A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation 365834329X, 9783658343293

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Mechanik, Werkstoffe und Konstruktion im Bauwesen | Band 59

Marcel Berlinger

A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation

Mechanik, Werkstoffe und Konstruktion im Bauwesen Band 59 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-, Schädigungs- und Versagensverhaltens von Werkstoffen, Bauteilen und Tragstrukturen, die Entwicklung neuer Materialien, Produktionsverfahren und Gebäudetechnologien sowie deren Anwendung im Bauwesen unter Berücksichtigung sicherheitstheoretischer Überlegungen und der Energieeffizienz, konstruktive Aspekte des Umweltschutzes sowie numerische Simulationen von komplexen Stoßvorgängen und Kontaktproblemen in Statik und Dynamik.

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Marcel Berlinger

A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation

Marcel Berlinger Institut für Mechanik und Materialforschung Technische Hochschule Mittelhessen Gießen, Deutschland Die vorliegende Schrift wurde als Dissertation am Promotionszentrum für Ingenieurwissenschaften am Forschungscampus Mittelhessen unter Federführung der Justus-Liebig-Universität Gießen in Kooperation mit der Technischen Hochschule Mittelhessen am 14. Mai 2021 angenommen. 1. Gutachten: Prof. Dr.-Ing. habil. Stefan Kolling 2. Gutachten: Prof. Dr. Sangam Chatterjee Tag der Einreichung: 11. November 2020 Tag der mündlichen Prüfung: 14. Mai 2021 Gießen 2021 Zeughaus

ISSN 2512-3238 ISSN 2512-3246  (electronic) Mechanik, Werkstoffe und Konstruktion im Bauwesen ISBN 978-3-658-34329-3 ISBN 978-3-658-34330-9  (eBook) https://doi.org/10.1007/978-3-658-34330-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 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

Affidavit I declare that I have completed this dissertation single-handedly without the unauthorized help of a second party and only with the assistance acknowledged therein. I have appropriately acknowledged and cited all text passages that are derived verbatim from or are based on the content of published work of others, and all information relating to verbal communications. I consent to the use of an antiplagiarism software to check my thesis. I have abided by the principles of good scientific conduct laid down in the regulations of the leading University which were delivered to me in carrying out the investigations described in the dissertation.

Gießen, 11.11.2020

Marcel Berlinger, M.Sc.

Danksagung In diesem Moment liegt vor mir eine beinahe vollendete Dissertationsschrift. Zum Verfassen dieser Zeilen erinnere ich mich zurück an den mitunter steinigen Weg, der hierher führte. Ich denke an all jene, die sein Beschreiten erst ermöglichten. Die Arbeit entstand im Rahmen meiner Tätigkeit als wissenschaftlicher Mitarbeiter am Institut für Mechanik und Materialforschung der Technischen Hochschule Mittelhessen. Sie basiert auf den Ergebnissen des Forschungsprojekts SimPlex – Entwicklung einer Simulationsmethodik zur Berechnung des Crashverhaltens von Automobilverglasungen aus Plexiglas® . Dieses Projekt mit der Nummer 526/17-08 wurde im Rahmen der Innovationsförderung Hessen aus Mitteln der Landes-Offensive zur Entwicklung Wissenschaftlich-ökonomischer Exzellenz (LOEWE) in der Förderlinie 3 KMU-Verbundvorhaben finanziert und fachlich mit vielen wertvollen Anregungen und Diskussionen vom Fachgremium Simulation von Verbundglas der Forschungsvereinigung Automobiltechnik (FAT) begleitet. Der Projektpartner Evonik Performance Materials GmbH stellte dazu ein beträchtliches Kontingent an Prüfmaterial zur Verfügung. Mein Dank gilt allen, die zum Erfolg dieser Arbeit beigetragen haben. Ein ganz besonderer Dank geht an Professor Stefan Kolling, der mir die Chance zu diesem Promotionsvorhaben gab. Durch seine Förderung gelang es mir sowohl fachlich, als auch menschlich zu wachsen. Ich blicke zurück auf eine sehr wertvolle Zusammenarbeit. Ebenso möchte ich mich herzlich bei Professor Jens Schneider vom Institut für Statik und Konstruktion der Technischen Universität Darmstadt für seine fachliche Begleitung bedanken. In zahlreichen, fruchtbaren Diskussionen half er von der ersten Stunde an, die Arbeit reifen zu lassen. Auch bedanke ich mich bei Professor Sangam Chatterjee vom Physikalischen Institut der JustusLiebig-Universität Gießen für seine Betreuung und die Unterstützung des Promotionsvorhabens. Sein Blick fiel aus neuen Winkeln auf die Arbeit. Weiterhin bedanke ich mich vielmals bei meinen Kollegen und den studentischen Hilfskräften am Institut, die mir stets mit Rat und Tat zur Seite standen. Abschließend möchte ich mich bei meiner fürsorglichen Familie und meinen tollen Freunden bedanken, auf deren bedinungslose Unterstützung ich immer vertrauen konnte. Ihnen ist diese Arbeit gewidmet.

Lich, im November 2020

Marcel Berlinger

Abstract Acrylic glasses, as well as mineral glasses, exhibit a high variability in tensile strength. That is a crucial uncertainty factor in the dimensioning of structural parts. The current work presents the modeling proposal on a rate-dependent stochastic failure criterion for numerical simulation. Scope of application is on an automotive rear-side window in context of a head-impact crash scenario. For a common basis on statistics, an overview of established probability estimators and the relevant families of probability distributions is provided. Important goodnessof-fit tests are discussed, and a generalization of the Anderson-Darling test proposed. An algorithm is prepared to compute the new test’s significance levels by Monte-Carlo simulation. Whenever reasonable, the test is applied in the statistical analyses in this work. Experimental database for statistics are fracture strains, which are gained from uniaxial tensile tests in an amount hitherto unseen for acrylic glasses. Regarding two PMMA materials, a wide span of strain rates is examined. The requirements for the experimental test systems are accordingly versatile. The gathered samples are inspected critically. For each sample, the probability distribution of fracture strain is estimated using the introduced tools. Based on these, a rate-dependent modeling of the fracture strain distribution is developed, from which a stochastic failure criterion is deduced for finite-element simulation. It is adopted in simulation of an experimental head impact test on the automotive rearside window. On the resulting dispersion of the head injury criterion, the relevance of statistical material-characterization for product safety is discussed.

Kurzfassung Acrylgläser weisen, genau wie Mineralgläser, eine hohe Streuung in ihrer Zugfestigkeit auf. Dies ist ein kritischer Unsicherheitsfaktor in der Auslegung von Strukturbauteilen. Die vorliegende Arbeit stellt den Modellierungsansatz eines ratenabhängigen, stochastischen Bruchkriteriums für die numerische Simulation vor. Anwendungsfall ist eine automobile Heckseitenscheibe im Zusammenhang mit einem Kopfaufprall-Crashszenario. Für eine allgemeine Grundlage zur Statistik wird ein Überblick über etablierte Wahrscheinlichkeitsschätzer und relevante Familien von Wahrscheinlichkeitsverteilungen gegeben. Wichtige Goodness-of-Fit-Tests werden besprochen und eine Generalisierung des Anderson-Darling-Tests vorgeschlagen. Ein Algorithmus zur Berechnung der Signifikanzniveaus des neuen Tests durch Monte-Carlo-Simulationen wird vorbereitet. Stets wenn es sinnvoll ist, wird der Test auf die statistischen Analysen dieser Arbeit angewendet. Experimentelle Grundlage für die Statistik sind Bruchdehnungen, die in einem bisher unvergleichbaren Umfang für Acrylgläser aus einachsigen Zugversuchen gewonnen werden. Unter Betrachtung zweier PMMA-Materialien wird eine weite Spanne an Dehnraten untersucht. Entsprechend vielseitig sind die Anforderungen an die experimentellen Prüfsysteme. Die erlangten Stichproben werden kritisch untersucht. Für jede Stichprobe wird die Wahrscheinlichkeitsverteilung der Bruchdehnung unter Zuhilfenahme der vorgestellten Instrumente geschätzt. Darauf aufbauend wird eine ratenabhängige Modellierung der Bruchdehnungsverteilung entwickelt, aus der sich ein stochastisches Versagenskriterium für die Finite-Elemente-Simulation ableitet. Dieses wird in der Simulation eines experimentellen Kopfaufprallversuchs auf die automobile Heckseitenscheibe aufgegriffen. Anhand der resultierenden Spreizung des Kopfverletzungskriteriums wird die Relevanz einer statistischen Materialcharakterisierung für die Sicherheit von Produkten diskutiert.

Contents Glossaries

xi

1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Achievements of this Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6

2 Fundamentals 2.1 Poly(methyl methacrylate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Material Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Probability Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 13 16 20 23

3 Generalized Anderson-Darling Test 3.1 Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Chi-Squared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Kolmogorov-Smirnov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Cramér-von Mises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Anderson-Darling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Monte-Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 29 31 32 33 34 37

4 Experimental Investigations 4.1 Test Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Specimen Machining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Strain Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 47 51 54 58

5 Sampling 5.1 Removal of Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rate Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Data Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 67 70

xiv

Contents

6 Statistical Modeling 6.1 Probability Distribution Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Quantile Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Fracture Strain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 81 86

7 Stochastic Simulation 93 7.1 Head Impact Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.3 HIC Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8 Experimental Basis for Model Enhancements 111 8.1 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.2 Stress-State Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9 Summary

123

References

127

A Test A.1 A.2 A.3 A.4

Statistic Derivation 141 Cramér-von Mises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Anderson-Darling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Generalized Anderson-Darling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Lower-Tail Generalized Anderson-Darling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B Tensile Test Results 147 B.1 Strain-Time Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.2 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 C Fracture Strain Samples 151 ® C.1 Uniaxial Tension Plexiglas 8N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 C.2 Uniaxial Tension Plexiglas® Resist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 C.3 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Glossaries Abbreviations 2D 2PW 3D 3PW AD BLW BMW CAD CDF CoD CS CvM DIC DMA DMTA DPW ECE EDF

two dimensional two-parameter Weibull three dimensional three-parameter Weibull Anderson-Darling bilinear Weibull bimodal Weibull computer-aided design cumulative distribution function coefficient of determination chi squared Cramér-von Mises digital image correlation dynamic mechanical analysis dynamic mechanical thermal analysis Department of Public Works Economic Commission for Europe empirical distribution function

finite element generalized Anderson-Darling goodness of fit head injury criterion high-strength steel Japan New Car Assessment Programme KS Kolmogorov-Smirnov MIT Massachusetts Institute of Technology OEM original equipment manufacturer PC polycarbonate PDF probability density function PMMA poly(methyl methacrylate) SUV sport utility vehicle TSG tempered safety glass USA United States of America WPE Weibull’s probability estimator WRSS weighted residual sum of squares FE GAD GoF HIC HSS JNCAP

Subscripts 0 ∞ crit curr el eng ex

initial, residual infinite critical current elastic engineering expected

f G h lat LT max MC

fracture generalized hauling lateral lower tail maximum Monte-Carlo

xvi

min norm occ orig pl

Glossaries

minimum normal occurred original plastic

rand

random

true

true, Hencky

UT

upper tail

vM

von Mises

i, j k

indices total number of classes, image index length eigenvalue, wavelength, location parameter triaxiality mean value normal force normal vector total number, normal vector component degrees of freedom cumulative distribution function occurrence probability, pressure weight function strain vector displacement Cauchy’s stress tensor stress, true stress, standard deviation rotation tensor roughness parameter coefficient of determination fracture mirror radius temperature, thickness dimension time, thickness shear stress stretch tensor voltage elongation, value of cumulative distribution function velocity

Symbols 1 A A2 Am a α b β C D Δ E E erf ε ε ε˙ F F f G g γ H H0 HIC η I

identity matrix area Anderson-Darling test statistic fracture mirror constant resultant acceleration, lower integration bound significance level polynomial coefficient, upper integration bound decay coefficient, shape parameter stress-optical coefficient Kolmogorov-Smirnov test statistic phase difference Hencky strain tensor Young’s modulus error function strain tensor strain, true strain strain rate deformation gradient force frequency, probability density shear modulus linear regression line shear strain, location parameter humidity, height dimension null hypothesis head injury criterion viscosity, scale parameter invariant

l λ m μ N n n ν P p ψ s s σ σ R R R2 rm T t τ U U u v

xvii

W W2 WRSS w  X x

width dimension Cramér-von Mises test statistic weighted residual sum of squares width, mixture weight position vector in initial configuration position vector in current configuration

x

abscissa of Cartesian coordinate system

χ2

chi-squared test statistic

y

ordinate of Cartesian coordinate system

z

applicate of Cartesian coordinate system

ω2

Cramér-von Mises criterion

9

1 Introduction 1.1 Motivation The weight of a vehicle is an essential factor for its fuel consumption, or rather energy demand when we think of electric cars. Reason are driving resistances like inertial or tractional forces the engine has to overcome. In the year 2008, the Massachusetts Institute of Technology (MIT) published a report [122] quantifying the potential for reducing fuel consumption by weight savings. Regarding the average car in the United States of America (USA), 100 kg less weight effectuates a decrease in gasoline consumption by 0.40 L/100km, regarding the average light truck even by 0.49 L/100km. Efficiency is a subject that ecologists, economists, and engineers alike appreciate. Today’s ways to reduce a vehicle’s total mass are manifold. Primarily, a substitution of structural steel parts is realized by aluminum, high-strength steel (HSS), magnesium, glass-fiber or carbon-fiber reinforced polymer composites. A subsystem that has received little attention so far is the glazing. To the present day,

Figure 1.1 Plexiglas® granulate (left) – the raw material of injection molded components [HA Hessen Agentur GmbH – Jan Michael Hosan]. As procduct of chemical syntheses, its basic compounds are hydrogen cyanide and acetone [54], which both are obtainable from one and the same plant, such as the European beech (right). Hydrogen cyanide from the fruit [170] and acetone by dry distillation of wood [11].

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_1

2

1 Introduction

automotive glazing is almost entirely made from mineral glasses. Its percentage on the total mass of a passenger vehicle is about 3–5 %. A promising substitute is acrylic glass. Since, its density is lower than the one of mineral glass, weight reductions up to 50 % are possible. A beneficial side effect of reducing the glazing’s mass is the lowering of the vehicle’s center of gravity, resulting in better driving dynamics, what in turn allows weight savings in the chassis construction. Besides, acrylic glasses feature additional convincing properties for use in automotive glazing. They show a high light transmission and clarity, with a simultaneous reduced transmission of infrared radiation, causing less heating of the interior of the car. Unlike polycarbonate (PC), its optical properties are highly weather resistant and show no aging effects as yellowing. Compared to mineral glasses, the higher failure strains provide more effective resistance against stone damage. In order to gain a comparable scratch resistance, hardcoatings are applied. Another advantage in occupant comfort is the acoustic damping produced by the viscous material behavior. In addition to these mechanical properties, the manufacturing also offers advantages. Figure 1.1 shows the raw material. In the processing of acrylic glasses, unlimited free-form designs with all kinds of coloring are thinkable. In comparison with reference systems as single-layer safety glass, laminated safety glass, or PC, its life cycle assessment is rated positive. By chemical conversion into its starting materials, it can be completely recycled. Thus, sustainability is ensured. By all these benefits, the adoption of acrylic glasses is highly requested by automotive original equipment manufacturer (OEM). So far, the use is mostly limited to racing sports. For an integration into street-legal cars, there still are open issues. The development of new cars is almost entirely carried out via finite element (FE) simulation. Along with the stability of structural parts, important aspects

Figure 1.2 A real car’s glazing is regarded (left). It is the model for the studied rear-side window prototypes made from acrylic glasses (right). When made from conventional mineral glass, the window’s mass is 1.6 kg, whereas the substitutes achieve 0.8 kg.

1.2 State of the Art

3

are the safety for occupants and pedestrians in a crash scenario. For that purpose, simulation models on the material’s stress-strain behavior are already introduced, though a substantial experimental and theoretical research for a reliable prognosis on the point of failure is still missing. By the current work, a capable approach is presented. As scientific research object, rear-side window prototypes of a sport utility vehicle (SUV), shown in Figure 1.2, are considered.

1.2 State of the Art The subject of this work is categorized into the topics of polymer mechanics and statistical analyses regarding poly(methyl methacrylate) (PMMA), which is not to be confused with paramethoxymetamphetamine (PMMA) [89]. A groundwork on the polymer’s mechanical side is [92]. A good overview on experimental research and constitutive modeling of solid polymers is provided by [94], where among others, acrylic glass is considered. In the work of [20] the general methodological approaches are further deepened. Moreover, [102] is another example for a practical adoption. Application of polymers ranges from quasi-static load conditions up to ballistic load cases [148], having numerous different requirements on the material. Thus, the state of the art is divided into PMMA’s stress-strain and failure behavior. Furthermore, focus is placed on materials with already established stochastic failure models and onto general statistical fundamentals.

Stress-Strain Behavior of PMMA PMMA is known to show certain response to mainly three influencing factors, which are the type of load case, the temperature, and the rate a load is applied with. Individual considerations are conducted on the stress-strain behavior’s rate dependence under compression [128], rate dependence under tension [177], and rate dependence under shear [65]. For high compression rates with load application times in the range of 20 microseconds, commonly split Hopkinson pressure bar setups are utilized [95]. Also, modifications of the split Hopkinson pressure bar for dynamic tensile tests on PMMA are developed [37]. Furthermore, over a wide range of temperatures and strain-rates the yield behavior under compression [17, 142, 143], and likewise under tension [18] is examined, yet. A characterization of PMMA in dynamic mechanical analysis (DMA) is provided by [64]. Besides, approaches are introduced for a thermo-mechanical coupling of the influencing factors in a constitutive modeling [10, 27].

4

1 Introduction

Failure Analyses of PMMA Beyond stress-strain relations, there are several studies on the fracture behavior. On one side, there is the analysis of PMMA plates and PMMA-aluminum sandwich structures in dynamic impact penetration tests [50, 112, 131, 145]. In those tests no information is gained on fracture strain. Comparable low-velocity penetration and quasi-static puncture tests are performed by [134], though without digital image correlation (DIC) analysis for receiving a continuous strain field. Impact and compression damage on PMMA spheres with focus on the occurred fracture types are studied for a wide range of velocities in [75], again without a strain information. Focus is placed on the forms of failure. A study on the mechanical fracture modes applied to PMMA is provided by [46]. Concentrating on Mode I, [13, 106] analyze the time-temperature dependent fracture toughness, with regard to crazing mechanisms. Cracking and crazing of tensile specimens within different liquid environments are examined in [9]. In further experimental approaches, modifying a split Hopkinson pressure bar for four-point bending test, the fracture toughness is determined for different rates [172]. Unfortunately, the amount of measurements is not sufficient for an information on fracture statistics. Applications of PMMA Acrylic glasses are applied in a wide field. Already valued in glazing for their prominent optical properties, they still exhibit potential for further optimization, as shown in the refraction index modifications of [110]. The material has good processing capabilities, and is effectively to be joined by welding, as the fracture studies of [114] on welded interfaces state. [26] was able to detect a certain failure distribution underlying a PMMA-PMMA connection. A general analysis on the statistical height fluctuations of fracture surfaces is conducted by [156]. Besides adoption in mechanical and civil engineering structures, especially in the medical sector there is a lot of research on the material. Self-curing surgical bone cements based on PMMA are utilized to attach metallic prostheses, like hip and knee replacements, to living bone. Since, fracture of the bone cement would lead to function failure of the whole device, its statistical strength is very closely examined and characterized in Weibull probability distributions [141, 168, 169]. The bone cements origin lies in the developments for dental surgery. Nowadays, as substitute for ceramics, or for gold or cobalt-chrome-molybdenum alloys with or without veneer, PMMA-based polymer materials are widely established in temporary and even permanent dental restorations [79, 91, 147, 176, 181]. In dental application as well, a component failure must be prevented to protect the patient. In this case against swallowing or aspirating fragments of a failed prosthesis. Impact-resistant fiber-reinforced PMMA composites are conceivable to enhance strength [167]. Besides, Weibull

1.2 State of the Art

5

probability distributions are adopted to predict the material’s statistical strength [74, 138, 185] and fatigue [83, 179]. The first FE modeling and simulation of dental prothesis made from PMMA is [157, 158, 159]. Failure Statistics of Glass Weibull statistics are commonly used in practice to describe the stochastic failure of brittle materials. Primarily, two nearly ideal-brittle materials are considered. These are ceramics [44, 101, 164], and mineral glasses. The glasses show a distinct distribution of flaws, surface defects and cracks [30], reducing their theoretical maximum strength [132, 133], and provoking their stochastic failure [180]. Special attention is paid to glasses’ edge strength [55, 93, 111]. Since, critical flaws occur statistically, they are the more probable, the larger the specimen or component is. Thus, the fracture stress distribution is normalized to an experimentally examined reference volume or area [15, 136]. With regard to automotive glazing, which is main scope in this current work, the reference to stochastic failure of windscreens is given [29, 31]. Contrary to the research on PMMA, for mineral glasses in that field, failure models exist [30]. They base on linear-elastic fracture mechanics and are therefore not to be transferred to PMMA. Statistical Fundamentals In practice, order statistics play an important role. For the user they are quick and easy to handle. Experimental sample points are ordered by their magnitude and assigned to an estimated occurrence probability, providing coordinates on which distribution models are fitted [16, 81, 115]. In the simplest way, the fit is conducted as linear regression in a space of particularly transformed abscissa and ordinate. Beneficial in linear regression is the gain of confidence and prediction limits according to [175]. The probability estimator being most sufficient is controversial discussed. [22] did an early comparison of the thitherto proposed representatives. [116] first gave a mathematical argumentation of Weibull’s probability estimator being the only valid one. That provoked a discussion of critical responses [38, 39, 80] and defenses of the initial statement [117, 118, 119, 120, 121], which unfortunately leaves the uninitiated user in confusion. As outlined in the previous paragraphs, the Weibull probability distribution is widely established in fracture analysis. Though, excluding other distribution families right from beginning would be a mistake, since in some fields preferences are historically evolved and have to be scrutinized, as invoked by [52]. For the Weibull distribution [165, 173] there are several fit procedures and modifications that must be regarded. A good overview for fit techniques is provided by [45]. A bias in common maximum-likelihood estimation of Weibull parameters is stated by [36], providing an analytic bias-corrected

6

1 Introduction

maximum-likelihood estimation. On the other hand, [21] introduces a weighted linear regression fit for Weibull parameters, that corrects the error made by simple linear regression in a Weibull plot. As already discussed for flaw distribution in mineral glasses, the Weibull distribution is to normalize on reference volume or length [109, 129], which is related to weakest-link theory [107, 139]. Common modifications of the Weibull distribution are either the extended versions, as the Marshall-Olkin extended Weibull distribution discussed in [35, 40, 70, 178, 183], and the truncated Weibull distributions for cropped populations [68]. A comprehensive collection of generalized Weibull distributions is [104]. Another important field are the tests for goodness of fit (GoF), whose most important members are introduced later in this work. These tests show an asymptotic convergence for infinite samples. Since, in practice finite samples are the base for analyses, works are published providing tables for a sample-size optimized significance level allocation to the test statistics, as for the two-sample Cramér-von Mises (CvM) test [6]. Alternatively, the test statistic itself is adapted in dependence on the sample size [63, 161, 162, 163]. However, due to the massive increase in today’s computing capacity, asymptotic results are determined without the restrictions of researchers had to deal with in the past [108].

1.3 Achievements of this Work The scientific added value involves three essential milestones, material characterization, statistical groundwork and stochastic simulation. A hitherto unseen experimental database for PMMA materials is collected. More than 850 tensile tests are performed and individually analyzed for local fracture strain, in order to compile sample sizes that enable statistic profound conclusions. On top of that, about 100 additional shear and puncture tests are gathered, likewise individually analyzed. In the course of the experimental studies, test setups and techniques are proposed, each specifically adapted on the properties of the material. A guideline is demonstrated to produce fracture strain samples for subsequent statistical analyses. A material independent description on the statistical methodologies for probability distribution modeling is provided. Therefore, the most relevant families of probability distribution functions are introduced, and an inquiry is made, in order to collect all the probability estimators developed over the years. The application of the most important GoF tests is explained, and their advantages and disadvantages discussed. Based on the popular Anderson-Darling test, a new generalization is proposed that enables the consideration of arbitrary plotting positions offside the assignments of the empirical distribution function (EDF). A Monte-Carlo algorithm is established to calculate the significance levels of the new test statistic.

1.3 Achievements of this Work

7

For practical application, a model is proposed to describe the strain rate dependent stochastic fracture behavior of PMMA. Weight is set to the distribution function’s tails to enhance its relevance for structural designs that require a profound information on the material’s strain above which failure might occur, or information on the maximum expected deformation capacity of a component. The model provides a robust failure criterion that is directly to be added to existing non-stochastic FE simulation setups. The approach is material independent. On a concrete example, the importance of a statistical material characterization for product safety is demonstrated.

Publications of the author: - M. Berlinger, P. Schrader, S. Kolling: Analyses on the strain-rate dependent fracture behaviour of PMMA for stochastic simulations. Proceedings of the 15th German LS-DYNA Conference (2018). - C. Brokmann, M. Berlinger, S. Kolling, P. Schrader: Fraktographische Bruchspannungs-Analyse von Acrylglas. ce/papers 3.1 (2019): pp. 225-237. - M. Berlinger, S. Kolling, J. Schneider: Statistical characterization and stochastic finite-element simulation of PMMA. Proceedings of the 4a Technology Day (2020). - S. Kolling, M. Berlinger: Stochastic Fracture of Polymer Materials. Carhs Automotive CAE Companion (2020), pp. 45–47. - M. Berlinger, S. Kolling, J. Schneider: A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation. Proceedings of the 16th International LS-DYNA Conference (2020). - M. Berlinger, S. Kolling, J. Schneider: A generalized Anderson-Darling test for the goodness-of-fit evaluation of the fracture strain distribution of acrylic glass. Glass Struct. Eng. (2021).

9

2 Fundamentals Before the research topic is discussed, in this chapter the necessary groundwork is compiled. Conventions are established and definitions made, that are consistently used in further considerations. To begin with, the examined acrylic glasses are classified in a chemical context. Basic understanding for the micro processes that cause their mechanical material behavior are given. Model conceptions are introduced to describe these properties in the relationship of stress, strain, and time. The terms stress and strain are discussed in detail, and the principles of optical strain measurement explained. Finally, for statistical treatment of the collected experimental data, a selection of distribution functions is presented and a methodology for the fit of their parameters introduced. This chapter is based on the information from several fundamental books [59, 66, 77, 78, 87], user guides [73], papers [15, 23], and PhD theses [136, 150], to which the reader is referred to.

2.1 Poly(methyl methacrylate) Polymer materials are high-molecular organic compounds. They are the product of chemical reactions in which many (Greek πoλ´ υ , poly) low-molecular monomers are bonded into chains. Polymers are classified into three groups, the thermosetting plastics, the thermoplastics, and the elatsomers. Thermosetting plastics consist of a tight cross-linked mesh of molecular chains. The cross-links prevent the chains of sloughing, providing a stiff and strong material that is infusible for reprocessing. Elatsomers are also cross-linked, but in a wider mesh with fewer links, as schematically illustrated in Figure 2.1. The intermolecular forces are comparatively weak. Thus, elastomers are capable of high strains. Between temperatures below 0 ◦C and their respective decomposition temperature, they show a rubber-like elastic behavior. Often the term rubber is used synonymously for elastomers. Thermoplastics exist in semicrystalline and amorphous structures. With regard to PMMA, the main focus will be placed on the latter modification. An amorphous state is the statistical distribution of molecular chains without any chemical bonds connecting these chains, as shown in Figure 2.2. It is to be imagined as clew without any ordering. The clew is the most favorable state in terms of entropy. An important factor for the material properties is the clew density. Closer approaching chains experience higher intermolecular forces, leading to a higher stiffness and strength, © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_2

10

2 Fundamentals

Figure 2.1 Representation of the microstructure of elastomers. The slightly cross-linked polymer chains (left), return after elongation (middle) into the initial form (right). Red dots mark the crosslinks.

but difficult processing properties like high viscosity of the melt. Due to their transparent and clear appearance, amorphous thermoplastics are called organic glasses. Visually, they only differ slightly from inorganic mineral glasses. They are processed above fusion temperature, and can in principle be fused and re-molded as often as required. The reactions for polymerization usually take place at the manufacturer of the raw material. For the definition of polymer materials, the terminology of constitution, configuration, and conformation is introduced. The constitution is the chemical structure given in the structural formula. It provides information on the existing bonds between atoms in the molecule, its molar mass, type and length of the arborizations, type of end groups, and on the integration of foreign atoms and molecules. Configuration defines the permanent spatial arrangement of atom bonds within one molecule. In contrast, conformation is the spatial arrangement of all atoms within one molecule. It is defined by the degrees of rotational freedom around single bonds within the molecule. Thus, a molecule’s shape is not fixed. Two molecules of same constitution are of different configuration if they cannot be transferred into each other by rotation around articulated bonds, i.e. by conformational change. All three characteristics of the polymer’s structure affect the properties of the material. The energy in the bond of atoms, i.e. covalent bond energy, increases the resistance to high temperatures. A low molar mass causes a low fusion temperature and low fusion viscosity, allowing lower processing pressures. High intermolecular forces and a conformation allowing less rotatability of atoms increase stiffness and glass transition temperature. Furthermore, conformation and configuration of substituents, which are single atoms or a group of atoms replacing one or more hydrogen atoms in the molecule, influence polarization, and therefore the chemical and impact resistance. To some degree, all polymers show elastic, viscous and plastic behavior. As stated, the molecular chains seek the state of highest entropy. Elasticity describes the process by which a system returns to this initial state after being deformed by an external force. The sudden load with such external force leads to a time dependent response of the material. The strain viscosity arises from the inhibited mobility of

2.1 Poly(methyl methacrylate)

11

Figure 2.2 Representation of the microstructure of thermoplastic materials. The tangled polymer chains (left), return after elongation (middle) only partially into the initial form (right). The residual deformation is either permanent, or decaying.

the chains, their inhibited ability of sloughing. Compared to the covalent bonds inside a molecular chain, the intermolecular forces between chains, which mainly are dispersion forces that are part of the van-der-Waals force, are weak. The energy that has to be overcome intermolecular forces is dependent on the distance between molecules, their entanglement, and the temperature. When deformed over the yield limit, plasticity occurs, which is the permanent rearrangement of chains. By their orientation, the molecular chains define isotropy and anisotropy of the material. We distinguish between compression, leading to a change of volume, and shrinkage as the volume-constant re-clewing of orientated molecules. In most monomers, carbon atoms bond the macro molecules. That is true for the methyl methacrylate, too. Its structural formula is given to CH3

H

.

