Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests [1st ed.] 978-3-658-27112-1;978-3-658-27113-8

Table of contents :
Front Matter ....Pages i-x
Introduction (Nikolai Sygusch)....Pages 1-4
State of the Art (Nikolai Sygusch)....Pages 5-13
Mechanical Testing (Nikolai Sygusch)....Pages 15-39
Statistical Analysis (Nikolai Sygusch)....Pages 41-58
Material Modeling (Nikolai Sygusch)....Pages 59-72
Numerical Results (Nikolai Sygusch)....Pages 73-92
Summary and Outlook (Nikolai Sygusch)....Pages 93-95
Back Matter ....Pages 97-145

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Mechanik, Werkstoffe und Konstruktion im Bauwesen | Band 52

Nikolai Sygusch

Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-PointBending Tests

Mechanik, Werkstoffe und Konstruktion im Bauwesen Band 52 Reihe herausgegeben von Ulrich Knaack, Darmstadt, Deutschland Jens Schneider, Darmstadt, Deutschland Johann-Dietrich Wörner, Darmstadt, Deutschland Stefan Kolling, Gießen, Deutschland

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

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Nikolai Sygusch

Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests

Nikolai Sygusch Beselich, Germany Vom Fachbereich 13 – Bau- und Umweltingenieurwissenschaften der Technischen Universität Darmstadt zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Dissertation von Nikolai Sygusch aus Gießen 1. Gutachten: Prof. Dr.-Ing. Jens Schneider 2. Gutachten: Prof. Dr.-Ing. habil. Stefan Kolling Tag der Einreichung: 23.11.2018 Tag der mündlichen Prüfung: 24.01.2019 Darmstadt 2018 D17

ISSN 2512-3238 ISSN 2512-3246  (electronic) Mechanik, Werkstoffe und Konstruktion im Bauwesen ISBN 978-3-658-27112-1 ISBN 978-3-658-27113-8  (eBook) https://doi.org/10.1007/978-3-658-27113-8 Springer Vieweg © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Vieweg imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Abstract Increasing regulatory standards in the vehicle development create the demand for new computer aided engineering methods that improve the predictability of the simulations. A method for incorporating and comparing stochastic scatter of macroscopic parameters in crash simulations is developed in the present work and applied on a 30 wt.% short glass fiber reinforced polypropylene. The application of fiber reinforced polymers is of great interest for different crash worthiness and pedestrian protection load cases. However, the orientation and strain rate dependent material behavior has to be considered and the brittle rupture behavior has to be taken into account as well. A statistical testing plan on the basis of three point bending tests with 30 samples for each configuration is carried out in this work. The tests are conducted at 0◦ , 30◦ , 45◦ and 90◦ orientation angles and at strain rates of 0.021 s−1 and 85 s−1 . The obtained results are evaluated statistically by means of probability distribution functions. The normal and lognormal distributions are able to model most of the data accurately, however, the Weibull distribution provides better results for some of the configurations. An orthotropic elastic plastic material model is utilized for the numerical investigations. Monte Carlo Simulations with variations in macroscopic parameters are run to emulate the stochastic rupture behavior of the experiments. The stochastic rupture behavior in terms of the probability density functions can be recreated very well for the 0◦ orientation. Data analysis of the stochastic simulations shows that only two of the initial seven parameters can be identified as main influential parameters. These are the maximum principal strain at rupture and the specimen width. This finding is verified in a validation study that proves the validity of the introduced method.

