from Chapter 4 and the stochastic strength determined in Chapter 5, stochastic simulations with regard to automotive windscreens are performed in Chapter 6 and Chapter 7. Chapter 6 focuses on the stochastic fracture behavior of automotive windscreens by displacement-controlled impact experiments. Windscreens are tested on a unique test rig with an electrical cylinder at different constant impact velocities. Windscreens in convex and concave orientation are tested and monitored by an acoustic emission localization device to calculate the origin of the fracture. Subsequently, the stochastic scatter of fracture from experiments is numerically reproduced with the stochastic fracture model. The scatter of displacement done from the impactor until fracture and the location of fracture are compared. Chapter 7 provides an investigation on dynamic head impact tests on automotive windscreens. The chapter is divided into two parts. The first part investigates head impact replacement tests. These tests consist of a free flying impactor and an impact velocity according to the EuroNCAP testing protocol. The difference between EuroNCAP experiments and the test done within this investigation, is the four-point-support of the windscreens. Afterwards, the experimental results are compared to stochastic simulations using the stochastic fracture model. Afterwards head impact simulations were performed under real crash test conditions. A pedestrian head impact on an automotive windscreen is utilized. The aim is to

6

1 Introduction

calculate the stochastic scatter of the HIC. A stochastic distribution of the HIC is presented. The main findings of the present investigation are summarized in Chapter 8. Based on the observations during this investigation, several topics for further investigations are suggested. The structure of the present thesis is drawn in Fig. 1.1. Introduction (Chapter Background (Chapter Introduction (Chapter1)1)and and Theoretical theoretical Background (Chapter 2) 2)

A Stochastic Fracture Model for (Chapter Glass (Chapter 3) Stochastic Fracture Model 3) Parameter for Windscreen WindscreenSimulations Simulations ParameterEstimation Estimation for Mechanical Mechani Parameter ameter Quantification (Chapter4)4) Quantification (Chapter

Stochastic Stochastic Strength Strength ofofa a Windscreen Windscreen (Chapter (Chapter 5)5)

Stochastic Simulations Stochastic Simulations Displacement Controlled Displacement Controlled Windscreen (Chapter6)6) Windscreen Tests Tests (Chapter

Free-Flying Impact Free-Flying Head Head Impact (Chapter 7) (Chapter 7)

Summary (Chapter8)8) Summaryand and Outlook Outlook (Chapter

Figure 1.1

Structure of this thesis.

1.4 Achievements of This Study In summary, the achievements of this thesis are the following: • A new model for the stochastic fracture behavior of glass • Strength distributions for an automotive windscreen with regard to free surface, silkscreen, edge and surface in contact with the poly(vinyl butyral) (PVB) interlayer • Comparison of estimation methods for the truncation point within LTW distributions • Head injury criterion distribution considering the stochastic fracture behavior of glass for a pedestrian head impact

1.4 Achievements of This Study

7

• Fracture process of the windscreen during pedestrian head impact • Influence of environmental conditions on subcritical crack growth parameters The main achievement of this thesis is the stochastic fracture model. The model is capable of reproducing and predicting the stochastic fracture of glass. The necessary input parameters are physically motivated and are commonly found in the pertinent research literature. The model calculates initial flaw sizes from a strength distribution obtained from standardized tests. These flaws are distributed over the FE and grow subcritically depending on the applied stress. After a certain combination of stress and flaw size is achieved, fracture occurs. This enables us to take into account influences on glass fracture strength, such as element sizes, different stress rates and stress states. In order to perform stochastic simulations on automotive windscreens, the strength of the windscreen type is necessary. For this purpose, 612 coaxial ring-on-ring tests for the surface strength and 62 four-point-bending tests for the edge strength were performed. The strength of the windscreen with regard to free surface, silkscreen, edge and surface in contact with the PVB interlayer was determined. It is shown that the strength for these populations differs significantly. The former assumption of an overall strength distribution for windscreens is therefore only a coarse approximation. To describe the experimental strength values by means of a stochastic distribution, the LTW distribution is utilized. Several methods for determining the lower bound for glass strength were reviewed. It is shown that most methods overestimate the lower bound significantly, especially for samples of small size. A distribution of the HIC for a pedestrian head impact on an automotive windscreen is calculated by using the stochastic fracture model. Although the stochastic strength of glass is well known, it is not yet considered for pedestrian head impact tests on automotive windscreens in regulatory testing protocols. This may be attributed to the enormous experimental and financial efforts required for a certain amount of full-vehicle head impact experiments. Within this investigation, stochastic simulations of a pedestrian head impact on the exterior windscreen side with an impact velocity of 36 km/h are performed. For identical conditions with regard to the stochastic fracture behavior of glass, the head injury criterion varies between 411 and 1292. The calculated values are subjected to an extreme value distribution with the highest density around 480. 1.6 % of the values were above the recommended upper bound for the HIC of 1000. It is also observed that the calculated fracture process differs from the expected process. For laminated safety glass and an identical strength for both glass plies, the interior side should fracture first during an impact on the exterior side. This was the case for 235 out of 250 simulations. Six simulations had a reversed fracture sequence where the exterior ply failed before the interior ply. Eight simulations calculated only exterior glass fracture while the interior glass ply stayed intact. Within one simulation, no glass fracture occurred. Furthermore it is shown that the influence of temperature and humidity on the crack growth parameters has a non-negligible influence. While the environmental

8

1 Introduction

influence on glass has already been investigated in detail, the influence on the parameters for the linear approximation of subcritical crack growth still constitutes a central research desideratum. The importance of the environmental influence is shown by the fact that, for example, the terminal crack velocity v0 increases by a factor of two between a humidity of 30 %rh and 70 %rh at a temperature of 35◦ C. The dependence of the parameters on the humidity is fitted by a 2nd order polynomial. Parts of the results of this study have already been published and thus been made available to an international audience of experts:

Journal Articles • C. Brokmann, C. Alter, and S. Kolling, “Experimental determination of failure strength in automotive windscreens using acoustic emission and fractography”, Glass Structures & Engineering, vol. 4, no. 2, pp. 229–241, 2019 • C. Brokmann, S. Kolling, and J. Schneider, “Subcritical crack growth parameters in glass as a function of environmental conditions”, Glass Structures & Engineering, vol. 6, no. 1, pp. 89–101, 2021 • S. Müller-Braun, C. Brokmann, J. Schneider, and S. Kolling, “Strength of the individual glasses of curved, annealed and laminated glass used in automotive windscreens”, Engineering Failure Analysis, vol. 123, p. 105 281, 2021

Book Contributions • C. Brokmann, M. Berlinger, P. Schrader, and S. Kolling, “Fraktographische Bruchspannungs-Analyse von Acrylglas”, Glasbau Jahrbuch 2019, pp. 225– 237, • C. Brokmann, S. Kolling, and J. Schneider, “Subkritisches Risswachstum in Glas in Abhängigkeit der Umgebungsbedingungen”, Glasbau Jahrbuch 2021, pp. 283–294,

Conference Proceedings • C. Brokmann and S. Kolling, “Comparison of failure stress distributions in automotive windscreens by experiment and simulation”, 2018 • C. Brokmann, L. Aydin, and S. Kolling, “Robustness evaluation of the pedestrian head impact on an automotive windscreen”, in Proceedings of the 17th Weimar Optimization and Stochastic Days Conference, 2020 • C. Brokmann and S. Kolling, “A Model for the Stochastic Fracture Behaviour of Glass. Proceedings of the 16th LSTC World Conference”, 2020 • S. Kolling, M. Berlinger, C. Brokmann, and C. Alter, “Strain-rate dependent distribution functions for stochastic fracture simulation of glass and glassy polymers. cae grand challenge”, 2020

2 Theoretical Background 2.1 Glass 2.1.1 Definition of Glass The definition of glass is not uniformly regulated. The German standard DIN 1259-1:2001-09 [85] defines glass as „an inorganic product of fusion which has been cooled to a rigid condition without crystallizing“. [86] gives the definition of glass as a frozen super-cooled liquid in the physic-chemical sense. Although the atomic structure of glass is comparable to that of a liquid, the mechanical properties are those of a solid. When liquid glass cools, no phase transition is achieved. The transition from liquid to solid takes place in a temperature range, beginning at the glass transition temperature [87, 88]. No crystallization occurs during the entire cooling process. A definition of glass as materials that have a glass transition temperature would include organic glasses, for example PMMA. The quest for a general definition is going on worldwide. Although the general usage of the word glass has changed considerably over the last centuries, in a scientific context, it usually depends on whether the macro- or microscopic behavior is considered [37, 86, 89, 90]. Most of the glasses produced nowadays are soda-lime glasses [91], which belong to the group of silicate glasses and are inorganic non-metallic glasses. All glasses in the group of silicate glasses have in common that their network is formed mainly from silicon dioxide (SiO2 ). Due to the irregular and non-crystalline atomic structure, silicate glasses are homogeneous and isotropic within a macroscopic scale [37]. A further classification of silicate glasses is carried out, which categorizes oxide as the second most abundant type in terms of quantity. The most known and most produced silicate glasses are soda-lime, borosilicate, lead and aluminosilicate glass. A special case for silicate glasses is fused quartz glass, which is made from chemically-pure silica. Within this investigation, a FE model to simulate the stochastic fracture behavior of glass for engineering applications is developed. For engineering applications, glassy materials are defined by a brittle and linear elastic behavior at a macroscopic level [37, 92].

2.1.2 Properties and Application The properties of glass are as numerous as its possible applications. The usage of glass ranges from simple window glass, glass as an electrical insulator to glass © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_2

9

10

2 Theoretical Background

fiber deep-sea cables for the World Wide Web. Most properties are manipulated by changing the chemical composition of the glass during production. By adding lead as a second oxide to silicate glass, for example, an isolation for nuclear radiation is achieved. The most important properties of glass are transparency and optical properties as well as chemical and electrical isolation [88]. Within this investigation, the macro-mechanical properties of glass are of central importance. For more detailed information about general properties and the fields of applications for glass, the interested reader is referred to more specific literature for general glass properties [86, 88, 92, 93], the production of glass [94] or the possible applications [37, 90]. The main macro-mechanical property is the stress response to an external applied force. Most glasses show a linear elastic behavior until fracture at room temperature [86]. Due to the random molecular structure of glass, there are no slip planes that would allow plastic deformation [8]. Exceptions are extreme load cases like the Vickers indentation test, where densification underneath the indentation zone occurs [9]. The linear elasticity of glass is based on the change of equilibrium spacing of atoms by an external force. The Young’s modulus is the proportional measure of external applied force and the change in atomic bonding forces [95]. The Young’s modulus for soda-lime silicate glasses is often assumed to 70 GPa [13]. The strength of glass is subject to enormous statistical variations. Glass has a high theoretical strength compared to the strength that can be determined for practical applications. [96] estimated the theoretical maximum strength σmax of glass up to 45 GPa. He calculated the maximum strength by r γE σmax = , (2.1) r0 with a Young’s modulus of E = 72.1 GPa, a fracture surface energy of γ = 4.56 J/m2 and the atomic spacing of r0 = 0.162 nm. Due to microscopic flaws on the glass surfaces, a much lower strength is reached for practical applications. Local stress concentrations at these flaws are not reduced as it is the case for other materials such as steel. The reason for this is the non-existent plasticity of glass. These microscopic flaws are distributed pseudo-randomly over the entire surface of the glass and grow subcritically when an external stress is applied. It is possible to reduce flaw lengths by heat treatment [97]. Due to their different depth and geometries, these flaws cause a statistical distribution of fracture stresses which has not yet been adequately investigated [45]. Flaws are either surface- or volume-distributed. Due to modern production and quality control technologies, glass flaws are nearly exclusively located at the surface. [98] gives an insight into several flaw geometries and their origins. An overview of flaws indented by grinding is given in [99]. For more information on the countless types of flaws and their origins, the reader is referred to the excellent guide of fractography for ceramics and glasses [100]. Fig. 2.1 shows a selection of idealized crack morphologies from indentation contact [37, 101]. A Vickers crack system imaged by using an electron scanning microscope is shown in Fig. 4.2.

2.1 Glass

(a) Lateral

11

(b) Radial

(c) Median

Figure 2.1 Schematic representation of possible types of contact cracks on glass surfaces [101]. Pictures from [37].

2.1.3 Production The following section gives an insight into basic glass production techniques. It is by no means complete, but it provides the necessary information to understand the contents of this thesis. In the course of industrialization, glass went from being a luxury product to being suitable for mass production. Today, global glass production is estimated at 194,445.3 kilotons in 2020, with a forecast of 255,616.5 kilotons for 2027 [102]. Glass is usually the result of a melting process. The various starting raw materials are mixed and melted at temperatures depending on the glass composition. The float process is the most widely used manufacturing process and represents about 90% of modern glass production [62]. Within the float process, the molten glass is continuously poured onto a bath of liquid tin. The glass floats on the tin and forms smooth surfaces during cooling. The advantages of the float process are the quality of the glass and the high throughput [90, 94]. One disadvantage is the high CO2 output. Another noteworthy glass production process is the overflow downdraw or fusion method. Within this method, molten glass overflows from a V-shaped container, forming two glass streams that connect with each other at the bottom of the container. Different glass thicknesses can be realized by changing the drawing speed. The advantage of this method is that the glass surfaces are not in contact with molten tin and that a high planarity of the glass is achieved [90]. The fusion method is mostly utilized for very thin glasses [90, 94]. Within this investigation, automotive windscreens made of laminated safety glass are of great importance. Laminated safety glass consists of two or more glass plies bound together by an intermediate layer of plastic [103]. The interlayer keeps the glass layers bound even when the glass plies fracture and it prevents the glass from breaking into large and sharp pieces. One disadvantage of a plastic interlayer is that the description of the mechanical behavior is an ongoing research topic, especially for laminated safety glass [104–106]. Laminated safety glass for automo-

12

2 Theoretical Background

tive windscreens is nowadays made of two glass plies and one intermediate layer, mostly PVB. Modern interlayers for the usage in automotive windscreens consist of several layers, which have an acoustic insulation [107, 108]. The most common manufacturing method for laminated safety glass is the hot-press method [109]. The hot-press method uses heated metal rollers to create a pre-bond of the glass plies with the interlayer. In the main bonding process, a stable bond of glass and film is created in an autoclave. The autoclave contains a pressure of about 12 bar to 14 bar and a temperature of about 140◦ C [37]. The interlayer binds the enclosed air molecules during autoclaving to obtain a completely transparent windscreen. The autoclave process takes between 90 min and 360 min [37].

2.2 Mechanical Behavior of a Crack The stochastic fracture model developed in this thesis is based on simulating subcritical crack growth in glass. In the following, the fundamental behavior of cracks is examined to provide the reader with a basic understanding of the central assumptions of the stochastic fracture model. For more detailed information about continuum and linear elastic fracture mechanics, the reader is referred to more specific literature [110–114].

2.2.1 Stress Concentration 2.2.1.1 Stress at Circular Holes Consider an infinite body under a uniform tensile stress. When introducing a flaw into the body perpendicular to the uniaxial stress, the stress field around the flaw will change. In 1898, Ernst Gustav Kirsch developed a mathematical solution for stresses around a circular hole in an infinite plate [115]. He assumed a uniaxial tension σ∞ in the direction of φ = 0. Kirsch calculated a stress concentration of three at the hole under uniaxial loading by σ = σ∞ [1 − 2cos(2φ)] .

(2.2)

The main drawback of Kirsch’s solution is that only a circular hole is considered. 2.2.1.2 Stress at Elliptic Holes In 1913 Charles Inglis developed a solution for non circular cracks by considering an ellipse within an infinite body under tension [116]. He assumed an ellipse of the length 2a and width 2b within an infinite body under uniaxial stress σ∞ , shown in Fig. 2.2. His solution for the near crack stress is given by  r  a σ = σ∞ 1 + , (2.3) ρ

2.2 Mechanical Behavior of a Crack

13

2b

σ∞

Figure 2.2

2a

σ∞

Ellipse within a infinite body under uniaxial stress according to Inglis [116].

where the curvature of the ellipse ρ is expressed as b2 . (2.4) a A linear elastic material behavior was assumed. Inglis’s solution relates the stress at the crack to the distance to the crack tip a and the curvature ρ. The disadvantage of his solution is the assumption that for cracks with a  b the stress becomes infinite. In the event of a crack, fracture would have to occur immediately even at very low stresses, which is not the case. ρ=

2.2.1.3 Energy-Based Fracture Criterion Based on the model by Inglis’s, Alan A. Griffith proposed an energy-based analysis of cracks in 1920 [117]. One key insight of his fracture criterion is that energy is needed for crack growth. Crack growth is the creation of new surfaces by separating existing atomic bonds. The energy EBond necessary to separate atomic bonds as a function of the crack length a and the crack width b can be expressed as EBond = 2γs ab.

(2.5)

The constant value γs gives the energy required to separate atomic bonds per unit surface area created by the crack. When further considering a body under uniaxial tension σ∞ , the strain energy U for a linear elastic material behavior can be expressed as 2 V σ∞ , (2.6) 2E where V is the volume and E the Young’s modulus of the body. Using Griffith’s solution, the strain energy release rate for different crack geometries is

U=

U=

2 V σ2 σ∞ − ∞ πba2 . 2E 2E

(2.7)

14

2 Theoretical Background

crack growth: stable unstable

energy E

sum of both

strain energy

atomic bond energy crack length a Figure 2.3 [118].

Griffith’s energy balance depending on the crack length a under constant applied stress

The expression in Eq. (2.7) gives the strain energy of Eq. (2.6) with an additional part of strain energy release necessary for crack growth. Griffith derived his fracture criterion based on the observation of the total energy within a body. The total energy Etotal of the energy necessary to separate atomic bonds and the strain energy can be expressed as Etotal = EBond + U = 2γs ab +

2 V σ∞ σ2 − ∞ πba2 . 2E 2E

(2.8)

The fundamental insight of Griffith’s fracture criterion is that energy is needed for crack growth. Crack growth is the formation of new surfaces by separating atomic bonds. A graphical illustration of Eq. (2.8) is drawn in Fig. 2.3. Fig. 2.3 shows the difference between stable crack growth and fracture by the sum of bond energy and strain energy. For short crack lengths the total energy of the body increases with increasing crack length. Additional energy must be applied to the material to cause crack growth. The crack is stable. After reaching the maximum of the total energy curve an increase in crack length leads to a decrease in total energy. The crack can grow without additional applied energy. The crack growth is unstable and leads to fracture. With this observation, fracture can be expressed by σ2 dEtotal = 2γs b − ∞ πba = 0, da E

(2.9)

2.2 Mechanical Behavior of a Crack

15

y r

φ crack

Figure 2.4

x

Definition of the coordinate system ahead of the crack tip.

which leads to a critical stress value σf for a certain crack length a by r 2γs E σf = . πa

(2.10)

The expression 2γs can be replaced by the Griffith’s critical energy release rate Gc . Griffith’s criterion is used to estimate the critical fracture stress for a given crack length within an elastic body. Griffith’s energy-based fracture criterion is considered to be the beginning of the field of linear elastic fracture mechanics [117, 118]. 2.2.1.4 Stress Field at the Crack Tip The next step in the history of fracture mechanics is the solution for the stress field at the crack tip. Westergaard developed a complex function for the stress field surrounding a crack [119]. By using the coordinate system shown in Fig. 2.4, he developed a description of the stress field as σx = Re(Z) − y Im(Z 0 ) σy = Re(Z) + y Im(Z 0 ) τxy = −y Re(Z 0 )

(2.11)

where Z is a function of crack size a and applied stress. For cracks within an infinite body under tension σ, the function Z depends on z = x + iy by σ Z(z) = q 1−


a 2 z

.

(2.12)

The advantage of Westergaard’s solution is that it is directly valid for cracks and not for an ellipse approaching a crack in the limiting case. The disadvantage of his solution is the calculation of the real and imaginary parts. Without modern

16

2 Theoretical Background

computers, this was not practical at the time. However, Westergaard’s solution provided the fundamentals for the development of the stress intensity factor K, which is commonly used today. The stress intensity factor forms the basis for the stochastic fracture model developed in this thesis. For more information and more detailed derivations the interested reader is referred to more specific literature [112–114, 118, 120].

2.2.2 Stress Intensity Factor The stress intensity factor was developed by George R Irwin in 1957 [121]. Irwin used Westergaard’s solution and simplified it by substituting z from Eq. (2.12) by z = a + r exp(iφ). Using the coordinates from Fig. 2.4, Irwin achieved a solution for the stress field around the crack tip in an infinite body under tension σ∞ by      φ 3φ   1 − sin − sin 2 2    √ σx   φ  ∞ πa φ 3φ   σy  = σ√ cos (2.13) 1 + sin 2 − sin 2  2       2πr τxy φ 3φ sin 2 − sin 2 p The original solution did not contain the π/π term. Irwin recognized that the state of stress at the crack tip only depends on the applied stress σ∞ and the crack size a for r → 0. He introduced the formulation “stress intensity factor“ to describe the stress at the crack tip by √ K = σ∞ πa. (2.14) Comparing the stress intensity K with the critical energy release rate Gc from Eq. (2.10), one can obtain a formulation for the critical stress intensity or fracture toughness Kc by p √ Kc = Gc E = σf πaf . (2.15) The fracture toughness can be interpreted as the necessary combination of critical crack width af and critical stress σf which leads to unstable crack growth. Literature values √ for the fracture √ toughness of soda-lime silicate glass are ranging from 0.7 MPa m [122] to 0.84 m [123]. The considerations explained so far always assumed an uniaxial stress field at a perpendicular crack within an infinite body. The stress field acting on a crack is more complex under real-life conditions. In the pertinent research literature, the stress necessary for crack growth is divided into three cases. These cases are tensile stress, in-plane shear stress and out-of-plane shear stress. Fig. 2.5 illustrates all three crack-opening modes. The mode of loading is considered when using the stress intensity factor by KI for mode I, KII for mode II and KIII for mode III. This distinction is mandatory because the different modes have different effects on the crack. The three modes are not additive: X K 6= KI + KII + KIII . (2.16)

2.2 Mechanical Behavior of a Crack

Mode I

17

Mode II

Mode III

Figure 2.5 The three modes of crack opening. Mode I: pure tension, Mode II: in-plane shear and Mode III: out-of-plane shear.

The effect of different crack-opening modes on the fracture of glass is the subject of ongoing research [124–129]. Because shear stress plays only a minor role within the fracture of glass, generally only mode I stress is considered. The fracture toughness of glass is therefore usually referred to as KIc [37].