C

C

C

H

O CH3

O

Methyl methacrylate is the monomer from which through polymerization the polymer poly(methyl methacrylate) is created. Polymerization is an exothermic reaction. The polymers are in a state of lower energy as the unsaturated monomers. For PMMA, the process is started by a radical R that induces an unpaired electron into the molecule. Active principle is the radical polymerization H R

+

CH3 C

C

R X

H

H

CH3

C

C

H

X

,

where X is in case of methylmethacrylat the substituent C(O)OCH3 . The unpaired electron attacks another molecule’s carbon atom, and so the chain

R

H

CH3 H

CH3

C

C

C

C

H

X

H

X

H +

CH3 C

H

C X

12

2 Fundamentals

R

H

CH3 H

CH3 H

CH3

C

C

C

C

C

C

H

X

H

X

H

X

grows. The polymerization is terminated when two chain ends meet with their unpaired electrons. The resulting PMMA polymer is then ⎤ ⎡ H CH3 ⎥ ⎢ ⎥ ⎢ ⎢C C ⎥ ⎦ ⎣ H C n O O CH3 with n as the degree of polymerization, giving the number of monomers in the molecular chain. In general, polymers’ degree of polymerization lies in a range of 100 to 4 million. Plexiglas® 8N In the present work, two different acrylic glasses are examined. Both PMMA materials are products of the Evonik Performance Materials GmbH under the registered trade name Plexiglas® . Both acrylic glasses are developed for the use in automotive applications. One is the Plexiglas® 8N, for which the manufacturer claims high mechanical strength, surface hardness, abrasion resistance, and light transmission, as well as a very good weather resistance. Information on further material properties are given in Table 2.1. A detailed information on the chemical structure is not publicized, only the classification as amorphous thermoplastic. Its polymer chains might be conformed of equal molecules, i.e. homopolymerization. In that case, additional polymerization would follow the mechanism of the abovementioned chain reaction. Application examples are optical waveguides, luminaire covers, automotive lighting, instrument cluster covers, optical lenses, displays, etc. The raw Plexiglas® 8N material is provided either as extruded plates of 3 mm

Table 2.1

Manufacturer information on Plexiglas® 8N [57].

density

Young’s modulus

fracture strain

1190 kg/m2

3300 MPa

5.5 %

glass transition luminous temperature transmittance 117 ◦C

92 %

2.2 Material Models

13

thickness, or as injection molded tensile specimens following ISO 527-2 Typ 1A, cf. Figure 4.6. The commercial product is sold as granulate. Plexiglas® Resist (AG 100) The second examined material is an acrylic glass from Evonik Performance Materials GmbH specially developed for automotive glazing. It has approval to ECE R43 [53], which is required for certification for this application. The Plexiglas® Resist AG 100 is an PMMA modified with elastomers to increase impact resistance. The manufacturer states a fracture strain of 45 %, cf. Table 2.1, which is over eight times the fracture strain of 8N. The precise method for this modification is not publicized. A copolymerization is unlikely. In this process foreign atoms, or molecules are included in the polymer chains. The tacticity is either of statistical, or of regular manner. A categorization as PMMA would no longer be given. Another possibility is the Resist AG 100 being a polymer blend. Again foreign molecules are used to influence the mechanical properties, but they are not part of the molecular chain, only added to the melt. A physical blend of PMMA polymers with elastomers is thinkable. Both approaches have in common that they increase the impact resistance of the basic thermoplastic. In either case, the polymerization is processed at the manufacturer. The raw material is again sold as granulate. As research material, extruded plates of 3 mm thickness are provided. Table 2.2

Manufacturer information on Plexiglas® Resist AG 100 [58].

density

Young’s modulus

fracture strain

1160 kg/m2

2200 MPa

45 %

glass transition luminous temperature transmittance 112 ◦C

91 %

2.2 Material Models Main aim of this work is the characterization of PMMA’s stochastic fracture. This is the continuation of the investigations of [150]. In dependence on strain rate, load case, and temperature the PMMA was modeled in its stress-strain behavior, but without the regard of statistical effects for fracture. Before the research on the latter is started, first the adopted engineering models are introduced briefly. Plexiglas® 8N is primarily a visco-elastic material. In common definition, viscoelasticity is defined as the observation of rate dependent stress-strain behavior, involving creep and relaxation. Is the material set under sudden loading and stress is hold constant, the strain will continuously increase toward a limit εt=∞ . This is

14

2 Fundamentals

the effect of creep. On the other hand, relaxation is the decrease of stress towards a limit σt=∞ when strain is hold constant after sudden loading. The difference between the instantaneous stress σt=0 and this limit σt=∞ is called overstress. For a perfect visco-elastic material the viscous parts completely decay at a relaxation period of t = ∞. The underlying mechanism can be approximated by rheological models, which describe the visco-elastic material behavior by a system of springs and dashpots. By neglecting plastic effects, the whole stress response is modeled by the combination of linear-elastic springs, i.e. Hooke elements, and linear dashpots, i.e. Newton elements. The series connection between both elements is the Maxwell model, which is shown in Figure 2.3. In order to increase complexity in the model, an arbitrary amount of standard Maxwell elements are parallelized with a single Hooke element to become a generalized Maxwell model. The combination of a single Hooke element with a single Maxwell element is the standard linear solid model. For a 5-parameter generalized Maxwell model the stress response is given by the Prony series  



 E2 ε E1 ε + η2 ε˙ 1 − exp − . (2.1) σ(ε, ε) ˙ = E∞ ε + η1 ε˙ 1 − exp − η1 ε˙ η2 ε˙ This system accords with the developed material model used in this work. Also a full linear visco-elastic model of eighteen parallel Maxwell elements is introduced, derived by the results of a dynamic mechanical thermal analysis (DMTA). With regard to computation performance in later stochastic simulations, the five-parameter engineering model is preferred, whose parameters are fitted to experimental tensile tests at various strain rates. To comply the requests of the FE solver, the system is described by shear moduli Gi instead of Young’s moduli Ei for the spring ele-

ı

ı

ı ...

Ș

E

Eᅹ

Ș1

Șn

E1

En

Gᅹ

ȕ1

ȕ2

G1

G2

...

ı

ı

ı

Figure 2.3 Maxwell model (left), n-parameter generalized Maxwell model (middle), and Plexiglas® 8N material model (right).

2.2 Material Models

15

Figure 2.4 Fracture patterns of Plexiglas® 8N (top) and Plexiglas® Resist (bottom) tensile specimens. For the latter, crazing can be observed.

true stress σ

ments, and by decay coefficients βi = Gi /ηi replacing the pure viscosities ηi for the dashpots of the generalized Maxwell. Plexiglas® Resist is due to contained elastomer particles not to describe by a visco-elastic approach. Besides reversible elastic strains, permanent deformations occur when exceeding a distinctive strain limit. These become visible for highly strained areas by a change in material appearance, from crystal clear to a whitish discoloring, which is called crazing. It is the rise of plenty small surface cracks, hardly visible by the eye. In Figure 2.4 crazing is demonstrated at a Resist tensile specimen. The difference in material behavior becomes obvious in the comparison to a Plexiglas® 8N tensile specimen, which fractures brittle. Another characteristic of the Resist is necking. In the stress-strain curve that is a decline of the stress level, followed by a stabilization phase, as shown for the experimental curve in Figure 2.5. Regarding the illustration of Figure 2.1, that effect is explained by orientation of the polymer chains into a predominant direction. As approximation

experiment material model Hookean

true strain ε Figure 2.5 Visco-plastic material model for an increasing strain rate compared to quasi-static experimental stress-strain curve.

16

2 Fundamentals

of the material behavior for numerical simulation, a visco-plastic model is adopted. Young’s modulus is set independent of the strain rate, and fitted via linear regression on the quasi-static stress-strain curve, considering all points up to a assessed yield stress σyield . Above σyield a hardening curve is specified. In contains the deviation between total strain ε and linear-elastic strain ε given by Hooke’s law σ = E ε.

(2.2)

ε = εel

(2.3)

Total strain is defined as

for ε ≤ εyield , and ε = εel + εpl =

σyield + εpl E

(2.4)

for εyield < ε. As Young’s modulus, the hardening curve is fitted to quasi-static experiments. Since the use within a numerical material model is aimed, one aspect must be regarded. The FE solver prohibits negative gradients in hardening curves. In consequence, necking cannot be reproduced. Thus, the local stress peak is not considered in the modeled hardening curve, as demonstrated in Figure 2.5. Unlike for Young’s modulus, rate effects are taken into account for hardening. As engineering approach, for higher strain rates the hardening curve is scaled, as exemplarily shown in Figure 2.5. Later for FE simulation, the hardening curves are provided at reference strain rates, between which the FE solver interpolates arbitrary rates.

2.3 Stress and Strain An essential part in this present work is the examination of stresses and strains. One is a quantity directly to measure, whereas the other is derived with several assumptions. To begin with, the nomenclature shall be defined. We start with the one-dimensional concept of stresses. Generally, stress is the force F referred to an area. Is the force orthogonally orientated to this area, it is denoted a normal force N . For a infinitesimal small area stress is

ΔN dN (2.5) = σ = lim dA ΔA→0 ΔA

2.3 Stress and Strain

17

as asserted by the Cauchy principle. The referred area is either in initial configuration A0 , or current configuration A. The engineering approach refers force to A0 of the undeformed state. For a defined body, the engineering stress becomes N . A0

σeng =

(2.6)

Positive is the simple determination of this stress, since the area must not be viewed in the deformation process. But an error is accepted when in reality the area changes. The stresses in this work are all referred to the current area A of the deformation process. Is is the true stress σtrue =

N . A

(2.7)

In practice, the current area is usually not precisely to be determine, because not all the necessary strain information are provided. Hence, models are adopted. For the calculations in this work, the applied model is always stated.

The second quantity is strain. With regard to stress definition, it is the link between undeformed and deformed area. Its definition is raised from an one dimensional perspective. We imagine a body that is deformed in direction x. It has the initial length l0 and the current length l in deformed state. Again, there is an engineering approach that defines strain as the sum of infinitesimal length change du referred to an initial length increment dx by εeng :=

du dx

⇒

εeng =

Δl l0

(2.8)

for the body in total. For higher strains, the measure of εeng shows certain deviation to a bodies actual strain. Hence, another definition is raised, the one of Hencky’s strain, or true strain. It is the integration of an infinitesimal change of elongation dl related to the current length l, as dl dεtrue := l

⇒

εtrue = ε



l ˜ l dl dε = . = ln ˜l l0

(2.9)

l0

This is the strain measure used from this point on. Since hitherto only one dimension is considered, it is generalized to all three dimensions. The complexity of the strain tensor increases from a scalar to a 3x3 matrix. On the diagonal, strains are listed in direction of the coordinate system xyz, whereas on the secondary diagonals the shear components are listed, with the shear strains γij = 2εij . This

18

2 Fundamentals

coordinate system is arbitrary, hardly allowing a comparison of strain information with a body in a different configuration. Therefore, the strain tensor ⎡

εx ε = ⎣εxy εxz

εxy εy εyz

⎤ ⎡ εxz εx εyz ⎦ = ⎣ 12 γxy 1 εz 2 γxz

1 2 γxy εy 1 2 γyz

⎤ 1 2 γxz 1 ⎦ 2 γyz εz

(2.10)

is transformed into its principal directions. The treatment is entirely mathematical like for any other tensor. Basis is the constraint that the eigenvalues of a tensor must satisfy the equation !

det(εε − λ 1) = 0

(2.11)

of the characteristic polynomial. The solution of this so-called eigenvalue problem is given by the Cayley-Hamilton theorem. It states λ3 − I1 λ2 + I2 λ − I3 = 0 .

(2.12)

with the invariants I1 = tr(εε) = εx + εy + εz

(2.13a)

I2 = εx εy + εy εz + εx εz − ε2xy − ε2yz − ε2xz I3 = det(εε) = εx εy εz + 2εxy εyz εxz − εx ε2xy − εy ε2xz − εz ε2xy .

(2.13b) (2.13c)

There are three solutions for the third-degree polynomial of Eq (2.12), the eigenvalues λ1 , λ2 , λ3 . They are arranged that λ1 ≥ λ2 ≥ λ3 . Resuming to the strain tensor, its principal strains are given to ⎡ λ1 ε=⎣0 0

0 λ2 0

⎤ ⎡ ε1 0 0⎦=⎣0 λ3 0

0 ε2 0

⎤ 0 0⎦. ε3

(2.14)

This simplified expression, is independent of a local coordinate system, enabling the definition of material properties for material modeling. A nice visualization of the relationship between principal strains and the shear components of a rotated coordinate system is Mohr’s three-dimensional circle, here in the form of a strain circle instead of the more popular stress circle. Figure 2.6 gives an example. Three circles are constructed, each from two of the three three principal strains. Their center points are given by (εi + εj )/2, and their radii by (εi − εj )/2 for εj ≤ εi . The circle’s tops mark the maximum shear strain εij , respectively the bottoms with reversed sign the minimum ones. In this plot, all possibly strain points, which result from an arbitrary rotation of the strain tensor’s

2.3 Stress and Strain

19

shear strain γ/2

εxz,max εxy,max

εyz,max 0 −εyz,max

−εxy,max −εxz,max

ε3

0

ε2

ε1 normal strain ε

Figure 2.6 Mohr’s three-dimentional strain circle. In shaded area, blue the positive and red the negative shear strains possible from rotation of the reference coordinate system.

reference coordinate system, lie on or around the 1-2 and 2-3 circles, and on or inside the 1-3 circle. A rotation of the reference coordinate system is to imagine as a slanted cut through a cube defined in principal coordinates. The cut plane is orthogonal the unit normal vector

(2.15) |n| = n21 + n22 + n23 = 1 . On the plane, normal strain is calculated from the principal strain components by ε = ε1 n21 + ε2 n22 + ε3 n23 .

(2.16)

The definitions of the strain vector, s = ε n and |s | = relationship ε2 +

γ2 = ε21 n21 + ε22 n22 + ε23 n23 4

↔



ε2 + γ 2 /4, provide the

γ = ± ε21 n21 + ε22 n22 + ε23 n23 − ε2 , 2

(2.17)

from which the shear strain γ/2 derives. Figure 2.6 demonstrates the position of all the possible shear strains as shaded area. The transformation approaches for the strain tensor are to adopt for the stress tensor σ , too. Analogously to the eigenvalues of ε , those of the stress tensor ⎡

σx σ = ⎣τxy τxz

τxy σy τyz

⎤ τxz τyz ⎦ σz

→

⎡ σ1 ⎣0 0

0 σ2 0

⎤ 0 0⎦ σ3

(2.18)

20

2 Fundamentals

Table 2.3

Triaxiality m for the different load cases.

biaxial compression

uniaxial compression

pure shear

uniaxial tension

biaxial tension

σ1 = −σ σ2 = −σ σ3 = 0

σ1 = −σ σ2 = 0 σ3 = 0

σ1 = σ σ2 = −σ σ3 = 0

σ1 = σ σ2 = 0 σ3 = 0

σ1 = σ σ2 = σ σ3 = 0

-2/3

-1/3

0

1/3

2/3

are determined. By these principal stresses an additional information on the predominant load case is deduced. The relation of hydrostatic pressure p over von Mises stress σvM is defined as triaxiality m=−

p σvM

=

1 3 (σ1 + σ2 + σ3 ) 1 2

[(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 ]

.

(2.19)

This expression is a measure of the stress state within a body. In case of uniaxial tension, the principal stresses σ2 , σ3 become zero, and the triaxiality is calculated to 1/3. Is tension equally in two principal directions, the triaxiality is 2/3 with σ1 = σ2 = σ. The corresponding compression states are with negative sign. Differently for pure shear, σ1 and σ2 are reversely signed, resulting in m = 0. An overview is listed in Table 2.3. Besides these main cases, all intermediate forms are possible. The hydrostatic states σ1 = σ2 = σ3 = ±σ are left out.

2.4 Digital Image Correlation Basic principle of DIC is the comparison of a bodies current deformed state to a reference state, which is mostly chosen to be the undeformed body. Images are taken either from a dual camera system, or a single camera system. Both types are utilized in this work and are depicted in Figure 2.7. The dual camera system enables three-dimensional analyses based on triangulation. There are several ways to prepare the specimens surface for the DIC software to recognize points, or patterns from one image to another. There are point markers that can be adhered and tracked for their displacement. For a more continuous analysis of surface strain, a stochastic paint pattern is applied. The surface is primed with white and speckled with black color, or vice versa. The succession of coloring depends on the characteristics of the specimen and the desired accuracy. For a material with matt

2.4 Digital Image Correlation

21

Figure 2.7 3D measuring system with 12 megapixels resolution (left) and single high-speed camera with 1 megapixels resolution (right).

black, or matt white surface, the priming can be omitted when the speckle color is chosen to be the respective contrast color. Matt colors are important to reduce reflection. All specimens in this work are primed white, since the examined PMMA is clear, and speckled black, because the utilized water-based acrylic spray paint produces smaller speckles than the white one. Smaller speckles combined with a high resolution camera system produce high accuracy. Important is to begin the experiments before the paint is dried-out. That would cause a brittle layer, tearing apart with higher strains of the surface. The optimum condition is a dry skin with flexible paint underneath. Thus, as soon as the paint becomes matt on its surface the specimens are tested. To recognize and track patterns, the software lays so-called facets over the speckled surface. These are square areas, within which the speckle pattern is unique, due to the stochastic spray mist of the spray paint. With a standard facet size of 19x19 pixels and allocation of a gray value from 0 to 255 there are 25619·19 possible constellations. Due to this uniqueness, facets can be identified in the images from one camera of the three dimensional (3D) DIC system to the other. By the principle of triangulation, a three-dimensional information is gained. Figure 2.8 illustrates the facets in over-sized form as green rectangle in the undeformed state and in the deformed state. In reality the facets are overlapping for higher surface resolution. Each facet has a center point, which is the interception of the diagonals. Three adjacent center points form a triangular surface. Six triangles are combined to one hexagonal surface that one facet center point becomes the apex of six triangles. In such hexagon the facet center points are projected onto one averaging plane. The positions of the projected points are used for continuum

22

2 Fundamentals

Figure 2.8 Principle of digital image correlation analysis. Tracking the facets and their center points from the reference image (left) to a deformed state (right).

mechanical strain calculation. The deformation gradient F of the hexagonal surface is determined as derivative F=

∂x  ∂X

(2.20)

 of the of the vector x of the deformed configuration with respect to the vector X initial configuration from the reference image. Thus, the deformation gradient F contains both, the rotational and the stretch part, which is needed for strain calculation. By polar decomposition, the deformation gradient is divided in F = R U,

(2.21)

the rotation tensor R and the right stretch tensor U. The product of the deformation tensor and its transpose then is FT F = UT RT R U = UT 1 U = U2 , leading to the calculation of the right stretch tensor as √ U = FT F .

(2.22)

(2.23)

Finally, the strain tensor for Hencky strain, or true strain is calculated by E = ln(U) .

(2.24)

2.5 Probability Distribution Functions

23

For strain calculation and deformation gradient determination a single triangle surface would be sufficient. By considering an averaging hexagonal plane with seven points, an overestimated equilibrium for the deformation gradient is given. A weighting of the involved points dependent on the examined position and an overlap of many hexagonal planes over the specimen’s surface, increases the strain measurement accuracy. Thus, the DIC analysis is not directly comparable to an FE analysis. Neither the facet size, nor the triangular surfaces accord with FE elements.

2.5 Probability Distribution Functions The probability distribution of occurrences can be reproduced by multi-parametric distribution functions. These functions take two forms. Once, there is the probability density function (PDF) f (x). In case of the popular normal distribution its function curve has the characteristic bell-shape. On the other hand, there is the cumulative distribution function (CDF) P (x), which is the integral x P (x) =

f (˜ x)d˜ x

(2.25)

−∞

of the PDF. While a PDF’s diagram gives more a visual impression of the frequency of single occurrences, the CDF plot offers the readout of explicit occurrence probabilities. In the present study, the distribution of PMMA fracture strains εf,i (i = 1, 2, 3, ..., n) is tested for agreement with selected probability distribution functions from different distribution families, that are listed in Table 2.4. Since tensile tests are performed on specimens clear of residual stresses [33], no negative fracture strains are expected to occur. Hence, all distribution functions are considered for an argument εf ∈ [0, ∞). The function parameters are consistently defined as -

mean value μ ∈ R, standard deviation σ ∈ R+ , shape parameter β ∈ R+ , scale parameter η ∈ R+ , and location parameter γ ∈ R.

The most common distribution function in science is the normal, or Gaussian distribution. It is defined by two parameters, which give a tangible information on the constellation of the sample. Its parameter μ is simultaneously expected value

24

2 Fundamentals

Table 2.4 List of regarded distribution functions, featuring the families of normal, Weibull, logistic, Gumbel, and Cauchy distribution.

name

cumulative distribution function  

Normal

P (εf ) =

1 1 εf − μ + erf √ 2 2 σ 2

Log-Normal

P (εf ) =

1 1 ln(εf ) − μ + erf √ 2 2 σ 2

2-Parameter Weibull

P (εf ) = 1 − exp −

Log-Weibull

P (εf ) = 1 − exp − exp

(2.26)



 (2.27)

    β εf η



3-Parameter Weibull

P (εf ) =

Left-Truncated Weibull

P (εf ) =

Bilinear Weibull

⎧ ⎨0

 

(2.28)



εf − γ η

⎩1 − exp − ⎧ ⎨0  ⎩1 − exp −

εf − ln(η) 1/β

(2.29) for 0 ≤ εf < γ

β 

β εβ f −γ



for γ ≤ εf

 for 0 ≤ εf < γ for γ ≤ εf

ηβ

    ⎧ εf β1 ⎪ ⎪ 1 − exp − for 0 < εf ≤ ε∗f ⎨ η1     P (εf ) = ⎪ ε β2 ⎪ for ε∗f < εf ⎩1 − exp − f

(2.30)

(2.31)

(2.32)

η2

Bimodal Weibull

        ⎧ εf β2 εf β1 ⎪ ⎪ − exp − ⎨1 − exp − η η 1 2    P (εf ) =  β2  β1 ⎪ ε ε f f ⎪ + ⎩+ exp − η1



 

Gumbel

P (εf ) = exp − exp −

Logistic

P (εf ) =



 

1 + exp −





Log-Logistic

P (εf ) =

εf 1+ exp(γ)

Cauchy

P (εf ) =

1 1 + arctan 2 π

η2

εf − γ η

εf − γ η



(2.34)

−1 (2.35)

−1/η −1 

(2.33)

εf − γ η

(2.36)

 (2.37)

2.5 Probability Distribution Functions

25

and median of the sample, and its parameter σ is the square-root of the variance, i.e. the standard deviation of the sample. The corresponding CDF is given in Eq. (2.26), where the expression erf(x) is the error function, which is defined by x 2 erf(x) = √ exp(−˜ x2 )d˜ x. (2.38) π 0 Its probability density is symmetric about μ, showing the familiar bell-shaped curve. A relative is the log-normal distribution in Eq. (2.27), with μ and σ as median and standard deviation of the natural logarithmized random variable εf . In analyses on the stochastic fracture behavior of glasses and ceramics, typically Weibull distributions are applied. The CDF of the two-parameter Weibull (2PW) distribution is given in Eq. (2.28). It has the scale parameter η, which is approximately to be interpreted as the 63th percentile, since P (η) = 1 − exp[−1] ≈ 0.63, and the parameter β. In technical application, β, or Weibull modulus is often used to describe the dispersion of fracture stress; the larger β becomes, the smaller the dispersion. Why this interpretation is not valid for the examined samples of fracture strain is exemplarily demonstrated in Section 6.2 on a real sample. A more general form of the 2PW is obtained by adding a location parameter γ, resulting in the three-parameter Weibull (3PW) distribution of Eq. (2.30). The parameter γ describes a lower limit, below which no observations are expected, i.e. only in excess of the null strain εf = γ fracture appears. Similarly, the so called left-truncated Weibull distribution in Eq. (2.31) provides a lower limit γ, as well, but by truncation of the 2PW distribution’s probability density function. That changes the interpretation of the lower limit: fracture at strains below γ is possible, but not represented in the sample. Furthermore, the CDFs of the log-Weibull, the Gumbel, the logistic, log-logistic, and Cauchy distribution are provided in Eq. (2.29,2.34–2.37), cf. [3, 66, 144]. For the logarithmic members, the variable εf is distributed log-normal, log-logistic, or log-Weibull when respectively ln(εf ) is normal, logistic, or 2PW distributed. In case the population features two modes, as potentially indicated by the empirical data, the bilinear Weibull (BLW) distribution in Eq. (2.32) offers the reasonable approach of splitting the sample in two sets and assuming an 2PW distribution for each. The intersection ε∗f of the two 2PW distributions derives from equating both functions and solving for  ε∗f

=

η1β1 η2β2



1 β1 −β2

.

(2.39)

At transition the differentials of both 2PW distributions differ, unless η1 = η2 and β1 = β2 , which is trivial. For a smoother transition, the bimodal Weibull (BMW)

26

2 Fundamentals

distribution of Eq. (2.33), see [15], merges the two separate functions together. Its parameter values differ from the ones of a corresponding BLW distribution and can not simply be transferred. Biggest difficulty in the fit of probability distributions upon a sample is the probability assignment to each occurrence since the population is unknown. Over time many probability estimators were introduced for various applications. An overview of these estimators is given in Table 2.5. It is noteworthy that Eq. (2.42) gives the mean rank and Eq. (2.44) the median rank plotting positions [24, 45, 174].

Table 2.5

Overview of the present probability estimators

year

reference

probability estimator

1914

Hazen [82]

pi =

1923

California DPW [34]

1939

Weibull [174]

1943

Beard [19]

pi =

i − 0.31 n + 0.38

(2.43)

1953

Bernard & Bos-Levenbach [24]

pi =

i − 0.3 n + 0.4

(2.44)

1958

Blom [25]

pi =

i − 0.375 n + 0.25

(2.45)

1962

Tukey [166]

pi =

3i − 1 3n + 1

(2.46)

1963

Gringorten [76]

pi =

i − 0.44 n + 0.12

(2.47)

1975

Filliben [61]

i − 0.5 n i pi = n i pi = n+1

pi =

Cunnane [42]

0.365 ⎪ ⎩ n +1/n

i−c pi = n − 2c + 1 pi =

1979

Landwehr et al. [105]

1991

McClung & Mears [125]

2001

Yu and Huang [182]

i − 0.4 n + 0.2

i − 0.35 n i − 0.4 pi = n i − 0.326 pi = n + 0.348

pi =

(2.41) (2.42)

⎧1 − p i=1 i ⎪ ⎨ i − 0.3175 0.5

1978

(2.40)

i = 2, 3, ..., n − 1

(2.48)

i=n for 0 ≤ c ≤ 1 (2.49) (2.50) (2.51) (2.52)

2.5 Probability Distribution Functions

27

In case of the estimator in Eq. (2.41) published by the California Department of Public Works (DPW) the leading author is unknown, so typically it is referred to as the California method. As discussed later, this estimator is the one commonly used for goodness-of-fit evaluations. Cunnane showed a generalized estimator, to which many of the others can be attributed to by an appropriate choice of c in Eq. (2.49). He suggested c = 0.4. Their application is simple. The n measured fracture strains are sorted in ascending order εf,1 ≤ εf,2 ≤ ... ≤ εf,n that position i = 1 holds the minimum fracture strain and position i = n the maximum one. Then, only dependent on position i in the sorting a corresponding occurrence probability pi is assigned using one of the probability estimators. The generated coordinates (εf,i | pi ) are the plotting positions. To fit the CDFs from Section 2.5 on the empirical CDF of the plotting positions, the function parameters that minimize the weighted residual sum of squares (WRSS) WRSS =

n 

  [pi − P (εf,i )]2 ψ P (εf,i )

(2.53)

i=1

P

y = ln[− ln(1 − P )]

are determined, in analogy to the later discussed weightened goodness-of-fit tests [7, 160]. In Eq. (2.53) pi is the ith plotting position, P (εf,i ) is the function value for the corresponding ith fracture strain and ψ(u) is a weight function, for which examples are given in Section 3.2. The fit via WRSS is adopted for all introduced distribution functions, even for normal, and log-normal distribution, to have the same conditions for all functions. Indeed, by appropriated transformation of the coordinate system the 2PW, BLW, log-Weibull, Gumbel, logistic, log-logistic and Cauchy distribution can be brought into a linear form. In such transformed coordinate system their

εf

x = ln(εf )

Figure 2.9 Cumulative distribution function of a 2PW distribution (left) transformed into a corresponding Weibull plot within the same axes limits (right).