2.2.3 Geometry Factor The load cases considered so far assumed a crack in an infinite body. In-service cracks deviate from this assumption. This is taken into account by means of the geometry factor Y as an extension of the stress intensity to √ K = Y σ πa. (2.17) A related solution is the geometry factor of an edge crack within a semi-infinite plate. This configuration can be achieved by cutting the infinite body in Fig. 2.2 half horizontal. The crack is then less restrained than a through crack. This leads to an increase of 12 % for the stress intensity [114]. The geometry factor for an edge crack in a semi-infinite plate is thus Y = 1.12. A simple solution for the geometry factor is not possible through the different crack-opening modes and crack geometries. An overview of the most common load cases and the related geometry factor can be found in [130, 131] or with detailed derivation in [114]. In the context of this thesis, the solution of the geometry factor from Newman and Raju [132] is utilized. This solution is based on empirical results from different investigations [133–136]. Their solution represents an empirical stress-intensity factor equation for a surface crack as a function of parametric

18

2 Theoretical Background

σt M 2b

θ

a

2c

t

σb 2h

a

2c M σt Figure 2.6

Surface-cracked plate subjected to tension σt and bending σb load [132].

angle, crack depth, crack length, plate thickness and plate width. Geometry factors estimated from different crack lengths and load cases were used. The utilized notation for the following equations is drawn in Fig. 2.6. The solution from [132] is only valid for 0 < a/c ≤ 1, 0 ≤ a/t < 1, c/b < 0.5 and 0 ≤ θ ≤ π. The angle θ defines the location for the geometry factor within the crack. [132] divided the stress acting on the crack into bending σb and tension σt by √ KI = Y (σt + Hσb ) πa. (2.18) The geometry factor is given by F Y =√ , Q

(2.19)

with Q as geometric function of crack width to crack depth from [137] by  a 1.65 Q = 1 + 1.464 . (2.20) c The function F depends on the geometry of the crack and the surrounding body. F is defined in such a way that the boundary conditions for tension are considered. The product of H and F gives the boundary correction for bending. The functions H and F can be found in Appendix D.

2.3 Statistical Treatment of Glass Strength Glass has a disadvantageous variation in strength caused by the statistical distribution of surface flaws and their relationship to the material strength. These flaws oc-

2.3 Statistical Treatment of Glass Strength

19

cur randomly in size and geometry due to the production and handling of the glass. The relation between strength and flaws can be described by linear elastic fracture mechanics. Therefore, the strength of glass is also stochastically distributed. Because the origin of these flaws is different for each glass ply, the strength of glass cannot be assumed as a material-specific value. Common distribution functions for modeling the fracture strength of glass are shown in this section. The fracture strength of an automotive windscreen is investigated in Chap. 5. The calculated strength scatters from 42 MPa up to 622 MPa under identical test conditions. This yields a factor of nearly 15 for the strength of glass under identical load.

2.3.1 Probability Distribution Functions The probability of fracture or other stochastically distributed values can be reproduced by distribution functions. These functions show the occurrence of several events xi (i = 1,2,...,n) by a probability density function p(x) or cumulative distribution function P (x). The relationship between these two functions for values greater than or equal to zero is Z xn P (x) = p(x)dx. (2.21) 0

The probability density function (PDF) gives an impression of the frequency of the measured values, while the cumulative distribution function (CDF) shows the probability for each individual value. The assignment of a probability to each individual value within a sample pi is done by the Weibull probability estimator within this thesis [32]. By doing so each of the sorted values (i = 1,2,...,n) is assigned to a probability by pi =

i . n+1

(2.22)

Many probability estimators are available in literature for various applications. An overview of common probability estimators is provided by [31]. Among the numerous statistical distributions available, the Weibull, log-Normal and other extreme value distributions are commonly used to describe the strength of glass. By far the most common is the 2PW distribution [38, 39]. One disadvantage of the 2PW distribution is the problem of capturing experimental data at the lower bound for glass. [45] investigated this with different forms of the Weibull distribution. It was concluded that the LTW distribution can represent the strength of glass more accurately than the 2PW distribution. The present investigation follows the recommendation of [45]. A short introduction to the 2PW distribution and two truncated Weibull distributions, the three-parameter and the mentioned LTW distribution is given. This serves the purpose of being able to follow the procedure within this thesis. It is assumed that the strength sample originates from experiments with identical stress

20

2 Theoretical Background

fields. In order to compare values from different stress fields or values from samples with different geometries, further considerations have to be made [45]. For a detailed explanation of the different Weibull distributions and the methods for parameter estimation, more specialized literature is recommended[32, 38, 39, 138– 140]. 2.3.1.1 Two-Parameter Weibull Distribution The 2PW distribution is widely used in fracture data analysis. It consists of two parameters, the shape parameter β and the scale parameter η. The shape parameter is also known as Weibull modulus. The 2PW CDF is defined by "   # β x P (x; η, β) = 1 − exp − , (2.23) η with the parameter defined as: - scale parameter η ∈ (0, +∞) - shape parameter β ∈ (0, +∞). The scale parameter η is a value for the centrality of the estimated distribution. From a visual point of view, 63.21 % or 1 − exp(−1) of the measured values are below the value of the scale parameter. The other Weibull parameter, the shape parameter β, is an indicator for the occurrence variance. A low shape parameter means that the measured values have a higher scatter around the mean value than it would be the case with a higher shape parameter value. 2.3.1.2 Three-Parameter Weibull Distribution By introducing a third parameter γ to the 2PW distribution, the formulation for the three-parameter Weibull (3PW) distribution by "  β # x−γ P (x; η, β, γ) = 1 − exp − (2.24) η can be obtained. Analogous to the parameters from the 2PW distribution, the parameters for the 3PW distribution are defined by: - scale parameter η ∈ (0, +∞) - shape parameter β ∈ (0, +∞) - location parameter γ ∈ (0, min(x)]. Changing the location parameter γ when other parameters stay constant will result in a parallel shift of the CDF plot. Contrary to the 2PW distribution, the scale parameter η of the 3PW distribution is not the 63.21 %-quantile of the sample.

2.3 Statistical Treatment of Glass Strength

21

The value of the 63.21 %-quantile is given by the sum of the scale and location parameters. The location parameter γ suggests that there is a lower bound for the strength of glass. This assumption may be correct due to modern manufacturing processes and quality control mechanisms which prevent cracks above a certain size. However, a lower bound creates further complications as well. On the one hand, there is the fact that the fracture stresses for glass must always be seen in relation to the critical crack length. The material-specific critical stress intensity in Eq. (2.15) is defined as the combination of critical stress and critical crack length. On the other hand, fracture stresses are mostly estimated under laboratory conditions within a constant environment and a constant stress rate. [45] discusses the possibility of a lower limit for the strength of glass in detail. He concluded that there are experimental values observed so far which show that natural aging and the resulting reduction in strength do not fall below a certain value. This assumption is supported by the investigation of sandblasted glass [46]. No further reduction in the strength of glass samples was observed after a certain duration of sandblasting. 2.3.1.3 Left-Truncated Weibull Distribution Another Weibull distribution with a lower bound is provided by the LTW distribution. The LTW distribution can be expressed by "   β # β τ x P (x; η, β, τ ) = 1 − exp − . η η

(2.25)

The scale η and shape β parameters from the 2PW distribution, are extended by the truncation point τ . The LTW distribution is defined by the following parameters: - scale parameter η ∈ (0, +∞) - shape parameter β ∈ (0, +∞) - truncation point τ ∈ (0, min(x)]. The advantage of the LTW distribution is in the scaling of the sample with regard to size and stress effects. Shape and scale parameters are comparable with those of the 2PW distribution. The LTW distribution is recommended by the detailed investigation on a lower bound for glass by [45]. Other investigations showed that the lower bound for glass should follow a power law from a micro-mechanical point of view, which is given by the LTW distribution [48, 49]. This investigation follows the recommendation of using a LTW distribution for the strength of glass.

2.3.2 Parameter Estimation Methods Some procedures for estimating the parameters of the Weibull distributions are briefly discussed. The focus of this study is on the point estimation of distribution

22

2 Theoretical Background

parameters. The idea behind this section is to provide an insight into the most commonly used methods for parameter estimation regarding the strength of glass. Many estimation methods are available, each of which has its specific advantages and disadvantages. From the viewpoint of statistical theory, no method can be assumed as the best with regard to all statistical characteristics [138]. For information beyond the point estimation of parameters, see more specialized literature in the field of statistics [139, 141] or literature dealing with the statistical treatment of strength [112, 138].

2.3.2.1 Least-Square Method The simplest and most straight forward method of distribution parameter point estimation is the group of least-square (LS) methods. The LS method minimizes the summed square of residuals S between the sample data probability pi and the distribution function P (xi ) with the estimated parameters p1 , . . . , pm by S=

n X

pi − P (xi ; p1 , . . . , pm ).

(2.26)

i

One possible benefit of the result is given by the coefficient of determination R2 , which ranges between zero and one. A value towards zero indicates a low correlation of probability distribution function with the plotting positions, whereas an outcome near one indicates a good agreement. The coefficient of determination R2 is defined as R2 = 1 −

Pn 2 i (pi − P (xi )) P . n 2 ¯) i (pi − p

(2.27)

The main advantage of the LS method is its simplicity. The coefficient of determination is a dimensionless quantity and has a simple interpretation between zero and one. One major drawback of this method is the assumption that all measured values have the same deviation from the underlying distribution. The LS method handles each data point equally, which can cause errors if the assumed distribution fails to represent the data within a certain range [141]. This is especially true for small sample sizes [142]. Another drawback is that the coefficient of determination provides no information on whether the model was assumed correctly. Nonlinear relationships between the variables may contain important explanatory information, even if the coefficient of determination has a value close to zero [143]. A former investigation has been able to show that a high coefficient of determination yields no information on the underlying model [144].

2.3 Statistical Treatment of Glass Strength

23

2.3.2.2 Weibull Plot Another possibility to estimate the parameters of a LTW distribution is a linear regression within the Weibull plane. The Weibull plane is achieved by transforming the axis of the LTW CDF diagram to x → lnx and P → ln [G − ln (1 − P (x))]. The cumulative probability function P (xi ) is thereby transformed into the linear form of f (x) = mx + b by ln [G − ln (1 − P (x))] = β ln(x) − βln(η), | {z } |{z} | {z } | {z } f(x)

m

x

(2.28)

b

with the value G defined as  β τ G= . η

(2.29)

The value of G must be estimated iterative, so that the best linear alignment of the sample points is achieved [43]. One possible procedure consists in varying G until the coefficient of determination from Eq. (2.27) is maximized. The scale η and shape β parameters can then be calculated by   b β = m and η = exp − (2.30) β from the coefficients from the linear regression. One disadvantage of this method is that there is no consistent resolution within the double-logarithmic representation. A unwanted weighting of residuals is obtained. This approach contradicts the assumption that the model needs to be fitted to the measured sample and not vice versa [34]. Another disadvantage is the overestimation of the truncation point τ . An estimation of τ > min(x) is possible due to the graphical approach without direct limitation of the truncation point. 2.3.2.3 Maximum Likelihood The maximum likelihood estimation methods are another possibility of estimating distribution parameters [145]. In the context of this study, only a brief introduction to the procedure for a logarithmic likelihood parameter estimation is provided. For more information on possible likelihood functions and other possible statistical values regarding the maximum likelihood estimator (MLE) methods, more specific literature is recommended [138, 146, 147]. The MLE is used to calculate which parameters for the given sample of the population would occur with the greatest probability. The unknown parameter is thus the maximum of the function relating to the calculated probabilities. The likelihood function is obtained by the measured sample xi and the parameters of the presumed underlying distribution bi to L (x1 , . . . , xn ; b1 , . . . , bm )) =

n Y i

f (xi ; b1 , . . . , bm ).

(2.31)

24

2 Theoretical Background

The maximum likelihood is often extended to the logarithmic likelihood method. The log-likelihood function L is defined as the natural logarithm of the likelihood function L as L = lnL =

n X

lnf (xi ; b1 , . . . , bm ).

(2.32)

i

The advantage of the logarithmic representation is the need to calculate the sum and not the product of the derivatives. The partial derivatives of the log-likelihood function for m parameter b1 , . . . , bm are denoted by ∂L ∂L = 0, . . . , = 0. ∂b1 ∂bm

(2.33)

The solution for the obtained system of likelihood equations in Eq. (2.33) is given by the parameters bi . For the CDF of the LTW distribution in Eq. (2.25) with the parameters η, β and τ the derivatives of the log-likelihood function leads to P n h x i
β x i i n ln τ X i τ ∂L n xi i + = 0 → 0 = − P h
β ln (2.34) n x ∂η β τ i −1 i i τ and " n #1  β ∂L 1 X β β =0→η= xi − τ . ∂β n

(2.35)

i

Eq. (2.34) can be solved for the shape parameter β by using a standard iterative method. With the known shape parameter and Eq. (2.35), the scale parameter η can also be estimated. As it is evident, the parameter τ is needed for both equations. The problem of estimating the parameters of a LTW distribution by the log-likelihood method is that the partial derivation of the log-likelihood function L with respect to the truncation paramter τ by ∂L =0 ∂τ

(2.36)

has no unique solution. The parameter τ is equal to the smallest possible value within the complete population xmin . For in-use conditions the approximation by the smallest observation to τ = min(xi )

(2.37)

can be made. Since min(xi ) > xmin , the bias might be quite large towards the real truncation point for small samples.

2.4 Acoustic Emission Localization

25

2.3.2.4 Modified Maximum Likelihood [47] proposed a solution for estimating the truncation point for the LTW distribution by means of a log-likelihood function. This modified maximum likelihood estimator (MMLE) is obtained by substituting Eq. (2.36) by the expected value of the smallest possible observation E(xmin ). After some mathematical work, the equation for τ as replacement for Eq. (2.37) is obtained to

τ=

η1

nτ β exp η 

β

n

  β  nτ 1 Γ ;1+ , η β

(2.38)

where Γ is the incomplete gamma function. Within the original research [47], it is shown that the bias for the truncation-point estimate is reduced for small-sized samples by using Eq. (2.38) instead of Eq. (2.37).

2.4 Acoustic Emission Localization Due to the random fracture pattern of laminated safety glass windscreens after fracture, finding the origin of the fracture by an optical examination of the fracture pattern is a complex and time consuming task. For the purpose of identifying the origin of the fracture, acoustic emission analysis is utilized. The acoustic emission localization method measures acoustic waves passing through the observed medium. There are several possible algorithms to determine the signal source [148, 149]. In this thesis, the time difference of arrival (TDOA) is used. In the following, a short introduction into the acoustic emission localization with the TDOA method is provided. For a deeper insight into acoustic emission localization methods, see more specific literature on fracture localization within automotive windscreens [76, 150] or on acoustic emission localization methods in general [151, 152]. The TDOA method measures the incoming signal by the arrival time at the sensors, and it is especially useful for events of unknown initial event time. By the time difference ∆t = |ti − tj | of the incoming signal between two acoustic emission sensors i and j, multiplied with the speed of sound vs of the corresponding medium, a difference in distance ∆d can be calculated. The speed of sound for the utilized windscreens was estimated within a masters thesis of a student at approximately 1500 m/s. For a sensor pair i and j, the signal source must be within the measured distance difference. This can be expressed mathematically by a hyperbolic function as r ∆di,j =

y2 +



x i 2 − x+ 2

r

 x j 2 y2 + x + , 2

(2.39)

26

2 Theoretical Background

assuming that the locations of the acoustic emission sensors si and sj are at si (xi |0) and sj (xj |0) with xi = −xj or ∆x = 2x. With some simplifications, Eq. (2.39) can be rearranged to



x2 2 − 

∆d 2

y2 ∆x 2

2

−



∆d 2

2 = 1,

(2.40)

which is similar to the canonical form of a hyperbola. The total number of hyperbolas nhyp increases with the number of sensors n by nhyp =

n X

(i − 1).

(2.41)

i=1

Note that interference factors and measurement uncertainties do not lead to an exact intersection of all hyperbolas. To solve this issue, the calculated hyperbolic functions are iterated by adjusting the measured arrival times and minimizing a location uncertainty to determine a possible intersection point. This iteration requires that more sensors than needed measured an acoustic signal. For a planar localization, a total of three sensors is required. In order to obtain a qualitative value of the localization accuracy, a location uncertainty value (LUCY) is determined by sP n 2 1 (∆di,j − ∆pi,j ) LU CY = , (2.42) n−1 where ∆d is the first calculated distance and ∆p the adjusted distance between the sensors. The calculated values are obtained by a gradient algorithm in which the measured arrival times are adjusted. Accordingly, a high LUCY value indicates a high difference between the measured and calculated values. The validity of LUCY as a location uncertainty criterion has already been investigated [153].

2.4 Acoustic Emission Localization

27

s5

y-axis

s1

s4

s2 s3 x-axis

Figure 2.7 Example of acoustic emission localization via TDOA method using five sensors on an automotive windscreen with calculated fracture origin and corresponding location uncertainty (red).

3 A Stochastic Fracture Model for Glass 3.1 Introduction While the strength of most materials is considered as a material constant, the strength of glass depends on microscopic flaws on the glass surface. These flaws are created during the production and handling of the glass. When mechanical stress is applied, these flaws grow subcritically. Depending on the initial crack depth, the fracture stress of glass shows very large scatter. Chap. 5 will show that the strength between two samples can vary by more than a factor of 15 under identical test conditions. The dependence of the fracture strength on these microscopic cracks and their stochastic distribution leads to the conclusion that the strength of glass must be considered as a function of its loading and manufacturing history. The stochastic fracture model developed in this thesis uses a stochastic distribution of fracture strength. The fracture strength distribution originates from standardized material tests, for example [13]. The model determines the crack length at fracture and calculates the flaws back to their initial geometry. The calculated flaws are then distributed to each FE and will grow during simulations until fracture. An overview of the explained procedure is illustrated in Fig. 3.1.

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_3

29

30

3 A Stochastic Fracture Model for Glass

Failure stress distribution from experiments

Calculate failure stresses back to initial crack length

Distribute cracks on FE element randomly

Simulation of crack growth Figure 3.1

Procedure of the stochastic fracture model.

3.2 Subcritical Crack Propagation The strength of glass and other nearly ideal brittle materials is characterized by microscopic flaws. In the case of glass, these defects are almost exclusively distributed on the surface. The occurrence of volume defects is nowadays reduced to a minimum by means of modern production technology and quality controls. Due to their different depth and geometry, these flaws cause a statistical distribution of fracture stresses. Furthermore, these microscopic flaws grow subcritically when an external stress is applied. Depending on the growth time and growth velocity, subcritical crack growth is a significant influence on the fracture strength of glass [7]. If the crack growth velocity is represented as a function of the stress intensity KI in a double logarithmic scale, four different regions can be detected [7]. The subcritical crack growth through all four regions is illustrated in Fig. 3.2. The crack growth takes place above region 0, in which no crack growth is assumed [154]. In region I, the crack growth is mainly driven by the humidity at the crack tip in combination with an external mechanical load. Growth during region I has an almost linear dependence on humidity. The relation between crack velocity and stress intensity in region I is often expressed by a power law [155] with the parameters n and v0 as  n da KI v= = v0 , (3.1) dt KIc where KIc is the critical stress intensity. The physical processes during crack growth in region I can be explained as a stress-enhanced thermal activation of

crack velocity log(v)

3.2 Subcritical Crack Propagation

31

III lin. approx. from Eq. (3.1) 0

I

II inert conditions

KI,2 KIC Kth stress intensity log(KI ) Figure 3.2 Idealized relationship between subcritical crack velocity and stress intensity and the linear approximation from Eq. (3.1).

a dissociative hydrolysis reaction of the silica-oxygen network with free water molecules [156]. As the rate of crack growth increases, the crack velocity becomes almost independent from the stress intensity towards region II. This is due to a lack of additional water molecules available at the crack tip. Despite the fact that region II can only be measured in a very low range, region II is still only responsible for a small part of crack growth. When reaching region III, the crack propagation becomes nearly independent of the surrounding humidity and temperature. A small influence of the temperature on crack growth in region III could be measured by [157]. This shows that a change in temperature shifts the linear crack velocity versus the stress intensity curve in region III on the abscissa. Compared to region I, crack growth in region III also occurs only in a very small proportion, similar to region II. Region III exhibits such a small range that the measurement of region III is almost only possible in inert conditions [158] or by examining crack arrest when reaching the critical stress intensity [159]. After region III, when the critical stress intensity KIc is reached, the mechanical energy reaches a critical point. At this point the crack growth becomes unstable, which marks the end of subcritical crack growth. For further information on subcritical crack growth see a much more detailed overview in [7, 160]. For example, subcritical crack growth plays only a minor role for the pedestrian’s head impact on an automotive windscreen due to very high stress rates. High stress rates lead to small crack growth because less time is available for cracks to grow. Its role increases significantly with lower stress rates. Regardless of

32

3 A Stochastic Fracture Model for Glass

the stress rate, the environmental influence is assigned a greater role within the framework of the stochastic fracture model. This is due to the reverse-calculation of coaxial ring-on-ring test (CRRT) to initial crack lengths as the starting point of upcoming simulations. In [161], a revision of observed mechanisms on subcritical crack growth by modern methods was summarized and discussed to give an insight into the physical nature of subcritical crack growth. In order to be able to quantify the influencing factors on subcritical crack growth, a short overview of the main influencing factors for reproducing the fracture of glass is given in the following: • Temperature and humidity Because in region I and region II, the growth of cracks is mainly controlled by water molecules in combination with an applied mechanical stress, temperature and humidity have a significant influence on the fracture strength. The influence of the surrounding environment on crack growth was first observed by [7] via direct observation of crack growth using double cantilever beams. At higher humidity with more water molecules at the crack tip glass normally fails faster. In contrast to that, [162] showed that glass stored in water for 112 h at 88◦ C obtained a strength enhancement of around 25 %. This may be caused by water molecules penetrating into the glass network and creating a compression zone. In [77], the influences of humidity and temperature on the crack growth parameters n and v0 from Eq. (3.1) between 30 %rh to 70 %rh and temperatures between 15◦ C and 35◦ C were investigated. Crack growth parameters were obtained by dynamic fatigue tests and verified by in-situ observations. The method of direct observation of subcritical crack growth with a microscope was introduced by [163]. Fig. 3.3 shows the dependency of n and v0 on temperature and humidity for the considered soda-lime silica glass, fitted by a 2nd order polynomial. • Chemical composition of the glass In [158], the subcritical crack growth according to the type of glass was investigated. This is illustrated in Fig. 3.4. It can be seen that the chemical composition also has a significant influence on the crack velocity. Crack growth parameters found in literature are often subject to wide scattering. Even frequently used glass types such as soda-lime silica glass are subject to compositional fluctuations caused by different manufacturers. It is recommended to always consider the chemical composition in the context of the subcritical crack growth parameters. [77, 164] both indicated the chemical composition of the glass used to determine the subcritical crack growth parameters. Despite a small deviation in the chemical composition and a very good agreement of the crack growth exponent n, the crack growth parameter v0 was about four times higher in [77] compared to

3.2 Subcritical Crack Propagation

16 15

33

12

25°C 35°C

10 v0 [mm/s]

n [-]

14 13

8

12 6 11 10 30

Figure 3.3 [77].