28

2 Fundamentals

CDF is simply fitted to the plotting positions by linear regression. Advantageous are the provided confidence and prediction limits of the linear regression. But by a transformation of the coordinate system, an unwanted weighting of residuals is brought into the fit. Impressively, that is visualized in Figure 2.9. The 2PW distribution is brought into the form β ! εf → ln[− ln(1 − P )] = β ln(εf ) −β ln η . (2.54) P = 1 − exp − " #$ % "#$% " #$ % " #$ % η y

b1

x

b2

which corresponds to the line definition y = b1 x + b2 . A CDF diagram on transformed axes like x = ln(εf ) and y = ln[− ln(1 − P )] is called Weibull plot. One could perform a linear regression as fit of the parameters b1 and b2 , but as indicated by the diagram’s grid and the length of line dashes, the resolution is not consistent anymore, and so is the weighting of residuals. Makkonen[116] warns against this fit of observations onto a model, instead of fitting a model onto the observation. So, to avoid that instance and keep the weights comparable, all CDFs are fitted by minimization of the WRSS.

9

3 Generalized Anderson-Darling Test The quality of a modeled distribution function for describing the distribution of a set of random variables is evaluated in statistical hypothesis testing. A null hypothesis H0 is assumed, which claims the sample and the fitted probability distribution to have the same population [8]. Thereby, two errors can be made. A type I error, which is the rejection of a true null hypothesis, and a type II error, which is the non-rejection of a false null hypothesis. In goodness-of-fit tests the type I error is quantified, one would make by rejecting the null hypothesis [149]. Typically, the outcome of these tests, the so-called test statistic, is compared to tabulated critical values, which are dependent on sample size and significance level [163]. The significance level α is the probability for the rejection of a true H0 . The procedure is explained in detail for the different goodness-of-fit tests in this section. As an illustrative example, a real sample of 50 fracture strains, coming from tensile tests on Plexiglas® 8N specimens, is analyzed, cf. [23]. Using the approaches from Section 2.5, the occurrence probability pi for each of the fracture strains εf,i (i = 1, 2, 3, ..., n) is estimated by choosing one of the equations from Table 2.5. Null hypothesis H0 is the fracture strains to have the population of a given probability distribution function P (εf ).

3.1 Goodness-of-Fit Tests 3.1.1 Chi-Squared A commonly used testing method is the so-called chi squared (CS) test [41] that verifies whether a sample arises from an assumed CDF. The sample is sorted in ascending order and divided into k classes. The choice of each classes’ width should provide a similar number of expected observations nex among them, which are determined by integration of the PDF in the corresponding interval, and should not undercut an amount of five [136]. For each j th class the amount of occurred observations nocc is compared to the number of expected observations and summarized into k  (nocc,j − nex,j )2 χ2 = . (3.1) nex,j j=1

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_3

30

3 Generalized Anderson-Darling Test

This test statistic χ2 is a measure for the height of deviation between expectation and occurrence. One might use this measure as rating of probability distributions functions among each other. To make an absolute statement about a goodness-offit, the CS distribution has to be considered. Dependent on the degrees-of-freedom ν = k − 1, the CS distribution provides the probability density for a particular test statistic χ ˜2 by ˜2 /2) χ ˜ν−2 · exp(−χ , (3.2) ˜2 ) = fν (χ 2ν/2 · Γ(ν/2) where Γ is the so-called gamma function [66]. The probability PH0 for a random sample of the population to generate a test statistic greater or equal to χ2 is determined by ∞ 2 ˜2 )dχ ˜2 . (3.3) PH0 (χ ) = fν (χ χ2

probability density fν [-]

Typically, when PH0 exceeds a significance level α = 5%, the null hypothesis, which claims a correlation between measurements and assumed distribution, is accepted. In statistical terms, the null hypothesis is falsely rejected, which is the type I error, to a probability of PH0 . A notional example of this test evaluation is visualized in Figure 3.1. A disadvantage of the CS test is the vague test statistic χ2 for small sample sizes, since for those the choice of the classes’ width affects significantly the test result. With increasing sample sizes, this effect declines. In order to generate a distinct test statistic, a database of about 50 measurements is relatively small and thus prone to the mentioned binning effect.

0.1 f ν (χ ˜2 )

0.08

χ2 α = 5%

0.06 0.04 0.02 0 0

5

10

15

20

25

30

test statistic χ ˜2 [-] Figure 3.1 Chi-squared distribution for ν = 10 degrees-of-freedom. The significance level is α = 0.05. The notional test statistic χ2 has a probability PH0 > α, thus the test outcome is statistical significant.

3.1 Goodness-of-Fit Tests

31

3.1.2 Kolmogorov-Smirnov

A testing method that is more suitable for small sample sizes is the KolmogorovSmirnov (KS) test [162]. Here, the experimental data is considered as empirical CDF Pn (εf ), i.e. EDF, which is defined through Pn (εf ) =

number of observations ≤ εf , n

(3.4)

cf. [7]. Thus, for the measured fracture strains εf,i the occurrence probabilities Pn (εf,i ) are just the ones of the California method from Eq. (2.41). Meaning that in all EDF statistics the single occurrence probability pi = Pn (εf,i ) is estimated by i/n. Figure 3.2 displays the EDF of a sample of PMMA fracture strains. For each probability step of the EDF the starting and the ending point are compared to the CDF P (εf ) at the corresponding abscissa-position (D+ and D− ). The maximum occurred deviation is defined as test statistic D with & & &i & (3.5) D+ = max && − P (εf,i )&& 1≤i≤n n and

& & &i−1 & & D = max & − P (εf,i )&& 1≤i≤n n −

(3.6)

as   D = max D+ , D− .

(3.7)

Similar to the CS test, the null hypothesis H0 in the KS test claims that the EDF’s population is the CDF. In Figure 3.2 the examined CDF is a normal distribution. Having sample sizes greater than 35, the critical deviation Dcrit for rejecting H0 is approximately calculated by ' 1 α Dcrit = − ln , (3.8) 2n 2 where n is the total number of observations and α is the statistical significance [153]. Exact critical values one might take from [43]. For the subject of this work, this test statistic is not suitable, since only EDF’s one one point of maximum deviation to the CDF is considered, colored red in Figure 3.2. The EDF’s overall progression within the critical limits is disregarded. Furthermore, at the distribution’s tails the KS test is less sensitive, as Figure 3.2 indicates.

3 Generalized Anderson-Darling Test

occurrence propability P (εf ) [-]

32 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.02

fracture strain EDF normal CDF critical deviation max. deviation

0.025

0.03 0.035 0.04 fracture strain εf [-]

0.045

0.05

Figure 3.2 The Kolmogorov-Smirnov test’s deviation limits for a statistical significance of α = 0.05. The PMMA fracture strain’s empirical distribution function is tested against a normal distribution.

3.1.3 Cramér-von Mises Contrary to the KS test, the CvM test [41] considers in equal measure the EDF in every step. Thus, it generates the test statistic W 2 . The smaller W 2 gets, the better is the empirical data reproduced by the CDF. The calculation follows ∞ W = nω = n 2

2

[Pn (εf ) − P (εf )]2 dP (εf )

−∞

1 = + 12n

n   i − 0.5 i=1

n

2 − P (εf,i ) ,

(3.9)

in which Pn (εf ) represents the EDF and P (εf ) the CDF of the assumed probability distribution. Using the midpoint rule, the integral in Eq. (3.9) is to be converted into a sum, which significantly simplifies the calculation for the test statistic W 2 . The occurrence probability pi for each observation is again estimated according to Eq. (2.41), and not Eq. (2.40) as one could assume. The best possible test statistic obviously equals the error term 1/(12n). A derivation of the described conversion is given in Appendix A.1. The test statistics asymptotic and modified borders for not rejecting H0 are again given by [43], regarding various types of distribution functions. The mandatory use of the California method’s probability estimator in the CvM test, given by the EDF, becomes problematic when the plotting positions, initially used to fit the CDF, based on a different estimator, because consequently they are not considered in the test statistic. Thus, for the comparison of probability

3.1 Goodness-of-Fit Tests

33

estimators in order to detect their effect on the fitted CDF, the conventional CvM test is not a proper evaluation method. A consistent use of the plotting positions for fit and goodness-of-fit analysis is not possible.

3.1.4 Anderson-Darling An enhancement to the CvM test was introduced in 1954 [7], later known as the Anderson-Darling (AD) test. The purpose was to bring in the goodness-of-fit rating higher weight to the tails of the distribution. Therefore, a weight function ψ(u) =

1 u(1 − u)

(3.10)

the shape of which is shown in Figure 3.4 was added to the residual squares, as ∞ A =n

[Pn (εf ) − P (εf )]2 ψ [P (εf )] dP (εf )

2

−∞ ∞

=n −∞

[Pn (εf ) − P (εf )]2 dP (εf ) . P (εf )[1 − P (εf )]

(3.11)

Since, the function values of a CDF represent an occurrence probability, they always have to result in values between 0 and 1. Hence, this is the very interval for the argument in Eq. (3.10), with an approach of the function value towards infinity at the limits. As before, the integral expression of the test statistic is transformed into a sum, for what it is first simplified by substituting the continuous CDF P (εf ) = u and dP (εf ) = du. Then the single integral is divided into an integral for each probability step P (εf,i ) = ui . Following the California methods estimation, the EDF is Pn (εf,i ) = pi = i/n for an ascend ordering of εf,1 ≤ εf,2 ≤ ... ≤ εf,n . With p0 = 0 and pn = 1 the fraction of the first and of the last integral reduces. Simple integration and collecting of terms result in ⎧u u2 ⎨ 1 u2 (p1 − u)2 2 A =n du + du ⎩ u(1 − u) u(1 − u) u1 0 ⎫ u3 1 ⎬ (p2 − u)2 (1 − u)2 + du + ... + du ⎭ u(1 − u) u(1 − u) u2

= −n +

un

n  i=1

1 − 2i [ln(ui ) + ln(1 − un+1−i )] . n

(3.12)

34

3 Generalized Anderson-Darling Test

A detailed derivation of the AD test is provided in Appendix A.2. Critical values of the test statistic A2 for an acceptance or rather not rejection of the null hypothesis in dependence on sample size and probability distribution can also be taken from [43]. Similar to the CDF test, the AD test uses a mandatory probability estimator. That becomes obvious by examining the resulting sum in Eq. (3.12), which only contains the CDF positions ui , and no longer the EDF positions pi . Thus, by choosing a probability estimator differing to the one of the California method to fit a particular distribution function, the conventional EDF based AD test is likewise the CvM test not suitable for the goodness-of-fit rating.

3.2 Generalization Since the conventional AD test requires the California methods probability estimator, we propose the generalized Anderson-Darling (GAD) test. The derivation is similar up to the point, where the California method’s probability estimation replaces the general approach. That is omitted to keep pi unspecified. Collecting the integrated terms leads to + A2G = n −1 − ln[un (1 − u1 )] +

n−1  i=1

p2i ln



ui+1 ui





1 − ui+1 − (pi − 1) ln 1 − ui 2

, .

(3.13)

Replacing pi in Eq. (3.13) by the California methods estimator would lead back to the conventional AD test statistic of Eq. (3.12). Appendix A.3 shows the derivation in detail. The important advantage of the GAD test is now the free choice of the probability estimator and thus the enabling of a consistent use for function fit and afterward goodness-of-fit examination. With this test the conventional EDF statistics are left. It is worth noticing, that in the conventional AD test the probability p0 equals zero and pn equals one. Otherwise, Eq. (3.11) is indefinite. Hence, the GAD test enables to change the examined plotting positions apart from the nth one. The last plotting position is still adopted from the EDF. Basically, this is still an inaccuracy, because of the inconsistent use of the nth plotting position in CDF fit and goodness-of-fit test when the initial probability estimator is not i/n. That inaccuracy increases for small sample sizes, since for those the last plotting position gets a higher deviation from one, as exemplarily shown in Figure 3.3 for Weibull’s probability estimator (WPE) from Eq. (2.42), which is the one with lowest estimate for pi .

3.2 Generalization

35

nth probability pn [-]

1 0.9 0.8 0.7 n pn = n+1 p50 = 0.98

0.6 0.5 0

10

20

30

40 50 60 70 sample size n [-]

80

90

100

Figure 3.3 Probability distribution of the nth estimated plotting position using Weibull’s probability estimator. In addition, the corresponding probability value for a sample size of 50.

To deviate even with the last plotting position pn from the EDF, the approach of [160] can be utilized. They split the weight function ψ of the conventional AD test to gain one test statistic putting higher weight to the lower plotting positions and one putting higher weight to the upper ones. Adopting their lower-tail weighting ψ(u) =

1 u

(3.14)

in the approach of Eq. (3.11), bringing the integral into sum form, and simultaneously keeping the plotting positions pi as argument, provides a new test statistic. It

50 weighting ψ(u) [-]

ψ(u) =

40

ψ(u) =

1 u(1−u) 1 u

30 20 10 0 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 occurrence propability u [-]

0.9

1

Figure 3.4 Function ψ putting weight to the plotting positions dependent on the magnitude of their occurrence probability. In solid line the weighting of the AD and the GAD test and in dashed line the weighting of the lower-tail GAD test.

3 Generalized Anderson-Darling Test

occurrence propability P (εf ) [-]

36

1

0.95

0.9

CDF pi = pi =

0.85 0.04

0.045

0.05 0.055 fracture strain εf [-]

i n

i n+1

0.06

Figure 3.5 Comparison of the deviation from the CDF for the nth stairstep of the conventional EDF and the nth stairstep coming from a probability estimation by Weibull’s probability estimator.

is named the lower-tail GAD test A2G,LT . In Appendix A.4 the detailed derivation of its sum + 1 2 − 2pn (1 − un ) − p2n ln(un ) AG,LT = n 2 ,

n−1  ui+1 2 + pi ln , (3.15) + 2pi (ui − ui+1 ) ui i=1

is shown. The weights of both tests are compared in Figure 3.4. The more the estimated fracture probabilities tend to zero, or one in case of Eq. (3.10), the higher is their weight in the test statistic determination. Now with the lower-tail GAD, the nth plotting position is arbitrary, as Eq. (3.15) states. However, the condition pn < 1 has a relevant effect on the deviation integral between the nth step of the staircase function and the CDF. For εf → ∞ the occurrence probability for another observation is still pn < 1 and the deviation integral tends to infinity, except when a probability estimator is chosen that defines the nth plotting position as one. Due to simultaneously decreasing weight towards zero, the impact on the magnitude of the test statistic A2G,LT is negligible especially for larger sample sizes. But it is the reason why no upper-tail weighting with ψ(u) = 1/(1 − u) is possible for the GAD. The nth deviation integral would tend to infinity, making an upper-tail GAD become A2G,UT = ∞ as well. In Figure 3.5 the problem is illustrated. Hence, for investigations on goodness-of-fits in this work, only the conventional AD, the GAD, and the lower-tail GAD are compared.

3.3 Monte-Carlo Simulation

37

3.3 Monte-Carlo Simulation In GAD and lower-tail GAD test the consideration of arbitrary step functions independent of the EDF is enabled. Analogously to the conventional AD test, weight is put either to both or only to one tail of the distribution, which for many applications are the areas of major importance. To complete a goodness-of-fit analysis, the results of GAD and lower-tail GAD must be classified. Typically, the test result is compared to critical values for not rejecting the null-hypothesis H0 . Besides providing overflowing tables, here the methodology is shown to calculate corresponding p-values via Monte-Carlo simulation. The p-value of a goodnessof-fit test statistic describes, how probable a test outcome that is higher than the observed result would be if H0 was true [43]. The p-value for itself gives no information about the truth of a null hypothesis [171]. It has to be tested against a chosen significance level α, which in commonly set to 5 %. The significance level is the probability for the given study to falsely reject the null hypothesis, i.e. type I error. Is p-value > α then the test result has statistical significance and H0 is typically accepted. Only that means, the probability density of all possible test outcomes has to be known. Fortunately, the permanent increase in computer processing power allows the carryout of extensive Monte-Carlo simulations, in order to estimate this probability density with great accuracy. The procedure is as follows: having a distribution function fitted to the sample of 50 fracture strains, the goodness-of-fit test provides a distinct test statistic. The detected distribution function is postulated to be the population of the experimental data. The composition of the present sample is made by chance alone. So, different compositions are to expect, if the 50 tensile tests were repeated again and again, but all following the population. Conversely, based on the population it is possible to create samples with random compositions that are representative for sets of real experiments. The generation of random variables from a distribution function is enabled by the so-called method of inverse transform sampling. The inverse of a given CDF P (εf ) is received by solving for εf = P −1 (u), in which for u random uniformly distributed numbers are generated in the interval (0, 1). Basis is the concept of probability integral transform, which states that a random variable εf with CDF P (εf ) entails the uniform distributed variable u = P (εf ), and vice versa. In other words, the CDF performs the transformation from a variable of particular distribution into a variable of the standard uniform distribution [146]. For example, 3PW distributed random fracture strains are received by transforming Eq. (2.30) as described into εf,rand = P −1 (u) = γ + η[− ln(1 − u)]1/β ,

(3.16)

38

3 Generalized Anderson-Darling Test

START 50 experimental fracture strains εi with probability distribution P (εf )

end

for j=1 to 107 step 1

next

calculate test statistic A2orig

generate 50 uniform random numbers ui and calculate εf,rand,i = P −1 (ui ) fit CDF parameters on random sample εf,rand,i calculate test statistic A2MC,j determine p-value as P (A2MC,j > A2orig ) Figure 3.6 Monte-Carlo algorithm for determining the p-value, exemplarily demonstrated for a sample of 50 fracture strains.

with u as uniformly distributed variable. The generation formulars for popular distribution functions are given by [66]. For the BMW distribution the generation becomes more extensive, due to the fact that its CDF is not to inverse and therefore by given u a numerical solution of Eq. (2.33) for ε has to be conducted. In this way, random sets of each 50 fracture strains are generated. For each set the parameters of the original distribution function are fitted again and a goodness-of-fit test is performed, in order to simulate all possible test outcomes that result from the underlying population. It was found appropriate to create at least 10 million of these sets, resulting in 10 million stimulated test statistics A2MC,j (j = 1, 2, ..., 107 ) for a p-value that is reproducible to the third decimal place. The p-value arises from the percentage of test statistics from Monte-Carlo simulation being greater than the original test statistic A2orig with p-value = P (A2MC,j > A2orig ) ,

(3.17)

where P is the probability measure. An overview of the described procedure is illustrated in Figure 3.6 as flowchart. An interpretation of the test statistic A2orig

normalized frequency f [-]

3.3 Monte-Carlo Simulation

39

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

A2MC A2orig α = 5%

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2

test statistic A [-] Figure 3.7 Probability density of 107 test statistics A2MC gained by Monte-Carlo simulation. The significance level is α = 0.05. The test outcome is statistical significant, since the p-value of the original test statistic has a probability P (A2MC,j > A2orig ) > α.

and the significance level α is visualized in Figure 3.7. The 107 Monte-Carlo simulated test statistics A2MC are clustered in bins of 0.01 width. The number of occurred test statistics in each bin is divided by the maximum occurred number, providing a normalized frequency plot of the test statistics A2MC . Figure 3.7 shows the results of a sample test for 2PW distribution that are to be rated in Table 5.1 in Section 5.1. The p-value for AD and GAD test becomes 0.170 for a test statistic of A2orig = 0.6842, following Eq. (3.17). Thus, the probability limit of a significance level α = 0.05 is exceeded and the test result statistical significant.

9

4 Experimental Investigations The data basis for stochastic analyses in this work is collected in ad-hoc performed laboratory testings. In this chapter, the individual test setups are introduced. Since primarily uniaxial tensile tests are considered, focus is placed on the three different types of applied testing machines. Further setups for shear and puncture tests are separately introduced in Chapter 8. Service testings, as the head impact test in Chapter 7, are briefly discussed in the respective section. The following experiments have all be performed at room temperature. As known from [151, 152], acrylic glasses are sensible for temperature deviation. Hence, during all tests the ambient temperature is monitored and documented. As fracture criterion in the tensile tests, strain is preferred rather than stress, since a strain value is always unique in the material’s non-linear stress-strain behavior. As seen in Figure 2.5, a stress value might be referred to two or more strain values. Besides, strain is a directly measurable kinematic quantity, to be determined in laboratory tests, whereas the quantity stress is always a derivative and underlies model assumptions. All the presented studies are conducted on Plexiglas® 8N specimens. For production method evaluation of the specimens in Section 4.2 and Section 4.3, only injection-molded specimens made from 8N are provided by the manufacturer, precluding an examination of Resist. Though, stress examination by photoelasticity is likewise realized for Resist specimens to ensure their freedom of residual stresses. A photoelasticity analysis of Resist in deformed state is not possible, due to the loss of transparency by crazing effects, as shown in Figure 2.4. Last, the fractography analysis of Resist specimens is prevented, since it is no brittle material and therefore does not develop fracture mirrors, cf. Section 4.5. In Section 4.3 Photoelasticity the utilized configuration of the test setup is specified and discussed. For a detailed derivation of the stress-optic law and a closer description of the polariscope the reader is referred to [1, 51, 135]. Section 4.5 Fractography is based on [33], transferring the results into the context of this work.

4.1 Test Setups Uniaxial tensile tests are performed on material testing machines featuring three different working principles. A system with common electro-mechanical actuation1 1 Inspekt

Table Blue 5 kN by Hegewald & Peschke MPT GmbH

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_4

42

4 Experimental Investigations 3

force F [kN]

2.5 2 1.5 1 0.5 0 0

0.5

1 1.5 2 displacement s [mm]

2.5

3

Figure 4.1 Force-displacement curves of 50 tensile tests on Plexiglas® 8N with 6 mm/min hauling velocity. Nicely the test’s reproducibility is demonstrated, along with the material’s high variation in tensile strength.

is utilized for quasi-static and moderate hauling velocities. Its working interval is between 0.01 mm/min and 1000 min/min. The system is equipped with an internal displacement sensor with a resolution less than 1 μm, and a load cell with 5 kN nominal load, which is a proper interval for the PMMA specimens. Is the expected maximum force in the test below the nominal load, the resolution interval is adapted, as for the measurement set shown in Figure 4.1, where the force-signal’s maximum of 10 V is related to 3.5 kN. The testing machine provides the output of its force and displacement signals. Externally recorded they are synchronized with the DIC for strain measure, as discussed shortly. Always before testing, both signals are calibrated to a linear relationship between voltage and force, as well as voltage and displacement. When closing the pneumatic clamping jaws, a compressive force of about 100 N is applied to the specimens. As compensation, the traverse is lifted until the force is back to zero. The second material testing machine2 features a servo-hydraulic actuation, realizing hauling velocities up to 1 m/s. This system is equipped with a load cell of 25 kN nominal load, which is far beyond the expected maximum force. Therefore, the load cell is calibrated with defined weights for its lower working range. Again, the force is recorded externally. The hydraulic clamping jaws are closed by slow increase of pressure in order not to pre-damage the specimens. The highest velocities are realized at a droptower impact system3 , allowing impact velocities up to 24 m/s. In the current configuration, as shown in Figure 4.2, a weighted striker drops from a defined height to drag the lower clamping of the 2 Bionix

Tabletop 25 kN by MTS Systems Corporation 9350 by Instron GmbH

3 CEAST

4.1 Test Setups

43

Figure 4.2 Droptower test setup for realization of high strain rates (left). In close-up view (right), a mounted specimen (A) focused by single high-speed camera (B) and cold light (C) for 2D DIC analysis. Above, the resting striker (D).

specimen. The clampings are screwed down hand-tight and feature a spike holding the specimen in position. To enhance inertia of the slide carrying the striker, weights are added for a total mass of 67.09 kg. The upper clamping is fixed to a force sensor with 15 kN nominal load, whose signal is recorded by the measuring board of the system. Displacement information are not provided. As discussed in Section 8.2, two different camera systems are utilized for strain measurement via DIC, a dual camera system4 for 3D DIC analysis and a single high-speed camera5 for two-dimensional (2D) DIC analysis. The high-speed camera is adopted where the recording frequency of the DIC system is not sufficient for recording at least 100 images of the ongoing test. The camera systems are mechanically uncoupled from the material testing machines. For both camera systems, a cold-light ensures a proper illumination without heating the specimen. Though, the specimens are not applied until shortly before the test. The reference images of unloaded specimens are recorded with only one clamping jaw closed, in order to ensure stress-freedom. Focus and visual axis of the cameras are set with open aperture right onto the surface of the specimen’s center. For maximum depth of field at a given exposure time that is defined by the required recording frequency, the aperture is closed till the amount of light is just sufficient for a proper DIC analysis. For the dual camera system, which is equipped with two light sources, polarization filters prevent reflections on the specimen’s surface. An exemplary configuration of the dual-camera system is depicted in Figure 4.3 for a shear test, 4 Aramis

12M by GOM GmbH N4 by IDT Inc.

5 MotionXtra

44

4 Experimental Investigations

Figure 4.3 Variable dual-camera DIC system uncoupled from the testing machine, therefore combinable with any test setting giving view onto the specimen. Photo of a shear test.

that is discussed in Section 8.2. The image series for DIC analysis provides strain information at the very moment of image recording. Usually, the point of specimen failure occurs somewhere between the last picture of the intact and the first picture of the fractured specimen. Using the higher measuring frequency of the load cells, the last determined local strain is linearly extrapolated to the actual moment of failure in order to receive fracture strain. Despite, an amount of 100 images per test showed a sufficient resolution. The strain amendment is marginal. The measurement precision of the DIC system is not specified in a conventional way. There is no explicit interval labeled by the manufacturer. The accuracy is strongly dependent on the requirements of the test laboratory, such as ambient temperature, illumination, vibrations, etc. In order to quantify the error, two approaches are at hand. One is to determine the noise in the generated strain field. For this purpose, an unloaded specimen is recorded by the DIC system and the resulting strain field is analyzed. Since the specimen is unloaded, no strain should occur. Present strains are assumed to be measuring noise. Exemplary, this procedure is applied to a series of 100 images in an actual setup. Minimum and maximum strain in the strain field are determined for each image. Their progression is demonstrated in Figure 4.4, together with a dashed horizontal line marking the average. Their magnitude is considerably high, although in a single point whose position is not varied, the strain variation is far less. The minimum and maximum noises primarily occur in the corners of the defined strain field, where, due to the miss of adjacent facets, the redundancy in strain calculation is reduced. Hence, the uncertainty is assumed to be more on the level of the point noise. This uncertainty measure is the only choice for the 2D DIC. For the 3D DIC system a second measure is possible. The system is calibrated by

4.1 Test Setups

true strain ε [-]

0.002

45

minimum

maximum

single point

0.001 0 −0.001 −0.002 0

10

20

30

40 50 60 image index k [-]

70

80

90

100

Figure 4.4 Measuring noise within a strain field. The minimum strain averaged over the series is εmin = −0.001151, the maximum one εmax = 0.001109. The averaged punctual strain in a random point located at the specimen’s center becomes εp = 0.000145. Dashed lines mark the strain average.

recording several reference images of a calibration plate. Based on reference points on the plate, in an internal calibration process the sensor configuration is defined in camera distances, angles, foci, lens distortions, etc. Furthermore, the intersection deviation between the two observation rays of the cameras is computed, which is a measure for uncertainty in the analysis. The mean value of the intersection deviations in all captured reference points is provided as calibration deviation. The system manufacturer recommends the aim of a calibration deviation in interval of 0.01 to 0.04 pixels. [72]. The object width of a BZ tensile specimen with 12 mm width, cf. Figure 4.6, is about 650 pixels. That corresponds with a pixel size of 12 mm/650 px = 0.0185 mm/px, leading to a maximum uncertainty in 3D position detection of 0.0185 mm/px · 0.04 px = 7.4 · 10−4 mm. From both evaluation methodologies for uncertainty, the manufacturer suggests the first approach, since a strain uncertainty is gained, instead of a displacement uncertainty. The dual-camera DIC system is applied at quasi-static and moderate tensile velocities. A decisive advantage of the system is the included measuring board, allowing the input of external signals. In combination with the electro-mechanical material testing machine, providing the output of its force and displacement measures, during the test both signals are recorded by the DIC system and internally synchronized in time with the images taken by both cameras. Thus, a synchronized force-strain-time information is gained for further analyses. For higher hauling velocities, the single high-speed camera replaces the DIC system, since the maximum possible measuring frequency of the latter is limited and does not allow the aimed for amount of about 100 pictures in each experiment during the test. The material testing machine is exchanged, as well, for the servo-

46

4 Experimental Investigations

selected experiment

3 voltage U [V]

force F [kN]

3

2

1

2 1 0

0 10.2

10.4

10.6

time t [ms]

10.8

11

10.2

10.4 10.6 time t [ms]

10.8

11

Figure 4.5 Outcome of a droptower test. Force signal (left) and camera sync-out (right) lie on the same timeline.

hydraulic system. Again, the force signal is tapped. But without the measuring board of the DIC system, the setup has to be revised. A transient recorder is added as replacement, recording the force signal against time with a frequency of 100 kHz. For the tensile tests performed in this setup, the highest recording frequency of the high-speed camera is 10 kHz. Consequently, more data points are gained for force against time than for strain, and no common timeline is provided. The high-speed camera is equipped with a ring buffer, giving a sync-out signal for the moment of image recording. The sync-out is sent to the transient recorder, which is set up with an internal counter. Whenever an image is taken by the high-speed camera, the transient recorder counts up. In result, on the timeline of the transient recorder an explicit pairing of force and image, i.e. strain information, is provided. A third setup is adopted for tests with the droptower system. Its internal measuring board records force against time with a frequency of 1 MHz. Again, the single high-speed camera is applied. By reduction of exposure time and framesize, just capturing the specimen’s measuring zone for DIC, a maximum recording frequency of 20 kHz is gained. Both systems are triggered right before impact by the signal of a photocell, which is activated by the slide carrying the drop-yoke. The sync-out signal of the camera is recorded by the droptowers measuring board. The obtained result looks like shown in Figure 4.5. Force and sync-signal are recorded against time, though without an allocation of distinct images, as for the counter of the transient recorder. The allocation is realized manually. The moments of image recording in the test are determined from the falling edges in the sync-signal. The first image in the image series showing the failed specimen is referred to the syncsignal’s falling edge at the moment of force drop. Due to the limited amount of

4.2 Specimen Machining

47

images, the allocation is well-defined. The prior images are then reversely allocated to respective moments in the force signal.