40 50 60 humidity [%rh]

70

4 30

40 50 60 humidity [%rh]

70

Influence of humidity and temperature on subcritical crack growth parameters n and v0

[164]. • Corrosive media and pH value The pH-value of the surrounding medium also affects the crack growth parameters. It also affects the crack growth threshold between the crack growth regions 0 and I. [165] measured crack growth in NaOH, HCl and pure water as surrounding medium and observed a drop in crack growth towards the lower crack growth threshold Kth . Because the crack growth threshold only has a small influence on the dynamic fracture behavior of glass, this influence can be neglected for non quasi-static simulations. Long-term applications with a surrounding medium consisting of a different pH-value than strength values from laboratory tests should use a very accurate crack growth threshold for correct fracture prediction.

34

3 A Stochastic Fracture Model for Glass

100

v [mm/s]

10−2 Soda-lime

10−4

Borosilicate

10−6 0.2

Figure 3.4

Silica

0.3

Aluminosilicate 0.4 √ KI [MPa m]

0.5

0.6

Influence of chemical composition of the glass on crack velocity [158].

3.3 Stochastic fracture Model 3.3.1 Initialization Procedure During the initial procedure, flaws and their geometry are calculated for each FE. This section is divided into the derivation of the flaw calculation and a part in the consideration of the non-linearity of the flaw geometry. 3.3.1.1 Estimating Initial Flaw Sizes The following model distributes initial flaws among the FE stochastically in length and geometry. The physical background for a numerical model makes a wide range of numerical applications possible. In order to calculate initial flaws from a given fracture strength, it is assumed that the fracture strengths originate from CRRT. This has the advantage that the stress rate, the tested surface size and the triaxiality are all well-known. Assuming that the differential equation for crack growth from Eq. (3.1) is valid for the whole stress intensity range (which neglects the crack growth threshold) and inserting the stress intensity formulation leads to  √ n Y σ πa da = v0 dt. (3.2) KIc The assumption that Eq. (3.1), which describes subcritical crack growth in region I, is also valid for region II and region III is a frequently used approximation for engineering problems. This is particularly useful, because the crack only grows for a comparatively short time in the last two growth regions. Most of its growth occurs

3.3 Stochastic fracture Model

35

in the region I of the subcritical crack growth in Fig. 3.2. The linear approximation could already be used successfully in several numerical models, e.g. for structural glass elements [62], float glass under consideration of the stochastic fracture behavior [63], for automotive windscreens [52] or CRRT by considering subcritical crack growth [68], to name a few of the numerous numerical fracture models for glass. By integrating Eq. (3.2) from the initial crack size ain at the time t = 0 into the crack size at fracture af when t = tf it follows that Z af Z tf n −n n a− 2 da = v0 KIc π2 σ(t)n Y (t)n dt. (3.3) ain

0

While the left side of Eq. (3.4) is simple to solve, the stress history σ(t) and the evolution of the geometry correction factor Y (t) are not known a priori. When assuming that the crack growth took place in a CRRT with a constant stress rate, the stress history σ(t) can be replaced by the stress rate σ˙ multiplied by the time to fracture tf , both of which are invariant over time. A more complicated situation is represented by the geometry factor. It undergoes a more complex evolution over time. The fact is that the necessary development of the geometry factor for the determination of the initial crack length is not known Rt a priori. To take the evolution of the geometry factor 0 f Y (t)dt into account, a shift factor fY is introduced by Z tf Y (t)dt = Y0 fY , (3.4) 0

where Y0 states the initial geometry factor of the calculated flaw. The determination of the correction function fY is derived in detail in the following section. With these assumptions, one can integrate Eq. (3.4) to  tn+1 n 2  1− n2 1− n f −n −af + ain 2 = v0 KIc (Y0 fY )n π 2 σ˙ n . n−2 n+1

(3.5)

Replacing the time to fracture tf to the power of n + 1 by the fracture stress σf to the power of n + 1 divided by the stress rate σ˙ and expanding with the stress rate, described as follows σ˙ σfn+1 σ˙ −1 = σ˙ n tn+1 , (3.6) f σ˙ the time to fracture in Eq. (3.5) can be replaced. This leads to  σfn+1 2  1− n2 1− n −n n n n −af + ain 2 = v0 KIc Y0 fY π 2 σ˙ −1 . n−2 n+1

(3.7)

By rearranging Eq. (3.7), a formulation for the initial flaw size ain is obtained to  ain =

n−2 1− n −n n n n v0 KIc Y0 fY π 2 σ˙ −1 σfn+1 + af 2 2(n + 1)

2 − n−2

.

(3.8)

36

3 A Stochastic Fracture Model for Glass

During the estimation of fracture stresses, for example by CRRT, one obtains fracture stresses as the sum of crack-opening stress and residual stress σ0 . Residual stresses may originate from tempering or chemical strengthening. In order to calculate initial flaw sizes through the crack-opening stress only, the residual stresses need to be subtracted from the fracture stresses in Eq. (3.8). If stress values from experiments without a biaxial stress field should be used, the type of stress field also has to be considered. While CRRT generate a biaxial stress field, a uniaxial stress field is present, for example during four-point bending tests. To take this into account, a correction factor χ for the ratio of major and minor principal stress is introduced. This factor was already suggested by [62] by χ = 1 for biaxial stress fields and χ = 0.83 for uniaxial stress fields. The determination of these values is examined in more detail in the following chapter. The factor χ must be adjusted depending on whether the fracture stress used to calculate initial crack lengths originates from CRRT or four-point bending tests. This operation and replacing the critical crack size af by the stress intensity formulation yields the final formulation to calculate initial flaw depths from fracture stresses by 
n+1 n−2 −n n n n ain = v0 KIc Y0 fY π 2 σ˙ −1 σf χ − σ0 + 2(n + 1) 2 (3.9)  1− n2 #− n−2 2 KIc . Y02 fY2 (σf χ − σ0 )2 π The main advantage of this formulation is that the necessary variables are commonly found in literature for most glasses and other brittle materials. As fracture stress σf in Eq. (3.9) a value from an inverse probability distribution can be taken. For the current development state of the stochastic fracture model, only Weibull distributions can be used. This is due to the fact that Weibull distribution can consider size effects by  η2 = η1

A1 A2

1

β

,

(3.10)

with η and β as the Weibull scale and shape parameters and different surface sizes Ai . This is particularly advantageous in the context of FE simulations, where different element sizes are often used. The initial flaw width can be estimated by the initial flaw depth ai , the initial geometry factor Y0 and the inverse function of the NewmanRaju [132] relationship. In order to consider anisotropic strength effects, each crack gets a crack angle φ. Because an-isotropic strength effects have not been investigated within this thesis, a uniformly distributed crack angle between zero and π degrees is taken randomly for each calculated crack. The influence of crack orientation depending on the manufacturing process constitutes an interesting topic for future research.

3.3 Stochastic fracture Model

37

3.3.1.2 Geometry Factor Shift For the purpose of modeling crack growth two-dimensionally by length and width, the evolution of the geometry factor Y also needs to be taken into account. The geometry factor gives a value for the relationship between crack length and width. In this investigation, the formula from Newman and Raju [132] was chosen for the geometry factor. It is widely applied for glass and based on empirical data. In order to calculate fracture stresses back to initial crack geometries, the evolution of crack geometry must be known in advance. The left side of Fig. 3.5 shows four different crack geometries evolving during growth. All four cracks lead to a fracture stress of 200 MPa. All initial crack geometries are tangent to a certain value, which could already be observed when observing the crack growth by means of microscopes of different cracks [163] and numerically [166]. With regard to the crack geometry evolution curve no solution has been found by the author and in literature. The curve depends on the fracture stress and the crack growth exponent. The crack growth exponent serves as an exponent for the geometry factor in Eq. (3.9) and thus has a significant influence on the initial flaw size. A consideration of evolution using the emergent values directly would be preferable in order to avoid uncertainties by fitting. The right picture in Fig. 3.5 shows the evolution of crack growth. While the 1.1 geometry correction factor Y [-]

geometry correction factor Y [-]

1.1

1

0.9

0.8

0.7

0.6

0

1 2 3 4 crack depth a [µm]

5

1

0.9

0.8 Y0 fY 0.7

0.6

0

1 2 3 4 crack depth a [µm]

5

Figure 3.5 Left: Evolution of crack geometry with different initial geometry factors. Note: With a stress rate of 2 MPa/s all cracks lead to a fracture stress of 200 MPa. Right: Evolution of the geometry factor during crack growth with the corresponding correction value fY for the initial geometry factor Y0 .

38

3 A Stochastic Fracture Model for Glass

initial geometry factor is Y0 = 0.69, the factor evolves towards fracture to Yf = 1. Most of the time between initial status and fracture, the geometry factor lies in the first region below Y ≈ 0.9. Because only a constant value can be taken into account in Eq. (3.9), the correction factor fY was introduced. It will modify the initial geometry factor Y0 to a weighted mean value. This serves to ensure that all regions of crack evolution are considered in relation to the associated growth duration. The correction factor fY has been estimated numerically by Eq. (3.9) to calculate an initial flaw depth for a certain fracture stress value. After the initial flaw depth has been estimated, the growth of the flaw was simulated until fracture by the linear approximation of [155]. The calculation of the growth of flaws could already be shown in [167]. The procedure was repeated with different correction factors until the initial stress value and the calculated value matched. The calculated values for the correction factor fY are shown in Appendix C

3.3.2 Fracture Calculation After initialization, the initially calculated cracks grow as a function of the applied stress state at the crack tip over time. The stress at the flaw σf law is calculated for each time step by the plane stress components and the stress angle φ by σf law =

σx + σy σx − σy + cos(2φ) + τxy sin(2φ), 2 2

(3.11)

where σx , σy and τxy are the in-plane components of the stress tensor, calculated by the linear elastic relation between stress and strain:     E  νE 0 2 2 σx x 1−ν 1−ν νE E  σy  =  0  (3.12)  1−ν 2 1−ν 2   y  E τxy 2 0 0 xy 2(1+ν) A representation of the crack geometry, crack orientation and the linear elastic stress components for crack-opening within a shell element is given in Fig. 3.6. It must be taken into account that the stress at the crack tip σf law is the sum of the crack-opening stress σcos and the residual stress σ0 . At this point, residual stresses, for example from thermal treatment or chemical strengthening can be considered during simulations. With the stress at the crack tip σf law and the residual stresses, the crack-opening stress σcos can be calculated to σcos = σf law − σ0 .

(3.13)

By assuming that the crack depth a is much smaller than the glass thickness t, only the surface stress components at the outer integration points during FE simulations are utilized for the crack-opening stress. With the known crack-opening stress σcos and the incremental crack growth da

3.3 Stochastic fracture Model

39

σy

τxy

2c

σx

σx

σy Edge Length

Figure 3.6

2c a

Thickness

τyx

σx z x

σx

φ

τxy τyx

σy

y x

Crack geometry and orientation within FE with corresponding stress components.

from Eq. (3.2), the new crack depth am and the crack width cm at the current time step m can be calculated by   √ Ya σcos,m πam−1 n am = am−1 + dam = am−1 + dtm v0 KIc (3.14)   √ Yc σcos,m πcm−1 n cm = cm−1 + dcm = cm−1 + dtm v0 KIc with dtm as the time that passed between the last and the current time step. The geometry correction factor for the crack depth Ya and the crack width Yc are calculated by the formulation from Newman and Raju [132] by Ya,m = Y (cm−1 , am−1 , t, b, 0.5π) Yc,m = Y (cm−1 , am−1 , t, b, 0), where t is the shell thickness and b the characteristic shell edge length. fracture occurs if the condition √ Ya,m σcos,m πam = KIc

(3.15)

(3.16)

is fulfilled, which is checked at the end of each cycle m.

3.3.3 Post-Fracture Behavior In order to model the crack propagation after initial fracture, an existing fracture model for glass [17] is modified for the combination with the stochastic fracture model. In addition to the original formulation in [17], fracture can only be triggered if a crack exists in the neighboring element or if the stochastic fracture model

40

3 A Stochastic Fracture Model for Glass

detects fracture in the current element. The aim of the combination of both models is to simulate crack propagation after initial fracture while initial fracture and its stochastic scatter are represented by the stochastic fracture model. After the stochastic fracture model has estimates the fracture or when a neighboring element contains a crack in the direction of the current element, an additional fracture stress formulation is activated. The additional formulation is expressed as

σf =

σ˜f , f (lel )

(3.17)

with a geometric function f (lel ), which scales the fracture stress depending on the element edge length lel to s lel ψ f (lel ) = . (3.18) l0 ξ 2 The geometric constants ψ and ξ are used to reduce the influence of the element size on the stress behavior in the surrounding area of a crack tip. These constants are derived in detail in [17]. The value l0 is used for regularization and is constantly set to l0 = 1mm. A detailed justification for this can also be found in the original research. The fracture stress σ˜f from Eq. (3.17) is divided into three parts: One part below the crack growth threshold Kth , one a part above the critical stress intensity KIc and one part between the two by  if σc < σth σth σ˜f = σc if σth ≤ σ˜f ≤ σIc (3.19)  σIc if σc > σIc where σth and σIc are the stress values from the critical stress intensity and crack growth threshold formulations: σth =

Kth KIc and σIc = . √ √ Yalt πaalt Yalt πaalt

(3.20)

The index alt refers to an alternative value for the parameters for the utilized fracture model for crack propagation. This will avoid confusion with the values for the stochastic fracture model. The utilized model for crack propagation holds both the crack length and the geometry factor constant. Within the stochastic fracture model, an evolution of both is simulated. For this reason separate values must be used for both models. The fracture stress between both cut-off conditions in Eq. (3.19) is given as σc = B σ˙ 1/(n+1) .

(3.21)

3.3 Stochastic fracture Model

41

with the parameter B according to [168] by B=

n 2(n + 1)KIc √ n (n−2)/2 v0 (n − 2)(Yalt π) aalt

!1/(1+n) .

(3.22)

fracture is finally triggered, when the first principal stress σ1 reaches σf . At this point, the angle between the principal stress system and the element system αmaj. by   σ1 − σx αmaj. = arctan , (3.23) τxy is stored to determine the crack orientation in the current element. The new crack within the current element is then oriented perpendicularly to the maximum principal stress at fracture. A standard square shell element has a maximum of eight possible neighbors to which the crack can propagate further. Fig. 3.7 shows the partition of a square shell element into the eight possible propagation directions.

42

3 A Stochastic Fracture Model for Glass

5 8π

1

3 8π

2

2 1

1 8π

lel

αmaj.

π 5




8 4

tan

lel

3

−1 8 π

7 5 4 y

6 −5 8 π

−3 8 π

3

x Figure 3.7 to [17].

Octagonal segmentation of crack propagation for eight neighboring elements according

3.4 Model Validation The validation of the stochastic fracture model is based on the initial fracture of glass. The post-fracture behavior, adopted from [17], has already been validated in the original publication. The main influencing factors on a possible calculation of fracture stress distributions by an FE model are the mesh dependency, the influence of the glass surface size and the change in crack growth by different stress rates. First, the mesh dependency is investigated by a so-called constant surface test (CST). This new test method is developed within this thesis. It is a test method in order to check if a stochastic fracture model can reproduce fracture distributions correctly with different element edge lengths. In this test the same surface size is discretized differently for several kinds of simulation. In the following, the influence of the simulated surface size is computed. Singleelement tests with different element sizes are performed. The results are compared to an analytical solution. The same will be done for the influence of different stress rates. Tab. 3.1 shows the selected input parameters for the following simulations besides the fracture distribution. The crack growth parameters n and v0 for the considered

3.4 Model Validation

43

glass in this article have already been determined in an earlier publication [77]. The stress rate σ˙ was chosen according to the guidelines for CRRT [13].

Table 3.1 Necessary input parameters besides the fracture distribution for the fracture model and the chosen values for the validation.

Parameter n v0 KIc Kth σ˙ Y0 χ

Value 15.098 10.22 0.75 0.26 2 [0.663:1.122] 1

Unit mm/s√ MPa√m MPa m MPa/s -

Source [77] [77] [169] [154] [13] [62]

3.4.1 Mesh Dependency To check the mesh dependency, a new test method is proposed. The so-called CST is a new test method to prove if a stochastic fracture model can reproduce fracture distributions without the influence of the element size. An example of a CST is illustrated in Fig. 3.8. The main advantage of this test is that the simulation input distribution parameters do not need to be converted in order to compare them with the numerical values. Different sizes of the total surface would result in different fracture stresses. This is due to the different probability of smaller or larger flaws in smaller or larger surfaces. For all CST in this thesis, a surface of 100 mm2 is equally discretized by 1×1,

Figure 3.8 CST discretized with one (1x1), four (2x2) and sixteen (4x4) elements. Through the same surface size and a constant stress rate, fracture distributions can be determined for a constant surface area with different element edge lengths. The main advantage is the non-existent influence from different surface sizes and their impact on the calculated fracture distribution.

44

3 A Stochastic Fracture Model for Glass

2×2, 4×4 and 10×10 elements. These discretizations correspond to an element edge length of 10, 5, 2.5 and 1 mm. A 2PW distribution and a LTW distribution have been selected as input distribution. For the 2PW distribution, the range of the scale parameter η has been set between 50 MPa and 300 MPa. This covers the usual range of fracture stress values for glass. Here the scale parameter represents the pure crack-opening stress without any residual stresses. The shape parameter β has been set to 5, which is a frequent value for glass. The shape parameter defines the scatter of the fracture stress values obtained by the corresponding distribution. A value of β=5 yields fracture stress values in the whole range between the chosen scale parameters. In order to avoid unrealistically small fracture stresses for stochastic simulations, the Weibull distribution can be extended by a truncation-point τ . The distribution thus obtained is referred to as LTW distribution. In order to additionally prove whether this truncation point is mapped correctly, all simulations with a 2PW distribution are repeated with a truncation point value. The truncation point τ has been set to τ = 0.5η for the associated distribution. All possible distributions have been simulated with all mentioned element edge lengths 250 times each. The results are shown in Tab. 3.2. As illustrated in Tab. 3.2, the given fracture stress distributions can be reproduced with small deviations. The maximum deviation is around 6 % for the scale parameter η and 10 % for the shape parameter β. The deviation decreases with increasing scale parameter. Moreover, only random deviation with decreasing edge length can be observed, which leads to the conclusion that a systematic deviation of the element edge length can be excluded.

3.4.2 Surface Size In order to represent the stochastic fracture behavior of glass correctly, the size effect must also be considered as a kind of regularization. The fracture of glass is dominated by randomly distributed microscopic flaws on the glass surface. As the test area gets larger, the probability of testing a larger and therefore more critical crack increases. Accordingly, a larger test area correlates with a decreasing fracture stress. The change in fracture stress due to a different surface size can be considered analytically via the Weibull shift [32] by  σ2 = σ1

A2 A1

− 1

β

.

(3.24)

To check whether the stochastic fracture model can represent the influence of different surface sizes, one-element tests with different element sizes have been performed. The results are shown in Tab. 3.3. Each sample consists of 250 simulations and has been performed at a constant stress rate of σ= ˙ 2MPa/s. The calculated fracture stresses were evaluated for their statistical distribution using the 2PW distribution.

3.4 Model Validation

45

Table 3.2 Results from the CST simulations. A constant surface of 100 mm2 was discretized by 1x1, 2x2, 4x4 and 10x10 elements, in order to check a possible influence of element edge lengths. Each sample consists of 250 simulations.

discretization edge length [mm] η [MPa] β [-] 2PW distribution, η = 50 MPa, β = 5, τ = 0 MPa: 1x1 10 52.94 5.01 2x2 5 52.34 5.13 4x4 2.5 53.30 4.91 10x10 1 51.29 5.17 LTW distribution, η = 50 MPa, β = 5, τ = 25 MPa: 1x1 10 51.16 5.12 2x2 5 53.16 5.55 4x4 2.5 53.27 4.65 10x10 1 50.72 4.99

τ [-] -

28.15 28.15 26.35 26.53

2PW distribution, η = 100 MPa, β = 5, τ = 0 MPa: 1x1 10 101.77 5.04 2x2 5 102.52 5.29 4x4 2.5 102.16 5.03 10x10 1 107.23 4.99 LTW distribution, η = 100 MPa, β = 5, τ = 50 MPa: 1x1 10 103.84 4.78 51.14 2x2 5 103.60 5.12 51.14 4x4 2.5 102.44 5.14 52.19 10x10 1 99.64 5.31 50.87 2PW distribution, η = 300 MPa, β = 5, τ = 0 MPa: 1x1 10 306.15 5.16 2x2 5 309.65 4.91 4x4 2.5 302.75 5.05 10x10 1 303.12 4.78 LTW distribution, η = 300 MPa, β = 5, τ = 150 MPa: 1x1 10 316.70 5.39 166.01 2x2 5 304.60 4.71 164.01 4x4 2.5 314.08 5.23 161.01 10x10 1 302.49 5.20 156.61

46

3 A Stochastic Fracture Model for Glass

The maximum deviation between the numerical and the analytic shape factor η for the 2PW distribution is estimated to 6.71%. In addition, no systematic error is observed. This leads to the conclusion that the present fracture model can take different surface sizes into account. The total glass surface size is also a piece of an information which is not available for integration points during FE simulations. This problem is solved by distributing the fracture strength statistically on small areas, which together show the distribution of the total glass area in sum.

Table 3.3 Results of single element tests under constant stress rate with different surface sizes and the corresponding analytical solution according to Eq. (3.24). Each sample consists of 250 simulations.

Surface [mm2 ] 25 50 100 200 350 500 650 800 1000

ηanalytical 131.95 114.87 100.00 87.06 77.84 72.48 68.77 65.98 63.10

ηsimulation 135.99 120.32 96.90 92.16 78.94 74.13 71.76 70.72 64.19

∆η [%] 2.97 4.53 -3.20 5.54 1.40 2.23 4.16 6.71 1.70

3.4.3 Stress Rate A comparison of fracture stresses from different stress rates can be made on the basis of two assumptions: First, it is assumed that the power law for subcritical crack growth from Eq. (3.1) is valid over the whole subcritical crack growth range. The second assumption states that when both samples had the same initial flaw lengths. The relationship between two fracture stress values σf,1/2 and their corresponding stress rates σ˙ 1/2 can then be expressed by 

σf 1 σf 2



 =

σ˙1 σ˙2

1  n+1

,

(3.25)

where n is the crack growth exponent from Eq. (3.1). For a detailed derivation see, for example, [112]. If it is additionally assumed that the tested samples have the same population of initial flaws, fracture stress distributions can also be compared with Eq. (3.25). This can be done by substituting the fracture stress values σf,1/2 , for example with the Weibull scale parameter η1/2 and the stress rate for both fracture distributions. For the stress rate influence, single element tests with different constant stress rates are performed with stress rates. The stress rates chosen are between 0.02 and 200

3.4 Model Validation

47

MPa/s. For each stress rate, 250 simulations are performed. The resultant fracture stresses are examined for their statistical distribution by means of the 2PW distribution. The estimated shape parameter η was compared to the analytical solution from Eq. (3.25). Tab. 3.4 shows the resulting values. These are also illustrated in Fig. 3.9. It can be seen that the simulated shape parameter ηsim. is in the range of the analytical solution ηana. . At a stress rate of approximately 1,000,000 MPa/s, stagnation in the growth of fracture stress can be observed. The explanation for this is the inert strength of glass. The inert strength is the stress reached without subcritical crack growth. This mechanism is observed when the stress rate attains a magnitude where the critical stress intensity is reached without enough time for subcritical crack growth. For example, a fracture stress of 100 MPa at a rate of 1 TPa/s is reached in 0.1 ms. Even with very small time steps, no noticeable crack growth occurs.

Table 3.4 Comparison of numerical results to analytical solution of the stress rate influence on the fracture stress distribution. For each stress rate, 250 simulations of one FE were performed.

σ˙ [MPa/s] 0.02 0.06 0.2 0.6 2 6 20 60 200 600 2,000 6,000 20,000 60,000 200,000 600,000 2,000,000 6,000,000 20,000,000

ηsim. [MPa] 77.77 80.45 86.52 92.10 100.92 106.64 113.02 125.39 133.03 149.24 157.33 172.44 179.80 193.93 199.33 211.96 218.96 218.04 216.19

ηana. [MPa] 79.63 80.43 86.67 92.79 100.00 107.06 115.38 123.53 133.12 142.52 153.59 164.44 177.21 189.72 204.46 218.90 235.89 252.56 272.17

∆η [%] -3.52 -0.03 0.18 0.75 -0.92 0.39 2.05 -1.51 0.07 -4.71 -2.44 -4.87 -1.46 -2.22 2.51 3.17 7.18 13.67 20.57

scale parameter [MPa]

48

3 A Stochastic Fracture Model for Glass

220 190 160 130

numerical results analytical solution

100 70 10−2

10−1

100

101

102 103 104 stress rate [MPa/s]

105

106

107

108

Figure 3.9 Graphical illustration of the stress rate influence results from Tab. 3.4. Each point represents the Weibull scale parameter of 250 simulations with constant stress rate.

4 Mechanical Parameter Quantification 4.1 Introduction Quantitative studies of parameters describing the mechanical behavior of glass are important for several reasons. For one thing, there is the influence of the chemical composition of the glass which can exert a considerable influence on the mechanical behavior [86]. Often, mechanical parameters are found in publications in which the chemical composition of the investigated glass is not stated. Therefore, the parameters are subject to unknown uncertainties. Furthermore, there are different approaches to determining parameters. A pertinent example is the terminal subcritical crack velocity. Several methods of determining the terminal velocity can be found. such as in-situ observation [164], analytical solutions [155, 170] or comparsion of the two [77]. Due to the mentioned influences, mechanical parameters and their scatter should be investigated for simulations of each glass type. Small scatter result in large deviations, which can lead to major miscalculations and false predictions. In the following chapter, an overview of the input parameters for the stochastic fracture model is provided. Necessary parameters are determined experimentally and compared with existing values found in literature. Special attention is paid to the subcritical crack growth parameters. These are measured under different temperature and humidity conditions in this thesis. This demonstrates the influence of environmental conditions. All necessary parameters for the stochastic fracture model besides the fracture stress distribution are shown in Tab. 4.1. The investigated glass is soda-lime-silica glass. The chemical composition is determined by inductively coupled plasma optical emission spectrometry. The most common molecules in glass and their respective weight percentages wt% are listed in Tab. 4.2.

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_4

49

50

Table 4.1 tion.

4 Mechanical Parameter Quantification

Necessary parameters for the stochastic fracture model besides a fracture stress distribu-

symbol E υ KIc Kth n v0 χ Y

parameter Young’s modulus Poisson’s ratio fracture toughness subcritical crack growth threshold subcritical crack growth exponent terminal velocity for subcritical crack growth stress field factor geometry correction factor

Table 4.2 Chemical composition of the soda-lime-silicate float glass of the automotive windscreens examined in this thesis.

wt%

SiO2 70.02

Na2 O 14.04

CaO 9.49

MgO 3.66

Al2 O3 K2 1.34 0.58

Fe2 O3 SO3 0.535 0.266

TiO2 0.021

4.2 Fundamental Mechanical Parameters 4.2.1 Young’s modulus For mechanical applications, glass exhibits an almost linear elastic material behavior. Only under extreme load cases, such as the Vickers indentation test, can plastic deformation through densification be observed. A Young’s modulus E = 70 GPa is probably the most commonly used value for soda-lime-silicate glass and has already found its way into various standards, such as DIN EN 572-1. Under real conditions the chemical composition of soda-lime-silica glass varies depending on the manufacturer. This may cause a significant change in the Young’s modulus [86]. An increase in the Young’s modulus to 74 GPa already entails an increase of strain energy of nearly 6 % under the same deformation. By characterizing the fracture of glass by mechanical stress a possible error of this quantity has to be taken into account. An overview of common glass types and their Young’s modulus found in literature is summarized in Tab. 4.3. For the windscreens used in this thesis, uncurved glass samples from the identical glass type are made. These are tested by three-point bending tests for determining the Young’s modulus. Six tests are performed with a specimen thickness of 1.8 mm, a specimen width of 34 mm and a support width of 180 mm. The Young’s modulus is determined to a maximum value of Emax = 71.86 GP a and a minimum value of Emin = 69.81 GP a. Within this thesis and in the context of automotive windscreens, the mean value of all six tests, Emean = 70.72 GP a, is selected for simulations.

4.2 Fundamental Mechanical Parameters

Table 4.3

51

A selection of glass types and their Young’s modulus from literature.

glass type soda-lime-silicate

borosilicate glass alumina-silicate

E [GPa] 70.72 68.5 70 70 72-74 63 59-62.3 74

reference present thesis [171] [37] DIN EN 572-1 [172] [37] [171] [37]

4.2.2 Poisson’s Ratio Besides the Young’s modulus, the Poisson’s ratio υ is necessary to calculate the linear elastic relationship between stress and strain. The isotropic Poisson’s ratio is given by the transversal elongation divided by the axial compression. Similar to the Young’s modulus, the Poisson’s ratio also scatters depending on the chemical composition. Tab. 4.4 provides an overview of values for the Poisson’s ratio found in literature. For the influence of the Poisson’s ratio, DIN EN 1288-1 indicates that an error of 1.6 % is made within CRRT when using υ = 0.2 with an actual value of υ = 0.23. The deviation that is made when using an incorrect Poisson’s ratio can be easily calculated using the stress strain relationship from Eq. (3.12). [173] calculated the Poisson’s ratio for 30 glasses with different chemical compositions. This gives an overview of the influence of each molecule on the Poisson’s ratio. The manufacturer of the considered soda-lime-silica glass states that the Poisson’s ratio is υ = 0.23 in an unpublished material sheet. The Poisson’s ratio of υ = 0.23 is in a realistic range and is used for further simulations within this study.

Table 4.4

A selection of glass types and their Poisson’s ratio from literature.

glass type 100% SiO2 soda-lime-silica

borosilicate glass alumina-silicate

υ [-] 0.162 0.2 0.23 0.2-0.24 0.2 0.215

reference [173] DIN-EN 572-1 DIN-EN 1288-1 [173] [37] [37]

52

4 Mechanical Parameter Quantification

4.3 Subcritical Crack Growth Parameters The parameters n and v0 define the subcritical crack growth as a function of the stress intensity KI according to Eq. (3.1). A large scatter of both parameters is found in literature [77]. This may be due to different experimental methods, different chemical compositions of the investigated glass type or unconsidered influence from residual stress. The influence of the temperature on subcritical crack growth has already been shown in [158]. This influence has not been investigated yet for the frequently used linear approximation for subcritical crack growth. In the following section an investigates on the subcritical crack growth parameters as a function of environmental conditions is conducted. For this purpose, glass specimens are tested within a CRRT inside a climate chamber. The aim of this investigation is to draw conclusions about the environmental influence on subcritical crack growth. The investigation of the influence of environmental conditions on subcritical crack growth parameters has already published in [77]. Subsequently, an overview of subcritical crack growth parameters for soda-lime-silica float glass depending on the test environment and the test method found in literature is given.

4.3.1 Experimental Part 4.3.1.1 Specimen Preparation The tested glass is soda-lime-silica float glass, originally intended for use as automotive windscreen by Pilkington for the model Audi A3 in 2017. Glass plates with the dimension of 1480×1000×1.8 mm are cut out into circular samples with a diameter of 80 mm. A total of 390 samples are tested. All specimens are heated up to 520◦ C and cooled down to room temperature by a tempering furnace with a maximum of 2 K/min to remove residual stresses. Residual stresses before tempering are determined by a scattered light polariscope up to 8 MPa. In the following, all specimens are pre-damaged by means of the Vickers indentation test with an indentation force of 9.8 N and a holding time of 3 s to obtain nearly identical crack systems in all specimens. All indentations are examined for symmetry of the cracks. Specimens that did not develop four cracks perpendicular to each other are rejected. Identical flaws are necessary in order to obtain the same strength population without the need for a certain sample size. After indentation, all specimens are heated up to 520◦ C again and cooled down to room temperature by a maximum of 2 K/min to remove residual stresses from indentation. These are caused by densification during the indentation process. The influence of residual stresses generated by Vickers indentation is known [174]. In preliminary tests, it is also observed that the scatter of fracture stresses after Vickers indentation could be significantly reduced by heat treatment before and

4.3 Subcritical Crack Growth Parameters

53

centering ball

specimen

load ring

support ring

one of four temperature and humidity sensors

Figure 4.1

CRRT setup within the climate chamber.

after indentation. The absence of residual stresses is verified by a scattered light polariscope SCALP-5. 4.3.1.2 Dynamic Fatigue Experiments In order to determine subcritical crack growth parameters as a function of environmental conditions, dynamic fatigue tests are performed. These are carried out at several constant stress rates at a constant temperature and humidity within a climate chamber. As test setup, a CRRT is chosen as shown in Fig. 4.1. Tests are performed at 15, 25 and 35◦ C at 30, 40, 50, 60 and 70 % relative humidity for each temperature. The stress rates σ˙ = 0.6, 2, 6 and 20 MPas−1 are performed with six experiments per stress rate. The CRRT consist of a nearly biaxial plane stress field. 0n0 and the associTo calculate the crack growth parameters, the inert strength σin ated initial crack depth ain are necessary. For this purpose, specimens are sealed with silicone oil to prevent a reaction between the crack tip and water during test-

54

4 Mechanical Parameter Quantification

ing. The specimens are tested in CRRT with a stress rate of 450 MPas−1 . The inert strength of the samples with Vickers indentation is determined as the arith0n0 = 80.62 MPa with a standard deviation of s = metic mean of 20 samples to σin 2.24 MPa. One of the obtained fracture surfaces from testing sealed specimens at high stress rates is shown in Fig. 4.2. The initial crack depth is determined from these images to ain = 54.86 µm. Analogously, the inert √ crack depth can be calculated with the fracture toughness KIc = 0.75 MPa m and the geometry correction factor Y = 0.72 by KIc √ 0 0n σin Y π

ain =

!2

(4.1)

to ain = 53.14 µm. The geometry correction factor Y for Vickers indentation is obtained by measuring crack growth in-situ with a microscope [77]. The morphology of cracks in glass introduced by Vickers indentation is shown in [175]. Previous studies have shown that Vickers indentation creates a half-pennyshaped crack [101, 120].In these investigations, the crack was inserted into the sample by an indentation force of 90 N, and no information was given on the existing residual stresses before the indentation in [120]. [176] shows that at an indentation force of 9.8 N with no residual stress before indentation, no half-pennyshaped crack system is observable. It is assumed that the median crack and the lateral cracks connect to a half-penny-shape crack system at higher indentation loads. It could also be shown in [101], that below 10 N indentation load, no radial cracks are observable. These findings are in accordance with Fig. 4.2, in which radial cracks cannot be observed either.

4.3.2 Evaluation 4.3.2.1 Parameter Estimation The relationship between fracture stress and the applied stress rate is already shown in Eq. (3.25). This equation will be expanded by a further parameter λ0 as the ordinate intercept of the double-logarithmic stress versus stress rate diagram to 

σf 1 σf 2



  1 σ˙ 1 n+1 = λ0 . σ˙ 2

(4.2)

The crack growth exponent n is thus estimated by the relationship of fracture stress and stress rate [155]. The crack growth exponent n can be determined directly in a double-logarithmic representation of Eq. (4.2). One example of a double-logarithmic fracture stress versus stress rate plot is shown in Fig. 4.3. The condition for this is that the initial crack lengths of all samples are identical or

4.3 Subcritical Crack Growth Parameters

55

densification zone indentation

lateral crack median crack

Figure 4.2 Scanning electron microscope image of a Vickers indentation side after testing with 450 MPas−1 and silicone sealing of the crack tip.

have the same population. Vickers indentation significantly reduced the scatter of initial crack lengths. Nevertheless, even with Vickers-induced cracks, a certain scatter of initial flaw sizes remains. The second parameter v0 is determined by the ordinate intercept λ0 of the doublelogarithmic fracture stress versus stress rate plot. The initial flaw size ain and the 0n0 for the current flaw sizes are also necessary as corresponding inert strength σin well. The terminal crack velocity v0 can be calculated by 0

v0 =

0n c 2σin in . λ0 (n + 1)(n − 2)

(4.3)

For a detailed derivation of Eq. (4.3), the reader is referred to [170]. The estimation of the subcritical crack growth parameters by Eq. (4.2) is only valid, if the cracks grow by a certain amount during testing. The relation between initial flaw size ai and critical flaw size af should not exceed 

ai af

 n−2 2 < 0.01.

(4.4)

This is particularly important for experiments with low ambient humidity and high stress rates. [112] shows that if the relationship between fracture stress and

ln(σf ) ln[MPa]

56

4 Mechanical Parameter Quantification

4.4 4.2 4 −1

0

1 ln(σ) ˙

2

3

ln[MPas−1 ]

Figure 4.3 Logarithmic plot of the measured fracture stresses vs stress rate at 25◦ C and 50 % relative humidity. Linear fit with R2 = 0.98.

stress rate in a double-logarithmic plot is linear, the use of Eq. (4.2) is justified as well. For a relative humidity of 30 % and 40 % at 15◦ C, no clear linearity within the double-logarithmic plot can be observed. The coefficient of correlation for the linear fit is below R2 < 0.9. Due to the poor correlation the subcritical crack growth parameters for a relative humidity of 30 % and 40 % at 15◦ C are not used any further. 4.3.2.2 Evaluation of Environmental Influences The determined crack growth parameters from dynamic fatigue tests are shown in Fig. 4.4 and Tab. A.2. As expected, the resulting subcritical crack growth parameters increase with increasing temperature and humidity. Higher temperatures and thus a higher energy level within the water molecules at the crack tip leads to an accelerated reaction with the Si-O-Si molecule chains of the glass. In order to make a general statement about the influence of the environmental conditions on the crack growth parameters, both parameters are fitted as a function of humidity. The crack growth exponent n is fitted by a 2nd order polynomial to n(25◦ C) = −0.001564H 2 + 0.06987H + 14.78 n(35◦ C) = −0.001622H 2 + 0.05754H + 14.19,

(4.5)

with a coefficient of determination of R2 = 0.96 and R2 = 0.97. The relation between the terminal crack growth velocity v0 and the relative humidity is expressed by v0 (25◦ C) = 0.002236H 2 − 0.359H + 7.103 v0 (35◦ C) = 0.004236H 2 − 0.2703H + 10.1,

(4.6)

with a coefficient of determination of R2 = 0.97 and R2 = 0.99. A fit for the crack growth parameters at 15◦ C is omitted because there were only three measured val-

4.3 Subcritical Crack Growth Parameters

57

ues. In a general comparison of the measured values from 15◦ C to 35◦ C a comparable shape of all curves is observed for n and v0 . The polynomials are intended to show the general behavior of both crack growth parameters as a function of environmental conditions. In the following, the estimated parameters are compared to literature values, summarized in Appendix A. Only values with information on the environment during estimation and the test method are used. A comparsion of the subcritical crack growth parameters estimated within this thesis with values from literature yields the recognition, that the values determined here for the parameter n are in good agreement. The terminal crack velocity for subcritical crack growth v0 is often higher than the values commonly found in literature, yet it is still in a realistic range. The values for v0 in [177] are v0 = 4.51 mm/s for a summer environment and v0 = 8.22 mm/s for winter conditions. The maximum value of v0 found in literature with the information on the environmental conditions during testing which was not performed in water is v0 = 14.3 mm/s at 45 %rh in [176]. [164] compared soda-lime-silica glass (SiO2 = 72.3 %wt) and sodium-aluminosilicate glass. A terminal crack growth velocity of v0 = 2.6 mm/s could be estimated for soda-lime-silica glass and v0 = 21.8 mm/s for sodium-aluminosilicate glass. This deviation supports the assumption that the scatter in subcritical crack growth parameters is caused by the chemical composition. Within [164], the fracture stress of pre-damaged specimens is around 15 MPa higher than in this investigation. Both studies used the same Vickers indentation test configuration. This deviation is probably due to the fact that the residual stress field after indentation is eliminated by tempering [174]. Because the chemical composition obviously has a significant influence on the subcritical crack growth parameters, the chemical composition should always be estimated when determining these parameters. The influence of the chemical components on the subcritical crack growth has not been investigated yet and constitutes an interesting topic for further research. Subcritical crack growth velocity can significantly influence the fracture strength and thus the strength of glass could possibly be changed by manipulating the chemical composition.

4.3.3 Discussion The influence of humidity and temperature on the subcritical crack growth parameters n and v0 of the linear approximation proposed by [155] are determined. The subcritical crack growth parameters depending on the surrounding humidity could be mapped by a 2nd order polynomial. It is observable that the crack growth velocity increases faster above 50 %rh than in the area below 50 %rh. The importance of the environmental influence is demonstrated by the fact that, for example, the terminal crack velocity v0 increases by a factor of two between a humidity of

58

4 Mechanical Parameter Quantification

30 %rh and 70 %rh at a temperature of 35◦ C. While the parameter n is in good agreement with the values found in literature, the parameter v0 determined within this thesis is generally higher than the values available in literature. One possible explanation for this difference is that often Vickers indented specimens are used with residual stresses due to indentation. This is contradicted by the fact that another study took this into account and obtained lower values as well [164]. Comparing the determined parameters with the values found in literature yields the recognition that a certain part of the scatter of the literature values can be traced back to a wide range of environmental conditions. Nevertheless, some unexplainable variations of subcritical crack growth parameters in literature remain. This may be possible due to the chemical composition of glass. [164] also measured subcritical crack growth parameters for soda-lime-silica glass and stated the chemical composition. This differs slightly from the glass used in this investigation. Still, a factor of five is between both terminal crack growth velocities v0 while the crack growth exponent n is in good agreement with values from literature. Another possible influence may originate from a wrong initial flaw size within this study, which directly influences v0 . Although the initial crack length was measured by a scanning electron microscope and compared to in-situ values in [77], there is still the possibility of an unknown mistake. Because the values calculated within this thesis are in the expected range from literature, these are also utilized for further simulations. These are n = 15.098 and v0 = 10.22 mm/s at 25◦ C and a humidity of 40 %. The investigated glass also has the same chemical composition as the simulated windscreen glass.

4.3 Subcritical Crack Growth Parameters

59

T = 25◦ C n [-]

20 15 10 25

v0 [mm/s]

25

30

35

40

45

50

55

60

65

70

75

40

45

50

55

60

65

70

75

T = 25◦ C

20 15 10 5 25

30

35

T = 35◦ C n [-]

20 15 10 25

v0 [mm/s]

25

30

35

40

45 50 55 relative Humidity [%]

60

65

70

75

40

45 50 55 relative Humidity [%]

60

65

70

75

T = 35◦ C

20 15 10 5 25

30

35

Figure 4.4 Measured values for subcritical crack growth parameters n (left) and v0 (right) at different temperatures as a function of humidity. Solid and dotted lines are 2nd order polynomial fits and 95 % confidential intervals.

60

4 Mechanical Parameter Quantification

4.4 Fracture Toughness The fracture toughness KIc , also called critical stress intensity, defines the point at which the combination of stress and crack length triggers critical fracture. During the growth of a crack, atomic bonds of the considered body are separated to create new fracture surfaces. The fracture toughness defines the point where the stored strain energy is higher than the energy needed for subcritical crack growth. After reaching the fracture toughness, cracks accelerate rapidly towards their maximum velocity. When reaching the maximum velocity, the crack cannot increase the rate of conversion from strain energy to new fracture surfaces and will branch out into two new crack fronts. Tab. 4.5 gives an overview of fracture toughness values for soda-lime-silica glass √found in literature. Within this study, a stress intensity value of KIc = 0.75 MPa m is chosen. Although there are only small variations in the values for the fracture toughness found in literature, the influence of this scatter on results from the stochastic fracture model cannot be quantified yet. A study on the influence of the fracture toughness on numerical results achieved by the present stochastic fracture model is recommended for future investigations.

Table 4.5

A selection of glass types and their fracture toughness from literature.

√ KIc [MPa m] 0.75 0.74-0.75 0.75 0.74-0.84 0.70-0.72 0.75 (recommendation)

reference [178] [179] [120] [123] [122] [62]

4.5 Stress Field Factor The stochastic fracture model calculates initial flaw sizes out of fracture stress values during initialization. The derivation of the model is based on the assumption that the given fracture stress values are estimated by experiments with a biaxial stress field and pure mode-I crack opening. While within a biaxial stress field all cracks have the same crack-opening stress regardless of their orientation, a uniaxial stress field yields a crack-opening stress depending on the crack orientation. In some cases it may be necessary to use fracture stress data which originate from experiments with a uniaxial stress field, for example by four-point-bending tests. In order to compare fracture stress values from biaxial and uniaxial stress fields, a correction of the input strength data has to be made. [62] proposes a stress field correction factor χ. The author recommends χ = 1 for biaxial and χ = 0.83 for

4.6 Crack Growth Threshold

61

Table 4.6 Results for the Weibull scale factor η from 250 simulations and the deviation from the input parameter. Input values are a 2PW distribution by η = 100 and β = 5 and different stress field factors.

field factor [-] 1.00 0.90 0.84 0.83 0.82 0.80 0.70

resultant η [MPa] 121.47 109.11 102.95 102.21 99.70 96.37 85.82

deviation ∆η [%] 17.68 8.35 2.87 2.16 0.3 3.77 16.52

uniaxial stress fields. To quantify the influence of input stress values for the model from uniaxial testing, 250 simulations with a uniaxial stress field are performed. As the input a 2PW distribution with the parameter η = 100 and β = 5 is given. It is assumed that these values are estimated by a four-point-bending test. With these values, four-point-bending simulations are performed using different stress field factors. With the correct value of the factor, the resulting distribution of the simulations should be identical to the input distribution. The resultant scale parameter η from the simulation and the deviation from the original value to η = 100 for each scale factor is shown in Tab. 4.6. The recommendation from [62] with χ = 0.83 seems to be correct in order to take into account input stress values originating from uniaxial experiments. Although a value of χ = 0.82 provides a lower deviation, χ = 0.83 is recommended. This is justified by the fact that during the validation of the model in the previous chapter, fracture stresses were always overestimated to a small extent.