4.2 Specimen Machining







5





5 



Most of the provided Plexiglas® material is in form of extruded plates. These plates come from the same production batch having a mean thickness of 3.065 mm with ± 0.043 mm standard deviation [90]. Additionally, a limited amount of plates having 4 mm thickness is available. The specimens needed for uniaxial tensile tests are milled out. Mainly, the type BZ geometry for polymers by [20] is produced. It is a geometry optimized for high strain rates at moderate hauling velocities of the material testing machine, combined with an uniaxial stress state in the measuring zone, cf. Section 4.4. In addition to the extruded plates, finished tensile specimens are provided, produced by injection molding. These tensile specimens are in dimension of the type 1A geometry according to DIN EN ISO 527-2 [49]. Its dimensions and the one of the BZ geometry are defined in Figure 4.6. A comparison of their strain-time behavior is given in Figure 4.7. In initial phase they prove a similar strain increase, at a hauling velocity of 6 mm/min for the BZ specimen, and 10 mm/min for the 1A geometry. Strain is measured as surface average in the specimen’s parallel measuring zone, cf. Figure-4.6. With increasing strain, both curves drift apart. Reason are localizations in the strain field of the BZ specimen. In contrast to the one of the 1A specimen, the designated 12x12 mm measuring zone of the BZ specimen is not homogeneous, as revealed later in Figure 4.17. Hence, the concept of strain measurement as surface average is discussed critically in Section.4.4. With the production of milled specimens, an insertion of certain surface defects is assumed, which reduce the specimen’s tensile strength. Concern is that fracture

 



Figure 4.6 Dimensions in millimeters for BZ tensile specimen (left) and 1A tensile specimen according to ISO 527-2 (right).

48

4 Experimental Investigations

true strain ε [-]

0.04 0.03 0.02 0.01

BZ 1A

0 0

5

10

15 time t [s]

20

25

30

Figure 4.7 BZ and 1A tensile specimen compared in strain increase over time. True strain is measured as surface average in the measuring zone.

strains coming from the measurement of milled specimens are not to transfer to the material behavior of an automotive window manufactured for high quality surfaces. Hence, a study on the influence of the milling process is performed. With injection molded specimens available, an evaluation is possible. These specimens feature no machined surface and are therefore comparable to the end-product. A third type of specimens is added to the study. Basis are again specimens milled from extruded plates, but with additional treatment of the machining surface by flame-polishing. A side-view onto the respective specimen surfaces is depicted in Figure 4.8. As expected, the injection-molded specimen shows a widely homogeneous surface. So does the flame-polished specimen. In order to quantify the surface roughness, the ¯ z , and R ¯ max are determined on a roughness measuring station6 ¯a, R parameters R as mean of 20 specimens of each production method. Ra is the arithmetic mean of absolute height profile coordinates within a measuring section, cf. [48]. This parameter gives a good overall impression of the present roughness. For Rz , the measuring section is divided into several subsections, within which the difference between minimum and maximum profile height is detected. Rz is the mean of the subsection results, whereas Rmax is the value for the subsection with maximum height difference, cf. [48]. The detected roughness parameters for the three production methods are listed in Table 4.1. Unsurprisingly, the injection-molded specimens feature the least roughness, though flame-polishing is able to distinctly reduce the roughness parameters for milled specimens. For all three sets of specimens, tensile tests at 1 mm/min hauling velocity are performed and evaluated for stress-strain behavior and fracture strain distribution. The procedure for strain determination from DIC analysis is discussed in detail 6 MarSurf

GD 25 by Mahr

4.2 Specimen Machining

49

Figure 4.8 Side-view on specimen that is injection molded (left), milled (middle), and milled and flame-polished (right). The thickness is 4 mm each.

in Section 4.4. At this point, the experimental parameters current normal force N , initial specimen’s width w0 , initial specimen’s thickness t0 , and true lateral strain εtrue,lat are premised. Furthermore, it is postulated that the strain in the specimen’s width is equal to the strain in the specimen’s depth. That is necessary, since the DIC system views only one side of the specimen. Analogously to the definition in Equation 2.9, true lateral strain is defined as



w t = ln . (4.1) εtrue,lat = ln w0 t0 where w and t are the specimen’s current width and thickness. The ratio of current to initial cross-sectional area is . σeng A N A wt = = = exp 2 εtrue,lat = , A0 w0 t0 A0 N σtrue

(4.2)

in which the multiplication with the neutral element N/N produces the ratio between engineering stress σeng and true stress σtrue . Derived from this relationship, current true stress is expressed only in dependence of the current normal force and true lateral strain by σtrue =

. N exp −2 εtrue,lat , A0

(4.3)

50

4 Experimental Investigations

true stress σ [MPa]

70 60 50 40 30 injection-molded, tempered milled milled, flame-polished, tempered

20 10 0 0

0.01

0.02 0.03 0.04 true strain ε [-]

0.05

0.06

Figure 4.9 Study on the production method’s influence on the stress-strain behavior of PMMA tensile specimens. For each kind 20 tests have been performed.

since A0 is constant. For stress calculation in this study, the initial A0 is measured for every single specimen. In later research, due to the enormous amount of tensile tests, an averaged A0 is adopted. With tolerances coming from the milling process, the cross-sectional area in the measuring zone of the BZ geometry becomes averaged 36.94 mm2 with a standard deviation of ±0.57 mm2 . The resulting stress-strain curves in Figure 4.9 prove a high reproducibility of the test, with accordance among the three types of specimens. Hence, the production method does not influence the gained material’s stress-strain behavior. The three sets only differ in their fracture strain level. For an appraisal of the fracture strain distribution, a 2PW distribution is fitted to each set according to Section 2.5. Table 4.1 lists the corresponding distribution parameters and the function’s me-

Table 4.1 Roughness parameters as average of n specimens, measured at their side faces. Below, parameters of 2PW CDFs fitted to fracture strain samples of size n. Additionally, the CDF median.

production method

n

¯ a [μm] R

¯ z [μm] R

¯ max [μm] R

injection-molded, tempered

20

0.073

0.41

0.69

milled

20

2.908

15.15

20.06

milled, flame-polished, tempered

20

0.552

2.10

3.23

production method

n

β

η

median

injection-molded, tempered

19

5.852

0.04361

0.04096

milled

20

8.063

0.04527

0.04326

milled, flame-polished, tempered

16

3.933

0.02431

0.02215

4.3 Photoelasticity

51

dian. A massive reduction of fracture-strain levels for flame-polished specimens is revealed. The heat-treatment must be assumed as damaging for the material. The injection-molded specimens are reference in this examination, since they are most comparable to the end-product. Astonishingly, the conclusion is made that milled specimens are as suitable as the injection-molded specimens for material testings. Regarding the 63th percentile η and the median, their fracture strain distribution is very similar to the one of injection-molded specimens, i.e. fracture strain lies on the same level. Only the shape parameter β shows some deviation, though due to the comparably small amount of occurrences in the samples, a difference in dispersion is not to overestimate. Consequently, the validness for milled Plexiglas® specimens is attested. It is chosen as production method for all specimens examined in this work.

4.3 Photoelasticity No matter if injection molded or milled, the PMMA specimens have to be free of residual stresses. These stresses are provoked by uneven shrinkage of the material in the process of injection molding and extrusion. The resulting residual strains falsify the determination of fracture strain. Thus, the pre-test condition of the specimens is verified using the method of two-dimensional photoelasticity, which enables the visualization of stresses within a loaded body. Prerequisite is the material’s transparency and birefringence, which both is given for PMMA [2]. Due to birefringence a ray of light that enters a stressed specimen is divided into two component waves orientated vertically to the direction of light propagation, each with its plane of polarization parallel to the direction of the principal stresses. Dependent on the magnitude of the principal stresses σ1 and σ2 , the component waves feature different wavelengths, having a phase difference Δ. The relationship σ1 − σ2 =

λΔ 2πCt

(4.4)

is the stress-optic law, where t is the thickness of the body, λ the wavelength of the intruding light, and C the stress-optical constant of the respective material. Considering the introduced multiaxial stress-states in Section 2.3, the maximum phase difference occurs for pure shear, and the minimum one for biaxial loading. An instrument to visualize the stress field within the planar specimen is the polariscope. Its simplest configuration is the plane polariscope, consisting of two linear polarizer plates with the specimen in between, cf. Figure 4.10. For even illumination of the specimen a spot light rays a diffusing surface, which produces in every point light that propagates in all directions [135]. The plate positioned at the light source is called polarizer. The second plate, in its polarization plane

52

4 Experimental Investigations










 






 



 

Figure 4.10 Schematic drawing of the plane polariscope (top) and the circular polariscope (bottom) with diffuse light source. Adumbration of spot light (L), diffusing surface (D), polarizer (P), quarterwave plates (Q), specimen (S), analyzer (A), and camera (C).

orthogonal to the one of the polarizers, is called analyzer. In this setup, light that is polarized parallel to direction of residual stresses vanishes. The arising dark bands of light extinction are called isoclinics. Thus, by simultaneous rotation of polarizer and analyzer, the angle of the principal stresses can be tracked. Besides isoclinics, colorful bands occur for white light that is polarized out of principal stress direction. Due to the light’s retardation by birefringence into different wavelengths, the respective color in the light spectrum is produced. Fringe patterns of equal color, i.e. equal principal stress difference, are the isochromatics. The light extinction of isoclinics is obstructive for the analysis of residual stresses. A more favorable arrangement is the circular polariscope, which describes a setup in which

Figure 4.11 Polariscope applied to the tensile test of a 1A specimen (S). The diffusing surface (D) is generated by a white paper, covering polarizer and quarter-wave plate. Labels similar to Figure 4.10.

4.3 Photoelasticity

53

two crossed quarter-wave plates are added, as depicted in Figure 4.10. Through circular polarization of light, the additional plates prevent the vector components to extinguished when falling in principal stress direction. In consequence, no isoclinics develop, simplifying the detection of isochromatics. Polarizer and analyzer planes are set parallel for a light-field configuration. At stress-free areas within the specimen, where no wavelength retardation occurs, the emerging light is the same as the incident light. It allows a clearer image of the specimen’s edges, compared to a dark-field configuration, where the light in stress-free areas is extinct. Furthermore, white instead of monochromatic light is employed in order to visualize the stress pattern in continuous color. The experimental setup is shown in Figure 4.11. The examined specimens correspond to the 1A geometry, of which injection molded and milled specimens are available. The gained images are provided in Figure 4.12. Specimens that are adopted directly from the injection molding process exhibit clearly visible residual stresses, but after a tempering for 4 h at 80 °C and subsequent slow cooling most of these stresses can be relaxed. Still, a small coloring is visible. For the milled specimens, no coloring is noticeable, i.e. the specimen proves to be stress free. In addition, to show the stress pattern of a loaded specimen, an injection-molded and tempered specimen is loaded with 2 kN tensile force. The homogeneously stress distribution within the measuring zone of the specimen stands out. Though, towards the edges where the injection molded specimens feature a distinct radius

Figure 4.12 Stress analyses on 1A tensile specimens that are (left) injection-molded, (mid-left) injection-molded and tempered, and (mid-right) milled from plates. In (right) a injection-molded and tempered specimen is loaded with 2 kN tensile force.

54

4 Experimental Investigations

originated from the production process, a change in the birefringence is visible. The milled specimens do not exhibit such edge effect, which makes them particularly suitable for further application.

4.4 Strain Measurement In Section 4.2 the stress-strain relations in the tensile test are determined from force measurement of the load cell and strain computation via DIC, whose active principle is introduced in Section 2.4. The result from DIC on a speckled specimen is a continuous strain field, having certain variation over the surface, either due to local strain variation, or a distinct measuring noise, as discussed in Section 4.1. That raises the question of how to determine the specimen’s local fracture strain in the most accurate way. So far, for determination of the stress-strain relations in Section 4.2, strain is measured as an average along the sample’s width. Since for stress calculation the current surface A is necessary, averaging lateral strains should provide a more profound information on material contraction than a point measurement that is affected by local effects. In doing so, the same lateral contraction on the viewed surface plane is assumed for the contraction in depth direction, cf. Equation (4.1), enabling stress calculation following Equation (4.3). However, the fracture strain analyses of Section 4.2 base on strain measurements in a single point at the position of crack initiation with the aim to most accurately determining the local strain. In this section, these intentions are tested. In DIC strain measurement, four different approaches are reasonable, whose principles are demonstrated in Figure 4.13. One is the averaging of strains within a defined section, which gives the least resolution for local strain incidences. More local is the generation of a line cut in the strain field, along which strain is also

Figure 4.13 DIC analysis of the specimens surface. The fracture strain is taken one picture before failure in a single point at position of crack initiation (left), as average of a horizontal line-cut (midleft), as average of a rectangular surface (mid-right), and as percentage of an extensometer’s length deviation (right).

4.4 Strain Measurement

55

averaged. The cut is directly positioned on level of crack initiation. But sill, very local strain is not considered in such averaged strain quantity. Therefore, a single point is added to the considerations, representing the crack’s starting point. That gives the most detailed information on local incidences. As a downside, this approach is most affected by measuring noise. As stated in Section 2.4, instead of generating a strain field, DIC enables the displacement tracking of distinct points on the surface as well. Thus, strain can be defined as the length deviation of the distance between two surface points. The advantage of this approach is the steadiness of the strain information, even within measuring series with high image noise. A loss of triangular surface elements in the strain field, due to facet loss, is not uncommon. Especially the two-dimensional single-camera DIC is sensible due to missing redundancy. The connection between these two surface points is called extensometer. Since for the extensometer a length deviation to the initial length is determined, its measure is engineering strain. Analogously to the definition in Section 2.3, its strain is εeng = Δl/l0 , cf. Equation (2.8). As stated, in this work consistently true strain is employed. From true strain definition of Equation 2.9 arises the relationship





l l0 + Δl Δl (4.5) εtrue = ln = ln = ln 1 + = ln (1 + εeng ) , l0 l0 l0 which is utilized to translate the extensometer’s engineering strain into true strain. For evaluation of the four strain measurement approaches’ accuracy, a study is conducted. The 20 tensile tests for milled specimens from Section 4.2 are utilized to examine the different strain measurement approaches. For every test, the specimen’s fracture strain is determined by each of the four options. In result, four fracture strain samples are collected. Following Section 2.5, a 2PW distribution is fitted to each sample. Occurrence probability within a sample is estimated by WPE. The parameters of this 2PW distributions are listed in Table 4.2. Additionally, the test statistics of the GAD are calculated, and their p-values determined. Figure 4.14 depicts the corresponding CDFs in a Weibull plot. A certain deviation

Table 4.2 Shape and scale parameters of the 2PW distribution for different evaluation methodologies with the respective generalized Anderson-Darling test statistic and its p-value.

β

η

A2G

p-value

point

8.063

0.04527

0.3221

0.7099

line-cut

7.795

0.04536

0.3354

0.6676

surface

7.713

0.04533

0.3391

0.6558

extensometer

7.719

0.04533

0.3358

0.6662

methodology

56

4 Experimental Investigations

ln[− ln(1 − P )]

0.033

0.037

0.041

0.045

0.050

0.055 99.94

1

93.40

0

63.21

−1

30.78 cut point surf extenso

−2 −3 −4 −3.5

−3.4

−3.3

−3.2

−3.1

−3

12.66 4.85 1.81 −2.9

occurrence probability P (εf ) [%]

fracture strain εf [-] 0.030 2

ln(εf ) Figure 4.14 ologies.

Weibull plot of the 2PW distributions resulting from the four different evaluation method-

between the strain measurement approaches are at hand, though less pronounced than expected. The 63th percentile η is nearly constant, only the shape parameter β shows distinct variation. In result, the goodness-of-fit test is affected, too. One would expect a higher fracture strain level for a point measurement, since local maxima are captured. Despite, point analysis gives the lowest strain levels. Due to the results of this study, the assumption is made that all four approaches provide the similar results for a homogeneous strain field. Since this homogeneity is primarily a characteristic of the 1A and not the BZ geometry, cf. Figure 4.17, the point measurement is adopted for further examinations in this work. As mentioned in Section 2.4, the size of facets and their distance among each other define the resolution of the examined specimen’s surface. In order to evaluate this influencing factors, two further studies are conducted on one and the same tensile specimen. The analyzed specimen of 1A geometry has a width of 10 mm in the measuring zone, relating to 193 pixels in the gained images for DIC. In a first inspection, a strain field is generated with different facet sizes. The facet distance is defined as 80 % of the latter. The strain determined within the test is taken in a single point for highest sensibility to measuring noise. Figure 4.15 demonstrates the results. For the small facet sizes frequently triangular strain field elements get lost, affecting the accuracy of adjacent areas, and leading to noise in the field. Facet sizes larger than 15 pixels are reliably tracked. Beyond that facet size, differences in the resulting strains are hardly visible. As result, the facet size is chosen to be 19 pixels, which showed to be a good compromise and is about 10 % of the specimen’s width.

4.4 Strain Measurement

57

true strain ε [-]

0.04 0.03 0.02

5 px 10 px 15 px, 20 px, 25 px, 30 px, 35 px, 40 px, 45 px, 50 px

0.01 0 0

10

20

30

40

50

60

70

80

90

100

image index k [-] Figure 4.15 Variation of the facet size in DIC analyses. The point distance is kept constant at 80 % of the facet size. The specimens width of 10 mm equals 193 pixels.

Analogously, the facet distance is examined. Basing on a constant size of facets with 19x19 pixels, a strain field is generated with varying facet distances. The results are demonstrated in Figure 4.16. The constellations with most variance are plotted in colored, though even those are hardly differing from the rest of the strain curves. The facet distance is a less influencing factor compared to the facet size. For all upcoming analyses, a facet distance of about 80 % the facet-size is chosen, in order to have a facet overlap of 20 %. For the facet-size of 19 pixels that is 16 pixels. In case of frequent facet loss causes by image noise, the facet size is enlarged. In particularly critical cases the extensometer is applied. As can be seen in Figure 4.14 the strain field for the 1A geometry is very homogeneous. The whole surface is of equal color. The BZ geometry exhibits more

true strain ε [-]

0.04 0.03 0.02 4 px 7 px 10 px, 13 px, 16 px, 19 px, 22 px

0.01 0 0

10

20

30

40

50

60

70

80

90

100

image index k [-] Figure 4.16 Variation of the point distance in DIC analyses. The facet size is kept constant at 19x19 pixels. The specimens width of 10 mm equals 193 pixels.

58

4 Experimental Investigations İWUXH>@

WULaxialityP΀Ͳ΁

 

5333 _ 5067 _



4800 _ 

4533 _ 

4267 _ 4000 _ 3733 _





 ±





 



 

3467 _ 

3200 _ 2933 _



2667 B 

 

Figure 4.17 Triaxiality at 2.5 mm elongation (left) and analysis of local strains via DIC. Flags indicate either the position at which fracture strain is taken one picture before failure (middle), and the outer edges of the fracture pattern one picture later (right).

inhomogeneity, as shown in Figure 4.17, which is another argument for performing fracture strain measurement as local as possible. The crack that causes the specimen to fail is initiated at the specimen’s edge, even for injection molded specimens. Hence, the state of multiaxiality in this area has to be evaluated to ensure the intended uniaxial stressing of the material. For that matter, a BZ tensile specimen is numerically simulated. Young’s modulus and Poisson’s ratio are taken from [150]. Figure 4.17 illustrates the results of this linear-elastic FE simulation with respect to triaxiality. The triaxiality is determined analog to Equation 2.19 for which the classifications are provided in Table 2.3. The areas of turquoise color state an almost uniaxial stress state in the specimen’s measuring zone and on edge level. Thus, the validness of the fracture strain determination approach is attested. Fracture strain is taken locally with slight edge distance right at the position of crack initiation one picture before failure, as depicted in Figure 4.17. The cracks starting point lies in the tip of the v-shaped fracture pattern.

4.5 Fractography Determination of fracture stresses and strains based on DIC provides a good estimation of the actual material behavior. Though, especially for stress calculation, several assumptions are made, cf. Section 4.2, like a constant stress over the crosssectional area, and an equal lateral strain of width and thickness. That causes a certain error for the actual fracture stress. A methodology to analyze the actual

4.5 Fractography

59

C

C B

B A

A 1 mm

1 mm

Figure 4.18 Fracture mirror (A) for a mineral glass (left), and for Plexiglas® 8N (right). Both topographic patterns show mist areas (B) and hackle areas (C).

stress at the point of fracture is the fractography. Just by the topography of the fracture surface, information on the cause of fracture, crack propagation, and type of loading can be obtained. For brittle materials, a characteristic topographic pattern is the so-called fracture mirror, marking the position of failure origin, where an initial, nearly motionless crack is accelerated towards maximum propagation velocity [28]. Primarily, fracture mirrors are examined for brittle ceramics and mineral glasses. For the latter, an example is depicted in Figure 4.18. Visually, the glossy area of the semi-circular fracture mirror merges into a section of higher surface roughness, called mist area. The harsh transition is purely optical. The increase of surface roughness is even, though areas of a roughness magnitude below wave length of light optically appear glossy, whereas those above appear matt. The fracture mirror’s radius is defined as distance between the mirror’s center and this transition area. Its exact determination is standardized in [12]. Beyond the mist area, the crack proceeds in lance hackles, which show as lines on the surface. They indicate the direction of crack propagation [67]. With increasing velocity of the crack, instabilities develop at its tip, which are visible by a higher surface roughness. The mentioned fracture mirror radius rm is a useful measure. It provides information on the magnitude of actual stress σf causing the failure. Both quantities are related by −0.5 , σf = σ + σ0 = Am rm

(4.6)

where σ is applied stress, σ0 is residual stress, and Am is the fracture mirror constant, which is a material constant [137]. As stated, fracture mirrors occur for brittle materials. Especially for almost ideal-brittle materials they give quantitative information on the fracture stress. Brittle amorphous polymers, like Plexiglas® 8N, do similary develop fracture mir-

60

4 Experimental Investigations

rors, which are sometimes referred to as smooth regions [28]. An example is depicted in Figure 4.18. Analogously, the crack’s initiation point lies within the center of the semi-circular fracture mirror. The crack propagation proceeds in radial direction, accelerating towards its maximum velocity, when instabilities develop at the crack’s tip [60]. These are marked by arcuate mist areas of increased roughness. Unlike the glass’s topography, PMMA shows repeated mist areas. Acceleration and instabilization occur in an almost wavelike motion, as Figure 4.18 demonstrates. A possible reason might be micro-branching instability effects giving rise to large velocity oscillation [62]. Generally, the maximum crack propagation velocity of polymers is by far lower than for inorganic glasses, since much higher plasticity combined with a much lower Young’s modulus reduce potential energy of elastic deformation at similar load cases. In fracture mirror analyses, the mirror constants for glasses and ceramics are usually known [137], though for polymers only few experimental results exist. [14] √ provides a mirror constant of 8.5 MPa m for PMMA. In the work of [33] a mir√ ror constant of just 1.462 MPa m is determined for Plexiglas® 8N. The varying chemical structure and composition of the polymer materials, which is anew optimized for the respective application, prevents a universal validity of the mirror constant. Furthermore, the high plasticity of polymers compared to mineral glasses or ceramics, limits a linear-elastic fracture mechanics analysis. Hence, in this work fracture mirrors are simply used as indicators for crack initiation, as presented in Section 5.1.

9

5 Sampling A statistical sample is a set of possible outcomes of a probabilistic experiment [66]. The phrase probabilistic experiment motivates the analyses of this chapter. To gain statistically valid samples, the bound conditions in the experimental research have to be consistent for every test of a kind. Within these defined bound conditions the occurrences are assumed to be probabilistic. The stricter these conditions are, the more outliers will be detected from the experiments, and the more tests have to be performed in total for an adequate sample size. The closer the test specimens, or specimen tests are examined, the more irregularities will be noticed. In particular, those result from the machining tolerances of the specimens. Ideal specimens would only differ in the random distribution of polymer chains at their micro structure, which is not a realistic aim. Hence at some point, a compromise has to be made. In the following, the chosen containment for grouping the experimental results into representative samples is presented. That includes the screening of posttest specimens for validness, and differentiating between appearing viscous effects, where strain rate is the regarded measure.

5.1 Removal of Outliers The study in Section 4.2 on relevance of machining flaws confirms the use of milled specimens. Though, especially brittle polymers must be presumed as sensitive to local stress cracking [59]. In the findings of Section 4.5 fracture mirrors are recognized as indicator for the position of crack initiation for the brittle Plexiglas® 8N. In order to ensure a measurement’s validness within a sample, each 8N specimens is viewed after the test by microscope to locate the fracture mirror. That revealed some specimens to be not representative, due to different bound conditions. These are the group of outliers, whose detection is expounded in the following. The plastic Plexiglas® Resist does not develop fracture mirrors. However, by reason of the local plasticizing it must be assumed to be far less sensitive to stress cracking. The experienced observations of its fracture behavior support this postulate. The meaning of outliers within a sample for the empirical probability distribution and the fitted distribution functions is exemplarily shown on a sample of Plexiglas® 8N fracture strains from uniaxial tensile tests at 1 · 10−4 m/s hauling velocity. In Figure 5.1 the tested specimens of this sample are depicted for an im© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_5

62

Figure 5.1

5 Sampling

Tested and reassembled tensile specimens from Plexiglas® 8N.

pression on the fracture patterns. In many cases the fracture arises in the transition between parallel and radial area. The measurements are still valid since Figure 4.17 did show uniaxial tension in those areas. Without exception, all cracks start from the edge, where due to tapering of the cross-sectional area and the small length of the BZ specimen’s parallel measuring zone, stress concentrations develop. Another reason is the occurrence of uniaxial load mostly in these edge areas, as revealed in Figure. 4.17. As the studies in Section 8.2 will show, the material is less affected by multiaxial load and shear, than by uniaxial load. The tensile tests are analyzed via DIC. The resulting fracture strains εf,i (i = 1, 2, ..., 50) form the database of the untrimmed sample. The probabilistic density in the sample is shown in the histogram of Figure 5.2. The histogram, whose interval ranges from 0.02 to 0.05 divided into ten even bins, gives a rough idea of what the PDF of the population

frequency f [-]

10 8 6 4 2 0 0.02

Figure 5.2

0.025

0.03 0.035 0.04 fracture strain εf [-]

0.045

0.05

Histogram of 50 Plexiglas® 8N fracture strains at 1 · 10−4 m/s hauling velocity.

5.1 Removal of Outliers

63

might look like. Yet, it is uncertain whether the PDF features a unimodal or a bimodal trend. With the aid of the p-value, the in Table 2.4 introduced distribution functions are assessed for their ability to reproduce the experimental data concerning AD, GAD, and lower-tail GAD test. Each probability distribution function is fitted to the fracture strain sample. As stated in Section 2.5, fit criterion is the WRSS, for which the same weight ψ(u) = 1/u, cf. Eq. (3.14), is chosen as for the lower-tail GAD, because it provided the best capturing of the data points. The plotting positions are respectively generated for each of the probability estimators in Table 2.5, in order to detect the pairing between distribution function and estimator that results in minimum WRSS. In doing so, surprisingly one estimator shows a

Table 5.1 Ranking of the considered distribution functions by minimum WRSS. Fit criterion is the minimization of WRSS with ψ(u) = 1/u. In comparison the respective results of Anderson-Darling, generalized Anderson-Darling, and lower-tail generalized Anderson-Darling test

pos. 1 2 3 4 5 6 7 8 9 10 11 12 pos. 1 2 3 4 5 6 7 8 9 10 11 12

distribution

probability estimator

WRSS

Bilinear Weibull Bimodal Weibull 3-Parameter Weibull Gumbel Left-Truncated Weibull Log-Normal Log-Logistic Normal Logistic 2-Parameter Weibull Log-Weibull Cauchy

(i − 0.3)/(n + 0.4) (i − 0.5)/n i/(n + 1) i/(n + 1) i/(n + 1) i/(n + 1) i/(n + 1) i/n i/n i/n i/n i/(n + 1)

0.0464 0.0572 0.1148 0.1335 0.1354 0.1490 0.1916 0.2259 0.2684 0.2778 0.4095 0.5156

distribution Bilinear Weibull Bimodal Weibull 3-Parameter Weibull Gumbel Left-Truncated Weibull Log-Normal Log-Logistic Normal Logistic 2-Parameter Weibull Log-Weibull Cauchy

A2

p-val.