4.6 Crack Growth Threshold The crack growth threshold Kth defines the lower boundary at which crack growth occurs. As the barrier for subcritical crack growth it also sets the minimum fracture strength in connection with the corresponding initial flaw depth. While the crack growth threshold has a significant influence on long-term applications, it has only a small relevance for dynamic load cases. A detailed explanation of physical processes for the crack growth threshold can be found in [160]. When using the present fracture model with an initial flaw length for a fracture stress of 100 MPa and a stress rate of 100 MPa/s, the difference √ in fracture stress m and values between a crack growth threshold of K = 0.25 MPa th √ √ Kth = 0.2 MPa m reaches 0.014 %. The error made between K = 0.25 MPa m and Kth th √ = 0.3 MPa m is 0.25 %. Because the influence of the crack growth threshold decreases with an increasing stress rate, the influence of the crack growth threshold

62

Table 4.7

4 Mechanical Parameter Quantification

Literature review of crack growth threshold Kt h values for soda-lime-silica glass.

Kth 0.2 0.27 0.21 0.25 0.25 0.14-0.22 0.26 0.37

environment water, 25◦ C deionized water deionized water recommendation 0.2%rH water 50 %rH

reference [180] [165] [62] [181] [182] [154] [154]

is negligible for the following dynamic impact simulations. Tab. 4.7 shows values available in literature. For the following simulations, the √ recommendation of [62] is followed. A crack growth threshold of Kth = 0.25 MPa m is recommended. If the stochastic fracture model is used for long-term applications, this value should be critically reviewed.

4.7 Geometry Factor The original formulation for the stress intensity K is made for a crack in an infinite body. As cracks in glass mostly occur on the surface, a correction to the original formulation has to be made by the geometry factor Y . Several solutions can be found for the geometry factor depending on the crack width to crack depth relation and as well as the thickness and width of the surrounding component. An overview of possible flaw geometries is shown in Fig. 4.5. For this investigation the solution according to [132] is utilized. In this investigation, a solution based on empirical data for the geometry factor is presented. This solution is valid for 0 < a/c < 1, 0 5 a/t < 1 and c/b < 0.5, where a is the crack depth, c the crack width, t the thickness and b the width of the surrounding specimen. While [132] only fitted their equation to data with 0.2 < a/c < 1, this range is also used for this investigation. The geometry factor is only valid between the glass surface φ = 0 and the crack depth φ = π, assuming that the crack has a half-penny shaped geometry. The complete formula for the geometry factor according to [132] is shown in Sec. 2.2.3.

4.7 Geometry Factor

63

coplanar parallel crack

coplanar parallel crack linked with natural flaw

coarse zipper crack

V machining crack

zipper crack

coarse grinding parallel crack

Figure 4.5 Schematic representations of different parallel machining cracks within a fracture mirror from [100].

The evolution of the geometry factor during crack growth has already been discussed in Sec. 3.3.1.2. The problem is that only very little is known about the original state of crack geometries in glass. Depending on the origin of the crack, it can have a wide range of depth to width ratios. Fig. 4.5 shows an excerpt of possible crack geometries which are found on newly-formed glass surfaces after fracture. An overview of values for the geometry factor found in literature is shown in Tab. 4.8. Due to the lack of information on the initial crack geometry depending on the flaw origin, a uniform distribution of the initial geometry factor at the crack depth (φ = 0.5π) of Y = 0.6625 to Y = 1.06 is assumed within the following simulations. The range of Y = 0.6625 to Y = 1.06 states the maximum valid range of the conditions for the crack geometry in the solution according to [132].

64

4 Mechanical Parameter Quantification

Table 4.8 Overview of determined values for the geometry factor of surface flaws on glass. Most data have been taken from [62].

type of flaw glass on glass scratching Vickers indentation Vickers indentation half-penny in a semi-finite specimen half-penny on a flexure specimen quarter-circle crack on glass edges sandpaper scratching long, straight-fronted plane edge crack

geometry factor 0.564 0.666 0.72 0.637-0.663 0.713 0.722 0.999 1.120

reference [183, 184] [183, 184] [77] [185] [183, 184] [186]

4.8 Summary All necessary input parameters for the stochastic fracture model are estimated or obtained from literature. The estimation of the fracture strength distribution for an automotive windscreen will be the topic of the next chapter. The Young’s modulus E is determined experimentally to E = 70.72 GPa as the mean value of six three-point-bending tests. A value of E = 70 GPa is often used for soda-lime-silica glass and has already found its way into national and international standards. This allows E = 70.72 GPa to be classified as a realistic value. The Poisson’s ratio υ and the fracture toughness KIc are identified through literature values. After discussing the values found in literature, the Poisson’s ratio √ of υ = 0.23 and the fracture toughness of KIc = 0.75MPa m are selected for the considered soda-lime silica glass. Only minor deviations in the literature values are noted. Caution should be taken when using literature values, because the chemical composition of the considered glass plays a significant role for most material parameters. The subcritical crack growth parameters n and v0 are determined experimentally depending on the environmental influence within a climate chamber. This study has already been published in [77]. It is observed that the crack growth parameters could be described as a function of the humidity at the crack tip by a 2t extnd order polynomial. In addition, large deviations between subcritical crack growth parameters found in literature are observed. This may be explainable by different chemical compositions and test methods. A comparison of subcritical crack growth parameters found in literature and their chemical composition confirm this assumption. Within this investigation, the values n = 15.098 and v0 = 10.22 mm/s are used. These were estimated at a temperature of 25◦ C and a humidity of 40 %rh, which are comparable to conditions in unconditioned laboratories and during head impact tests. The crack growth threshold Kth , which defines the lower boundary for subcritical crack growth is only found with a certain scatter in literature. These values range

4.8 Summary

Table 4.9

symbol E υ KIc Kth n v0 χ Y

65

Necessary parameters for the stochastic fracture model besides fracture stress distribution.

value 70 0.23 0.75 0.25 15.098 10.22

unit GPa √ MPa√m MPa m mm/s

1 or 0.83 0.6625-1.06

-

parameter Young’s modulus Poisson’s ratio fracture toughness subcritical crack growth threshold subcritical crack growth exponent terminal velocity for subcritical crack growth stress field factor geometry correction factor

between Kth = 0.14 and Kth = 0.37, estimated outside of water. This thesis follows the recommendation of [62] with Kth = 0.25. For dynamic simulations the crack growth threshold is shown to be negligible by simulations, because the mistake made by varying the crack growth threshold in a certain limit was below 1 %. The stress field factor χ takes into account the origin of given fracture stress values. The fracture stress of glass depends on the state of stress because a biaxial stress field applies more stress to all flaws than a uniaxial stress field. As a recommendation from literature, which is verified by simulations, a factor of χ = 1 for stress values from biaxial testing and a factor of χ = 0.83 for stress values from uniaxial testing should be used. Frequently utilized test methods for glass are CRRT for biaxial stress fields and four-point-bending tests for uniaxial stress fields. The geometry factor Y takes finite-specimen and surface flaw geometries into account regarding the original stress intensity formulation. Although the development of the crack geometry during growth can be mapped, information on the original condition can be found only rarely. The equation used for the geometry factor in [132] gives a defined range of Y = 0.6625 to Y = 1.06 which is used for this investigation by a uniform distribution.

5 Stochastic Strength of an Automotive Windscreen 5.1 Introduction The strength of glass is defined by microscopic flaws. These flaws are significantly influenced in geometry and frequency by the manufacturing and handling of the glass. It is therefore a logical conclusion that with regard to different areas on automotive windscreens, different populations must be assumed for the fracture strength. Such different populations may originate from the silkscreen process, the cover of the PVB layer or the glass edge processing. Furthermore, the tin and air side from the float process is a non-negligible influence. In the following chapter, the different strength populations of an automotive windscreen are investigated. Parts of this investigation have already been published in [78]. The estimation of the windscreen strength samples is divided into two parts: the surface strength and the edge strength. The surface strength is estimated by CRRT. For this purpose, samples were extracted from windscreens by means of water jet cutting. The existing curvature of the windscreens and thus also the obtained specimens are investigated. The edge strength is estimated by four-point-bending tests. Special attention is drawn to the imprinted area up to the edge. This area ranges partly to the edge and influences the edge strength. To take this into account, all specimens are examined for their fracture origin. After all strength values have been determined and the associated distributions have been estimated, a comparison to fractographic fracture stress values is made. Fractographic stress values can be obtained by measuring the fracture mirror. This characteristic part of a fracture pattern can be found at the fracture origin. The size and geometry of the fracture mirror depend on the applied stress and stress state at fracture. Parts of the research presented in this chapter were not conducted by the author of this thesis. These parts were carried out by Steffen Müller-Braun at the Institut für Statik und Konstruktion at the TU Darmstadt, supervised by Prof. J. Schneider. This work has already been published in [78]. The work not done by the author of this thesis in the current chapter includes: • 314 of 612 CRRT • estimating the difference in principal stresses during CRRT • 62 of 62 four-point-bending tests © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_5

67

68

5 Stochastic Strength of an Automotive Windscreen

• • • •

estimation of specimen extraction points specimen extraction from windscreens separating approximately 50 % of the laminated glass specimens measuring of residual stress at the windscreen edge

5.2 Lower Bound for Glass Strength All four methods for estimating the parameters of a LTW distribution introduced in Sec. 2.3.2 are compared. The methods are the LS, the Weibull plane, the MLE and the MMLE method. Special attention is given to the determination of the truncation point for the LTW distribution. In the process of estimating the lower bound for glass, special care must be taken when determining this limit [45]. An overestimation would have devastating effects, especially for structural applications. While the lower bound is known within a sample of the size n → ∞, in-use samples are usually of discussable size. This is usually due to the high experimental effort, the associated experimental costs, or simply the small number of available specimens. The strength data from the working group CEN/TC129/WG8 of European Committee for Standardization (CEN) 1 is utilized [42, 45, 187]. The strength data was determined by samples from different European glass manufacturing plants. The samples were approximately 6 mm thick and were tested by means of CRRT [13]. This experimental program formed the basis for the evaluation of the characteristic value of glass strength indicated in European standards and guidelines. The data are randomly shuffled once because the provided data are sorted with regard to the fracture strength. Sample sizes of n = 20, 40, 60 and the whole sample of n = 741 values are used for parameter estimation. The main purpose of this procedure is to estimate the goodness of the truncation point estimation with regard to small sample sizes. This is especially important for the windscreens investigated in this thesis. The sample size of the windscreen strength distributions is limited by the experimental effort. An overestimation of the truncation point would lead to an overestimation of the windscreen fracture strength. The predicted probability of injury during a pedestrian head impact would be unrealistically high. The estimated parameters of the LTW distribution with regard to the sample size are shown in Tab. 5.1. No distinction between tin and air side is made. Comparing the truncation point τ in Tab. 5.1 with regard to the sample size, it is clearly evident that the LS, graphical and MLE methods overestimate the truncation point. The smallest strength value according to the sample size is xmin (n = 20) = 45.22 MPa, xmin (n = 40) = xmin (n = 60) = 44.07 MPa and xmin (n = 741) = 28.22 MPa. The minimum value within the whole sample is obtained at test number 158. The minimum measured strength before test number 158 is 44.07 MPa. By a sample size of up to 157 values, the truncation point 1 The

data was kindly provided by G.R. Carfagni and G. Pisano

5.3 Differentiation of Strength Populations

69

Table 5.1 Comparison of LTW parameters estimated by data from the working group CEN/TC129/WG8 of CEN. Sample sizes of n = 20, 40, 60 and the whole sample of n = 741 values are used. The LS, the Weibull plane, the MLE and the MMLE methods are utilized.

n = 20 LS Graph MLE MMLE

η 67.30 69.33 70.96 77.60

β 2.68 2.96 3.30 4.50

τ 45.78 45.22 45.22 22.61

n = 40 LS Graph MLE MMLE

η 75.29 75.97 72.12 82.87

β 3.43 3.37 3.12 3.97

τ 44.97 44.07 44.07 22.04

n = 60 LS Graph MLE MMLE

η 69.48 73.32 72.12 78.90

β 3.09 3.37 3.12 4.10

τ 46.87 44.07 44.07 22.04

n = 741 LS Graph MLE MMLE

η 73.50 82.01 82.13 82.87

β 2.82 4.46 3.81 3.97

τ 45.66 28.22 28.22 14.11

would be overestimated by a factor of nearly two by the LS, graphical and MLE method. The MMLE method proposed by [47], estimated a truncation point that is closer to the minimum measured strength within the complete sample. Within this thesis the MMLE method is utilized to estimate the parameters for the LTW distribution. The LTW distribution is proposed by [45] for the strength of glass. In order to describe the fracture strength distribution of an automotive windscreen, the LTW distribution is utilized almost exclusively within this investigation.

5.3 Differentiation of Strength Populations An automotive windscreens consists of two glass plies connected by a PVB layer. The exterior glass ply is referred to as number one and the interior glass ply as number two. Each ply has two surfaces, and each surface undergoes a different treatment during handling and production. This results in a total of four differently treated surfaces. These are numbered from one to four starting with the outer, exterior surface. Because side two and four are also imprinted a total of six populations is distinguished. In this thesis, windscreens for an Audi A3 manufactured in 2017 are used. One of the windscreens is shown in Fig. 5.1. The glass plies are 1.8 mm thick while the PVB layer consists of an acoustic PVB with three inter-layers and a total thickness of 0.76 mm. To obtain samples with a uniform curvature in all directions, the windscreens are optically scanned three-dimensionally. The measured points are fitted to calculate the local curvature. Locations with a minimum difference between the two main axis radii are determined for sample extraction. The sample extraction points and the calculated curvature can be seen in Fig. 5.2. A more detailed derivation of the estimation of the specimen extraction points is given in [78]. The name definitions

70

5 Stochastic Strength of an Automotive Windscreen

ply two - interior side

imprinted area

Figure 5.1

pvb interlayer ply one - exterior side

Imprinted area and structure of an Audi A3 windscreen manufactured in 2017.

of all strength populations are shown in Tab. 5.2.

Figure 5.2

Specimen extraction positions [78]. The color scheme shows the local curvature.

5.4 Experimental Part

Table 5.2

71

Series names according to their origin and the number of total experiments.

series name

glass side

PVB cover

imprinted

side-one-clear side-two-silk side-two-clear side-three-clear side-four-silk side-four-clear edge-outer edge-inner

one two two three four four ply one ply two

no yes yes yes no no no no

no yes no no yes no -

total (sum: 674) 79 71 147 157 41 117 30 32

valid (sum: 321) 45 38 45 61 22 48 30 32

5.4 Experimental Part The experimental part is divided into the estimation of surface and edge strength distributions. CRRT are utilized for the surface strength. For this purpose the influence of curvature on the results is investigated. For the edge strength four-point-bending tests are performed.

5.4.1 Specimen Preparation The necessary specimens are cut out of windscreens by water jet cutting. The glass surface that is in contact with the machine support during the cutting process is excluded for further testing. To separate both laminated glass plies from the PVB interlayer, a self-made shearing device is used. The device is shown in Fig. 5.4. Within this device, each of the two glass plies is clamped into an aluminum frame. While one of the frames is rigid, the second frame can move freely vertically. After inserting the specimen, the whole frame was heated to 120◦ C in order to reduce the shear modulus of the PVB interlayer. After approximately 30 min the vertically free aluminum frame and the clamped glass ply slide down due to gravity. The PVB layer decomposes at approximately 260◦ C [188]. An influence of the temperature on the properties of the PVB layer is excluded for the following investigations. The PVB serves as a protective layer for the glass and will not be tested separately. Its stiffness is also negligible compared to the stiffness of glass. After separating the two glass plies, half of the PVB layer remains on the glass specimens. A total of 612 specimens for CRRT and 62 specimens for four-point-bending tests could be obtained. One of the specimens for CRRT before and after separation is shown in Fig. 5.3.

72

5 Stochastic Strength of an Automotive Windscreen

glass

glass

pvb

remaining pvb glass (a) before separation Figure 5.3

(b) after separation

Glass specimen cut out of windscreen before and after separation.

5.4.2 Surface Strength The surface strength is estimated by CRRT. These are standardized in DINEN-1288. A modification with a support ring radius of r2 = 15 mm instead of r2 = 30 mm is used. The advantage of this modification lies in a higher rate of valid tests [15]. The fracture stress σf can be estimated by the maximum applied force Fmax as     3(1 + υ) r2 (1 − υ)(r22 − r12 ) Fmax σf ail = ln + . (5.1) 2π r1 h2 (1 + υ)(2r32 ) A Poisson’s ratio of υ = 0.23, a load ring radius of r1 = 6 mm and a support ring radius of r2 = 15 mm are used. The circular specimens have a radius of r3 = 33 mm and a thickness of h = 1.8 mm. A total of 612 tests are performed of which 259 are valid. A CRRT can be considered as valid if fracture occurs within the load ring. Fig. 5.5 shows a CRRT specimen with fracture inside the load ring. Special attention is paid to the curvature of the specimen. The specimens have a certain curvature because they are cut out of ready-to-use windscreens. Specimens from all extraction sites are analyzed for their curvature by a probe system MarSurf LD 130. For this purpose the specimen surface is measured four times with an angle of 45◦ to each measurement. Every measurement goes through the specimen center point. The measured curvature of one specimen is displayed in Fig. 5.6. After measuring the curvature of each extraction point FE simulations are performed. The purpose of these simulations is to identify the influence of the curvature on the resulting fracture stress. With non-uniform curved glass specimens, the state of stress and the stress rate are different from standard CRRT. This complicates a comparison with standardized fracture stresses and needs to be investigated further. Within the performed simulations, the maximum stress rate within the CRRT is calculated. With this information the measured stress σmeasured after

5.4 Experimental Part

Figure 5.4 at 120◦ C.

73

Shearing device for separating the glass samples from the PVB interlayer due to gravity

Eq. (5.1) is corrected to the real stress σtrue which would be estimated at 2 MPa/s. The relation between both is given by  σtrue = σmeasured

2 σ˙ curvature

1  n+1

.

(5.2)

The crack growth exponent is estimated in the previous chapter to n = 15.098. The correlation between the measured stress and the true stress is displayed in Fig. 5.7. A maximum stress rate of 2.15 MPa/s and a minimum stress rate of 1.99 MPa/s are calculated. Another important difference between CRRT without curved specimen and curved specimen is the stress field. During standard CRRT a biaxial stress field with two identical first principal stresses is present. This leads to a uniform crack-opening stress which is equal for all cracks within the load ring. With a second principal stress that is lower than the first principal stress, not all cracks obtain an equal crack-opening stress. The deviation of both principal stresses is considered to ∆σp = 1 −

σp,min . σp,max

(5.3)

Fig. 5.8 shows the calculated deviation for all extraction points in concave and convex specimen orientation. Accordingly the deviation for concave positioning is negligible with a maximum deviation of nearly 4 %. ¨For a convex positioning the deviation is up to 90 %. Within this thesis deviations for convex testing up to 20 % during load and up to 10 % at fracture are accepted. These values represent a compromise between the usability of the specimens and a

74

5 Stochastic Strength of an Automotive Windscreen

load ring

fracture origin

support ring r1 r2 r3

Figure 5.5 Specimen from a valid CRRT with fracture inside the load ring. The radii are r1 = 6 mm, r2 = 15 mm and r3 = 33 mm.

nearly uniform biaxial stress field. Convex positioning of the specimen is necessary for testing side one and three of windscreens.

5.4.3 Edge Strength The edge strength of windscreens is of great importance for the simulation of a pedestrian head impact. Former investigations showed that fracture of the exterior glass ply originates from the windscreen edge [52]. The edge strength is determined by four-point-bending tests. The utilized setup is shown in Fig. 5.9. For specimen extraction, glass plies are taken from the production line before lamination. In order to obtain the maximal stress at the edge, the specimens are tested upright. For the test setup, a support span of 124 mm and a load span of 30 mm are chosen. The specimen exhibits a thickness of 1.8 mm, a height of 15 mm, and a width of 130 mm. To prevent the specimens from tipping over during the test lateral holders are arranged. These are coated with Teflon to reduce friction. Examining the fracture patterns from the four-point-bending tests leads to the observation that not all specimens fail exclusively at the location of maximum bending stress. Fig. 5.10 shows the location of two different fracture origins. In

5.4 Experimental Part

75

0.2

height [mm]

0.1 0 −0.1

0◦ 45◦ 90◦ 135◦

−0.2 −0.3

0

10

20

30 40 surface axis [mm]

50

60

70

Figure 5.6 Curvature of a specimen from extraction point 18. The surface was measured through the center of the sample at a separation of 45◦ .

Fig. 5.10a, fracture occurred directly at the outer edge at the point of maximum bending stress. For most materials, the fracture at the point of maximum stress is a logical conclusion. For glass, the combination of stress, stress history and initial flaw geometry leads to fracture. Regarding the edge strength of windscreens initial flaws are introduced mainly by edge processing and the silkscreen process. The silkscreen is applied up to the edge for specimens from the exterior glass ply. The interior glass ply is imprinted with an distance from the edge of approximately 5 mm. Fig. 5.10b shows a sample where the origin of fracture lies in the transition between edge and silkscreen. For this fracture origin it is impossible to distinguish between edge and silkscreen. Tab. 5.3 shows all identified origins depending on the localization of initial fracture. A distinction is made between the non-imprinted edge, the transition between edge and silkscreen, and the silkscreen. The exterior glass ply, imprinted to the edge, fails mostly at the transition between edge and silkscreen. The interior glass ply which is not imprinted up to the edge fails mostly at the edge. This leads to the conclusion that the silkscreen process reduces the strength to a greater extent than edge processing. Further attention is given to the distance between the outer edge and the fracture origin. The equation to determine fracture stresses by four-point-bending assumes fracture directly at the outer edge, the point of maximum stress. The distance between the fracture origin and the outer edge is estimated by a microscope. The estimated distance is then considered to determine the correct fracture stress value depending on the fracture location. The distinction of the fracture origin between silkscreen and edge processing is irrelevant for the following simulations. This is because the sum of the two mecha-

76

5 Stochastic Strength of an Automotive Windscreen

concave

convex 600 true stress [MPa]

true stress [MPa]

600

400

200

0

0

200 400 measured stress [MPa]

600

400

200

0

0

200 400 measured stress [MPa]

600

Figure 5.7 Relation between measured stress during CRRT and simulated true stress for all 19 extraction points.

nisms yields the population of the edge strength. For simulations, the combination of both is necessary. A differentiation is made only between inner and outer glass ply.

Table 5.3

Classification of fracture origins of the specimens for the determination of the edge strength

origin edge transition silkscreen unknown

exterior glass ply 4 23 2 1

interior glass ply 24 8 0 1

5.4 Experimental Part

77

convex

concave 100

5

80 ∆σp [%]

∆σp [%]

4 3 2 1

60 40 20

0

Figure 5.8

200 400 600 calculated stress [MPa]

0

0

200 400 600 calculated stress [MPa]

Relation between calculated stress and deviation between the first principal stresses.

load pins

support pins

specimen

tipping support

Figure 5.9 Experimental setup for four-point-bending tests to estimate the edge strength. The specimen was taken from an interior glass ply, which is not covered by silkscreen at the edge.

78

(a) failure at outer edge.

5 Stochastic Strength of an Automotive Windscreen

(b) failure between edge and silkscreen.

Figure 5.10 fracture at outer edge (a) and at the transition between edge and silkscreen (b) within four-point-bending tests.

5.5 Residual Stress

79

5.5 Residual Stress The stochastic fracture model separates the stress values estimated by coaxial ringon-ring or four-point-bending tests into crack-opening and residual stresses. Residual stresses result from the cooling process after the float-process. Uneven cooling of the surface and of the core of the specimen causes compression stresses on the surface. They must be determined and subtracted from the experimental determined values. For this purpose, specimens of each extraction point are investigated for their surface pressure with a scattered light polariscope (SCALP-5). According to the manufacturer, a measurement uncertainty of less than 5 % must be considered. The residual stresses are measured to a mean stress of σ0 = 7.89 MPa with a standard deviation of 0.49 MPa for extraction points 1-11 and 16-19. For the imprinted specimens from extraction point 12-15, a mean stress of σ0 = 7.01 MPa with a standard deviation of 1.12 MPa is determined by the scattered light polariscope. To verify the measured data, one sample from extraction point twelve is additionally measured by an external institute. The maximum tensile stress within the volume of the specimen is estimated to 3.75 MPa in comparison to 4.09 MPa from the scattered light polariscope. Problems occur when measuring the residual stresses at the edge. The windscreen is imprinted up to the edge and measuring directly at the edge is not possible because of the curvature. High differences between the residual stress from the edge comparing to the residual stress from the surface are to be expected. While the windscreen plies are shaped, they are supported by a 20 mm wide metal frame on the outer edge. The high temperature difference between the glass ply and the metal frame causes rapid cooling at the points of contact. This is similar to the process of manufacturing toughened safety glass. For this reason, windscreens are manufactured without imprinted areas at the edge. These windscreen edges are examined with a scattered light polariscope (Edge-Stress Meter GES-100-MWA). Residual stresses between 42.98 MPa and 79.63 MPa are measured. Repetitions of these measurements using the SCALP-5 polariscope resulted in residual stresses of 33 MPa. Only two measurements with the SCALP-5 were possible due to the curvature. A residual stress of 79.63 MPa is above some of the measured strengths during the four-point-bending tests. Because the measured fracture stress is the sum of the residual stress and the crack-opening stress, a residual stress above the measured fracture stress is not possible. After excluding unrealistic values a mean residual stress of 39.82 MPa remains. Being aware of the uncertainty of this value, I nevertheless decided to use it for further investigations. With regard to the numerical modeling of the fracture behavior of windscreens, a future investigation of the residual stress at the edge of the windscreen is recommended. For the following simulations of a pedestrian head impact the uncertainty of the residual stresses at the edge can be neglected. The residual stresses mainly exert an influence on the initiation of subcritical crack growth. This is particularly

80

5 Stochastic Strength of an Automotive Windscreen

important for static simulations. A detailed explanation of the estimated residual stresses can be found in [189] (in German). An overview of the measured values is shown in Tab. 5.4. Table 5.4

Residual stresses of the samples extracted from the windscreen.

extraction point

1 - 11, 16 - 19 12-15 (imprinted) A-D A-D

residual stress σ0 [MPa] 7.89 7.01 58.99 33.01

std. dev.

0.49 1.12 14.05 -

measuring method SCALP-5 SCALP-5 GES-100-MWA SCALP-5

5.6 Statistical Evaluation The estimated fracture stress values from the CRRT and four-point-bending tests are examined for their statistical distribution. Former investigations discussed a lower bound for the strength of glass and concluded that a lower bound for the strength of glass exists [45]. This investigation has shown that the frequently used 2PW distribution fails to interpret experimental fracture stress values at a low probability. Through modern manufacturing processes and quality controls, flaws larger than a certain value are avoided. This suggests that freshly produced glass has a lower bound for fracture stresses. If a windscreen is used within a car, the lower bound can be reduced, for example by chips of stone, potholes or thermal shock. Other investigations with glass predamaged by sandblasting [46, 190] also showed that the strength can be reduced, but only up to a certain limit. To model the lower boundary for glass, the left-truncated or the three-parameter Weibull distribution are applied. [45] suggests that the LTW distribution appears to interpolate fracture stress values from glass better than the three-parameter Weibull distribution. For further investigations within this thesis, this recommendation is adhered to. The LTW distribution can be formulated by  P (σf ; η, β, τ ) = 1 − exp

τ σf − η η

β ,

(5.4)

with η and β as the shape and the scale parameter similar to the 2PW distribution. The lower bound is characterized by the truncation point τ . The parameters are estimated by the MMLE for the LTW distributions [47].

5.7 Fractographic Verification

81

Table 5.5 Estimated parameters for the LTW distributions. Parameter fitting is performed with strength data excluding residual stresses. Each sample consists of n values.

series name ws-one-clear ws-two-silk ws-two-clear ws-three-clear ws-four-silk ws-four-clear ws-edge-outer ws-edge-inner

η [MPa] 99.00 67.02 381.14 359.85 86.51 148.96 48.53 84.30

β [-] 1.97 2.97 4.43 5.18 3.85 3.19 3.04 6.75

τ [MPa] 48.26 40.49 55.65 147.52 44.44 56.48 21.88 51.99

n [-] 45 38 45 61 22 48 30 32

σ0 [MPa] 7.89 7.01 7.89 7.89 7.01 7.89 39.82 39.82

5.7 Fractographic Verification To validate the influence of curvature and stress rate on fracture stresses, the fractured specimens are investigated for their fracture mirror. A fracture mirror marks the origin of fracture. Fig. 5.11 shows two fracture mirrors and their diameter used for fracture stress calculations. Fracture stress σf and fracture mirror radius r are proportional to each other by the fracture mirror constant A to σf = Ar−0.5 + σ0 .

(5.5)

With fracture mirrors only the crack-opening stress can be determined. No conclusions on the residual stress σ0 can be drawn. The mirror constant is assumed to be a material constant [98, 100]. Influences from the chemical composition of the glass are not investigated. Within the pertinent research literature, a wide range of values can be found for the mirror constant [100]. The mirror constant for the utilized Audi A3 windscreen glass was determined within the framework of this study. For this purpose, 50 coaxial ring-on-ring samples are heat-treated to remove the residual stresses σ0 . The specimens are heated up to 120◦ C and cooled down with a maximum of 2 K/min. The specimens are tested with CRRT according to DIN EN 1288-5. The mirror constant is estimated by a linear regression without y-intercept between the fracture strength σf and −0.5 . The mirror constant is estimated the inverse square root √ of the mirror radii r to A = 1.7232 MPa m with a coefficient of determination of R2 = 0.9965 by a least square algorithm. The development of the fracture mirror occurs after reaching the critical stress intensity. The crack rapidly accelerates towards terminal velocity. While accelerating, the crack tip becomes more and more unstable and the fracture surface roughness increases. At the transition between the mirror region and the surrounding region, the so-called mist, the roughness reaches the wavelength of light. For a human eye, this transition is seen as the end of the mirror region. If a higher strain energy is available at fracture the crack will also have a higher

82

(a) Fracture mirror at curved edge.

5 Stochastic Strength of an Automotive Windscreen

(b) Fracture mirror at plane surface.

Figure 5.11 Fracture mirrors from soda-lime-silica glass. Fig. 5.11a shows a four-point-bending specimen from the edge of a windscreen. A fracture mirror from a CRRT is displayed inFig. 5.11b.

acceleration through a higher energy transfer. This leads to a smaller fracture mirror. The knowledge about the fracture mirror mechanics also allows conclusions to be drawn on the stress state at fracture. Fig. 5.11b shows a fracture mirror that is open towards the specimen thickness. This may be due to the fact that the applied stress decreases towards the center of the specimen. The fracture mirror in Fig. 5.11b is obtained from a CRRT in which a neutral stress fiber is present in the center of the sample during load. At higher stresses and thus higher accelerations of the crack, the opening of the fracture mirror no longer occurs. Within this thesis, the radii of fracture mirrors are measured at the line of the highest applied stress. In order to determine the radius, the diameter of the fracture mirror is measured. The estimated parameters for the LTW distribution by experimental and fractographic fracture stress values are shown in Tab. 5.6. Considering the deviations in Tab. 5.6, several observations are made. First, a good agreement of estimated parameters for series without silkscreen is present. The highest deviation of all non-imprinted series is series ws-four-clear with a deviation of the shape parameter by 14.05 %. One possible reason for this is that only 33 out of 48 fracture mirrors were found in this series. In addition, it is also noticeable that the truncation point undergoes a large deviation. The truncation point estimated by fractographic values is always higher than that of experimental values. As described in Sec. 2.3.2, the estimation of the truncation point is heavily impacted by the lowest measured value. Because the fractographic sample size is always smaller than the experimental sample size, this can be one possible explanation for the deviation of the truncation point. When comparing the fitted parameters for the imprinted samples it becomes evident that they exhibit a much larger deviation. Fig. 5.12 shows the experimental and fractographic values of a non-imprinted (ws-three-clear) and a imprinted

5.7 Fractographic Verification

83

Table 5.6 Estimated parameters and their deviation for the LTW distributions by values obtained experimentally and fractographically.

series name η [MPa] β [-] τ [MPa] Parameter estimated with experimental values: ws-one-clear 99.00 1.97 48.26 ws-two-silk 67.02 2.97 40.49 ws-two-clear 381.14 4.43 55.65 ws-three-clear 359.85 5.18 147.52 ws-four-silk 86.51 3.85 44.44 ws-four-clear 148.96 3.19 56.48 ws-edge-outer 48.53 3.04 21.88 ws-edge-inner 84.30 6.75 51.99 Parameter estimated with fractographic values: ws-one-clear 107.53 1.91 49.97 ws-two-silk 96.71 3.62 77.47 ws-two-clear 389.33 4.38 93.73 ws-three-clear 347.10 4.77 175.41 ws-four-silk 97.29 11.73 76.44 ws-four-clear 173.32 3.14 84.27 ws-edge-outer 64.22 4.56 51.68 ws-edge-inner 99.50 12.09 67.48 Percentage deviation experimental - fractographic [%] ws-one-clear 7.93 -3.14 3.42 ws-two-silk 30.70 17.96 47.73 ws-two-clear 2.10 -1.14 40.63 ws-three-clear -3.67 -8.60 15.90 ws-four-silk 11.08 67.18 41.86 ws-four-clear 14.05 -1.59 32.98 ws-edge-outer 24.43 33.33 57.66 ws-edge-inner 15.28 44.17 22.96

n [-] 45 38 45 61 22 48 30 32 33 38 18 28 21 33 28 30

33/45 38/38 18/45 28/61 21/22 33/48 28/30 30/32

(ws-two-silk) series. The imprinted sample has a horizontal shift. One possible explanation could be a different fracture mirror constant in the imprinted area. This is contradicted by the fact that other imprinted samples also exhibit a large deviation for the scale parameter β. As this value characterizes the scatter of the sample, deviations cannot be explained solely by an abscissa shift through a different fracture mirror constant. This deviation is also observable in the samples from four-point-bending tests. During the four-point bending tests no different stress rate and stress state have to be considered in comparison to the curved samples from CRRT. This leads to the final conclusion that estimating fracture stresses by fractographic means may be incorrect for imprinted samples. For

84

5 Stochastic Strength of an Automotive Windscreen

1

1

0.8

0.8 probability [-]

probability [-]

further investigations, the experimentally determined samples will be used.

0.6

0.4

0.4

0.2

0.2

0

0.6

200 400 fracture stress [MPa]

(a) ws-three-clear (non-imprinted area)

600

0

50 100 150 fracture stress [MPa]

(b) ws-two-silk (imprinted area)

Figure 5.12 Comparison of experimental (red) and fractographic (blue) fracture stresses values (marks). Solid line represents fit by a LTW distribution.

5.8 Summary In the present chapter, the different strength populations of an automotive windscreen have been investigated. Parts of this chapter have already been published in [78]. For the characterization of the statistical strength of the utilized windscreen, the windscreen is divided into eight different strength populations. These parts are designated according to whether the samples are imprinted, PVB-covered or from the edge of the windscreen. In the course of this investigation, it became clear for the first time how different the strengths are in different areas of an automotive windscreen. For determining the surface strength, samples were cut out of the windscreen by means of water jet cutting. These samples were tested by CRRT. The influence from pre-existing curvature of the samples was considered using FE analysis. In order to estimate the edge strength of the windscreen four-point-bending tests were performed. Fracture often occurred within the imprinted area, while the maximum stress was at the outer edge. The four-point-bending specimens are imprinted up to the edge. This leads to the conclusion that the silkscreen process reduces the

5.8 Summary

85

strength further than edge processing. Until now it was assumed that edge grinding reduces the strength the most [52]. After determining residual stresses within the specimens the samples were examined statistically. Parameters of a LTW distribution were estimated. It could be shown that the different strength populations vary significantly. The highest strength values were observed in samples with PVB cover. It is possible that the cover reduces the subcritical crack growth by reducing the amount of humidity reaching to the crack tip. Another possibility is that the PVB protects the glass surface during production, handling and transport. A clear reduction of strength can be observed for imprinted samples. This also includes imprinted samples with PVB cover. It is logical that the silkscreen process reduces the strength of the windscreen. During the silkscreen process, ceramic particles are pressed onto the glass surface. To verify the estimated strength samples a fractographic investigation was performed. All specimens were examined for their fracture mirror. Non imprinted strength values of experimental and fractographic means were in acceptable agreement. Imprinted samples showed a clear, non-systematic deviation. No fractographic investigations on imprinted glass fracture could be found in literature. This may be an interesting topic for further studies.

6 Displacement-Controlled Windscreen Tests 6.1 Introduction The present chapter focuses on the stochastic fracture behavior of automotive windscreens by displacement-controlled impact. Windscreens are tested on a test rig with an electrical cylinder at different constant impact velocities. Windscreens in convex and concave orientation are tested and are monitored by an acoustic emission localization device to calculate the origin of fracture. Subsequently, the stochastic scatter of fracture from experiments is numerically reproduced with the stochastic fracture model. The scatter of displacement effected by the impactor until fracture and the location of fracture are compared. Input parameters for the model are determined in Chap. 4. The stochastic strength of the utilized windscreens is estimated in Chap. 5.

6.2 Experimental Part 6.2.1 Experimental Setup A head impact test with a displacement-controlled impactor is utilized as shown in Fig. 6.1. The setup consists of a massive steel frame with a four-point support for the windscreens. Each support consists of a half-sphere made of poly(vinyl chloride) (PVC) with a diameter of 36 mm. The purpose of the selected support is to reduce influences of the adhesive from glued-in windscreen conditions. The time and location of the initial fracture are investigated. The behavior after the initial fracture is not considered. To reduce further influences, no additional components such as rear-view mirrors or a heating wire are installed. The windscreens originate from the model Audi A3 manufactured in 2017. The head impactor consists of a rubber skin which is mounted on a metal hemisphere. The rubber skin is chosen according to the European new car assessment program (NCAP) testing protocol [4]. The hemisphere geometry of the impcator is 136 mm in diameter, including the 14 mm thick rubber skin. The head impactor gets in contact with the windscreens 425 mm below the top edge, in the horizontal middle of the windscreen. The location of impact is shown in Fig. 6.2. The ve© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_6

87

88

6 Displacement-Controlled Windscreen Tests

force transducer impactor

Figure 6.1

3/5 acoustic sensors

Experimental setup for displacement-controlled windscreen testing.

locity of the impactor is realized by an electrical cylinder. The velocity is set to a constant value in each experiment. The cylinder is rigidly connected to the metal frame. All experiments with windscreen orientation and impact velocity are listed in Tab. 6.1. A concave orientation is defined as impact on the interior side. Accordingly, during a convex impact, the impactor encounters the exterior glass ply. A distinction is made between convex and concave orientation in order to influence the fracture origin. The purpose is to have the origin of fracture located in different regions. The numerical fracture model should also calculate the different fracture origins correctly. The windscreens are further monitored by an acoustic emission system. With the acoustic emission system, the origin of fracture can be calculated for each experiment. A detailed explanation of the acoustic emission system and the utilized localization algorithm is provided in Sec. 2.4.

6.2.2 Experimental Results The measured force versus displacement curves of experiments at 0.1 m/s impact velocity are shown in Fig. 6.3. The signals are cut off when initial fracture is detected. To recognize the displacement at the time of fracture more clearly, the last force value was set to zero. The stochastic scatter of the displacement effected by the impactor at windscreen

6.2 Experimental Part

Table 6.1 entation.

89

Displacement-controlled experiments depending on impactor velocity and windscreen ori-

orientation concave concave convex convex convex

impact velocity [m/s] 0.01 0.1 0.01 0.1 1

quantity 30 30 30 35 30

422 mm 700 mm

impact location

Figure 6.2

230 mm 58 mm

Audi A3 windscreen with impact position and four-point support placement (blue).

fracture is shown in Fig. 6.3. The displacement of the impactor at fracture varies between 64.82 mm and 137.45 mm for concave windscreen orientation and an impact velocity of 0.1 m/s. For convex orientation and an impact velocity of 0.1 m/s, the displacement ranges between 33.61 mm and 40.65 mm. For concave orientation an almost linear increase of the force versus displacement signal can be observed. For a convex windscreen orientation, a nearly linear increase of the force signal is measured as well. A rapid decrease of the force signal occurs after a displacement from the impactor of around 32 mm. A geometric instability may account for this. The displacement at fracture for convex-oriented windscreens accumulates after the geometric instability. The localized fracture origins are drawn exemplary in Fig. 6.4 for convex- and concave-orientated windscreens with an impact velocity of 0.1 m/s. For convex windscreen orientation and an impact from the exterior side, fracture solely occurs at the lower edge of the windscreen. The fracture origin is found at a distance of approximately 15 to 30 mm from the edge by visual examination of the fracture pattern. This matches the observation of a high residual stress at the edge.

force [N]

90

6 Displacement-Controlled Windscreen Tests

2,000 1,500 1,000 500 0

0

20

40

60 80 displacement [mm]

100

120

140

force [N]

(a) convex orientation, 0.1 m/s

2,000 1,500 1,000 500 0

0

5

10

15

20 25 30 displacement [mm]

35

40

45

(b) concave orientation, 0.1 m/s Figure 6.3 Force versus displacement curves from experiments for concave and convex windscreen orientation with 0.1 m/s impact velocity. The signals are cut off after initial fracture is detected.

The residual stress at the edge was investigated in Chap. 5. While the curvature is given to the windscreen during production, the windscreen is supported at the edge by a steel frame. Due to the high difference in temperature between glass and steel frame, a higher residual stress in comparison to the remaining windscreen is obtained in the contact zone between glass and steel frame. Fracture in experiments with concave windscreen orientation occurs mostly underneath the impact location. Fracture towards the upper windscreen region is also noticed several times. The performed experiments will be reproduced using the stochastic fracture model. Fracture location and impact displacement at the time of fracture will serve as possibilities for the comparison of experiments and simulations.

6.3 Numerical Part

91

500

500

250

250

0

0

-250

-250

-500

-500 -500

0

500

(a) convex orientation, 0.1 m/s

-500

0

500

(b) concave orientation, 0.1 m/s

Figure 6.4 Origin of fracture determined by sound localization for convex and concave windscreen orientation. Impact position at 0|-50.

6.3 Numerical Part 6.3.1 Finite Element Model The FE model for the numerical reproduction of the displacement-controlled experiments is introduced. The FE model for convex windscreen orientation is shown in Fig. 6.5.

Figure 6.5 FE model of Audi A3 windscreen tests. The windscreen is shown with imprinted, unimprinted and edge areas as well as the cylinder. The four-point support below the windscreen is not depicted.

92

6 Displacement-Controlled Windscreen Tests

6.3.1.1 Head Impactor The head impactor consists of a metal hemisphere and a rubber skin. The rubber model is adapted according to the European NCAP testing protocol [191] and was provided by Lasso Ingenieurgesellschaft mbH [192]. The rubber skin has a non-linear viscous-hyperelastic mechanical behavior. A rate-dependent hyperelastic model with tabulated data for the viscous part is provided by [193]. The model was improved later on by [194] by using data from rate-depended tensile tests. The impactor model used in this thesis has already been utilized in [17]. The rubber skin consists of 2535 quadratic 8-node solid elements. The average volume of each element ranges from 200 mm3 to 300 mm3 . The steel cylinder is modeled by 4290 quadratic 8-node solid elements as well. Linear elastic behavior is assumed for the steel hemoisphere with a Young’s modulus of 210 GPa. 6.3.1.2 Windscreen Automotive windscreens consist of laminated safety glass which is made of two glass plies connected by a PVB layer. The Audi A3 windscreens considered are made of two glass plies with a thickness of 1.8 mm each. The PVB layer is 0.76 mm thick. The windscreen is modeled as a shell-solid-shell laminate. The glass is described as fully integrated iso-parametric four-node shells with five integration points through thickness. The Gauss-Lobatto integration rule is selected for time integration. The geometric nodes from the glass shells are coupled with the nodes of the PVB interlayer solids. The shear transfer between the layers is realized by this method. This procedure is shown in Fig. 6.6. The PVB interlayer is modeled with selectively reduced, fully integrated eight-node solid elements. 6.3.1.3 Test Frame The utilized frame was characterized with regard to its stiffness in [195]. For this purpose, a head impactor was pressed onto a steel plate and placed on the test frame. A steel hemisphere without rubber was utilized as impactor. A resulting stiffness of k = 2705 Nmm−1 was calculated in a range between 0 and 2750 N. A spring is attached underneath each support to reproduce the stiffness. The stiffness for each spring is set to ki = k/4 = 687.5 Nmm−1 . The four-point support is assumed to be rigid.