A2G

p-val.

A2G,LT

p-val.

0.1743 0.1853 0.4028 0.5112 0.3296 0.4437 0.5660 0.5457 0.6382 0.6842 1.3663 1.3313

0.938 0.675 0.289 0.259 0.463 0.376 0.211 0.253 0.160 0.170 0.025 0.029

0.1749 0.1950 0.3550 0.4685 0.3190 0.4201 0.5424 0.5457 0.6382 0.6842 1.3663 1.3358

0.979 0.824 0.527 0.378 0.608 0.470 0.274 0.253 0.160 0.170 0.024 0.035

0.0778 0.0887 0.1571 0.1839 0.1937 0.2144 0.2746 0.3343 0.3978 0.4162 0.6040 0.7505

0.975 0.731 0.334 0.380 0.192 0.272 0.128 0.080 0.043 0.044 0.012 0.011

64

5 Sampling

continuously good performance. It is pi = i/(n + 1) by Weibull that produces in almost all cases the minimal or at least the second lowest WRSS. This conforms with the conclusions of Makkonen [116, 117], stating WPE to be the only valid approach for probability assignment. Ranking the distribution functions by WRSS, Table 5.1 lists the respective AD, GAD and lower-tail GAD test statistics and more important for comparability, their p-values. Noticeable is the inconsistent ascending order of the test statistics, though it is not astonishing, since the goodness-of-fit tests regard the continuous EDF and the WRSS single plotting positions. More important is the different ranking among the three goodness-of-fit tests regarding the p-values. Just for AD

ln[− ln(1 − P )]

0.027

0.033

0.041

0.050

2

99.94

1

93.40

0

63.21

−1

30.78

−2

12.66

−3

4.85 experiments BLW CDF

−4 −5

−3.8

−3.6

−3.4

−3.2

−3

1.81 0.67

occurrence propability P (εf ) [%]

fracture strain εf [-] 0.022

ln(εf )

frequency f [-]

120 100 80 60 40 20 0 0.02

0.025

0.03 0.035 0.04 0.045 fracture strain εf [-]

0.05

Figure 5.3 Weibull plot of the bilinear Weibull distribution’s cumulative distribution function (top). Analogous to Table 5.1 Pos. 1 the plotting positions are estimated by pi = (i − 0.3)/(n + 0.4) and weighted by ψ(u) = 1/u. In addition, the associated probability density function within the same interval limits (bottom).

5.1 Removal of Outliers

65

and GAD, which both possess the same weighting, the difference shows, that the consistent choice of plotting positions is not trivial. The BLW distribution, which ranks first in Table 5.1, is illustrated in Figure 5.3, choosing the Weibull plot for visualization. Attribute of the Weibull plot, which is gained by transforming the abscissa by x = ln(εf ) and the ordinate by y = ln [− ln (1 − P (εf ))], is the display of a 2PW CDF as straight line. In case of Figure 5.3 the plotting positions clearly show two line-ups, i.e. two 2PW distributions, what explains the top ranking of the BLW distribution. The occurrence of two separate 2PW distributions in the sample leads to the conclusion of two different fracture criteria to be present.

ln[− ln(1 − P )]

0.027

0.033

0.041

0.050

2

99.94

1

93.40

0

63.21

−1

30.78

−2

12.66

−3

4.85 experiments mixture CDF

−4 −5

−3.8

−3.6

−3.4

−3.2

−3

1.81 0.67

occurrence propability P (εf ) [%]

fracture strain εf [-] 0.022

frequency f [-]

ln(εf ) 90 80 70 60 50 40 30 20 10 0

mixture PDF mixture components

0.02

0.025

0.03 0.035 0.04 0.045 fracture strain εf [-]

0.05

Figure 5.4 Weibull plot of the mixture distribution’s cumulative distribution function (top). The plotting positions are estimated by pi = i/(n + 1) and weighted by ψ(u) = 1/u. In addition, the associated probability density function within the same interval limits (bottom).

66

5 Sampling

Hitherto, all sample points are assumed to follow the same population. But based on the prominent performance of the bimodal distribution functions, this assumption is to question. The superposition of two effects for the failure of the PMMA samples appears more likely. In response to the indication of two 2PW distributions P1 (εf ) and P2 (εf ), the distribution fit is repeated with consideration of a mixture distribution P (ε) = w1 P1 (εf ) + w2 P2 (εf ) = w1 P1 (εf ) + (1 − w1 )P2 (εf ) ,

(5.1)

where w1 and w2 are the weights for the respective mixture component, such that w1 + w2 = 1. In result, the fit produces the best WRSS so far with WRSS = 0.0463. This fit is shown in Figure 5.4. The left mixture component weights with w1 = 0.209 and the right accordingly with w2 = 0.791. Compared to Figure 5.3 the trend of the PDF in Figure 5.4 features a smooth transition between the two modes and so corresponds more to a physical behavior. That raises the question, whether there is a mechanical explanation for the two populations in the sample. As pronounced, the fracture patterns of the tensile specimens are inspected more closely with search of the fracture mirrors. These showed for some specimens a crack origin within prominent notch roots coming from the milling process, as pictured in Figure 5.5. Since, the stress state in such notch roots is multiaxial instead of uniaxial as in the experiments intended, these specimens were defined as outliers, without respect to their magnitude of fracture strain. The original sample is again plotted in Figure 5.6 with marking of the identified outliers. It is evident how all the lower fracture strains are neglected from the sample. Based on the 27 retaining experiments a linear regression line,

Figure 5.5 Fracture mirror of Plexiglas® 8N tensile specimen (left). Fracture surface (A) 200x magnified with highlight of the specimen’s geometric edges. The fracture mirror (B) reveals the cracks initiation point to be within a notch root (C). As illustration of the regarded fracture surface, the example of a 1A tensile specimen (right).

5.2 Rate Effects

67

ln[− ln(1 − P )]

0.027

0.033

0.041

0.050

2

99.94

1

93.40

0

63.21

−1

30.78

−2

12.66

−3

ponounced notch remaining experiments 2PW CDF

−4 −5

−3.8

−3.6

−3.4

−3.2

−3

4.85 1.81 0.67

occurrence propability P (εf ) [%]

fracture strain εf [-] 0.022

ln(εf ) Figure 5.6 Position of the outliers in the sample. Analogous to Figure 5.3 the plotting positions are estimated by pi = (i − 0.3)/(n + 0.4). A linear regression line is added to the retaining experiments.

i.e. 2PW distribution, is added, visually demonstrating a high level of agreement. To quantify the goodness of a 2PW fit, the introduced procedure is reapplied on the new sample. Due to its changed composition, the probability for each occurrence is re-estimated using WPE. The gained plotting positions are weighted by ψ(u) = 1/u, cf. Eq. (3.14), for an accordance with the previous analyses. As a result, the 2PW CDF is fitted with a WRSS to 0.0942, gaining a p-value of 0.577 in the lower-tail GAD test, which is chosen to regard consistent weights. Compared to the common significance level of α = 0.05, c.f. Section 3.3, this p-value is a strong indication for the 2PW distribution to reproduce the population. Certainly, in all experiments micro notches from machining initiate the fracture by stress concentration, but particularly prominent notches are showed to significantly reduce the strength of the material, causing an additional distribution of fracture strain. Thus, a criterion for outlier definition is established that is applied consistently to all Plexiglas® 8N specimens. Furthermore, the statistical analyses on the adjusted sample indicate a 2PW distributed population for the fracture strain in Plexiglas® 8N.

5.2 Rate Effects Both Plexiglas® materials are known to behave viscous. The velocity a load is applied with influences the ability of polymer chains to slough, as explained in Section 2.1. Two velocities are of major interest. Once, it is the quasi-static loading. Reconsidering the rheological models in Section 2.2, ideal static loading

68

5 Sampling

Tested hauling velocities vh in uniaxial tensile tests with respective setup configuration.

Table 5.2

hauling velocity 3 · 100 1 · 100 1 · 10−1 1 · 10−2 1 · 10−3 1 · 10−4 1.66 · 10−5

m/s m/s m/s m/s m/s m/s m/s

testing machine

DIC system

drop-tower system servo-hydraulic system servo-hydraulic system servo-hydraulic system servo-hydraulic system electro-mechanical system electro-mechanical system

2D highspeed camera 2D highspeed camera 2D highspeed camera 2D highspeed camera 2D highspeed camera 3D dual camera 3D dual camera

max. frame rate 20 kHz 11 kHz 5 kHz 700 Hz 100 Hz 2 Hz 1/3 Hz

is infinitely slow that only the spring elements are addressed, not the dashpots. In reality that is not feasible, but by proper choice of slow velocities the viscous effects are negligible small. For Plexiglas® 8N the quasi-static tensile velocity is set to 1 mm/min. For Plexiglas® Resist no set is collected at this velocity. Reason is the vast test duration of about half an hour for a single tensile test. Its quasi-static velocity is set to 6 mm/min = 1 · 10−4 m/s, which will be shown as sufficient. The second important velocity is the one at the level of application. Since, the acrylic glasses are examined as mineral glass substitutes in automotive glazing, there are velocities to test which are representative for a crash scenario. In common practice that is gained in tensile tests of about 3 m/s. This is adopted as upper set for both Plexiglas® materials. A list of the regarded velocities is provided in Table 5.2 with information on the utilized test setup from Section 4.1. More than 60 tensile tests are performed for each hauling velocity vh of both acrylic glasses, having a good reproducibility, similar to the demonstrated force-

true stress σ [MPa]

100 80 vh = 3E0 m/s vh = 1E0 m/s vh = 1E − 1 m/s vh = 1E − 2 m/s vh = 1E − 3 m/s vh = 1E − 4 m/s vh = 1 mm/min

60 40 20 0 0

Figure 5.7

0.01

0.02

0.03 0.04 0.05 true strain ε [-]

0.06

0.07

0.08

Uniaxial stress-strain behavior of Plexiglas® 8N under seven different hauling velocities.

5.2 Rate Effects

69

true stress σ [MPa]

80 60 vh = 3E0 m/s vh = 1E0 m/s vh = 1E − 1 m/s vh = 1E − 2 m/s vh = 1E − 3 m/s vh = 1E − 4 m/s

40 20 0 0

Figure 5.8

0.1

0.2

0.3 0.4 true strain ε [-]

0.5

0.6

0.7

Uniaxial stress-strain behavior of Plexiglas® Resist under six different hauling velocities.

displacement curves in Figure 4.1. From one representative test of each set, a respective stress-strain curve is determined following the approaches introduced in Section 4.2. The pronounced rate effects are clearly visible in the compilation in Figure 5.7 for Plexiglas® 8N. With increasing hauling velocity, the material behaves stiffer and the level of maximum strain decreases. The first effect is modeled in the approaches from Section 2.2, whereas the latter is topic of Chapter 6.

Young’s modulus E [MPa]

Analogously, in Figure 5.8 the stress-strain curves are provided for Plexiglas® Resist. Similarly, they show a increasing stiffness and a decreasing fracture strain height. The maximum strain of the quasi-static curve even lies above the 45 % stated by the manufacturer. That affirms the choice of a hauling velocity of

6000 vh = 3E0 m/s vh = 1E0 m/s vh = 1E − 1 m/s vh = 1E − 2 m/s vh = 1E − 3 m/s vh = 1E − 4 m/s vh = 1 mm/min lin. reg. line

5000

4000

3000 −10

−8

−6

−4

−2

0

2

strain rate ln(ε) ˙ [s Figure 5.9

4 −1

6

8

]

Rate dependence of Young’s Moduli for Plexiglas® 8N tensile specimens.

10

12

70

5 Sampling

Young’s modulus E [MPa]

1 · 10−4 m/s for this set. The upscaling from Section 2.2 appears as good engineering approach. A closer look is taken to Young’s moduli of both materials, starting with Plexiglas® 8N, whose adopted rheological model is able to reproduce rate dependence, as proven in [150]. Figure 5.9 plots Young’s moduli against the initial strain rate, which both are gained as average over all experiments of each set from Table 5.2. Here, strain rate is the slope of the strain-time curve in interval of the Hookean. The logarithmic abscissa in the plot produces a linear relationship between both quantities. That sparks the idea for the upcoming treatments in Section 6.2. Again, the same is applied for Plexiglas® Resist. The plot is given in Figure 5.10. The same stiffening for increasing strain rates shows. As stated, the applied material model does not reproduce this behavior. But in regard of the short linear interval compared to the total curve progression, the error is accepted. Above the yield stress, lying at about 25 MPa, the hardening curves consider rate effects. 3000 2800 vh = 3E0 m/s vh = 1E0 m/s vh = 1E − 1 m/s vh = 1E − 2 m/s vh = 1E − 3 m/s vh = 1E − 4 m/s lin. reg. line

2600 2400 2200 2000 1800

−8

−6

−4

−2

0

2

4

strain rate ln(ε) ˙ [s Figure 5.10

−1

6

8

10

12

]

Rate dependence of Young’s Moduli for Plexiglas® Resist tensile specimens.

5.3 Data Filters Hauling velocity is no adequate measure for a time dependent stimulation of materials. The geometry of the tested specimen too much affects the strain proportion that is necessary to compensate the movement, as tensile tests on the introduced specimen geometries show, c.f. Figure 4.7. The strain behavior is not linear over test time for a constant hauling velocity. That is caused by the initially good ability of polymer chains to untangle, followed by a stiffing when the chains are orientated

5.3 Data Filters

71

0.05

vh = 1.66 · 10−5 m/s

true strain ε [-]

true strain ε [-]

0.06

0.04 0.03 0.02 0.01 0 −400 −300 −200 −100

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

vh = 1 · 10−4 m/s

−150

−100

−50

0

time t [s]

time t [s]

Figure 5.11 Strain displayed over time for the quasi-static sets of Plexiglas® 8N (left) and Plexiglas® Resist (right). The deep blue coloring of the curves’ ends marks the interval for strain rate measure.

and less mobile. Therefore, the hauling velocity is replaced by the universal quantity strain rate, defined as ε˙ =

dεtrue . dt

(5.2)

The comparability of occurrences within a sample has to be ensured. One aspect is a similar strain rate at the point of failure, since the acrylic glasses sensitively respond to rate effects, as shown in Section 5.2. For a first visual verification, all experimental strain curves of a set are plotted into one diagram. The curves’ endpoints are shifted to time t = 0, by what the other curve points become negative. In these diagrams the end slopes are to compare best. Sets of vast differing curves are excluded. The respective plots are provided in Appendix B.1. Figure 5.11 is the plot for both materials’ quasi-static sets. In the colored curve section, the slope, i.e. strain rate, is determined by linear regression. For Plexiglas® 8N a representative section of about 15 % the length of the longest curve is chosen. For Resist it is about 10 % of the longest curve’s length. Each individual experiment is viewed and the section for strain rate measure adjusted when the slope is not constant in this interval. In the end, for each experiment a paring of fracture strain and strain rate is gained. In Figure 5.12 the samples’ fracture strains are plotted over the gained strain rates on logarithmic abscissa. Noticeable is the decreasing level of the fracture strains’ height, combined with a reduction of its dispersion. Though, a certain variance is recognized for the strain rates within a sample, which disagrees with the demand for comparability. For that reason, a filter is defined. The deviation of an experiment’s strain rate to the mean value of the sample is not allowed to exceed 20 %. When an experiment is excluded, the mean value is again calculated

72

5 Sampling

0.06

vh = 1.66 · 10−5 m/s

fracture strain εf [-]

vh = 1 · 10−4 m/s vh = 1 · 10−3 m/s

0.05

vh = 1 · 10−2 m/s vh = 1 · 10−1 m/s

0.04

vh = 1 · 100 m/s vh = 3 · 100 m/s

0.03 0.02 0.01 10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

strain rate ε˙ [s−1 ] Figure 5.12

Samples of Plexiglas® 8N fracture strains related to strain rate.

0.06

vh = 1.66 · 10−5 m/s

fracture strain εf [-]

vh = 1 · 10−4 m/s vh = 1 · 10−3 m/s

0.05

vh = 1 · 10−2 m/s vh = 1 · 10−1 m/s

0.04

vh = 1 · 100 m/s vh = 3 · 100 m/s

0.03 0.02 0.01 10−5

10−4

10−3

10−2

10−1

100

101 −1

strain rate ε˙ [s

102

103

104

]

Figure 5.13 Samples of Figure 5.12 filtered by exclusion of fracture strains featuring strain rates with 20 % deviation from mean value.

5.3 Data Filters

73

0.8 vh = 1 · 10−4 m/s

fracture strain εf [-]

0.7

vh = 1 · 10−3 m/s

0.6

vh = 1 · 10−2 m/s

0.5

vh = 1 · 100 m/s

vh = 1 · 10−1 m/s vh = 3 · 100 m/s

0.4 0.3 0.2 0.1 0 10−3

10−2

10−1

100 strain rate ε˙ [s

101 −1

102

103

]

Figure 5.14 Samples of Plexiglas® Resist fracture strains. Filtered by exclusion of fracture strains featuring strain rates with 20 % deviation from mean value.

as average from the remaining tests, and the inspection is restarted. In doing so, the filtered samples in Figure 5.13 are gained. Analogously, the samples of Resist are filtered, resulting in Figure 5.14. Here, the decreasing dispersion and lowering of the fracture strain level is shown even more clearly. Still, the probabilistic composition of the samples is to assess. A nice visualization is the scatter plot for a by-eye inspection. Each of the samples from Figure 5.13 is now plotted separately on unscaled axes, giving the picture of Figure 5.15. The same is provided for Resist in Appendix B.2. A linear relationship between the two quantities is not indicated. But for a more profound judgment, the coefficient of determination (CoD) / [εf,i − g(ε˙i )]2 , R = 1 − /i ¯f,i )2 i (εf,i − ε 2

(5.3)

where g(ε˙i ) is a linear regression line, is calculated for each sample. A CoD outcome near one would be a strong hint for linear dependency, whereas a outcome near zero would decline such dependence. Resulting R2 for all the samples are provided in Table 5.3. The maximum occurred CoD takes R2 = 0.09403, which tells against a linear relationship between fracture strain and strain rate. Thus, a probabilistic composition is assumed for the samples.

5 Sampling

fracture strain εf [-]

fracture strain εf [-]

74

vh = 1.66 · 10−5 m/s

0.055 0.05 0.045 0.04 0.035 0.03

0.16

0.17

fracture strain εf [-]

strain rate ε˙ [ms

0.03 0.025

1.6

0.02 0.016 0.018 0.02 0.022 0.024

0.03 0.025 vh = 1 · 10−1 m/s

0.9

1

1.1

fracture strain εf [-]

strain rate ε˙ [s−1 ]

0.02 vh = 3 · 100 m/s

35

40

45

strain rate ε˙ [s Figure 5.15

−1

1.9

2 −1

50 ]

Scatter plots of the filtered 8N samples.

2.1 ]

vh = 1 · 10−2 m/s

0.035 0.03 0.025 0.16

0.18

0.04

−1

0.2 ]

vh = 1 · 100 m/s

0.035 0.03 0.025 0.02 3

3.5

4

strain rate ε˙ [s−1 ]

0.025

0.015

1.8

strain rate ε˙ [s

0.035

0.8

0.04

]

0.04

0.02

1.7

0.14

fracture strain εf [-]

fracture strain εf [-]

strain rate ε˙ [s

vh = 1 · 10−4 m/s

0.03

strain rate ε˙ [ms

0.035

−1

0.035

]

vh = 1 · 10−3 m/s

0.04

0.04

0.18

−1

fracture strain εf [-]

0.15

0.045

4.5

5.3 Data Filters

75

Table 5.3 Coefficients of determination R2 for the filtered samples of Plexiglas® 8N and Plexiglas® Resist fracture strains from Figure 5.13 and Figure 5.14.

reference 3.0 · 100 m/s 1.0 · 100 m/s 1.0 · 10−1 m/s 1.0 · 10−2 m/s 1.0 · 10−3 m/s 1.0 · 10−4 m/s 1.66 · 10−5 m/s

coefficient of determination Plexiglas® Resist Plexiglas® 8N 3.394 · 10−3 4.298 · 10−2 2.505 · 10−3 2.535 · 10−2 3.448 · 10−2 6.032 · 10−3 2.480 · 10−5

6.817 · 10−3 2.975 · 10−3 1.295 · 10−2 4.858 · 10−3 1.821 · 10−2 9.403 · 10−2 -

Finally, each sample is assigned to a strain rate, which determines as mean value of the included experiments, i.e. the same mean value as used for the data filters. Thus, the 8N samples are from now on referenced by strain rate ε˙1 – ε˙7 , the one of Resist by ε˙1 – ε˙6 . Their corresponding values are listed in Table 5.4. In addition, the necessity for use of strain rates instead of hauling velocities becomes clear by examination of ε˙6 and ε˙7 of 8N. Whereas the hauling velocity for ε˙6 is six times faster than the one for ε˙7 , the strain rate is over 10 times higher, due to the geometry change from BZ to CAMPUS. Table 5.4

Strain rates ε˙1 – ε˙7 at failure for both Plexiglas® materials.

ref.

hauling velocity [m/s]

specimen geometry

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6 ε˙7

3 · 100 1 · 100 1 · 10−1 1 · 10−2 1 · 10−3 1 · 10−4 1.66 · 10−5

BZ BZ BZ BZ BZ BZ CAMPUS

strain rate at failure [s−1 ] Plexiglas® 8N Plexiglas® Resist 4.2 · 101 3.7 · 100 9.4 · 10−1 1.6 · 10−1 2.0 · 10−2 1.9 · 10−3 1.6 · 10−4

1.5 · 102 1.4 · 101 3.8 · 100 4.5 · 10−1 5.0 · 10−2 5.3 · 10−3 -

9

6 Statistical Modeling This chapter is the link between experimental measurements and stochastic simulation. A model is introduced that is able to be integrated directly into a conventional non-stochastic FE simulation. For that matter, certain requirements have to be fulfilled. Based on a given occurrence probability, a fracture strain must be provided. As seen in Chapter. 5, PMMA is a highly strain-rate dependent material. This dependency has to be reproduced in a fracture model for application in FE simulation, as well. Figure 6.1 schematically illustrates the required process. Input in the model is a probability between zero and one that is either provided by a random number generator, or as fixed default value by the user. Upon that, the statistical model defines a relationship between fracture strain and strain rate. The output function regards the occurrence probability in its parameters, but no longer as argument. In the running FE simulation, the current strain is surveyed for exceedance of the given fracture strain dependent on the current strain rate. In all the diagrams of this chapter, the occurrence probability is set as ordinate, which is the common illustration for probability distribution functions. Nevertheless, sought output quantity for numerical simulation is fracture strain. occurrence probability P ∈ (0, 1) Figure 6.1

statistical model εf (P, ε) ˙ → εf (ε) ˙

FE simulation εcurr ≥ εf (ε˙curr ) ?

Input and output of the statistical fracture strain model.

6.1 Probability Distribution Fit Based on the sampling from the previous chapter, the filtered sets of fracture strains are examined for their probability distribution. At first, the samples for each averaged failure strain rate are viewed separately. The fracture strains are sorted in ascending order. Following the probability estimator by Weibull from Table 2.5, each fracture strain is assigned to an occurrence probability in dependence of its position in the sorting, providing the empirical distribution of the measurements. At this point the term EDF is avoided, since it would mislead to the definition of Eq. (3.4). But with the use of WPE the classical EDF is left, c.f. Section 3.2. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_6

78

6 Statistical Modeling

As fit criterion for probability distribution functions a weighted residual sum of squares is chosen. Following Eq. (2.53) with the weight function from Eq. (3.10), weight is put equally to both function tails. That decision is made concerning many applications, in which these are the areas of major interest. Usually, the important information is, under which strain a component fails, and up to which strain a remaining lifting capacity must be expected. The analyses in Section 5.1 showed a dominant capability of the 2PW in reproducing the fracture strains of Plexiglas® 8N. As quantification for correlation of the cumulative distribution curve with the empirical data, the CoD is preferred over the WRSS. The WRSS is not a proper comparative measure for samples with different sizes. The bigger the sample is, the closer the plotting positions at the distribution tails tend to zero or one. Caused by the weight function from Eq. (3.10) that makes the resulting sum strongly sensitive to the tail residuals. The CoD / [pi − P (εf,i )]2 2 , (6.1) R = 1 − i/ ¯)2 i (pi − p

occurrence probability P (εf ) [-]

which was initially proposed for linear regression models, can also be adopted to non-linear fits [103]. As stated in Section 5.3, a great benefit is the intuitive interpretation of its numeric value that ranges between zero and one. An outcome near zero tells against a correlation of probability distribution function with the plotting positions, where an outcome near one indicates a good agreement. Conversely, the quantity 1 − R2 is interpreted as proportion of unexplained variation [130].

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.01

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6 ε˙7

0.015

0.02

0.025 0.03 0.035 0.04 fracture strain εf [-]

0.045

0.05

0.055

Figure 6.2 Empirical fracture strain distribution for each averaged strain rate with fit of a 2PW distribution. The examined material is Plexiglas® 8N.

6.1 Probability Distribution Fit

79

Table 6.1 Coefficients of determination R2 of a probability distribution fit for each sample of measured fracture strains ε˙1 –ε˙7 .

reference

Plexiglas® 8N 2-parameter Weibull

Plexiglas® Resist 2-parameter Weibull 3-parameter Weibull

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6 ε˙7

0.976 0.977 0.944 0.982 0.966 0.945 0.985

0.972 0.955 0.981 0.990 0.982 0.966 -

0.994 0.974 0.988 0.986 0.990 0.964 -

Mean:

0.968

0.974

0.983

occurrence probability P (εf ) [-]

In Figure 6.2, 2PW distribution functions are fitted to the samples of Plexiglas® 8N fracture strains. The averaged strain rates ε˙1 − ε˙7 are the ones in Table 5.4. The corresponding CoDs are given in Table 6.1. For all samples, the 2PW distribution shows high correlation. In worst case, the unexplained variation is only 5.6 %. As a summarizing quantity for the 2PW distribution’s reproduction of the material’s fracture behavior, the mean of the R2 -values is determined. An averaged variation of just 3.2 % is not explainable by the probability distribution function. That mean value is set to be the benchmark for a statistical model to reach. In the plot of Figure 6.2, one conspicuity occurs. The distribution of sample

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

ε˙6 ε˙1 ε˙2 ε˙3 ε˙4 ε˙5

0

0.1

0.2

0.3

0.4 0.5 0.6 fracture strain εf [-]

0.7

0.8

0.9

Figure 6.3 Empirical fracture strain distribution for each averaged strain rate with fit of a 2PW distribution. The examined material is Plexiglas® Resist.

80

6 Statistical Modeling

occurrence probability P (εf ) [-]

ε˙4 intersects with with samples ε˙3 and ε˙5 , violating the homogeneous image of the remaining distribution functions. Due to the detailed examination of all specimens, as described in Chapter 5, no differences could be detected. The effect could by chance result from the particular composition of the sample, and a second sample at equal strain rate would show another progression. However, the 30 sample points of ε˙4 should provide a good estimation of the population. For the time being, the investigation is continued. In Chapter 9 the subject is discussed once again. The same applies to Plexiglas® Resist. Analogously, a fit of 2PW distributions is performed for the Plexiglas® Resist. Despite the comparatively small variation for this material, indicated by the resulting CoDs in Table 6.1, the distribution function is not effective in reproducing the steep rise in occurrence probability paired with a gentle convergence towards one, that is shown in Figure 6.3. It fails in capturing the lower-tail plotting positions, in particular. Unlike the progression of a 2PW distribution, the empirical distributions seem to converge toward a lower limit. That becomes obvious especially at the quasi-static sample ε˙6 . In consequence, further probability distribution functions from Section 2.5 are tested, especially the ones, which feature a lower limit ε0 , or are a able to take an extremely asymmetric progression. The best fit is realized with the 3PW distribution, depicted in Figure 6.4. The worst unexpected ¯ 2 becomes 1.7 %, cf. Tavariation 1 − R2 is 3.6 % and the samples’ mean 1 − R ble 6.1. That seems to be not much of an improvement compared to the results for the 2PW, but by inspection of the mean WRSS a reduction of 24.1 % is gained, from WRSS = 0.451 to WRSS = 0.342. That is caused by a better capturing of the higher weighted distribution limits, which is confirmed by visual comparison of 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6

0

0.1

0.2

0.3

0.4 0.5 0.6 fracture strain εf [-]

0.7

0.8

0.9

Figure 6.4 Same empirical fracture strain distributions as in Figure 6.3, but with fit of a 3PW distribution.

6.2 Quantile Interpolation

81

Figure 6.4 and Figure 6.3. Hence, the 3PW distribution is preferred in the following investigations, giving a benchmark CoD of 98.3 % for the statistical model to reach.