6.3.2 Interlayer Stiffness Evaluation The Audi A3 interlayer consists of an acoustic PVB triple layer. Within the PVB triple layer, there are two layers of standard PVB and one layer of an acoustic foil in between. The purpose of the triple layer is to make the cabin of the vehicle quieter. The mechanical behavior can be described as visco-hyperelastic. The influence of

6.3 Numerical Part

glass

5

5

pvb

1 glass

1

8

8 4

4

93

6

6 2

2

7

t t

7

coupling

3

5 1

8 4

6 2

7 3

thickness

5 1

8 4

6 2

7 3

3

Figure 6.6 Procedure for coupling shared nodes of laminated safety glass by a shell-solid-shell laminate within FE simulations. Method proposed by [52].

humidity, the stress rate and the type of stress field are of importance as well. For upcoming simulations the one-parameter material model for rubber-like materials according to P. Blatz and W. Ko [196] is utilized. It is implemented within the FE solver LS-Dyna as MAT-007. The material law consists of a strain energy functional W by   G 1 C) = W (C I1 − 3 + (I3−β − 1) , (6.1) 2 β where C is the right Cauchy Green tensor and G the shear modulus. Ii is the ith invariant of the right Cauchy Green tensor. The parameter β is expressed as β=

υ = 6.2567.... 1 − 2υ

(6.2)

The Poisson’s ratio is set to a constant value of υ = 0.463. Nearly incompressible rubber-like materials have a Poisson’s ratio of υ → 0.5. This value would lead to an infinite stress during compression within the FE algorithm, for example. An instability occurs, for example, with υ = 0.5 and an infinite bulk modulus K to K=

E . 3(1 − 2υ)

(6.3)

Within the BlatzKo rubber material law, the Cauchy stress σ is calculated by a derivative of the second Piola-Kirchhoff stress tensor S as h i ∂W S =2 = G 1 − I3−β C −1 , (6.4) ∂C

94

6 Displacement-Controlled Windscreen Tests

Table 6.2

Shear modulus for the PVB layer within the simulations.

orientation concave concave convex convex convex

impact velocity [m/s] 0.01 0.1 0.01 0.1 1

shear modulus [MPa] 0.7 1.3 1.3 2.25 9

to σ = J −1F SF T =

G B − I3−β 1 ). (B J

(6.5)

Within Eq. (6.4) and Eq. (6.5), the deformation gradient F and the relative volume J are defined as ~ x and J = detF F. F = ∇~

(6.6)

The deformation in the current reference configuration is given as ~x. As seen in the derivation of the Cauchy stress tensor only the shear modulus G is needed as material parameter to calculate the Cauchy stress. To fit the force versus displacement behavior to the experimental data, the shear modulus is estimated by reverse engineering. The shear modulus is varied until the numerical force signal lies within the experimental scatter. The estimated shear moduli are drawn in Tab. 6.2. The numerical force versus displacement curves in comparison to the experimental data are shown in Fig. 6.7. For convex-oriented windscreens, the mentioned geometric instability is also observable within the numerical curves. The convex simulation curves are within the experimental scatter with increasing deviations by higher test velocities. Oscillations are visible in Fig. 6.7e. Within this sample, the numerical reproduction of the force versus displacement curves is not in the same agreement as curves from lower velocities. For convex windscreen orientation and an impact velocity of 1 m/s the deviations should be kept in mind when comparing experimentally determined results with results from stochastic simulations.

6.3 Numerical Part

95

force [N]

1,500 1,000 500 0

0

20

40

60 80 displacement [mm]

100

120

force [N]

(a) Concave windscreen orientation, 0.01 m/s impact velocity.

2,000 1,500 1,000 500 0

0

20

40

60 80 displacement [mm]

100

120

140

25

30

35

force [N]

(b) Concave windscreen orientation, 0.1 m/s impact velocity.

2,000 1,500 1,000 500 0

0

5

10

15 20 displacement [mm]

force [N]

(c) Convex windscreen orientation, 0.01 m/s impact velocity.

2,000 1,500 1,000 500 0

0

5

10

15 20 25 displacement [mm]

30

35

40

(d) Convex windscreen orientation, 0.1 m/s impact velocity.

force [N]

3,000 2,000 1,000 0

0

5

10

15

20 25 30 displacement [mm]

35

40

45

(e) Convex windscreen orientation, 1 m/s impact velocity. Figure 6.7 Comparison of force versus displacement from experiments (blue) and simulation (red) during displacement-controlled head impact experiments.

96

6 Displacement-Controlled Windscreen Tests

6.4 Comparison of the Results In the following, the numerical and experimental results from displacementcontrolled impact tests are compared. The displacement effected by the impactor until the fracture of the windscreens and the localized fracture origins are compared.

6.4.1 Impactor Displacement The impactor displacement between first contact and fracture of the windscreen and its cumulative probability are shown in Fig. 6.8. The experimentally measured fracture displacement for concave-oriented windscreens lies within the numerical calculated range. The accumulation of experimental fracture lies between 80 and 125 mm for an impact velocity of 0.01 m/s, shown in Fig. 6.8a. Most numerically reproduced values are also within this range. The upper and lower limits from simulations are wider than the experimental values. The greater sample size of 30 experimental to 150 numerical data points may be one possible reason. The fracture for concave oriented windscreens at a constant impactor velocity of 0.1 m/s in Fig. 6.8b is comparable to the values from 0.01 m/s. The displacement at fracture is slightly shifted towards higher values. This has been expected because less time for subcritical crack growth is present during experiments with higher impact velocities. The comparison of numerical and experimental displacement at fracture for convexoriented windscreens shows more deviation than comparing both with concaveoriented windscreens tests. Experimental fracture mostly occurs directly after a decrease in the force signal which is probably due to a geometric instability. This is illustrated in Fig. 6.7. Numerical fracture also accumulates shortly after the geometric instability for an impact velocity of 0.01 m/s and 0.1 m/s with convexoriented windscreens. This is shown in Fig. 6.8c and Fig. 6.8d. For convex oriented windscreens tested with an impact velocity of 1 m/s, the numerical scatter is larger than the experimental scatter. This is especially observable towards lower displacement values. In addition, the numerical values do not match the upper bound of experimental values at a displacement of 46 mm and an impact velocity of 1 m/s. The discrepancy between numerical and experimental data for this configuration may be caused by the mismatched force versus displacement curve from Fig. 6.7e. The numerical force signal is mostly higher than the experimentally measured curves. This may be the reason for the premature numerically calculated glass fracture for this configuration.

6.4.2 Fracture Origin The calculated origin of fracture is compared between experiments and simulations. The origin of fracture is determined by acoustic emission localization during

6.4 Comparison of the Results

97

experiments. The method of sound localization is explained in Sec. 2.4. The localized fracture origin from experiments and simulations for each sample is shown in Fig. 6.9. The fracture origin from simulations lies within the experimental range for concave-oriented windscreens. A few of the experimental origins at the edge could not be reproduced by the stochastic fracture model. The main accumulation of fracture occurred underneath or besides the impact location during concave experiments. This is also observed during simulations. Fracture took place almost exclusively at the lower and upper windscreen edge at the imprinted area for experiments with convex-oriented windscreens, as shown in Fig. 6.9c to Fig. 6.9e. In contrast to this, numerical fracture takes place mostly underneath the impact position. Four possible reasons for the observed deviation are: 1) The strength distribution of windscreen side four is estimated as too low. 2) The edge strength is estimated as too high. 3) The force transferred between impactor and windshield is concentrated too much onto a small area. 4) The geometric instability and its effect on the fracture of glass cannot be reproduced correctly by the considered FEM. Point one might be true due to the fact that the sample for the imprinted interior glass side consists of only 22 values. A too small sample may lead to a wrong strength distribution. This could also explain the deviations in localization for concave-oriented windscreens. The cause for point two might also be a too small sample. The estimation of the residual stress at the edge may also be wrong for point two. Point three might be true due to the aging of the impactor. During the first experiments the aging of the impactor was not considered correctly. The rubber of the head impactor ages and can thus lose its stiffness, which leads to a greater stress field. A wider distributed transition of the force between impactor and windscreen would lead to a delayed fracture. This is not taken into account by the numerical model. The geometric instability from point four might also have an current unknown influence for convex oriented windscreen. Because the results from experiments with concave oriented windscreens agree better with the numerical values, this seems to be a realistic possibility.

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6.5 Summary The present chapter contains an investigation on displacement-controlled impact experiments and their numerical reproduction. Windscreens were impacted until fracture by a head impactor with different constant velocities. The constant velocity was realized by an electric cylinder. The windscreens were tested in convex and concave orientation. 155 quasi-static windscreen experiments were performed. The experiments were monitored additionally by an acoustic emission measurement system in order to calculate the origin of fracture. FE simulations to reproduce the stochastic fracture behavior of the windscreens were carried out. The mechanical behavior of the PVB layer within the windscreens was determined by reverse-engineering. For this purpose the force signals from experiments were fitted by varying the shear modulus of the PVB layer. The localized origin of fracture and the displacement effected by the impactor until fracture are compared between experiments and simulations. The stochastic scatter of fracture for windscreens in concave orientation was reproduced in acceptable agreement. The displacement of the impactor and the fracture origin are - within some expected deviations - possibly caused by a small experimental sample size. The comparison of experiments and simulations for convex-oriented windscreens revealed more deviations. The location of fracture for convex simulations differs from the experimental data. During experiments, the fracture was localized almost exclusively at the windscreen edge. During simulations the fracture occurred mostly underneath the impact position. One possible reason might be that the sample size of the imprinted interior glass side is relatively small with 22 values. The aging process of the impactor rubber skin or an observed geometric instability of the convex oriented windscreen are further possible influences. In summary, the stochastic fracture model was able to reproduce windscreen fracture with the mentioned deviations. If further research is conducted with displacement-controlled windscreen impact experiments, the geometric instability during convex windscreen orientation should be investigated. An increase of the sample size for the mentioned glass sample populations is advised as well.

7 Free-Flying Head Impact 7.1 Introduction The following chapter provides an investigation on dynamic head impact tests on automotive windscreens. This chapter is divided into two parts. The first part investigates head impact replacement tests. These tests consist of a free-flying impactor and an impact velocity according to the European NCAP [191]. The difference between European NCAP experiments and the test conducted within this investigation, is the four-point-support of the windscreens. Subsequently, numerical results are calculated using the stochastic fracture model. The experimental and numerical results were compared and discussed afterwards. In the second part, a pedestrian head impact is simulated under in-service conditions. The aim is to calculate the HIC, taking into account the stochastic fracture behavior of glass. A stochastic distribution of the HIC is presented.

7.2 Head Injury Criterion The HIC is an empirical value indicating the likelihood of head injury during impact. It is calculated by the acceleration acting on the center of gravity of the impactor over time. The HIC is calculated as ( ) 2.5 Z t2 1 HIC = max ares (t)dt (t2 − t1 ) , (7.1) t2 − t1 t 1 by varying the time interval boundaries t1 and t2 . Within Eq. (7.1), the resulting acceleration ares on the impactor as multiple of the gravity g in m/s2 and the time in seconds is utilized. A high acceleration may be tolerated for small time intervals. The HIC can be defined as the probability of an injury of a certain severity. A head injury value of HIC = 1000 is equivalent to a probability of 18 % for a severe injury, a probability of 55 % for a serious injury and a probability of 90 % for a moderate head injury [197]. Nowadays, the scale in Fig. 7.1 is universally used as an agreed ranking for the severity of injury [70].

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_7

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head injury criterion [HIC] Figure 7.1 Probability of head injuries of different severities for given head injury values according to [197]. Each region represents a value on the abbreviated injury scale (AIS). AIS values range from 0 (no injury) to 6 (fatal injury) [198].

7.3 Head Impact Replacement Test 7.3.1 Experimental Part Ten experiments with a free-flying impactor are conducted. In accordance with the experiments carried out in Sec. 6.2.1, a four-point supported windscreen is utilized as well. Fig. 7.2 shows the experimental setup used. Each support consists of a half-sphere made of PVC with a diameter of 36 mm, identical to the support used in Sec. 6.2.1. The main purpose of the selected support is to reduce possible influences of the adhesive from glued-in conditions. A further reason is the cost factor compared to glued-in windscreen tests. The location of the four-point support is shown in Fig. 6.2. The head impactor is identical to the adult impactor in the European NCAP testing protocol [191] based on the recommendations of the European Union [4]. The contact between windscreen and impactor takes place 422 mm above the lower windscreen edge in the horizontal middle of the windscreen. Windscreen of the type Audi A3 are utilized. The acoustic emission localization method is used within the free-flying head impact experiments. All calculated fracture origins are within the impact position. The windscreen are oriented with the exterior ply facing towards the impactor. An impact velocity of 10 m/s is chosen. Tab. 7.1 shows all calculated head impact criterion values with the corresponding time interval. Additionally, the observed initial fracture of windscreens tini.f rac. is given. This value is estimated by review-

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impactor data cable

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Figure 7.2

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Four-point support testing setup for experiments with free-flying impactor.

ing the high-speed recordings with a frequency of one frame per millisecond. A maximum acceleration of 136.89 g is measured. All acceleration curves are shown in Fig. 7.3. The data in Tab. 7.1 show the influence of the stochastic fracture behavior of glass on the head injury probability. The calculated HIC varies between 418.71 and 562.47. The initial fracture tini.frac. of the conducted experiments varies between one and nine milliseconds after impact. Although fracture occurs at nine milliseconds, the acceleration curve is almost zero, nine milliseconds after impact. The initial fracture of test number seven in Tab. 7.1 occurs two seconds after the right head injury boundary t2 . Due to the late fracture, only the stiffness of the windscreen defines the HIC value. Fig. 7.4 shows the fracture pattern depending on the time after impact for three tests. It can be seen that an earlier fracture results in a lower degree of fragmentation. While in Fig. 7.4a fracture occurred between zero and one second after impact, in Fig. 7.4c fracture occurred between eight and nine seconds after impact. Fig. 7.4b shows pictures of a head impact with fracture between two and three seconds.

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Table 7.1 Calculated HIC values with corresponding time interval. Initial fracture tini.frac. of windscreens detected by manual evaluation of the high-speed recordings with ∆t = 1 ms.

test no. 1 2 3 4 5 6 7 8 9 10

t1 [ms] 0.60 0.60 0.60 0.65 0.65 0.65 0.65 0.60 0.60 0.60

t2 [ms] 6.60 6.90 6.95 6.75 6.75 6.75 6.75 6.75 6.85 6.60

HIC [-] 492.47 482.99 418.71 554.11 562.47 561.82 559.88 566.03 518.48 508.26

tini.frac. [ms] 0 < tini.frac. ≤ 1 0 < tini.frac. ≤ 1 2 < tini.frac. ≤ 3 6 < tini.frac. ≤ 7 5 < tini.frac. ≤ 6 6 < tini.frac. ≤ 7 8 < tini.frac. ≤ 9 6 < tini.frac. ≤ 7 0 < tini.frac. ≤ 1 0 < tini.frac. ≤ 1

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7.3.2 Numerical Part The utilized model for FE simulations is shown in Fig. 7.5. The simulations are carried out with an unpublished version of the FE solver Radioss. The stochastic fracture model is currently being implemented for commercial use within this solver. The laminated glass is modeled as laminate with a shell-solid-shell discretization. While the shell elements describe the glass behavior, the solid elements represent the properties of the PVB interlayer. This modeling technique is similar to the method shown in Fig. 6.6. Five integration points through thickness with selective reduced integration points each are used for the shell elements and selective reduced solid elements. The necessary material parameters for the glass plies are outlined in Tab. 4.9. The input parameters set for the post-fracture behavior are identical to the values in [52].

7.3 Head Impact Replacement Test t = 1 ms

107 t = 5 ms

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(a) Initial fracture of test number 1 between 0 and 1 ms.

(b) Initial fracture of test number 3 between 2 and 3 ms.

(c) Initial fracture of test number 7 between 8 and 9 ms. Figure 7.4 Representation of different fracture times during the replacement tests. Each row represents a different test with the same conditions. The tests shown are number one, three and seven from Tab. 7.1.

As material law for the PVB interlayer, a hyperelastic material law is utilized. This is realized by LAW42 within Radioss with a strain energy density W according to W (λi ) =

5 X µp p

αp


K λ1αp + λ2αp + λ3αp − 3 + (J − 1)2 , 2

(7.2)

where λi is the ith principal stretch, J the relative volume and K the bulk modulus. The parameters αp and µp are coefficient pairs which have to be given as input. The parameter identification for the PVB interlayer was carried out in [195] and used for windscreen simulations in [17, 52]. Although the PVB interlayer used within the windscreens in this thesis is not identical to that investigated in [195], the parameters are used due to similar material properties and a high characterization effort. Tab. 7.2 shows the used parameter set. The FE model of the impactor has been validated for head impact simulations by [199] and is further developed by

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(b) back view

(a) front view Figure 7.5 side (b).

Table 7.2

α1 2

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Parameter set for the PVB interlayer (mat/ogden/). Unit system: kg,mm,ms

α2 -2

µ1 0.000482

µ2 -0.00412

ρ 1.19e-6

υ 0.495

[194] and finally by a tabulated Ogden approach by [193]. The four-point support is modeled as rigid. A total of 250 simulations are performed.

7.3.3 Comparison of Results The probability for time to fracture after impactor contact is drawn in Fig. 7.6. The experimentally determined time to fracture lies within the numerical range. Most numerical values are found within the first three seconds. The latest numerical fracture takes place at 13.869 ms, during experiments between eight and nine ms. The experimental acceleration, shown in Fig. 7.3, reaches zero after around nine ms. Fracture occurs within these tests when the impactor is no longer accelerated by the impact with the windscreen. This also happens in experiment number seven, shown in Tab. 7.1. If initial fracture occurs later than the calculated upper boundary t2 for the HIC, the HIC depends solely on the elastic response of the windscreen. This is the case for several of the chosen replacement tests and gives a statistical upper bound for the considered test configuration. The maximum HIC is 566.03 from experiments and 559.88 from simulations. The experimental and numerical HIC values are shown in Fig. 7.7. Most HIC values for the chosen test setup are accumulating at the upper bound. A putative normal distribution of head injury values is calculated between approximately 400 and 520. Five of ten experimental values are within this range, while the other five values are at the upper bound. Therefore, it can be concluded that the stochastic fracture model can reproduce the head impact replacement tests. The time between impact and initial fracture

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of the windscreen could be reproduced compared to the ten experimental values available. The HIC is also reproduced in the range of the experimental values. An upper bound is determined for a head impact on the chosen four-point supported windscreens.

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7.4 Head Injury Probability Distribution 7.4.1 Stochastic Head Impact Simulation The stochastic scatter of the HIC under in-service conditions is predicted within the following simulations. A pedestrian head impact on an automotive windscreen in glued-in condition is utilized. Former investigations also used an artificial frame as replacement for full-vehicle tests with a wooden [17, 52] or alumina frame [200]. A linear elastic frame is adopted for the following simulations. Fig. 7.8 shows the FE model of the described setup. The impactor simulates a pedestrian head according to the European NCAP [191]. The FE model of the impactor is provided by Lasso Ingenieurgesellschaft mbh and is identical to the impactor used for the replacement tests. In Sec. 7.3.2, the modeling of the laminated safety glass within the windscreen is described. A shared-nodes shell-solid-shell laminate is utilized for describing the mechanical properties of the considered windscreen. Tab. 4.9 shows the necessary mechanical parameters for the stochastic fracture model. Tab. 5.5 contains the determined stochastic strength distributions for the different windscreen regions. The windscreen frame is represented by a simple linear-elastic material law with a Young’s modulus of E = 5 GP a. The behavior of the adhesive is approximated by a strain rate dependent model [201]. Within this model, the linear elastic stress σelast. () is scaled with a logarithmic expression of strain rate ˙ by    ˙ σ(, ) ˙ = σelast. () 1 + Cln . ˙0

(7.3)

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Figure 7.8 FE model for the pedestrian head impact on a windscreen. The figure shows the impactor (blue and red), the adhesive (cyan), the elastic frame (brown) and the windscreen.

The parameters C and ˙0 were estimated by strain rate dependent experiments in [195] to C = 0.0473763 and ˙0 = 0.001 s−1 . The elastic behavior of the adhesive is calculated with an isotropic Young’s modulus of E = 0.5 GPa. The initial impactor velocity is set to 36 km/h. The impactor comes into contact with the windsreen 425.5 mm below the top edge at the horizontal middle. The windshield is subjected to the acceleration of gravity. 250 simulations are performed.

7.4.2 Head Injury Distribution Fig. 7.9 shows the numerically predicted HIC values calculated by Eq. (7.1). The values are scattered between 411.74 and 1292.01. An accumulation around HIC = 480 can be observed. 1.6 % of the calculated values are higher than 1000. Although there are no statutory regulations for a critical HIC value, an upper limit of 1000 is recommended by the European regulation 631/2009 [4]. The calculated values are similar to the values from literature in respect of the statistical range. No freely available database for the stochastic scatter of the pedestrian head impact could be found. This is complicated by the manufacturing-dependent, nonmaterial-specific strength of glass. As a result, each manufacturing line generates its own glass strength distribution and thus its own HIC distribution for a certain windscreen. Investigations concerning the stochastic nature of the HIC are, for example, [69], who performed 26 head impact tests on Seat Leon windscreens and obtained a HIC between 680 and 1752. Head impact values from Mercedes C-Class windscreens were determined by seven experiments in [17]. Three of the tests with impact on the exterior side resulted in a HIC between 296 and 539, while the remaining four tests with impact on the interior side resulted in a HIC between 295 and 575. All

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seven experiments were performed with central impact on the windscreen. [73] showed three identical head impact experiments with the HIC ranging from 410 to 1084. Apart from laminated safety glass, head impact simulations on an automotive side window made of PMMA obtained HIC values between 21 and 3300 [75]. Some varieties of PMMA have a similar stochastic fracture behavior compared to glass. Fig. 7.10 shows the time between impactor contact and fracture for all simulations. An accumulation around one ms is observed. Within one simulation, the windscreen stays intact; no fracture occurred. The latest fracture within simulation occurred 10.002 ms after impact. The earliest fracture occurred at 0.2475 ms. A further new insight of the simulated head impact experiments is the fracture process. The situation of no glass fracture during a pedestrian head impact also leads to the conclusion that only one of the two glass layers fails. [52] summarizes the fracture process to three stages. During the first stage, the elastic stage, no fracture occurs. Stage two begins with the fracture of the interior glass ply in the case of an exterior impact. The described fracture sequence is a logical conclusion for laminated safety glass with two glass plies with identical fracture strength. The measurement of this sequence is a great challenge due to the high crack speed of glass (vmax ≈1500 m/s [202, 203]), the small amount of available experimental data and the fast sequential glass ply fracture times. The two glass plies can have very different strengths due to the stochastic strength distribution of glass. The stochastic nature of glass strength may vary the fracture sequence compared to two plies with identical fracture strength. Tab. 7.3 shows the fracture sequence for all 250 simulations. While 235 out of 250 simulations followed the assumed fracture sequence, 15 simulations did not. Six simulations have a reversed fracture sequence where the exterior ply failed before the interior ply. Eight simulations calculated only exterior glass fracture while the interior glass ply stayed intact.

Table 7.3 Fracture sequence of the glass plies within the windscreen during 250 stochastic simulations with head impact on the exterior glass ply.

glass ply fracture sequence interior → exterior exterior → interior only interior only exterior no fracture

amount 235 6 0 8 1

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8 Summary and Future Research Topics 8.1 Summary The main objective of this thesis was to develop a numerical model that is capable of reproducing and predicting the stochastic fracture behavior of glass. The major findings are: • A stochastic fracture model for glass

A stochastic fracture model for glass was derived based on the principles of fracture mechanics and the theory of probability. The stochastic fracture model is able to model different load cases, variations in the glass surface size as well as possible arbitrary geometries. A possible mesh dependency was examined. No influence of the FE size could be identified. Only a handful of parameters are required as input. These parameters have a clear physical meaning and can be commonly found in literature or determined by considering international standards. This facilitates the applicability of the model. Furthermire, it is not only possible to reproduce experimental data, but also to predict the stochastic strength of glass for arbitrary geometries of glass components. The stochastic fracture model developed in this thesis was selected to be made publicly available in an upcoming version of the FE solver Radioss.

• The stochastic strength of an automotive windscreen

Different strength populations of an automotive windscreen were investigated. The windscreen was divided into edge and surface strength. The surface was further divided into four sides, two for each glass ply. The silkscreen and the influence of the PVB layer were also considered. Strength distributions were estimated by means of a LTW distribution. It could be shown that the different strength populations vary significantly. The highest strength values were observed in samples with PVB cover. A significant reduction of strength was observed for imprinted samples. This also includes imprinted samples with PVB cover. It is indispensable to mentaion the fact that not all the work of estimating the strength of an automotive windscreen was done within this thesis. A list of all pertinent studies structured according to the corresponding authors is shown in Chap. 5.

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6_8

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8 Summary and Future Research Topics

In order to describe the experimental strength values by means of a stochastic distribution, the LTW distribution was utilized. Several methods for determining the lower bound for glass strength were reviewed. It could be shown that most methods overestimate the lower bound significantly, especially for samples of a small size. • The stochastic distribution of the head impact injury criterion

The stochastic distribution of the head impact injury criterion was calculated. Although the stochastic strength of glass is well known, it is not considered for pedestrian head impact tests on automotive windscreens in regulatory testing protocols. This may be attributed to an enormous experimental and financial effort for a certain amount of full-vehicle head impact experiments. Within this investigation, 250 stochastic simulations of a pedestrian head impact on the exterior windscreen side with an impact velocity of 36 km/h were performed. It is shown that for identical conditions with regard to the stochastic fracture behavior of glass, the head injury criterion varies between 411 and 1292. The calculated values follow an extreme value distribution with the highest density around 480. 1.6 % of the values were above the recommended upper bound of 1000. In addition, it was also observed that the calculated fracture process differs from the expected process. For laminated safety glass and an identical strength for both glass plies, the interior side should first fracture during an impact on the exterior side. This was the case for 235 out of 250 simulations. Six simulations had a reversed fracture sequence where the exterior ply failed before the interior ply. Eight simulations calculated only exterior glass fracture while the interior glass ply stayed intact. Within one simulation, no glass fracture occurred.

• Subcritical crack growth parameters in glass as a function of environmental

conditions In the present investigation, crack growth parameters from the linear approximation of subcritical crack growth are determined at different temperatures and humidity within a climate chamber. The importance of the environmental influence is given by the fact that, for example, the terminal crack velocity v0 increases by a factor of two between a humidity of 30 %rh and 70 %rh at a temperature of 35◦ C. The dependence of the parameters on the humidity could be fitted by a 2nd order polynomial.

8.2 Future Research Topics Based on the observations made in this study, several topics for further investigations are suggested. For the pedestrian head impact on an automotive windscreen,

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the post-fracture behavior of laminated safety glass needs to be investigated. In particular, the stiffness depending on the degree of fragmentation after fracture is a topic that has not been fully investigated yet. The main aim of further research should be the possibility of optimizing windscreens with regard to passive pedestrian protection. The influence of all parts of the glass production process on the stochastic strength of glass should also be classified. This would enables insights into specifically manipulating the windscreen strength in order to lower the head injury criterion without affecting the overall strength of the windscreen. A further suggestions for a highly relevant research topic is the possibility to represent long-term load cases. This would enable an application in the field of civil engineering, especially with thermally strengthened safety glass. For civil engineering applications, the crack growth threshold plays an import role to avoid glass fracture and should therefore also be investigated in more detail. Another interesting topic for further research is the increasing usage of chemically strengthened glass. Improved manufacturing processes make chemically toughened glass cheaper and therefore more widely applicable. Mostly used for mobile phone displays, high-tech lenses and the glazing of fighter jet cockpits in the past, its use as a structural element is conceivable as well.

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Appendix A Subcritical Crack Growth Parameters A.1 Experimentally Determined Parameters Table A.1 Results for crack growth parameters of dynamic fatigue tests in dependence of the environmental conditions

T [◦ C] 15 15 15

H [%rh] 50 60 70

R2 0.91 0.93 0.94

n 21.361 20.645 17.372

v0 [mm/s] 4.86 5.34 7.83

25 25 25 25 25

30 40 50 60 70

0.97 0.96 0.98 0.93 0.97

15.431 15.098 14.751 12.961 12.263

9.54 10.22 10.47 13.95 15.99

35 35 35 35 35

30 40 50 60 70

0.95 0.97 0.97 0.98 0.99

14.356 14.013 13.263 11.347 10.453

11.18 11.18 13.40 17.87 22.30

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6

135

136

A Subcritical Crack Growth Parameters

Table A.2 95 % confidential intervals for crack growth parameters of dynamic fatigue tests in dependence of the environmental conditions

T [◦ C] 15 15 15

H [%rh] 50 60 70

95 %-CI-n [18.916;24.497] [18.168;23.851] [15.477;19.760]

95 %-CI-v0 [mm/s] [3.66;6.26] [3.96;6.98] [5.98;9.98]

25 25 25 25 25

30 40 50 60 70

[14.758;16.167] [13.702;16.794] [13.966;15.620] [11.327;15.090] [11.361;13.306]

[8.65;10.49] [8.17;12.56] [9.28;11.76] [10.09;18.63] [13.42;18.85]

35 35 35 35 35

30 40 50 60 70

[12.812;16.289] [12.992;15.197] [12.221;14.482] [10.506;12.323] [9.845;11.136]

[8.56;14.24] [9.41;13.14] [11.11;15.97] [14.96;21.13] [19.52;25.49]

A.2 Literature Values

137

A.2 Literature Values Table A.3

Literature review of subcritical crack growth parameters for soda-lime silicate float glass

Environment

Test Method

n

v0

22.7◦ C, 50 %rH Water 50 %rH Values from [164]: Soda-lime silicate glass: 27◦ C, 65 %rH

dynamic fatigue mod. double beam* mod. double beam*

14.22 15.44 16.66

2.2 2.92 0.83

in-situ

19.721.2 21.8 21.1

0.20.4 2.6 2.4

25.626.0 25.9 22.1

11.621.8 2.3 6.1

16.0 18.1 16.0 16.0

50.1 2.47 4.51 8.22

[205] [205] [177] [177]

17.7

10.7

[206]

27◦ C, 65 %rH dynamic fatigue 27◦ C, 65 %rH dynamic fatigue Sodium aluminosilicate glass: 27◦ C, 65 %rH in-situ 27◦ C, 65 %rH 27◦ C, 65 %rH Extract of the summary Water Air, 50 %rH Laboratory, Summer Laboratory, Winter, 2◦ C Water

dynamic fatigue dynamic fatigue from [62]: in-situ in-situ derived from [205] derived from [205]

Values from 9 laboratories Water dynamic fatigue Water dynamic fatigue Water dynamic fatigue Water dynamic fatigue Values from [7], converted by [210]: 25◦ C, Water double-cantilever 25◦ C, 100 %rH double-cantilever

Reference [167] [204] [204]

26±7 3.7×107 [207] 18±1 19±4 [208] 20.1±0.7 28.8±6.4 [209] 19.9±0.7 6.4±1.4 [209] 17.4 20.8

3.8 3.6

138

A Subcritical Crack Growth Parameters

25◦ C, 30 %rH double-cantilever 25◦ C, 10 %rH double-cantilever 25◦ C, 0.017 %rH double-cantilever Vacuum double-cantilever Extract of the summary from [210]: Water unkown 50 %rH unkown Water unkown 50 %rH unkown 25◦ C, 45 %rH dynamic fatigue, Vickers intended *fitted from displayed data

22.6 21.4 27.2 93.3

1.7 0.6 0.09 0.13

13.0 14.3 18.4 19.7 18.8

1.1 0.16 17.1 2.8 14.3

[211] [211] [183] [183] [176]

Appendix B Windscreen Strength Test Results B.1 Coaxial Ring-on-Ring Tests Table B.1 Determined surface fracture stress values from CRRT correct by curvature influence for each windscreen strength population. Shown values are in MPa.

Pos. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

one-clear 48.26 52.59 55.87 61.05 65.62 67.09 72.75 76.66 78.31 84.65 85.29 87.97 91.81 92.70 94.60 101.80 103.10 104.54 108.07 110.78 113.78 110.37 116.35 118.53

two-silk 42.44 42.82 44.23 44.75 40.49 46.39 46.56 46.95 43.00 48.02 49.24 57.82 54.62 59.15 64.79 60.41 65.51 66.82 62.39 69.34 75.18 79.62 81.04 81.76

two-clear 157.11 182.88 240.79 324.04 327.20 322.37 351.83 346.86 345.66 362.43 370.55 371.36 377.92 376.14 385.18 388.33 392.09 398.18 391.63 410.80 409.23 413.84 414.54 438.81

three-clear 221.30 230.64 249.44 252.82 258.25 270.76 276.91 284.51 290.27 295.71 301.62 309.27 319.70 311.43 327.24 332.60 336.99 338.27 338.49 342.39 345.74 348.19 354.92 355.25

four-silk 44.44 46.40 52.81 54.28 55.52 70.11 71.58 74.92 77.25 76.60 84.63 86.42 87.70 91.98 93.54 97.74 102.22 99.64 122.50 90.08 104.45 108.73 -

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6

four-clear 87.39 99.12 114.64 112.54 109.60 122.96 136.23 120.75 126.11 133.33 140.62 139.08 146.97 153.09 145.66 151.79 142.52 158.28 159.85 161.27 160.13 161.04 173.85 172.35

139

140

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

B Windscreen Strength Test Results

124.74 122.75 125.29 147.14 150.14 155.07 160.13 283.08 123.62 101.98 49.02 75.57 105.16 114.04 135.84 107.68 135.83 53.83 123.76 84.37 73.72 -

86.22 85.68 85.73 90.52 91.63 76.22 102.74 89.33 57.75 95.25 88.37 89.32 72.92 69.16 -

436.65 425.56 441.41 493.91 518.18 353.93 429.93 359.01 355.52 375.26 385.88 383.93 247.36 327.01 75.65 337.53 413.72 79.57 421.52 246.35 87.90 -

358.62 360.66 371.84 384.02 400.54 420.27 448.67 453.37 505.82 378.04 463.02 299.76 256.52 333.83 253.64 370.43 267.65 354.11 288.10 284.09 396.77 280.41 323.10 390.05 277.44 320.12 331.49 147.52 278.82 435.98 330.40 348.70 383.91 274.00 434.71 454.67 415.95

-

164.36 165.57 168.07 215.35 238.87 78.56 146.47 104.58 111.23 117.44 213.50 118.51 126.16 118.39 93.90 123.08 118.70 88.73 111.49 78.86 130.93 121.40 274.94 138.79 -

B.2 Four-Point-Bending Tests

141

B.2 Four-Point-Bending Tests Table B.2 Determined edge fracture stress values from four-point-bending tests for each windscreen strength population. Shown values are in MPa.

Pos. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

inner ply edge 112.50 146.09 140.84 136.47 123.03 114.34 116.85 128.76 123.30 138.69 128.94 103.14 116.91 106.04 123.86 110.55 124.57 118.88 136.06 115.04 119.23 115.98 114.89 111.77 91.99 116.27 110.65 131.30 118.91 121.49

outer ply edge 97.05 69.90 81.96 82.77 93.76 90.89 70.50 101.52 94.76 103.25 70.89 103.65 105.04 107.90 71.62 108.72 67.87 65.22 78.17 68.66 69.96 80.63 61.88 86.94 85.70 100.45 78.85 78.11 95.71 100.44

142

31 32

B Windscreen Strength Test Results

102.70 124.34

-

Appendix C Geometric Correction Factor Shift Necessary correction factor fy fit for crack width influence on reverse subcritical crack growth calculation depending on the initial geometry factor Y . Code written in Fortran90: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

if (sig_drb.lt. 75) then p1 = 3.854846e−04∗n_sim∗∗4 + −2.565276e−02∗n_sim∗∗3 . + 6.142425e−01∗n_sim∗∗2 + −6.200668e+00∗n_sim + 2.450664e+01 p2 = −1.368162e−03∗n_sim∗∗4 + 9.175738e−02∗n_sim∗∗3 . + −2.220515e+00∗n_sim∗∗2 + 2.272542e+01∗n_sim + −9.012056e+01 p3 = 1.787600e−03∗n_sim∗∗4 + −1.208241e−01∗n_sim∗∗3 . + 2.955620e+00∗n_sim∗∗2 + −3.070972e+01∗n_sim + 1.231156e+02 p4 = −1.020375e−03∗n_sim∗∗4 + 6.954743e−02∗n_sim∗∗3 . + −1.721442e+00∗n_sim∗∗2 + 1.820610e+01∗n_sim + −7.466173e+01 p5 = 2.130927e−04∗n_sim∗∗4 + −1.464948e−02∗n_sim∗∗3 . + 3.670756e−01∗n_sim∗∗2 + −3.957401e+00∗n_sim + 1.785698e+01 elseif (sig_drb.ge. 75.and.sig_drb.lt.125) then p1 = −8.014637e−05∗n_sim∗∗4 + 6.234394e−03∗n_sim∗∗3 . + −1.757171e−01∗n_sim∗∗2 + 2.165919e+00∗n_sim + −7.843583e+00 p2 = 1.196065e−04∗n_sim∗∗4 + −9.673449e−03∗n_sim∗∗3 . + 2.757151e−01∗n_sim∗∗2 + −3.450433e+00∗n_sim + 9.667971e+00 p3 = −3.978835e−05∗n_sim∗∗4 + 3.830078e−03∗n_sim∗∗3 . + −1.122047e−01∗n_sim∗∗2 + 1.423220e+00∗n_sim + 1.058530e+00 p4 = 2.550177e−06∗n_sim∗∗4 + −6.309084e−04∗n_sim∗∗3 . + 1.772278e−02∗n_sim∗∗2 + −1.612462e−01∗n_sim + −4.326689e+00 p5 = −1.793787e−05∗n_sim∗∗4 + 1.431770e−03∗n_sim∗∗3 . + −3.888580e−02∗n_sim∗∗2 + 4.324152e−01∗n_sim + 5.858975e−01 elseif (sig_drb.ge.125.and.sig_drb.lt.175) then p1 = −2.446033e−04∗n_sim∗∗4 + 1.828184e−02∗n_sim∗∗3 . + −5.015157e−01∗n_sim∗∗2 + 6.034614e+00∗n_sim + −2.490241e+01 p2 = 5.067851e−04∗n_sim∗∗4 + −3.734857e−02∗n_sim∗∗3 . + 1.003949e+00∗n_sim∗∗2 + −1.185853e+01∗n_sim + 4.575989e+01 p3 = −3.258463e−04∗n_sim∗∗4 + 2.332497e−02∗n_sim∗∗3 . + −5.974454e−01∗n_sim∗∗2 + 6.706583e+00∗n_sim + −2.038952e+01 p4 = 7.386059e−05∗n_sim∗∗4 + −4.882414e−03∗n_sim∗∗3 . + 1.054173e−01∗n_sim∗∗2 + −9.057073e−01∗n_sim + −2.098349e+00

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6

143

144 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

C Geometric Correction Factor Shift p5 = −3.790448e−05∗n_sim∗∗4 + 2.752424e−03∗n_sim∗∗3 . + −7.083969e−02∗n_sim∗∗2 + 7.759309e−01∗n_sim + −8.351173e−01 elseif (sig_drb.ge.175.and.sig_drb.lt.225) then p1 = −1.002426e−03∗n_sim∗∗4 + 7.487536e−02∗n_sim∗∗3 . + −2.056697e+00∗n_sim∗∗2 + 2.464744e+01∗n_sim + −1.066757e+02 p2 = 3.118627e−03∗n_sim∗∗4 + −2.316144e−01∗n_sim∗∗3 . + 6.316496e+00∗n_sim∗∗2 + −7.506691e+01∗n_sim + 3.214670e+02 p3 = −3.609067e−03∗n_sim∗∗4 + 2.667937e−01∗n_sim∗∗3 . + −7.231503e+00∗n_sim∗∗2 + 8.529346e+01∗n_sim + −3.613532e+02 p4 = 1.870434e−03∗n_sim∗∗4 + −1.378567e−01∗n_sim∗∗3 . + 3.720583e+00∗n_sim∗∗2 + −4.361605e+01∗n_sim + 1.826124e+02 p5 = −4.094512e−04∗n_sim∗∗4 + 3.029387e−02∗n_sim∗∗3 . + −8.210301e−01∗n_sim∗∗2 + 9.660467e+00∗n_sim + −3.938020e+01 elseif (sig_drb.ge.225.and.sig_drb.lt.275) then p1 = −1.661365e−03∗n_sim∗∗4 + 1.251400e−01∗n_sim∗∗3 . + −3.471542e+00∗n_sim∗∗2 + 4.204654e+01∗n_sim + −1.854847e+02 p2 = 5.550941e−03∗n_sim∗∗4 + −4.160687e−01∗n_sim∗∗3 . + 1.147311e+01∗n_sim∗∗2 + −1.379743e+02∗n_sim + 6.037335e+02 p3 = −6.851821e−03∗n_sim∗∗4 + 5.116860e−01∗n_sim∗∗3 . + −1.404451e+01∗n_sim∗∗2 + 1.679347e+02∗n_sim + −7.296794e+02 p4 = 3.731562e−03∗n_sim∗∗4 + −2.780340e−01∗n_sim∗∗3 . + 7.608298e+00∗n_sim∗∗2 + −9.060341e+01∗n_sim + 3.911483e+02 p5 = −8.004821e−04∗n_sim∗∗4 + 5.976458e−02∗n_sim∗∗3 . + −1.639088e+00∗n_sim∗∗2 + 1.955932e+01∗n_sim + −8.338423e+01 elseif (sig_drb.ge.275.and.sig_drb.lt.325) then p1 = −1.929460e−03∗n_sim∗∗4 + 1.471697e−01∗n_sim∗∗3 . + −4.141752e+00∗n_sim∗∗2 + 5.098240e+01∗n_sim + −2.294842e+02 p2 = 6.697166e−03∗n_sim∗∗4 + −5.079334e−01∗n_sim∗∗3 . + 1.419800e+01∗n_sim∗∗2 + −1.733844e+02∗n_sim + 7.736174e+02 p3 = −8.537941e−03∗n_sim∗∗4 + 6.448618e−01∗n_sim∗∗3 . + −1.793464e+01∗n_sim∗∗2 + 2.176802e+02∗n_sim + −9.643518e+02 p4 = 4.765638e−03∗n_sim∗∗4 + −3.590089e−01∗n_sim∗∗3 . + 9.951988e+00∗n_sim∗∗2 + −1.202820e+02∗n_sim + 5.297103e+02 p5 = −1.023363e−03∗n_sim∗∗4 + 7.720270e−02∗n_sim∗∗3 . + −2.143459e+00∗n_sim∗∗2 + 2.594356e+01∗n_sim + −1.131898e+02 elseif (sig_drb.ge.325.and.sig_drb.lt.375) then p1 = −1.839328e−03∗n_sim∗∗4 + 1.429385e−01∗n_sim∗∗3 . + −4.107221e+00∗n_sim∗∗2 + 5.173982e+01∗n_sim + −2.389880e+02 p2 = 6.608847e−03∗n_sim∗∗4 + −5.095222e−01∗n_sim∗∗3 . + 1.450735e+01∗n_sim∗∗2 + −1.808588e+02∗n_sim + 8.261478e+02 p3 = −8.661755e−03∗n_sim∗∗4 + 6.640298e−01∗n_sim∗∗3 . + −1.878160e+01∗n_sim∗∗2 + 2.323316e+02∗n_sim + −1.052092e+03 p4 = 4.935803e−03∗n_sim∗∗4 + −3.770376e−01∗n_sim∗∗3

C Geometric Correction Factor Shift

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

. + 1.061849e+01∗n_sim∗∗2 + −1.306599e+02∗n_sim + 5.876649e+02 p5 = −1.067185e−03∗n_sim∗∗4 + 8.159522e−02∗n_sim∗∗3 . + −2.300259e+00∗n_sim∗∗2 + 2.832827e+01∗n_sim + −1.262924e+02 elseif (sig_drb.ge.375) then p1 = −1.487196e−03∗n_sim∗∗4 + 1.192492e−01∗n_sim∗∗3 . + −3.542728e+00∗n_sim∗∗2 + 4.624246e+01∗n_sim + −2.215987e+02 p2 = 5.590149e−03∗n_sim∗∗4 + −4.422194e−01∗n_sim∗∗3 . + 1.294667e+01∗n_sim∗∗2 + −1.663471e+02∗n_sim + 7.845554e+02 p3 = −7.575999e−03∗n_sim∗∗4 + 5.938425e−01∗n_sim∗∗3 . + −1.720949e+01∗n_sim∗∗2 + 2.186274e+02∗n_sim + −1.018855e+03 p4 = 4.421525e−03∗n_sim∗∗4 + −3.445888e−01∗n_sim∗∗3 . + 9.921046e+00∗n_sim∗∗2 + −1.250843e+02∗n_sim + 5.777186e+02 p5 = −9.673316e−04∗n_sim∗∗4 + 7.539378e−02∗n_sim∗∗3 . + −2.170851e+00∗n_sim∗∗2 + 2.736578e+01∗n_sim + −1.251497e+02 end if fy = p1∗Y∗∗4. + p2∗Y∗∗3. + p3∗Y∗∗2. + p4∗Y∗∗1. + p5

145

Appendix D Newman and Raju Equation Stress-intensity factor equation for surfaces cracks from [132]. It is assumed that bending is not reduced within the geometric factor. Since all lengths values enter as quotients, care must be taken to use the same length dimension. Parameter definition: a: c: t: b: φ:

depth of the surface crack half-width of the surface crack surrounding continuum thickness half-width of surrounding continuum parametric angle of the elliptic crack

The geometric factor Y can be expressed by r 1 Y = F, Q with the shape factor for elliptical cracks Q as  a 1 Q = 1 + 1.464 .65 c and the stress-intensity boundary-correction factor as   a 2  a 4  F = (M1 + M2 + M3 fφ gfw . t t The parameters M1 , M2 and M3 as well as g can be expressed to: a M1 = 1.13 − 0.09 , c

M2 = −0.54 +

M3 = 0.5 −

0.89 , 0.2 + a/c

1 0.65 +

h  a i24
+ 14 1 − a c c

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 C. Brokmann, A Model for the Stochastic Fracture Behavior of Glass and Its Application to the Head Impact on Automotive Windscreens, Mechanik, Werkstoffe und Konstruktion im Bauwesen 63, https://doi.org/10.1007/978-3-658-36788-6

(D.1)

(D.2)

(D.3)

(D.4)

(D.5)

(D.6) 147

148

D Newman and Raju Equation

and   a 2  g = 1 + 0.1 + 0.35 [1 − sin(φ)]2 . t

(D.7)

The angular function from the embedded elliptical-crack solution fφ is given as   0.25 a 2 2 2 fφ = cos(φ) + sin(φ) . c

(D.8)

The finite width correction fw is expressed as 



πc fw = sec 2b

r 0.5 a . t

(D.9)