6.2 Quantile Interpolation With the function parameters defined, an approach is needed that is able to describe the probability distribution dependent on the strain rate, as requested by the process in Figure 6.1. To begin with, for the Plexiglas® 8N, the fitted function parameters β and η are viewed in Figure 6.5 over the logarithmic strain rate. One aspect stands out: with increasing strain rate, the function parameters level decreases. In terms of physical interpretation, it means the fracture strain level is lowered. A simultaneous statement on the fracture strain’s dispersion based on the shape parameter β is not possible. Reason is the varying scale parameter η. Only for 2PW distribution functions with equal η the shape parameter can be used as comparative measure of their dispersion. Demonstrated is the effect of a decreasing η on the 2PW PDF of sample ε˙7 in Figure 6.6, where β is kept constant. If β was an independent measure for dispersion, the PDF would just be shifted to the left. But instead, its dispersion becomes even smaller than the one of sample ε˙1 , which is not true. Hence, the shape parameter influences the distributions dispersion not alone. The decreasing progression of β in Figure 6.5 is plausible. The scale parameters η take a clearly linear progression in the logarithmic plot, whereas the shape parameters β have a comparably high variation from a linear curve. A linear regression of this point cloud is not adequate, since the individual function param0.05 scale parameter η [-]

shape parameter β [-]

9 8 7 6 5 shape parameters regression line

4 10−4

10−2

0.03

0.02 scale parameters regression line

0.01 100

strain rate ln(ε) ˙ [s Figure 6.5

0.04

102 −1

]

10−4

10−2

100

strain rate ln(ε) ˙ [s

Function parameters of the 2PW distributions from Figure 6.2.

102 −1

]

82

6 Statistical Modeling 140

ε˙1 ε˙7

frequency f [-]

120 100 80 60 40 20

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 fracture strain εf [-] Figure 6.6 Probability density function of the 2PW distribution for samples ε˙1 and ε˙1 . In addition, the shape parameter β of the Weibull distribtion for sample ε˙7 is kept constant, and its scale parameter η is step-wise decreased to the level of the one from sample ε˙1 .

occurrence probability P (εf ) [-]

eters would not be reproduced. In consequence, it is refrained from modeling the function parameters and a different approach is taken. As stated in the previous section, the function tails are in many applications the areas of major interest. As for statistical significance levels α, the 5 % and the 95 % quantiles are often important guidelines. Using the example of the 5 % quantile, it is interpreted as the strain at which 5 % of all possible specimens will have fractured. Contrariwise, 95 % of all possible specimens will feature a fracture 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.01

Figure 6.7 bility.

0.95

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6 ε˙7

0.015

0.02

0.025 0.03 0.035 0.04 fracture strain εf [-]

0.045

0.05

0.05 0.055

Quantiles of the 2PW distributions from Figure 6.2 for 5 % and 95 % occurrence proba-

6.2 Quantile Interpolation

83

strain above the 5 % quantile. In the present case, the 5 % and the 95 % quantiles are the fracture strains marking 5 % and accordingly 95 % occurrence probability of the CDF. In Figure 6.7 the positions of the quantiles are indicated. They are the abscissa values corresponding to the marked intersections with the CDF. Also in this view, the deviant progression of the purple curve from sample ε˙4 is particularly clear to see, as well. Expected would be a curve with higher dispersion for an even material behavior. That is one reason for the high variation of the shape parameters in Figure 6.5. But since the sample analyses from Chapter 5 resulted in this very constellation, and no conspicuities could be exposed for this sample, the curves are considered equivalently. To continue the function analysis, instead of the function parameters, now the 5 % and 95 % quantiles are plotted in Figure 6.8 over the logarithmized strain rates. In addition, to each plotting position error bars are added that indicate the 20 % deviation from the strain rate mean that a specimen is allowed to have in order to be included into the sample, cf. Section 5.3. Since the scale is logarithmic, the left error bar is optically wider than the right one. The plotting positions of the quantiles in this logarithmic diagram noticeably adapt to a straight line. That raises the idea of a new approach for the statistical modeling. The progression of the quantiles over the strain rate is described by a simple linear function, fitted by linear regression. Having these two linear regression lines defined, for every arbitrary strain rate the corresponding 5 % and 95 % quantiles of the 2PW distribution can be determined. Not only an interpolation between the measured strain rates is possible, but even an extrapolation, with all due precaution. But it is not only the

fracture strain εf [-]

0.06 95 % quantiles regression line 5 % quantiles regression line

0.05 0.04 0.03 0.02 0.01 10−5

10−4

10−3

10−2

10−1

100

101

102

103

strain rate ln(ε) ˙ [s−1 ] Figure 6.8 Linear regression of the 5 % and 95 % quantiles from Figure 6.7. Error bars are added in size of the 20 % tolerated deviation from the strain rate mean.

84

6 Statistical Modeling

quantiles themselves that the model provides. As its name says, a 2PW distribution is dependent on two parameters that have to be defined. The knowledge of the two quantiles gives two points of the CDF. Therefore, the function parameters can be calculated by solving the system of equations

β ! ε0.05 (6.2a) 0.05 = 1 − exp − η

β ! ε0.95 0.95 = 1 − exp − , (6.2b) η where ε0.05 and ε0.95 are the quantiles gained from the regression lines. For a better understanding, the principle is visualized in Figure 6.9. The plotting positions from Figure 6.8 are removed, since their only purpose was to define the regression lines. With these two quantile functions, for representative strain rates – here 10−4 s−1 , 10−2 s−1 , 100 s−1 , and 102 s−1 – the corresponding 2PW distribution is determined. Schematically, their real PDFs are added to the plot, with filled area of the lower and upper 5 % of occurrences. At this point, the view is taken to the second material, where the introduced approach is adopted for modeling the 3PW distribution of the Plexiglas® Resist. Analogously, the 5 % and 95 % quantiles are taken from each CDF, as illustrated in Figure 6.10. Since the fracture behavior of this material is described by a threeparametric distribution, another quantile has to be considered in order to define all three function parameters. The lower limit is chosen because it is just the

fracture strain εf [-]

0.06 95% quantiles 5% quantiles

0.05 0.04 0.03 0.02 0.01 10−5

10−4

10−3

10−2

10−1

100

101

102

103

strain rate ln(ε) ˙ [s−1 ] Figure 6.9 Probability density functions of the 2PW distributions defined by 5 % and 95 % quantile, given for 10−4 s−1 , 10−2 s−1 , 100 s−1 , and 102 s−1 strain rate.

occurrence probability P (εf ) [-]

6.2 Quantile Interpolation

85

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.95

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6

0

Figure 6.10

0.1

0.2

0.3

0.4 0.5 0.6 fracture strain εf [-]

0.7

0.8

0.05

0.9

Lower limits, 5 %, and 95 % quantiles of the 3PW distributions from Figure 6.4.

location parameter γ of the function. Thus, only the parameters β and η have to be calculated through the 5 % and 95 % quantiles by solving the system of equations

β ! ε0.05 − γ (6.3a) 0.05 = 1 − exp − η

β ! ε0.95 − γ 0.95 = 1 − exp − . (6.3b) η As before, the quantiles are plotted in Figure 6.11 over the logarithmic strain rate, and regression lines are added. Noticeable is an increased variation of the 95 % quantiles from a straight line. This is caused by the function fits, compromising between both function tail. The 3PW distribution has difficulties to capture the upper plotting positions, as seen in Figure 6.4, especially for the quasi-static strain rate. Another notable instance is the extrapolation to higher strain rates. At some point, the regression lines have intersections. In mathematical terms, Eq. (6.3b) is still solvable, but in a physical sense, the 95 % quantile is not allowed to be less than the location parameter and the 5 % quantile. In consequence, the proposed model should in maximum be applied for strain rates up to 2 · 102 , which is near the measured rates. For assumptions on higher rates, further samples should be tested, first.

86

6 Statistical Modeling

fracture strain εf [-]

1 95% quantiles regression line 5% quantiles regression line location parameters regression line

0.8 0.6 0.4 0.2 0 10−3

10−2

10−1

100

101

102

103

strain rate ln(ε) ˙ [s−1 ] Figure 6.11 Linear regression of location parameters γ, 5 %, and 95 % quantiles from Figure 6.10. Error bars are added in size of the 20 % tolerated deviation from the strain rate mean.

6.3 Fracture Strain Model The proposed approach satisfies all requested properties for a statistical model from Figure 6.1. For both materials, the probability distribution function of fracture strain is defined for a given strain rate. By default, input of a distinct occurrence probability a corresponding fracture strain is provided. For example, a fracture strain of Plexiglas® 8N is received from the inverse of the 2PW distribution’s CDF εf = η[− ln(1 − P )]1/β ,

(6.4)

where P is the default input and β, and η are determined by solving the system of equations Eq. (6.2b), which is dependent on the strain rate. Since the 2PW distribution is continuously defined over the strain rate by the introduced quantile interpolation from Figure 6.9, the correlation of the three quantities strain rate, fracture strain, and occurrence probability can be visualized as surface plot, which is shown in Figure 6.12. In order to follow the process from Figure 6.1, the surface is cut horizontally at the level of a given occurrences probability. The received cut line is the relationship between fracture strain and strain rate that is needed for FE simulation.

occurrence probability P [-]

6.3 Fracture Strain Model

87

1 0.8 0.6 0.4 0.2 0 5 0 −5

strain rate ln(ε) ˙ [s−1 ]

−10

0.01

0.02

0.03

0.04

0.06

0.05

fracture strain εf [-]

Figure 6.12 Relationship between fracture strain, strain rate, and occurrence probability gained from the proposed quantile model for Plexiglas® 8N.

occurrence probability P (εf ) [-]

The evaluation of the quantile model remains. For that purpose, the quality of the model in reproducing the initial distribution of empirical fracture strains is examined. The averaged strain rates for the samples are known, so the necessary probability distribution functions can easily be determined by the quantile model.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.01

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6 ε˙7

0.015

0.02

0.025 0.03 0.035 0.04 fracture strain εf [-]

0.045

0.05

0.055

Figure 6.13 Comparison of the initial empirical distributions to the modeled 2PW distribution functions at correspondent strain rate.

88

6 Statistical Modeling fracture strain εf [-] 0.018

0.022

0.027

0.033

0.041

0.050

0.061

1

93.40

0

63.21 ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6 ε˙7

−1 −2 −3 −4

−4.2

−4

−3.8

−3.6

−3.4

−3.2

−3

30.78 12.66 4.85 1.81 −2.8

occurrence propability P (εf ) [%]

ln[− ln(1 − P )]

0.015

ln(εf ) Figure 6.14

Transformation of Figure 6.13 into a Weibull plot.

In terms of the surface conception, it is a vertical cut at constant strain rate, where the cut line is the probability distribution of fracture strain. In Figure 6.13 the gained CDFs are plotted together with the empirical distributions. Through the performed linear regression of the initial quantiles, the image of the quantile modeled distribution functions is more even than in Figure 6.2. On the downside, that causes a certain deviation to the empirical distributions. An offset is produced. In Figure 6.14 the CDFs are transformed into a Weibull plot. The capturing of lower plotting positions seems challenging for the model. But a pure optical evaluation is difficult, so again the CoDs are calculated and provided in Table 6.2. The samples

Table 6.2 Comparison of the coefficients of determination R2 for the initial fits from Figure 6.2, and the ones for the modeled distribution functions from Figure 6.13.

reference

initial fit

quantile modeled

percentage deviation

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6 ε˙7

0.976 0.977 0.944 0.982 0.966 0.945 0.985

0.873 0.884 0.866 0.955 0.467 0.931 0.966

-

mean:

0.968

0.849

- 12.3 %

10.6 9.45 8.35 2.74 51.7 1.53 1.94

% % % % % % %

6.3 Fracture Strain Model

89

occurrence probability P [-]

ε˙4 , ε˙5 , and ε˙7 are reproduced with slight decline in quality. Samples ε˙1 , ε˙2 , and ε˙3 feature an increased unexpected variation. Sample ε˙5 however has a massive loss of quality and impairs with its CoD the mean over all samples. Reason is the evenness through the CDFs delivered with the quantile model. The bundle of the empirical distributions of samples ε˙3 , ε˙4 , and ε˙5 is hardly be reproduced from a model. For a better understanding of the material behavior at this sector further tensile tests with varying strain rate should be conducted. The same evaluation is made with the fracture strain distribution of Plexiglas® Resist. Analogously to the quantile model of Plexiglas® 8N, fracture strains are provided dependent on a default probability and the current strain rate. The 3PW quantile model is demonstrated as surface plot in Figure 6.15. The underlying inverse of the 3PW distribution, which is used to determine distinct fracture strains out of the quantile model, has already been adopted in Eq. (3.16) for Monte-Carlo simulation. For this material as well, the quantile modeled probability distributions are compared to the empirical distributions. Figure 6.16 is the classical CDF diagram, whereas Figure 6.17 is the transformation into a Weibull plot. Both forms have their benefits. In the classical plot, the unbiased relations are at hand. The viewer has a better sense for the size of the appearing residuals. Just by visual examination, it is obvious that the empirical data in this model is better reproduced than in the one for Plexiglas® 8N. Only minor offsets are visible, despite the challenging progression of the empirical distributions. In the Weibull plot, the lower

1 0.8 0.6 0.4 0.2 0 5 2.5 0 −2.5

strain rate ln(ε) ˙ [s−1 ]

−5

0

0.2

0.4

0.6

0.8

fracture strain εf [-]

Figure 6.15 Relationship between fracture strain, strain rate, and occurrence probability gained from the proposed quantile model for Plexiglas® Resist.

6 Statistical Modeling

occurrence probability P (εf ) [-]

90 1 0.8 0.6

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6

0.4 0.2 0 0

0.1

0.2

0.3 0.4 0.5 fracture strain εf [-]

0.6

0.7

Figure 6.16 Comparison of the initial empirical distributions to the modeled 3PW distribution functions at correspondent strain rate.

tails are much easier to examine. The quantile model shows a good capturing in this area. Clearly visible is the asymptotical convergency toward the lower limit. Besides the optical inspection, the CoDs are examined, too. Their results given in Table 6.3 are remarkable. The mean R2 is just 4.31 % less than the one for the ¯ 2 of 6 %. Thus, initial fits. This leads to an averaged unexpected variation 1 − R ® the quality of the Plexiglas Resist quantile model is attested.

fracture strain εf [-] 0.050

0.082

0.135

0.223

0.368

0.607

1

93.40

0

63.21

−1

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6

−2 −3 −4 -3.5

Figure 6.17

-3.0

-2.5

-2.0 -1.5 ln(εf )

Transformation of Figure 6.16 into a Weibull plot.

-1.0

-0.5

30.78 12.66 4.85 1.81

occurrence propability P (εf ) [%]

ln[− ln(1 − P )]

0.030

6.3 Fracture Strain Model

91

Table 6.3 Comparison of the coefficients of determination R2 for the initial fits from Figure 6.3, and the ones for the modeled distribution functions from Figure 6.16.

reference

initial fit

quantile modeled

percentage deviation

ε˙1 ε˙2 ε˙3 ε˙4 ε˙5 ε˙6

0.994 0.974 0.988 0.986 0.990 0.964

0.970 0.898 0.943 0.932 0.952 0.947

-

mean:

0.983

0.940

- 4.31 %

2.38 7.83 4.62 5.43 3.84 1.75

% % % % % %

9

7 Stochastic Simulation The introduced statistical model is now utilized in a practical application. In the studies of [113, 150] a head impact test on PMMA automotive rear side windows was chosen as validation test for the developed material models. As continuation of this work, the material models are extended by a stochastic failure criterion for FE simulation. Analyzed quantity is an injury risk, represented in the head injury criterion (HIC). The following investigations will be concentrated to windows made from Plexiglas® 8N. The impact resistant Plexiglas® Resist will show not to allow component failure in the conditions of the test. The examined rear-side windows are prototypes and were never employed in real cars. Hence, no measurements and no conclusions are transferable to existing vehicle models on the street. The head impact on a rear-side window is no official safety validation test of automotive designs. For street accreditation [56] typically the injury risk for pedestrians, who are hit frontally by a car and drop with their head onto the windscreen, is examined. The scenario of an impact onto a rear-side window is rather academic. Though, there are comparable tests established. In rating the hazard potential for pedelec cyclists in road traffic, insurance companies examine i.a. the injury risk in a crash against the side of motionless a car [69]. Especially for tall cars, such as the SUV from Figure 1.2, the impact point of the head is likely on the window. Another scenario is the automotive side crash, which is an actual validation test

Figure 7.1

CAD model of examined automotive rear side window from Figure 1.2.

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_7

94

7 Stochastic Simulation

[86]. Here, the motionless car is laterally hit by a crash cart. The sudden acceleration of the car effects the head of an occupant to be accelerated towards the side window. In modern cars, airbags prevent the contact to vehicle structures, but these are not existent in older models. For occupants in jump seats, which are located in the boot, airbags might not be available even today.

7.1 Head Impact Test The experimental component tests of [113, 150] are operated similar to established automotive validation tests. The chosen head impactor is the JHC06C child head form with 3.5 kg total mass, following the Japan New Car Assessment Programme (JNCAP) [123]. The choice of a child head form is motivated by the comparatively small surface of the window. The idea is to prevent edge effects, since the space between dummy and frame is bigger than with the adult head form. The test setup is shown in Figure 7.2. Analogously to the assembling at the vehicle, the windows are adhered to a frame providing the same bearing area as the coachwork. The dimensions are known from the computer-aided design (CAD) model in Figure 7.1. In orthogonal direction to the window surface, the head is accelerated up to a default contact velocity of 10 m/s at defined impact position. Within the center of gravity of the head impactor, a sensor measures the acceleration in all three dimensions. For documentation, a high-speed camera films the test, whose recordings of the impact are provided in Figure 7.3. The first picture is of a specimen made of tempered safety glass (TSG), which is the conventional material for side windows. Due to its thermally pre-stressed state, it fractures into numerous small fragments.

Figure 7.2 Test setup with child head form dummy and Plexiglas® 8N automotive rear side window (left), and positioning of the robot arm (right).

7.1 Head Impact Test

95

Figure 7.3 Head impact test on automotive rear side windows made from tempered safety glass (left), Plexiglas® 8N (middle), and Plexiglas® Resist (right).

The other two windows are made of Plexiglas® 8N and Plexiglas® Resist. The brittle material fractures in bigger fragments, whereas the impact resistant material does not feature any failure, at all. The effect of the different fracture behaviors on the head acceleration is tremendous. Figure 7.4 gives the corresponding acceleration within the head impactor. The acceleration is taken as resultant

(7.1) a = a2x + a2y + a2z . The signals are filtered by a SAE J211 CFC-1000, as specified by [154]. Compared to the Plexiglas® materials, the TSG is much stiffer and reaches its point of failure already at 2 ms with a resultant acceleration of about 200 times the gravitational acceleration g. At the beginning, the curves of 8N and Resist are very similar.

resultant acceleration a [g]

400 toughened safety glass Plexiglas® 8N Plexiglas® Resist

350 300 250 200 150 100 50 0 0

1

2

3

4

5

6 7 8 time t [ms]

9

10

11

12

13

14

Figure 7.4 Resulting acceleration within head dummy of the experimental impact tests in Figure 7.3, measured in multiples of the gravitational acceleration.

96

7 Stochastic Simulation

Table 7.1

material HIC

Resulting head injury criteria for the head impact tests of Figure 7.3.

tempered safety glass

Plexiglas® 8N

Plexiglas® Resist

426

121

7238

The one of 8N reaches its peak at 3 ms at about 150 g, which is a far smoother acceleration of the head as for the TSG. The absence of failure for Resist produces a massive peak of acceleration and a progression over a long period. That is a fatal exposure to the head. In order to quantify these exposures, the HIC is determined [127], which is a measure for the most critical constellation between acceleration’s height and duration time in the acceleration process. It is calculated by + , 2.5 t2 1 HIC = max a(t)dt (t2 − t1 ) , (7.2) t2 − t1 t1 with respective choice of the time limits for maximization of the function outcome. Dependent on the crash scenario, maximum time intervals are provisioned, e.g. 15 ms for testing a vehicle’s frontal protection systems [56]. Among other crash test scenarios, it is assessment criterion for occupant safety in the previous mentioned side crash tests [85], and in pedestrian head impact test on windscreens [56]. Dependent on the crash scenario, different critical values are defined. In this work, the critical limit of HICcrit = 1000 is adopted, since the setup is most comparable to a windscreen crash test, where also a dummy head is shot with similar velocity onto the outside of the window. The limit of 1000 results from studies on the effects of acceleration to the injury risk of a human head [126]. This limit ensured most of the population from a serious to fatal injury [127]. The gained HIC values for the present experiments are given in Table 7.1. As expected, the HIC for the Plexiglas® 8N is notably lower than the one for the TSG. The less stiff behavior, paired with a lower acceleration peak is positive for the head exposure. On the other hand, the Resist side window overshoots the limit with a HIC of 7238 by far. Such an exposure to the head would undoubtedly lead to exitus. But as already stated, these impact tests are academic evaluation tests and no automotive safety validation tests.

7.2 Simulation Model The constitutive relation for the Plexiglas® materials is prepared in [113, 150], as introduced in Section 2.2. The 8N is described by a generalized Maxwell model of springs and dampers. The statistical modeling in this work considers the fracture

7.2 Simulation Model

97

Figure 7.5 Simulation model of the head impact test on rear side window with Japanese child head form impactor JHC06C.

strains from uniaxial tensile tests, which were determined in tensile direction and equal first principal strain. Thus, failure criterion for erosion in FE simulation is chosen to be first principal strain, too. For the utilized type of visco-elastic material model the FE solver allows their direct input. In contrary, Resist bases on a visco-plastic material model. Here, the input of first principal fracture strains into the FE solver is not directly possible, because failure criterion is plastic strain. To overcome this drawback, the first principal fracture strains, i.e. total fracture strain εf , of the statistical model are translated into corresponding plastic fracture strains εpl,f = εf − εel = εf −

σyield E

(7.3)

by subtraction of the elastic part εel . The plastic fracture strains are always greater than zero, because σyield /E < εf for this very material. The simulation setting from [152] is rebuilt. Bearing dimensions and material model for the seal are adopted, but the side window is reworked. Having the CAD data from Figure 7.1, the FE window is re-meshed. For a plane contact between seal elements and the new window, the upper seal nodes are projected right onto the window surface. That concurs with the laboratory preparations, where the side windows were slightly pressed onto the adhesive on the bearing for continuous contact, but effecting a variable seal thickness corresponding to the respective gap between both components. The aluminum bearing is realized as rigid body. The head impactor is a commercial solution and embedded in the setup as depicted in Figure 7.5. Impact position is the same as in Figure 7.2. The impact velocity is taken from the experiments, where the default of 10 m/s is not reached. For test of the Plexiglas® 8N window that is 8.55 m/s. The mesh of the window is created for as much homogeneity in the impact area as possible. That is where

98

Figure 7.6 edge size.

7 Stochastic Simulation

Simulation model of the rear side window, featuring a finite element mesh with 10 mm

the fracture occurs. The material models from [152] are validated for a mesh size of 5 mm averaged edge length. Hence, this is the primary type that is considered in the following investigations. But in addition, two more versions are examined. A window meshed with elements of 2.5 mm averaged edge length, and one with 10 mm averaged edge length. The element mesh of the side window is shown in Figure 7.6, for a better visualization in the 10 mm version. In a more detailed view, Figure 7.7 compares all the three meshes side by side. Their effect on the HIC will be shown in Section 7.3.

Figure 7.7 Homogeneous finite element meshes near the contact zone with edge sizes of 2.5 mm (left), 5 mm (middle), and 10 mm (right).

7.2 Simulation Model

Table 7.2

99

Reference strain rates of Plexiglas® 8N erosion card.

mantissa

supporting points ε˙1 – ε˙90 ε˙91 – ε˙180 ε˙181 – ε˙270 ε˙271 – ε˙360 ε˙361 – ε˙450 ε˙451 – ε˙540 ε˙541 – ε˙630 ε˙631 – ε˙720 ε˙721 – ε˙810 ε˙811 – ε˙900

1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,

1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, 1.1,

1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2, 1.2,

1.3, 1.3, 1.3, 1.3, 1.3, 1.3, 1.3, 1.3, 1.3, 1.3,

... ... ... ... ... ... ... ... ... ...

,9.9 ,9.9 ,9.9 ,9.9 ,9.9 ,9.9 ,9.9 ,9.9 ,9.9 ,9.9

power of ten

unit

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102

s−1 s−1 s−1 s−1 s−1 s−1 s−1 s−1 s−1 s−1

In the course of this work, most important in the FE simulation’s setup is the integration of the developed statistical model. Both quantile models, the one for Plexiglas® 8N, and the one for Plexiglas® Resist, are employed to describe a rate dependent failure criterion for stochastic simulation. The process is the same as already indicated in Figure 6.1. The stochastic part is realized by random generation of occurrence probabilities, which are uniformly distributed numbers between zero and one. For the present study, 1000 random occurrence probabilities are generated with a uniform pseudo-random number generator [124]. Each occurrence probability defines a individual relationship between strain rate and fracture strain, i.e. defines the horizontal cut in the surface plot of Figure 6.12, or Figure 6.15. This ˙ given by the quantile model is continuous. The FE solver requests function εf (ε) tabulated data, so as supporting points of the function, fracture strains are calculated for reference strain rates, which are listed in Table7.2. For a most precise ˙ 900 supporting points are generated. Having 1000 random description of εf (ε), occurrence probabilities resulting in 1000 erosion cards, accordingly 1000 complete FE simulations of the head impact are run with this global failure criterion. Then, in post-processing the effects on the HIC are analyzed, cf. Section 7.3. To give an impression of the fracture behavior of the window with such failure criterion, in Figure 7.8 are the fracture patterns for a default occurrence probability of 5 % compared to the one for 95 %. In both cases, the elements under the impactor fail first, which is also symptomatic for all 1000 random simulations. In further progress, a element failure propagation in shape of a plus sign extends over the window. That is not that different to the experiment on Plexiglas® 8N, where also larger fragments developed. Though, the current fracture pattern is clearly influenced by the constellation of the FE mesh.

100

7 Stochastic Simulation

Figure 7.8 Fracture patterns right after failure for a default occurrence probability of 5 % (left), and 95 % (middle). In addition, the advanced fracture pattern at termination time for a default occurrence probability of 95 % (right).

150

resultant acceleration a [g]

resultant acceleration a [g]

The HIC calculation in the stochastic simulation is automated and follows Eq. (7.2). Figure 7.9 demonstrates the location of the HIC interval for a reproduction of the experimental head impact. The respective rate dependent fracture strain is determined via reverse engineering. In order to eliminate errors, every acceleration curve and HIC calculation is inspected in the stochastic simulation. In summary, for all runs the 8N windows failed, except one. This is the window with the least probable strain to endure. The occurrence probability of p = 0.99977 is interpreted as 99.977 % of the specimens from the population to have fractured at this strain. The failure criterion is shown in Figure 7.10. Integration follows Lobatto’s integration rule. Given by the geometry of the window and the impact

experiment simulation

100

50

0 0

1

2 3 time t [ms]

4

150

simulation HIC

100

50

0 0

1

2 3 time t [ms]

4

Figure 7.9 Head acceleration from FE simulation with reverse engineered fracture strain for reproduction of the experimental curve (left). Indication of the HIC for the simulated head impact (right).

7.2 Simulation Model

101

first principal strain ε1 [-]

0.05 0.04 0.03 0.02

critical integration point failure strain, p=0.99977 failure strain, p=0.95 failure strain, p=0.05

0.01 0 0

1

2

3 4 time t [ms]

5

6

7

Figure 7.10 Element analysis for Plexiglas® 8N window. Maximum principal strain at the window’s most affected integration point. In comparison, the rate dependent failure strain for different occurrence probabilities.

position of the head dummy, it is in most cases a specific integration point, lying on the outer tension stressed surface of the window, that fails first. The strain in this integration point is a good comparative for the failure criterion. The critical strain is reached for the 5 % and 95 % occurrence probability. But for 99.977 % the strain curve flattens before reaching the limit, and the window remains intact.

0.03 failure strain, p=0.05 critical integration point

0.1

0.025

0.08

0.02

0.06

0.015

0.04

0.01

0.02

0.005

0

plastic element strain εpl [-]

plastic failure strain εpl,f [-]

0.12

0 3

4

5 time t [ms]

6

7

Figure 7.11 Element analysis for Plexiglas® Resist window. Plastic strain at the window’s most affected integration point, falling far below of the rate dependent failure strain.

102

7 Stochastic Simulation

The undercut of the critical strain is dominant for all simulations of the side window made of Plexiglas® Resist. Again, the strain of the critical integration point is viewed. Since the FE solver gives the output of strain components in x-, y-, and z-direction and the shear components εxy , εyz , and εxz , the maximum principal strains, or rather the plastic strain for the Resist material model, has to be calculated as seen in Section 2.3. This results in the progression of Figure 7.11. Even for low occurrence probabilities, strain at the integration point is far below the failure strain. For a visualization in the same diagram, a second ordinate axis has to be added. Due to the absence of failure, in this very investigation on a head impact the stochastic failure criterion provides no additional information compared to a non-stochastic simulation of the experiment from Figure 7.4.

7.3 HIC Distribution As previously introduced, one thousand head impacts on a Plexiglas® 8N rear side window are simulated with a global failure criterion that bases on a random generated occurrence probability. The dispersive fracture strain causes different time intervals within which the head dummy is decelerated. As seen, the HIC is as quantity sensible for acceleration time. Thus, it is a good indicator to view the influence of a stochastic material behavior. In the current test, its calculation is automated in post-processing, which is experienced to work reliable, since the overall test time

1 0.9

frequency f [-]

200

0.8 0.7

150

0.6 HIC frequencies HIC EDF

100

0.5 0.4 0.3

50

0.2 0.1

0 0

Figure 7.12 suppressed.

100

200

300

occurrence probability P [-]

250

0 400 500 600 700 800 900 1000 1100 1200 head injury criterion HIC [-]

Head injury criteria of 999 simulated head impact tests. The 1000th HIC of 5634 is

occurrence probability P [-]

7.3 HIC Distribution

103 1

0.8 0.6 0.4 0.2 0 100

log-normal CDF mesh size 5 mm 5 % quantile 95 % quantile

101 102 103 head injury criterion ln(HIC) [-]

104

Figure 7.13 Empirical distribution of the HIC on logarithmic abscissa. In addition, the HIC for a default occurrence probability of 5 %, and 95 %.

undercuts the maximum time interval of 15 ms for HIC measurement [4] and no time limits are reached by the integral Eq. (7.2). In Figure 7.12 the resulting HICs are presented both as histogram with a bin width of 25, and as staircase plot of the empirical cumulative distribution. The plotting positions of the latter are the respective HIC with the initially generated occurrence probability for erosion card generation. In this diagram the biggest HIC of 5634 is left out to gain a reasonable resolution of the remaining HICs. A distribution with distinctive skewness towards the the lower limit becomes apparent. For a better view in its progression, in Figure 7.13 a logarithmic abscissa is chosen. In addition to the stochastic simulated HICs, the ones from Figure 7.8 for a default of 5 % and 95 % are added. These are of special interest in this work, because of their direct deviation from the quantile regression model in Figure 6.8. It is conspicuous that in this diagram the HIC distribution forms the well-known S-shape of a CDF, having a rather symmetric progression. A comparison of this distribution with the probability distribution functions of Table 2.4 is provoked. Drawing on experience, several eligible distribution functions are fitted and tested with the introduced goodness-of-fit tools. One function stands out. It is the normal distribution related to the logarithmic diagram, i.e. log-normal distribution for the unscaled HICs. For this type of function, the reproduction of the plotting positions can nicely be shown in a so-called Q-Q plot [71]. Based on the −1 of the cumulative distribution function from Eq. (2.26), the resulting inverse Pnorm quantiles qnorm,i (i = 1, 2, ...n) are calculated for a standard normal distribution

104

7 Stochastic Simulation 9 8

ln(HIC)

7

2981 1097

6

403.4

5

148.4

4

54.60

3

20.09

2

7.389

1

2.718

0

−4

−3

−2

−1

head injury criterion HIC [-]

8103 stochastic simulation linear regression line

1.000 0

1

2

3

4

standard normal quantile qnorm [-] Figure 7.14

Q-Q plot for the 1000 simulated HIC.

with mean μ of zero, and a standard deviation σ of one. Input probability Pi is the occurrence probability of the HIC CDF. With −1 [Pi |μ = 0, σ = 1] qnorm,i = Pnorm

(7.4)

the plotting positions in the Q-Q plot then become [qnorm,i | ln(HIC)]. Comparable to a Weibull plot, the data points take in case of normal distribution a linear form. In Figure 7.14 is the Q-Q plot for the logarithmic HIC distribution. As expected, the log-normal distribution is discovered to be a good approach. In Table 7.3 the results for the fit of a log-normal distribution, cf. Eq. (2.27), whose parameters are again determined via minimization of the WRSS of Eq. (2.53). The large sample size of 1000 HICs yields in a high WRSS, because for this measure the residuals are summarized without any normalization to the sample size. It gives no independent information on the goodness of the fit. Differently, the more general CoD. It calculates to a dominant score of just 1 − R2 = 0.95 % unexplained variation. However, the goodness-of-fit tests from Chapter 3 neglect the null hypothesis of the log-normal distribution to be the population of the HIC dis-

Table 7.3 Fit results for a log-normal probability distribution function to the stochastic simulated head injury criteria. Examined are weighted residual-sum-of-squares, coefficient of determination, and test statistics of Anderson-Daling, generalized Anderson-Darling, lower-tail generalized Anderson-Darling, and Cramér-von Mises test.

test

WRSS

R2

A2

A2G

A2G,LT

W2

statistic

4.633

0.9905

5.204

4.610

2.155

1.004

7.3 HIC Distribution

105

tribution, i.e. no statistical significance is attested. One problem for the weighted goodness-of-fit tests are the plotting positions at the function tails. Due to the large sample size, there are some plotting positions very close to zero and to one, effecting a vast weight of their residuals. As seen in Figure 7.14 especially these areas are not well captured. This is supported by the better outcome of the CvM test statistic, compared to those of the AD, GAD, and lower-tail GAD test. It considers no weighting and is therefore less effected. But still the CvM test attests no statistical significance. One thousand occurrences are an enormous sample and do not leave much space for a probability distribution model to estimate the population. Nevertheless, instead of continuing with the empirical HIC distribution the log-normal approach   ln(HIC) − μ 1 1 √ (7.5) P (HIC) = + erf 2 2 σ 2 is carried on, which is plotted in Figure 7.13. A more intuitive visualization for this type of distribution is the PDF   [ln(HIC) − μ]2 1 exp − , (7.6) p(HIC) = √ 2σ 2 σ 2π HIC analogous to [66]. Its bell-shaped curve on a logarithmic abscissa is depicted in Figure 7.15. This diagram is the most important result from this work. It nicely illustrates the frequency of the single HIC outcomes. Its mean is at a HIC of about 67.2. All occurrences above 1000 are colored deep red, symbolizing the violation

frequency f [-]

0.4 HIC < 1000 HIC ≥ 1000 experiment

0.3

0.2

0.1

0 100

101 102 103 head injury criterion ln(HIC) [-]

104

Figure 7.15 Log-normal probability density function of the HIC with indication of the critical limit and position of the experimental HIC.

106

7 Stochastic Simulation

of the raised limit. This refers to 0.61 % of the population. The interpretation of those 0.61 % is discussed in the following section. Thrilling is the classification of the experimental outcome in this context. The gained HIC of 121 leads in Eq. (7.5) to an occurrence probability of P = 0.707. Going all the way back to the start of the proposed process, that means for an input of a default occurrence probability of 0.707 into the quantile model from Figure 6.12 a relationship between fracture strain and strain rate is gained, which provokes as failure criterion in the FE simulation the very acceleration curve of Figure 7.9. An occurrence probability of P = 0.707 for the experiment confirms the plausibility of the stochastic simulation. A more extreme result, i.e. a location of the experimental HIC at the PDF tails, would have caused suspiciousness due to rare occurrence in these areas.

occurrence probability P [-]

Least, there is one influencing factor to examine. In the previous section, different FE meshes for the side window were presented. Main focus is placed on the one with 5 mm edge length since it is the validated mesh from [150]. Though, the 1000 runs of the stochastic simulation are repeated for the other two window versions. Again, the HIC distributions are determined. Their progression is in Figure 7.16 compared to the hitherto examined one. It reveals a noticeable effect. As one would imagine, the smaller elements, which provide a higher resolution of the strain levels and thereby fail earlier, reduce the height of the HIC. Respectively, the bigger elements blur the strain peaks and fail later, by what the head dummy is decelerated over a longer period and the HICs increase. The effect on the corresponding PDF is demonstrated in Figure 7.17. Both, standard deviation and mean of the distribution, are affected. That is a numerical issue that has to be addressed. A convergence is not visible in Figure 7.16. Thus, either the stochastic simulation has to be repeated for varying element sizes, in order to find a convergence point,

Figure 7.16

1 0.8

2.5 mm mesh 5.0 mm mesh 10 mm mesh

0.6 0.4 0.2 0 100

101 102 103 head injury criterion ln(HIC) [-]

104

Mesh dependence of the empirical HIC distribution on logarithmic abscissa.

7.4 Discussion

107

frequency f [-]

0.4

0.3

0.2

0.1

0 100

Figure 7.17

HIC < 1000 HIC ≥ 1000 2.5 mm mesh 10 mm mesh

101 102 103 head injury criterion ln(HIC) [-]

104

Mesh dependence of the log-normal HIC probability density function.

or a regularization for the failure criterion must be established. A recommendation is made for the latter. As referred in Section 1.2, several volume and area dependent regularization approaches exist for the Weibull distribution. In further investigations, a coupling with the element size and accordingly the arrangement of integration points must be developed.

7.4 Discussion Great advantage of the proposed procedure for stochastic simulation is its simple setup for the user. The statistical quantile model is once to prepare in laboratory testings for a respective material. Based on that, a stand-alone program is to implement having an internal random number generator, which automated creates erosion cards of given amount n. Then, an existing non-stochastic simulation is turned into a stochastic one by repeating the computation n times, once for each random erosion card. By automated post-processing, the additional effort for the user, compared to an ordinary simulation, is marginal. Only the calculation time increases with factor n. In a first attempt, the influence of the FE mesh’s element size on the received HIC distribution is examined for three different configurations, cf. Figure 7.16 and Figure 7.17. The observed sensitiveness is not to charge onto the stochastic fracture model. The element size’s influence on the resolution of calculated strains is an effect from numerics. The created FE elements feature a square-base of 4 integration points with 5 levels over thickness, i.e. 20 integration points in total per element. Their relative arrangement is constant. Consequently, for smaller

108

7 Stochastic Simulation

elements the integration points come closer together, by which local strain peaks in the component a captured more precisely. In Section 7.3, a suggestion is made for regularization of the failure criterion. Another way to overcome the uncertainty about a proper configuration of the simulation model is by additional measurements in the laboratory component tests. In the head impact test on the rear side window it was not possible to analyze the surface strain via DIC. The fracture strain in those experiments is only to be estimated by reverse engineering. However, the knowledge of the real local strains would enable a calibration of the FE model. With fracture strain and strain rate at failure known, the corresponding occurrence probability is received by Figure 6.12, from which in turn an erosion card is created. With that failure criterion, the element size of the FE mesh can be adapted in order to reproduce the experimental head acceleration, i.e. the HIC. Then, the gained FE model is the calibrated setup for stochastic simulation. The HIC distribution and the limit of 1000 is an abstract image. For a better sense of the consequences from such a distribution, a small demonstrative example is brought up. Every year in Germany the Kraftfahrt-Bundesamt publishes the registration numbers for the different vehicle classes and types. Looking at the numbers of the Volkswagen Golf during the production years of their type VI model, in sum 1,033,842 cars were registered, cf. Table 7.4. We take the fictitious instance that there was an equally successful vehicle built with registration numbers of about 1,000,000, and imagine that vehicle to have a similar Plexiglas® 8N rear side window with a stochastic fracture behavior as characterized in this work. If there was a head impact safety validation test for this car, the chances would be 99.39 % that in a single laboratory crash test the verification is passed. On the other hand, referred to a registration of 1,000,000 cars resulting in 2,000,000 rear side windows, the determined probability distribution predicts about 2, 000, 000 · 0.0061 = 1220 critical components that would be assembled in street legal vehicles. For a head impact away from the examined impact point and nearer to the solid coachwork structure of the vehicle, the injury risk even increases. The relevance for safety evaluations of structural parts is evident: for a material with high variation in its fracture behavior, a single experimental validation test carries the danger of underestimation of the risk potential. At volume production of automotive windscreens, or side windows it is most likely to have single units

Table 7.4 Registrations of Volkswagen Golf series in Germany during the production years of the Golf type VI. Numbers by the Kraftfahrt-Bundesamt [96, 97, 98, 99, 100].

year number

2008

2009

2010

2011

2012

total

172,461

287,534

195,068

193,253

185,526

1,033,842

7.4 Discussion

109

showing critically high HICs in a comparable test setup. Hence, we recommend the careful consideration, whether a statistical characterization of a materials failure interval would be a gain in application safety. With knowledge of the probability distribution a threshold interval, for example 99.997 % (4σ) of the expected occurrences, should then be the evaluation benchmark. A single evaluation test of a structural part made of a stochastic fracturing material might lead to false decisions for a design, since even small variations in the material behavior can result in high deflection of the evaluation criterion.

9

8 Experimental Basis for Model Enhancements This final chapter is an extended outlook on the subject of the present work. As known from numerous studies, cf. Section 1.2, the material behavior of PMMA is besides strain rate likewise sensible to the applied temperature and stress state. Hitherto, all experiments are at room temperature. The ambient temperature is monitored for a level of about norm temperature, i.e. 23 ◦C and 50 % humidity [47]. Due to laboratory conditions, in all tensile tests, a mean ambient temperature of 25 ◦C with a mean humidity of is 38.3 % achieved. An overview of the corresponding temperature minima and maxima with information on their standard deviation is provided in Table 8.1. Now in the subsequent testings, the material behavior is additionally examined at the temperature limits of automotive application. Then, in the second part of this outlook, test setups are introduced for material loading with shear and biaxial tension. In either case, purely experimental results are presented, with focus placed on the fracture strain distribution. Similar studies on the fracture behavior do not exist, yet. Though, the received empirical probability distributions state their necessity. Hence, this chapter might be an impulse for further investigation in this field. Table 8.1 Averaged ambient temperature and humidity for the tensile test sets at room temperature. In addition, the averages of each test set’s minimum and maximum.

temperature [◦C] mean value standard deviation

humidity [%]

T min

T

T max

H min

H

H max

24.2 1.42

25.0 1.49

25.6 1.69

36.8 5.78

38.3 6.63

40.0 7.77

8.1 Temperature Dependency The experimental data in this section bases on tensile tests that are performed on the same electro-mechanical system introduced in Section 4.1. Analogously, © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_8

112

8 Experimental Basis for Model Enhancements

Figure 8.1 BZ tensile specimen (A) fixed between two clamping jaws (B). The dual-camera system (D) films into the climate chamber (C).

the dual-camera system is applied for DIC analysis. As described, synchronization of force and image series is achieved by the DIC system. Additionally to the established setup, a climate chamber is added. The chamber features a refractionreduced side window for optical measurements. The entire setup is depicted in Figure 8.1. The DIC system is calibrated with applied climate chamber to reproduce the test configuration. The specimens are conditioned for minimum 12 hours in a climate cabinet. Afterwards, the whole set is placed inside the climate chamber of the testing machine and again conditioned for one hour. Since, for assembling of each individual specimen into the clamping jaws the front door of the climate chamber is opened, another conditioning time of 10 minutes is maintained until the test is started. Due to a catch at the clamping jaws for quick alignment of the specimens, the door is only opened briefly. With regard to internal automotive OEM test standards, temperature behavior of the material is examined at −30 ◦C and 85 ◦C. These temperatures are meant to ensure the operation of a vehicle likewise in a tundra, as under the desert sun. The setup’s applied climate chamber provides an adjustment control of ±5 ◦C. For the negative temperature, humidity is set to 10 % to prevent icing. To enhance comparability, for 85 ◦C humidity is set to 40 %, according with the room temperature average of Table 8.1. For the same reason, tensile velocity is set to 1 · 10−4 m/s.

8.1 Temperature Dependency

113

true stress σ [MPa]

100

+85 ◦C +23 ◦C −30 ◦C

80 60 40 20 0 0

0.1

0.2

0.3

0.4 0.5 0.6 true strain ε [-]

0.7

0.8

0.9

1

Figure 8.2 Uniaxial stress-strain behavior of Plexiglas® 8N at different temperatures for 1 · 10−4 m/s hauling velocity.

The choice for this quasi-static tensile velocity is made to reduce the material’s inner friction since effects of heating are not regarded in the current test setup. Representative stress-strain curves for both Plexiglas® materials at the tested temperatures are provided in Figure 8.2 and Figure 8.3. For an orientation, the already introduced curves at room temperature are added. Two effects stand out. For high temperatures, Plexiglas® 8N shows necking, similar to Plexiglas® Resist earlier. On the other hand, for low temperatures the Resist loses its necking and fractures in the same brittle manner as the 8N at room temperature. Though, the stress levels differ. The fracture strain distribution is affected alike. According to

true stress σ [MPa]

100

+85 ◦C +23 ◦C −30 ◦C

80 60 40 20 0 0

0.1

0.2

0.3

0.4

0.5 0.6 0.7 true strain ε [-]

0.8

0.9

1

1.1

1.2

Figure 8.3 Uniaxial stress-strain behavior of Plexiglas® Resist at different temperatures for 1 · 10−4 m/s hauling velocity.

114

8 Experimental Basis for Model Enhancements

ln[− ln(1 − P )] [-]

1

93.40

0

63.21

−1

30.78

−2

12.66

+85 ◦C +23 ◦C −30 ◦C

−3 −4 −4.5

−4

−3.5

−3

−2.5

−2

4.85 −1.5

−1

−0.5

1.81 0

occurrence propability P (εf ) [%]

fracture strain εf [-] 0.011 0.018 0.030 0.050 0.082 0.135 0.223 0.368 0.607 1.000 1.649 2 99.94

0.5

ln(εf ) Figure 8.4 Weibull plot of empirical fracture strain distribution for Plexiglas® 8N at different temperatures and 1 · 10−4 m/s hauling velocity. Regression lines are added dashed.

the definitions in Chapter 4, samples of fracture strains are generated. Different to the treatments in Chapter 5, they are kept unfiltered. Probability estimation follows WPE. All results are listed in Appendix C.3. At 85 ◦C the fracture strain level of 8N and Resist are very similar, as stated by Figure 8.4 and Figure 8.5, which is an enormous increase for 8N. At −30 ◦C the 8N fracture strain level is slightly beneath the one of Resist, which is explained by the elastomer additives within Resist. As discussed in Section 6.2, the slope of the data progression within the Weibull plot is no adequate measure for the dispersion of fracture strains. For a better sense on the results, in Table.8.2 the range between maximum and minimum fracture strain within the samples is provided. For 8N the dispersion of fracture strains increases with rise in temperature. That is what one would expect. However, Resist shows a different image. As for 8N, the dispersion at −30 ◦C is far less than at 85 ◦C. The polymer chains are hemmed in their movement and thus limit the interval of failure. Though, at room temperature the

Table 8.2

Range between maximum and minimum fracture strain εf,max − εf,min within a sample.

sample

Plexiglas® 8N

Plexiglas® Resist

85 ◦C 23 ◦C −30 ◦C

0.2638 0.0169 0.0110

0.1168 0.5985 0.0211

8.2 Stress-State Dependency

115

0.050

0.082

0.135

0.368

0.607

1.000

1.649 99.94

+85 ◦C +23 ◦C −30 ◦C

1 ln[− ln(1 − P )] [-]

0.223

93.40

0

63.21

−1

30.78

−2

12.66

−3

4.85

−4 −3.5

−3

−2.5

−2

−1.5

−1

−0.5

1.81 0

occurrence propability P (εf ) [%]

fracture strain εf [-] 0.030 2

0.5

ln(εf ) Figure 8.5 Weibull plot of empirical fracture strain distribution for Plexiglas® Resist at different temperatures and 1 · 10−4 m/s hauling velocity. Regression lines are added dashed.

Resist fracture strain dispersion is over 5 times higher than the one at 85 ◦C. Expected would be an outcome somewhere in between of the negative and the positive temperature. That effect could be explained by the elastomer particles being fully elastic at room temperature, enabling high strains, whereas the PMMA molecular chains are still limited in sloughing and provoke a pronounced fracture dispersion. In contrast, at 85 ◦C the highly orientated and maximum sloughed polymer chains might reduce room for variation in the fracture strain. The examination of these effects and their modeling in a stochastic PMMA failure criterion for numerical simulation, should be topic of further investigation.

8.2 Stress-State Dependency The manifold investigations in this work base solely on uniaxial tensile tests. Beside strain rate and temperature, the third important influencing factor for the PMMA’s material behavior is the type of load case. In Table 2.3 the conceivable stress states are already introduced. In the following, laboratory test setups for shear and biaxial tension are presented, and analogously to Section 8.1 the fracture strain distribution in these tests examined. In doing so, only Plexiglas® 8N is regarded. In the puncture tests, the impact resistant Plexiglas® Resist provokes forces beyond the nominal load of the available load cells and beyond the maximum force applied by the actuator. Therefore, no testing to the point of failure is possible and the material

116

8 Experimental Basis for Model Enhancements

Figure 8.6 Setup for puncture tests. A circular specimen plate (A) is loaded with a semi-spherical impactor that is attached to a load cell (B). A line sensor (D) detects the impactor displacement by a contrasted fin (D). On the underside the DIC system (E).

first principal strain ε1 [-]

is excluded from further examination. Though, the methodology is similarly to adopt with a respective test setup. Biaxial tension is considered in puncture tests. The setup depicted in Figure 8.6 is adopted from [88], and only discussed briefly. A semi-spherical impactor with 10 mm radius loads the specimen, which rests on a circular support. The specimen is a plate milled with 90 mm diameter from extruded plates of 3 mm thickness. The circular support features is diameter of 70 mm, with a radial edge of 1 mm. Force
true [-] 0.051

0.2 0.045 0.040

0.15

0.035

0.1

0.030 0.025

0.05

0.020 0.015

0 0

100

200 time t [s]

300

400

0.010 0.005 0.000

Figure 8.7 Strain-time behavor of Plexiglas® 8N in puncture tests, performed at 1.66 · 10−5 m/s (left). Strain is measured in the specimen’s center via DIC analysis (right).

8.2 Stress-State Dependency

117



ƒ



is measured by a 10 kN load cell applied to the impactor. Displacement is either captured internally by the actuator, and by a mechanically uncoupled line sensor directed to a fin on the actuator rod. Positioned on the underside of the setup, the DIC system captures the surface of the specimen. Force signal by the load cell and displacement signal by the line sensor are recorded by the DIC system’s measuring board for synchronization with the image series. In the tests on 8N, 35 valid experiments are gained. In DIC analysis, first principal strain, cf. Section 2.3, is determined for a measuring point right in the center of the circular specimen. A compilation of the strain progression over time is provided in Figure 8.7, together with an exemplary strain field of the DIC. The test shows good reproducibility of the strain behavior, combined with a wide dispersion in fracture strain. For shear tests, first a selection of two setups is evaluated. The first one is very similar to the setup for uniaxial tensile tests. The uniaxial movement of the machine is translated by the specimen’s geometry into a shear loading. The specimen’s dimensions are provided in Figure 8.8. In the following it is referred to as Z specimen. The basic idea is adopted from [20, 102], but with elongation of the clamping length for an insertion into common clamping jaws. The similarity to a simple tensile test is the great benefit of this setup. No further modifications have to be made. Figure 8.9 gives an impression on the setup. For displacement measure right at the top and bottom ends of the specimen’s free clamping length, as necessary to define boundary conditions in numerical analysis, fins are added to the clamping jaws, featuring a speckle pattern for DIC tracking. The surface of the specimen is speckled alike for a strain field computation. The DIC system records in 35 Hz. Therefore, the darkening polarization filters are removed, the apertures increased, the exposure time minimized, and the image size reduced. The planar surface of the specimen and the orthogonal positioning of the cameras allow the loss of depth of field. Either is the of loss polarization filters bearable, since reflections



 

Figure 8.8 Dimensions of the Z shear specimen. The specimens are milled from extruded plates, having 3 mm thickness.

118

8 Experimental Basis for Model Enhancements

Figure 8.9 Setup for shear tests with Z specimens. Analogous to uniaxial tensile tests, the specimen’s surface (A) is recorded with 35 Hz by a dual-camera system (B). For higher resolution at the point of failure, a additional high-speed camera (C) records with 10 kHz. Displacement is tracked by distance change of two fins (D).

are not an issue. Hauling velocity is 1.66 · 10−4 m/s. Shear strain is measured as average at a vertical line-cut between the tips of the two milled slots, demonstrated in Figure 8.11. An alternative setup is depicted in Figure 8.10. Specimen is an injection molded rectangular rod in the dimensions 80x10x4 mm (WxHxT), which will be referred to as RR specimen. It is inserted into a tight fitting mount, constraining all degrees

Figure 8.10 RR specimen (A) constrained by a mounting (C) and loaded with an penetration ram (B). Displacement is measured at a fin (D) attached to the ram.

8.2 Stress-State Dependency

119 İ[\

İ[\













!















!















Figure 8.11 Strain field of Z specimen (left) and RR specimen (right). Shear strain is determined as average over symbolized line-cut.

of freedom, except strain in width and depth direction, and deformation within a 20.04 mm gap. Through the gap, a 20 mm wide penetration ram shears the middle section of the specimen against the outer ends. Thus, two shear areas are gained, as indicated in the strain field in Figure 8.11. Shear is measured as averaging line-cut in these areas. The one with higher averaged strain is regarded. The positioning of the DIC system is already shown in Figure 4.3. Displacement of the penetration rod is also tracked via DIC. The installation of the whole setup is quite laborious, due to tight fittings and the angular alignment of all parts. For a strain rate comparable to the one at the first test setup, the penetration ram is lowered with 1.66 · 10−5 m/s velocity. For an evaluation on both shear setups, their strain is examined over time. Here, shear strain is the εxy component of the strain tensor. The local coordinate system is adapted for proper alignment with the x and y directions of the specimen. Figure 8.12 plots the test results. Both diagrams show good reproducibility, though the recording frequency of 35 Hz for the Z-specimen is not sufficient. Too much noise is in the gathered strain field. Noticeable is the difference in maximum strain that is gained with each setup. The Z specimens’ fracture strain is more than 10 times lower than the one of the RR specimens. That is provoked by the two milled slots, leading to early failure by the induced notch effect. As quick test for the initial shear stiffness of a material, the Z specimen is proposed. For the test until failure, the RR specimen seems to provide a better approach, which further studies should evaluate. At this point the fracture strains coming from the RR specimens are regarded in the statistical comparison.

8 Experimental Basis for Model Enhancements 0.15

0.006

Z specimen

shear strain εxy [-]

shear strain εxy [-]

120

0.004 0.002

RR specimen

0.1

0.05

0

0 0

1

2

0

3

20

time t [s] Figure 8.12

40 60 time t [s]

80

Results for the two shear test setups. The specimens are made from Plexiglas® 8N.

Figure 8.13 provides the empirical probability distribution for the three examined load cases. Probability is estimated using WPE. In the uniaxial tensile test, the lowest fracture strain level is gained. The highest fracture strains in puncture and shear test are quite similar, though in the puncture tests the fracture strain range of 0.1550 is considerably wider than the one of 0.0910 in the shear tests. The range calculation is analogue to Table 8.2. The RR shear specimens are injection molded and feature an accordingly high-quality surface. The circular plates in the puncture test are milled, but the crack arises from the top face in the specimen’s

fracture strain ε [-] 2

99.94

1

93.40

0

63.21

−1

30.78

−2

12.66 tensile test puncture test shear test

−3 −4

4.85

fracture propability P [%]

ln[− ln(1 − P (ε))]

0.033 0.041 0.050 0.061 0.074 0.091 0.111 0.135 0.165 0.202

1.81 -3.4

-3.2

-3.0

-2.8

-2.6 -2.4 ln(ε)

-2.2

-2.0

-1.8

-1.6

Figure 8.13 Weibull plot of empirical fracture strain distribution for Plexiglas® Resist at different load cases. Regression lines are added dashed.

8.2 Stress-State Dependency

121

center, where also an un-machined high-quality surface is at hand. Thus, a surface influence is eliminated. Whether the observed effects on the fracture strain distribution is a material response to the different load cases, or is to explain by mechanical boundary conditions, is an open topic. When the mechanisms are better understood, they might be modeled for inclusion into the proposed stochastic fracture strain model.

9

9 Summary With regard to two Plexiglas® materials, one brittle and one impact resistant, the statistical fracture behavior is modeled for stochastic FE simulation. First of all, a common ground of necessary fundamentals is prepared for in the fields of chemistry, mechanics, and statistics. For the latter, an overview of existing probability estimators and relevant families of probability distribution functions is compiled. Important GoF tests are explained, and their pros and cons discussed. Motivated by restrictions of the existing tests in plotting position consideration, a new generalized GoF test, the GAD test, is proposed. Since, so far no tabulated significance levels of the GAD test statistic exist a feasible Monte-Carlo algorithm is specified. For demonstration, the GAD test is applied in this work whenever reasonable. To characterize the acrylic glasses at a wide range of strain rate, several laboratory setups for uniaxial tensile tests are presented. All allow DIC analysis for surface strain calculation with synchronization between image series and the testing machine’s force signal. Studies are conducted on the influence of the specimen’s production method to the gained material properties. Injection molded specimens are compared to those that are milled from extruded plates, regarding their stress-strain behavior, fracture strain distribution and residual stresses. As a result, the choice is made for milled specimens. Evaluating different strain measurement techniques in DIC analysis, the materials stress-strain behavior is determined from line-cuts averaging over the specimen’s width, whereas fracture strain is determined in a single point at level of crack initiation to capture local effects. A uniaxial stress state in these areas is validated for the specimen’s geometries. In course of the experimental research, an unparalleled amount of PMMA fracture strains is collected. Differentiated by materials and hauling velocity, the fracture strains are combined to samples. The fracture pattern is taken as assessment criterion for validness of an individual experiment within a probabilistic sample. Furthermore, for each experiment the strain rate at failure is determined and compared to the maximum tolerated deviation from the sample’s mean. Thus, for each material fracture strain samples are gained for reference strain rates. In the subsequent statistical analyses, the Weibull distribution is discovered to best reproduce the empirical fracture strain distributions. Based on the 5 % and 95 % quantiles of the fitted CDFs, the © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9_9

124

9 Summary

statistical model is introduced. By linear regression on the plotting positions that are defined by logarithmic strain rate and corresponding quantile value, an interand extrapolation to arbitrary strain rates in enabled. From the linear curves, the parameters of the Weibull distributions are to calculate. Finally, a relationship between strain rate, fracture stain, and occurrence probability is obtained. Adopting this model as global failure criterion and using a random number generator for probability defaults, a head impact test on an automotive rear-side window is examined in stochastic FE simulation. In 1000 repetitions the head acceleration is evaluated for the resulting HIC. Another probabilistic relationship arises. On this HIC distribution and the HIC’s critical limit, the importance of a proper statistical material characterization in safety relevant domains is demonstrated. A single evaluation test of a structural part might lead to false decisions for design safety, since even small variations in the material’s fracture behavior can result in high dispersion of the evaluation criterion. Hence, in each individual case a careful consideration is suggested, whether a statistical material characterization is necessary, or not. In either way, compliance of a determined probability distribution with a critical quantile limit offers a higher safeguarding. In course of this work, at several research areas open issues for enhancements on the proposed model emerged. The quantile modeling of fracture strain distributions is a phenomenological approach. In examination of the experimental samples’ empirical probability distributions, overlaps of the data points occurred, making their progression not clearly to differentiate. It must be assumed that heating effects due to inner friction cause a certain temperature shift towards glass transitions temperature. Therefore, thermological analyses of tensile tests at the different strain rates will be a benefit. In this context, the integration of temperature as parameter within the fracture strain distribution model is to achieve. In analogy to the integration of strain rate, the 5 % and 95 % quantiles have been examined in their progression over the logarithmic temperature in Kelvin. A linear trend is not visible. The collection of more samples at various temperatures will clear the relationship. The herein provided experimental database is a starting point. One aspect will be the coupling of temperatures at which necking occurs, with those at which the specimens fail afore. Here, the concept of a bilinear, bimodal, or mixture Weibull distribution is a promising aim. Likewise, the influence of the induced stress state is to consider parametrically, for which the parameter of triaxiality is a good benchmark. As seen for uniaxial tensile tests, the fracture strain level lies beneath the one of biaxial tension and shear. Thus, when the proposed model is applied to a structural part under multiaxial load, a safety factor is added in case resistance of the material is relevant. For the examined HIC distribution, an increase in fracture strain level would cause a shift in direction of the critical limit. Hence, for general adoption of the model

125

the encouraged enhancements are essential. In that course, the hitherto neglected strain energy, accumulated within the specimen until the point of failure, is to analyze. For the different specimen geometries, a comparable strain energy at failure has to be assessed, or adaptions made. The regard of strain energy and triaxiality in the proposed statistical model will enhance its confidence and universal validity.

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[56] European Parliament and Council of the European Union: Regulation (EC) No 78/2009 on the type-approval of motor vehicles with regard to the protection of pedestrians and other vulnerable road users, amending Directive 2007/46/EC and repealing Directives 2003/102/EC and 2005/66/EC. 2009. [57] Evonik Industries AG: PLEXIGLAS® 8N - PMMA. CAMPUS Datasheet. 2018. [58] Evonik Industries AG: PLEXIGLAS® Resist AG 100 - PMMA-I. CAMPUS Datasheet. 2018. [69] Gesamtverband der Deutschen Versicherungswirtschaft e. V.: Sicherheitstechnische Aspekte schneller Pedelecs. Unfallforschung der Versicherer. 2012. [72] GOM GmbH: ARAMIS – Manual – Software. V6. Technical Documentation. 2007. [73]

GOM GmbH: Digital Image Correlation and Strain Computation Basics. V8 SR1. Technical Documentation. 2016.

[84]

Insurance Institute for Highway Safety: Roof Strength Crashworthiness Evaluation. Crash Test Protocol. 2016.

[85]

Insurance Institute for Highway Safety: Side Impact Crashworthiness Evaluation. Guidelines for Rating Injury Measures. 2014.

[86]

Insurance Institute for Highway Safety: Side Impact Crashworthiness Evaluation. Crash Test Protocol. 2017.

[90]

Joint Committee for Guides in Metrology: Evaluation of measurement data – Guide to the expression of uncertainty in measurement. JCGM 100:2008. 2008.

MANUALS, REPORTS, AND STANDARDS

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[96]

Kraftfahrt-Bundesamt: Neuzulassungen von Kraftfahrzeugen und Kraftfahrzeuganhängern nach Herstellern und Handelsnamen. Statistical Report. 2008.

[97]

Kraftfahrt-Bundesamt: Neuzulassungen von Kraftfahrzeugen und Kraftfahrzeuganhängern nach Herstellern und Handelsnamen. Statistical Report. 2009.

[98]

Kraftfahrt-Bundesamt: Neuzulassungen von Kraftfahrzeugen und Kraftfahrzeuganhängern nach Herstellern und Handelsnamen. Statistical Report. 2010.

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Kraftfahrt-Bundesamt: Neuzulassungen von Kraftfahrzeugen und Kraftfahrzeuganhängern nach Herstellern und Handelsnamen. Statistical Report. 2011.

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[154]

SAE International: Instrumentation for Impact Test, Part 1, Electronic Instrumentation. SAE J211/1. 2014.

Appendix A Test Statistic Derivation A.1 Cramér-von Mises The test statistic W 2 of the CvM test is defined as integral of the squared deviation between EDF Pn (εf ) and the CDF P (εf ) as ∞ W = nω = n 2

[Pn (εf ) − P (εf )]2 dP (εf ) .

2

−∞

Since both functions Pn (εf ) and P (εf ) represent probabilities and therefore are defined in the limits [0,1], the integral can also be expressed for the differential dPn (εf ) from zero to one, that 1 W =n

[Pn (εf ) − P (εf )]2 dPn (εf ) .

2

0

A method for solving an integral numerically is the midpoint rule. For a step size Δx =

b−a , n

where a is the lower and b the upper bound of an interval that is divided into n steps, the integral of a function f (x) becomes approximately b n

. . . .  f (x)dx ≈ Δx f a + 12 Δx + f a + 32 Δx + f a + 52 Δx + ... + f b − 12 Δx .

a

The error can be estimated as |error | ≤

& & (b − a)3 · max &f  (x)& . 2 a≤x≤b 24n

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9

142

A Test Statistic Derivation

Carried over to the integral of the test statistic W 2 , we substitute P (εf ) = u , P (εf,i ) = ui , Pn (εf ) = x , and dPn (εf ) = dx, that the integrand and its first and second derivative become f (x) = (x − u)2 , f  (x) = 2(x − u) , and f  (x) = 2 , for a step-wise calculation with step size Δx =

1 . n

Considering the occurrence probability of the EDF to be i/n for the ith step, the test statistic then becomes  







 1 1 − 0.5 2 − 0.5 3 − 0.5 n − 0.5 W2 ≈ n f +f +f + ... + f n n n n n +

2

2

2 1 − 0.5 2 − 0.5 3 − 0.5 1 − u1 + − u2 + − u3 + ... ≈n n n n n

2 !, n − 0.5 + − un n + n

2 , 1  i − 0.5 . − ui ≈n n n i=1

The correction with the error term leads to + n ,

2 1  i − 0.5 2 W =n − ui + |error| n n i=1 + n ,

2 1  i − 0.5 1 =n ·2 , − ui + n n 24n2 i=1

which is the familiar sum form 2 n   i − 0.5 1 + − P (εi ) W = 12n n 2

i=1

of the CvM test statistic.

A.2 Anderson-Darling

143

A.2 Anderson-Darling

The general definition of the AD test statistic is the weighted integral of the squared deviation between EDF Pn (εf ) and the CDF P (εf ) as ∞ A =n

∞ [Pn (εf ) − P (εf )] ψ [P (εf )] dP (εf ) = n

2

2

−∞

−∞

[Pn (εf ) − P (εf )]2 dP (εf ) . P (εf )[1 − P (εf )]

By substitution of P (εf ) = u , P (εf,i ) = ui , dP (εf ) = du , Pn (εf,i ) = pi , p0 = 0 , and pn = 1, the integral is to be divided in a single integral for each step of the EDF to ⎫ ⎧u u2 u3 1 ⎨ 1 u ⎬ 2 2 (p − u) (p − u) 1 − u 1 2 du + du + du + ... + du A2 = n ⎩ 1−u ⎭ u(1 − u) u(1 − u) u u1 u2 un 0 ⎧u ⎫ u2 u3 1 ⎨ 1 u ⎬ (1/n − u)2 (2/n − u)2 1−u =n du + du + du + ... + du , ⎩ 1−u ⎭ u(1 − u) u(1 − u) u 0

u1

u2

un

where the occurrence probability of the EDF is i/n for the ith staircase step. The integration and collecting of the single terms leads to a sum expression  A2 = n [−u − ln(1 − u)]u0 1  u + ( n1 )2 ln(u) − u − ( n1 − 1)2 ln(1 − u) u2 1  u + ( n2 )2 ln(u) − u − ( n2 − 1)2 ln(1 − u) u3 2

+ ... + [ln(u) − u]1un



= n {−u1 − ln(1 − u1 ) + 0 + ln(1 − 0) + ( n1 )2 ln(u2 ) − u2 − [( n1 ) − 1]2 ln(1 − u2 ) − ( n1 )2 ln(u1 ) + u1 + [( n1 ) − 1]2 ln(1 − u1 ) + ( n2 )2 ln(u3 ) − u3 − [( n2 ) − 1]2 ln(1 − u3 ) − ( n2 )2 ln(u2 ) + u2 + [( n2 ) − 1]2 ln(1 − u2 ) + ... + ln(1) − 1 − ln(un ) + un }

144

A Test Statistic Derivation

+ = n −1 +

n   1 − 2i

n2

i=1

,

1 − 2i + 2n ln(ui ) + , ln(1 − ui ) n2

which is the familiar sum form A = −n + 2

n  1 − 2i i=1

n

[ln(ui ) + ln(1 − un+1−i )]

of the AD test statistic.

A.3 Generalized Anderson-Darling As for the AD test statistic, starting point in the definition of the GAD test statistic is the weighted integral of the squared deviation between EDF Pn (εf ) and the CDF P (εf ) as ∞ A2 = n

∞ [Pn (εf ) − P (εf )]2 ψ [P (εf )] dP (εf ) = n −∞

−∞

[Pn (εf ) − P (εf )]2 dP (εf ) P (εf )[1 − P (εf )]

By substitution of P (εf ) = u , P (εf,i ) = ui , dP (εf ) = du , Pn (εf,i ) = pi , p0 = 0 , and pn = 1, the integral is to be divided in a single integral for each step of the EDF to ⎧u ⎫ u2 u3 1 ⎨ 1 u ⎬ 2 2 (p1 − u) (p2 − u) 1−u A2 = n du + du + du + ... + du . ⎩ 1−u ⎭ u(1 − u) u(1 − u) u 0

u1

u2

un

Instead of defining the occurrence probability of the EDF, they are kept unspecified. Again, the integration and collecting of the single terms leads to a sum expression  A2G = n [−u − ln(1 − u)]u0 1  u + p21 ln(u) − u − (p1 − 1)2 ln(1 − u) u2 1  u + p22 ln(u) − u − (p2 − 1)2 ln(1 − u) u3 2

+ ... + [ln(u) − u]1un



A.4 Lower-Tail Generalized Anderson-Darling

145

= n {−u1 − ln(1 − u1 ) + 0 + ln(1 − 0) + p21 ln(u2 ) − u2 − (p1 − 1)2 ln(1 − u2 ) − p21 ln(u1 ) + u1 + (p1 − 1)2 ln(1 − u1 ) + p22 ln(u3 ) − u3 − (p2 − 1)2 ln(1 − u3 ) − p22 ln(u2 ) + u2 + (p2 − 1)2 ln(1 − u2 ) + ... + ln(1) − 1 − ln(un ) + un } + = n −1 − ln(un ) − ln(1 − u1 ) + +

n−1  i=1

,  2  2 2 2 pi ln(ui+1 ) − (pi − 1) ln(1 − ui+1 ) − pi ln(ui ) + (pi − 1) ln(1 − ui ) .

Further simplifications then result in +

, n−1  ui+1 1 − ui+1 2 2 2 AG = n −1 − ln[un (1 − u1 )] + pi ln − (pi − 1) ln , ui 1 − ui i=1

which is the introduced GAD test statistic.

A.4 Lower-Tail Generalized Anderson-Darling Again, starting point in the definition of the lower-tail GAD test statistic is the weighted integral of the squared deviation between EDF Pn (εf ) and the CDF P (εf ) as ∞ A =n

∞ [Pn (εf ) − P (εf )] ψ [P (εf )] dP (εf ) = n

2

2

−∞

−∞

[Pn (εf ) − P (εf )]2 dP (εf ) . P (εf )

with the lower-tail weight function ψ(u) = 1/u. By substitution of P (εf ) = u , P (εf,i ) = ui , dP (εf ) = du , Pn (εf,i ) = pi , and p0 = 0, the integral is to be divided in a single integral for each step of the EDF to ⎫ ⎧u u2 u3 1 ⎬ ⎨ 1 2 2 2 (p − u) (p − u) (p − u) 1 2 n du + du + ... + du . A2 = n u du + ⎭ ⎩ u u u 0

u1

u2

un

146

A Test Statistic Derivation

The occurrence probabilities of the EDF are kept unspecified. Again, the integration and collecting of the single terms leads to a sum expression  u A2G,LT = n 12 u2 0 1 u  + p21 ln(u) + 12 u2 − 2p1 u u2 1  u + p22 ln(u) + 12 u2 − 2p2 u u3 2

+ ...  1  + p2n ln(u) + 12 u2 − 2pn u u n 0 = n 12 u21 − 0 + p21 ln(u2 ) + 12 u22 − 2p1 u2 − p21 ln(u1 ) − 21 u21 + 2p1 u1 + p22 ln(u3 ) + 12 u3 − 2p2 u3 − p22 ln(u2 ) − 12 u2 + 2p2 u2 + ... + p2n ln(1) + 12 − 2p2n ln(un ) − 12 un + 2pn un + =n

1 2

+

1

− 2pn + 2pn un − p2n ln(un ) +

n−1  i=1

,  2  2 pi ln(ui+1 ) − 2pi ui+1 − pi ln(ui ) + 2pi ui .

Further simplifications then result in + ,

n−1  ui+1 1 2 2 2 AG,LT = n , pi ln − 2pn (1 − un ) − pn ln(un ) + + 2pi (ui − ui+1 ) 2 ui i=1

which is the introduced lower-tail GAD test statistic.

Appendix B Tensile Test Results B.1 Strain-Time Diagrams The strain-time curves for all experiments of a sample are plotted within a diagram that shifts the curves’ endpoints to zero. In deep blue the curve section is visualized, at which strain rate is measured by linear regression. For Plexiglas® 8N the experimental results of the sets with hauling velocities vh = 1.66 · 10−5 m/s, 0.06

0.05

vh = 1.66 · 10−5 m/s

true strain ε [-]

true strain ε [-]

0.06

0.04 0.03 0.02 0.01 0 −400 −300 −200 −100

0.05

vh = 1 · 10−4 m/s

0.04 0.03 0.02 0.01 0 −30

0

−20

time t [s] 0.06 true strain ε [-]

true strain ε [-]

vh = 1 · 10−3 m/s

0.04 0.03 0.02 0.01 0

0

time t [s]

0.06 0.05

−10

−2

−1

0

0.05

vh = 1 · 10−2 m/s

0.04 0.03 0.02 0.01 0 −0.3

time t [s]

−0.2

−0.1

0

time t [s]

Figure B.1 True local strain of Plexiglas® 8N specimens. Fracture strain and strain rate at failure are considered in the data filters of Section 5.3.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9

148

B Tensile Test Results

vh = 1 · 10−4 m/s, vh = 1 · 10−3 m/s, and vh = 1 · 10−2 m/s are provided in Figure B.1. For each set two subsets of laboratory tests are performed. For the hauling velocities vh = 1 · 10−1 m/s and vh = 1 · 100 m/s the realized strain rates are not identical, cf. Figure B.2. The experiments of the main subsets, colored

vh = 1 · 10−1 m/s

vh = 1 · 10−1 m/s

0.04 true strain ε [-]

true strain ε [-]

0.04 0.03 0.02 0.01

0 −0.06 −0.04 −0.02

0.03 0.02 0.01 0 −0.06 −0.04 −0.02

0

time t [s] 0.04 vh = 1 · 100 m/s

true strain ε [-]

true strain ε [-]

0.04 0.03 0.02 0.01 0 −0.015

−0.01

−0.005

vh = 1 · 100 m/s

0.03 0.02 0.01 0 −0.015

0

time t [s] Figure B.2

−0.01

Deviation in strain rate and test duration in the second subset (red).

true strain ε [-]

−0.005

time t [s]

0.04 vh = 3 · 100 m/s

0.03 0.02 0.01 0

−0.8 −0.6 −0.4 −0.2

0

time t [ms] Figure B.3

0

time t [s]

Strain rate measures as linear regression over total curve progression.

0

B.1 Strain-Time Diagrams

149

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.5

vh = 1 · 10−4 m/s

−150

−100

true strain ε [-]

true strain ε [-]

blue, are kept, whereas the second subsets, colored red, are neglected. For set vh = 1 · 10−1 m/s a limited amount of images for DIC analysis is received, caused by the limited recording frequency of the high-speed camera. Strain rate is measured by linear regression of the whole curve, cf. Figure B.3.

−50

vh = 1 · 10−3 m/s

0.4 0.3 0.2 0.1 0

−10

0

true strain ε [-]

true strain ε [-]

0.6

vh = 1 · 10−2 m/s

0.3 0.2 0.1 0

−1

−0.5

0

time t [s]

time t [s] 0.4

−5

vh = 1 · 10−1 m/s

0.5 0.4 0.3 0.2 0.1 0

−0.15

0

time t [s]

−0.1

−0.05

0

time t [s]

0.3

true strain ε [-]

true strain ε [-]

0.1 vh = 1 · 100 m/s

0.2 0.1 0

−0.03

−0.02

−0.01

time t [s]

0.06 0.04 0.02 0

0

vh = 3 · 100 m/s

0.08

−1

−0.5

0

time t [ms]

Figure B.4 True local strain of Plexiglas® Resist specimens. Fracture strain and strain rate at failure are considered in the data filters of Section 5.3.

150

B Tensile Test Results

Whereas for Plexiglas® 8N the measuring section is consistently about 15 % of the length of the longest strain curve within a set, for Plexiglas® Resist the measuring sections have to be adapted in each set for the best reproduction of the end slopes. Within a set the measuring section is chosen as equal as reasonable.

vh = 1 · 10−4 m/s

0.6 0.4 0.2 4.5

5

5.5

6 −1

0.2 0.1 vh =

0

0.4

m/s

0.45

strain rate ε˙ [s 0.3

14

15

strain rate ε˙ [s Figure B.5

0.04 0.045 0.05 0.055 strain rate ε˙ [s−1 ] 0.4 0.3 0.2 0.1

−1

16 ]

vh = 1 · 10−1 m/s

0

4

strain rate ε˙ [s

0.1 13

0.1

]

0.2

12

0.2

3.5

vh = 1 · 100 m/s

11

0.3

0.5 −1

vh = 1 · 10−3 m/s

0.4

]

0.3

1 · 10−2

0.5

6.5

fracture strain εf [-]

fracture strain εf [-]

strain rate ε˙ [ms

fracture strain εf [-]

fracture strain εf [-]

0.8

fracture strain εf [-]

fracture strain εf [-]

B.2 Scatter Plots

0.1

4.5 −1

]

vh = 3 · 100 m/s

0.08 0.06 0.04 120 130 140 150 160 170 strain rate ε˙ [s−1 ]

Scatter plots of the filtered Plexiglas® Resist samples, analogously to Figure 5.15.

Appendix C Fracture Strain Samples C.1 Uniaxial Tension Plexiglas 8N ®

Table C.1

Samples ε˙1 – ε˙7 resulting from outlier definitions and data filters according to Chapter 5.

pos.

ε˙1

ε˙2

ε˙3

ε˙4

ε˙5

ε˙6

ε˙7

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

0.03215 0.03666 0.03698 0.03728 0.03853 0.04020 0.04114 0.04157 0.04199 0.04251 0.04322 0.04329 0.04484 0.04498 0.04634 0.04644 0.04816 0.04894 0.05077 0.05344 -

0.03044 0.03362 0.03401 0.03402 0.03492 0.03637 0.03651 0.03703 0.03773 0.03776 0.03793 0.03823 0.03823 0.03851 0.03868 0.03871 0.03993 0.03997 0.04220 0.04301 0.04374 0.04383 0.04407 0.04729 -

0.02087 0.02276 0.02363 0.02551 0.02573 0.02647 0.02690 0.02699 0.02807 0.02819 0.02886 0.02901 0.02923 0.02947 0.02966 0.03083 0.03124 0.03201 0.03214 0.03229 0.03231 0.03267 0.03276 0.03278 0.03286

0.02453 0.02481 0.02511 0.02519 0.02553 0.02830 0.02875 0.02904 0.02933 0.02936 0.02969 0.02996 0.03003 0.03058 0.03081 0.03098 0.03112 0.03156 0.03170 0.03171 0.03280 0.03291 0.03310 0.03422 0.03435

0.02187 0.02453 0.02535 0.02595 0.02597 0.02598 0.02657 0.02659 0.02675 0.02692 0.02718 0.02746 0.02759 0.02773 0.02774 0.02806 0.02890 0.02927 0.02988 0.03028 0.03084 0.03095 0.03201 0.03310 0.03407

0.01754 0.01941 0.02080 0.02108 0.02326 0.02372 0.02397 0.02421 0.02428 0.02447 0.02465 0.02516 0.02520 0.02674 0.02678 0.02715 0.02806 0.02857 0.02949 0.03110 0.03114 0.03115 0.03153 0.03159 0.03164

0.01508 0.01683 0.01732 0.01746 0.01778 0.01792 0.01824 0.01835 0.01902 0.01954 0.01966 0.02121 0.02149 0.02178 0.02201 0.02222 0.02225 0.02227 0.02265 0.02408 0.02448 0.02476 0.02482 0.02501 0.02778

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Berlinger, A Methodology to Model the Statistical Fracture Behavior of Acrylic Glasses for Stochastic Simulation, Mechanik, Werkstoffe und Konstruktion im Bauwesen 59, https://doi.org/10.1007/978-3-658-34330-9

152

C Fracture Strain Samples

pos.

ε˙1

ε˙2

ε˙3

ε˙4

ε˙5

ε˙6

ε˙7

26 27 28 29 30 31 32 33 34 35 36 37 38

-

-

0.03325 0.03330 0.03346 0.03370 0.03398 0.03399 0.03408 0.03415 0.03642 0.03697 0.03738 0.03757 0.04286

0.03441 0.03508 0.03607 0.03639 0.03855 -

0.03416 0.03440 0.03444 0.03466 0.03549 0.03709 0.04025 -

0.03260 0.03398 0.03666 0.03796 -

-

C.2 Uniaxial Tension Plexiglas Resist ®

Table C.2

Samples ε˙1 – ε˙6 resulting from outlier definitions and data filters according to Chapter 5.

pos.

ε˙1

ε˙2

ε˙3

ε˙4

ε˙5

ε˙6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.13391 0.13420 0.13870 0.14340 0.15274 0.15624 0.16784 0.16903 0.18135 0.20957 0.21861 0.21952 0.22188 0.22292 0.22810 0.23009

0.10418 0.11534 0.12587 0.12660 0.14842 0.15075 0.15833 0.16638 0.17569 0.17892 0.18341 0.20329 0.20671 0.21020 0.21410 0.21634

0.05984 0.07802 0.09359 0.10491 0.11736 0.12375 0.12675 0.13000 0.13011 0.15041 0.15519 0.15898 0.17755 0.17876 0.17915 0.18381

0.05240 0.07640 0.09058 0.09065 0.09171 0.09269 0.09291 0.10013 0.10741 0.10795 0.12713 0.12943 0.13357 0.13941 0.13942 0.14040

0.04803 0.05697 0.06594 0.06895 0.06976 0.07271 0.07412 0.07418 0.07505 0.07519 0.07903 0.07977 0.09964 0.10525 0.11231 0.11346

0.03625 0.03894 0.03964 0.04189 0.04506 0.04593 0.04794 0.04812 0.04832 0.05125 0.05319 0.05363 0.05792 0.05898 0.06151 0.06322

C.2 Uniaxial Tension Plexiglas® Resist

153

pos.

ε˙1

ε˙2

ε˙3

ε˙4

ε˙5

ε˙6

17 18 19 20 21 22 23 24 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

0.24473 0.24672 0.24678 0.24782 0.29276 0.33056 0.33176 0.33763 0.34761 0.42006 0.45299 0.45685 0.46547 0.47182 0.47872 0.48546 0.48771 0.51719 0.51753 0.52003 0.52946 0.54395 0.55618 0.57022 0.58190 0.59111 0.59724 0.61519 0.63364 0.63975 0.64959 0.66076 0.68309 0.73237

0.21817 0.22147 0.22306 0.22439 0.23506 0.24588 0.25317 0.25514 0.26031 0.26155 0.26236 0.27013 0.28105 0.30772 0.32393 0.33124 0.33153 0.33460 0.36628 0.37480 0.39093 0.40497 0.40803 0.42163 0.42695 0.42904 0.45795 0.46053 0.48651 0.49230 0.51119 -

0.20134 0.20635 0.21916 0.22214 0.22248 0.22775 0.22807 0.23309 0.23442 0.23529 0.23900 0.24395 0.25289 0.27326 0.28661 0.29960 0.30072 0.30331 0.30926 0.32538 0.33335 0.35671 0.36278 -

0.14114 0.14484 0.14771 0.14914 0.15218 0.16079 0.18275 0.18676 0.18827 0.19044 0.20282 0.20417 0.21421 0.21871 0.22396 0.22455 0.22690 0.23861 0.26474 0.28716 0.30952 0.31614 0.32001 0.32660 0.32844 0.33678 0.34423 0.35559 0.38019 -

0.12903 0.13926 0.15257 0.15427 0.17688 0.18452 0.18585 0.20790 0.21111 0.24040 0.25032 0.28530 -

0.06906 0.06988 0.07170 0.07669 0.08898 -

154

C Fracture Strain Samples

C.3 Further Results Table C.3

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 31 32 33 34 35

Unfiltered fracture strain samples from Chapter 8

Plexiglas® 8N puncture shear 0.05023 0.05105 0.05931 0.06040 0.06551 0.07048 0.07670 0.07705 0.08329 0.08437 0.08483 0.08495 0.09071 0.09121 0.09216 0.09250 0.09954 0.10019 0.10118 0.10439 0.10584 0.11463 0.11537 0.11608 0.11786 0.11900 0.13937 0.14004 0.14365 0.15475 0.16192 0.16423 0.18760 0.19074 0.20520

0.10933 0.12198 0.12290 0.12544 0.12621 0.13020 0.13212 0.13615 0.13868 0.14028 0.14396 0.14758 0.14998 0.15065 0.15509 0.18994 0.20028 -

Plexiglas® 8N T = −30 ◦C T = 85 ◦C 0.01871 0.01905 0.02062 0.02069 0.02093 0.02126 0.02214 0.02243 0.02250 0.02261 0.02277 0.02287 0.02346 0.02393 0.02403 0.02430 0.02494 0.02499 0.02527 0.02608 0.02608 0.02618 0.02626 0.02645 0.02661 0.02674 0.02705 0.02783 0.02788 0.02822 0.02836 0.02960 0.02969 -

0.80000 0.84282 0.85879 0.88506 0.90127 0.91419 0.92643 0.92956 0.93317 0.93763 0.94167 0.94425 0.94481 0.94822 0.95078 0.95645 0.96148 0.96822 0.97559 0.98102 0.98481 0.98578 0.99922 0.99993 1.00491 1.00986 1.01733 1.02025 1.02685 1.04101 1.04383 1.06163 1.06377 -

Plexiglas® Resist T = −30 ◦C T = 85 ◦C 0.03540 0.03546 0.03590 0.03661 0.03742 0.03836 0.03882 0.03965 0.03977 0.03978 0.03986 0.03996 0.04142 0.04235 0.04255 0.04257 0.04309 0.04347 0.04410 0.04474 0.04479 0.04489 0.04559 0.04572 0.04576 0.04599 0.04608 0.04647 0.04669 0.04705 0.04707 0.04710 0.04714 0.04716 0.04849

1.02521 1.04103 1.05164 1.05631 1.05822 1.06606 1.06691 1.07440 1.07695 1.07740 1.07765 1.08647 1.08894 1.09874 1.10236 1.10343 1.10816 1.11324 1.11608 1.12348 1.14202 -

C.3 Further Results

pos. 36 37 38 39 40

155

Plexiglas® 8N puncture shear -

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Plexiglas® 8N T = −30 ◦C T = 85 ◦C -

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Plexiglas® Resist T = −30 ◦C T = 85 ◦C 0.04891 0.04933 0.05167 0.05275 0.05647

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