Kurzfassung Steigende regulatorische Anforderungen in der Fahrzeugentwicklung erfordern neue computergestützte Berechnungsmethoden, welche die Vorhersagbarkeit der Simulationen verbessern. In der vorliegenden Arbeit wird eine Methode zur Integration und zum Vergleich stochastischer Streuung von makroskopischen Parametern in der Crash-Simulation entwickelt und an einem 30 gew.% kurzglasfaserverstärktem Polypropylen angewendet. Die Anwendung von faserverstärkten Polymerwerkstoffen ist von großem Interesse für verschiedene Crash-Lastfälle und für den Fußgängerschutz. Für eine umfassende Untersuchung des Materials ist es notwendig, das richtungs- und dehnratenabhängige Materialverhalten sowie das spröde Bruchverhalten zu berücksichtigen. In dieser Arbeit wird ein statistischer Versuchsplan auf der Grundlage von drei PunktBiegeversuchen mit je 30 Probekörpern für jede Versuchskonfiguration angewendet. Die Versuche werden unter 0◦ , 30◦ , 45◦ und 90◦ Winkeln und bei Dehnraten von 0.021 s−1 und 85 s−1 durchgeführt. Die statistische Auswertung der Ergebnisse erfolgt mit Hilfe von Wahrscheinlichkeitsverteilungsfunktionen. Die Normalverteilung und Log-Normalverteilung sind in der Lage, die meisten Daten präzise abzubilden, allerdings liefert die Weibull Verteilung für einige der Konfigurationen bessere Ergebnisse. Für die numerischen Untersuchungen kommt ein orthotropes elastoplastisches Materialmodell zum Einsatz. Es werden Monte-Carlo-Simulationen mit Variationen in makroskopischen Parametern durchgeführt, um das stochastische Bruchverhalten der Experimente nachzubilden. Die Ergebnisse zeigen, dass das stochastische Bruchverhalten in Bezug auf die Wahrscheinlichkeitsdichtefunktionen sehr gut für die 0◦ Orientierung reproduzierbar ist. Die Datenanalyse der Monte-Carlo-Simulationen macht deutlich, dass nur zwei der ursprünglich sieben Parameter als wichtigste Einflussgrößen identifiziert werden können. Dabei handelt es sich um die maximale Hauptdehnung beim Bruch und die Probenbreite. Diese Beobachtung wird in einer Validierungsstudie bestätigt, welche die Gültigkeit der vorgestellten Methode nachweist.

Contents 1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objective and Structure of the present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3

2 State of the Art 5 2.1 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Statistical Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Mechanical Testing 3.1 Studies on the Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Characterization of the glass fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 μCT scans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Macroscopic Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Specimen preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Tensile tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Three point bending tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 15 17 20 20 23 30

4 Statistical Analysis 4.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 General definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Goodness of fit tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Statistical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Quasistatic three point bending tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Dynamic three point bending tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 41 42 46 47 48 54

5 Material Modeling 5.1 Explicit Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Material theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Orthotropic modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Rupture condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 60 60 65 68

x

Contents

5.3

Contact Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Numerical Results 6.1 Deterministic Three Point Bending Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Stochastic Three Point Bending Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Evaluation and validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Influence of the explicit time integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 77 77 85 91

7 Summary and Outlook

93

Bibliography

97

List of symbols

105

A Appendix A.1 Technical data sheet Hostacom G3 R05 105555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Fiber length analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 μCT scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Surface roughness measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Three point bending test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Results Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6.1 Histograms of all design variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6.2 Scatter plots of all design variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 114 116 121 123 135 135 139

1 Introduction Today the vehicle development in the automotive industry is mostly driven by computer aided engineering practices. Currently, the application of advanced material models and comprehensive optimization and robustness studies are two of the challenges in the crash simulation environment in the vehicle development process [24]. New material models are developed that e.g. emulate the material behavior of polymers more physically than commonly used material models. Furthermore, the improvement of methods for rupture modeling plays an important role as well. One main focus is to model the stress state and load path dependent rupture behavior accurately [19], whereas another main focus is the use of capabilities of today’s increasing CPU power in order to run stochastic simulations for advancements in the field of non deterministic rupture modeling [67]. The present work is contributing to the field of CAE vehicle development because a new approach to stochastic rupture modeling for fiber reinforced polymers is introduced. The stochastic simulations are complemented by a large number of experiments that consolidate the conclusiveness of this work.

1.1 Motivation International and nationwide organizations like the Research Council for Automotive Repairs (RCAR) and the European New Car Assessment Programme (Euro NCAP) continuously advance the regulations for vehicle safety assessments. These promotions require the improvement of current practices in the field of CAE, which is one of the key disciplines in assessments of vehicles [64]. There is an active demand for the newest technologies that increase the simulation predictability of the vehicle structure in order to attain the best score in the different assessment disciplines. Two exemplary load cases are shown in Figure 1.1. The left image shows the RCAR insurance classification 10◦ structure test, whereas the Euro NCAP lower leg test is illustrated in the right image. The application of fiber reinforced polymers is of great interest for these specific load cases. Their low density combined with a high stiffness and low propensity of environmental stress cracking [18] make this class of materials important for applications in low velocity1 crash worthiness and pedestrian protection load cases. 1 Low

velocities in comparison to the velocities that occur in high speed crash worthiness load cases

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8_1

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1 Introduction

Figure 1.1 Two load cases in the vehicle development process that require accurate modeling of fiber reinforced polymers for assertions regarding the structural worthiness. Left: RCAR insurance classification 10◦ structure test. Right: Euro NCAP pedestrian protection lower leg test.

Two major challenges arise with the use of fiber reinforced polymers. On the one hand, the orientation and strain rate dependent material behavior has to be characterized experimentally and calibrated numerically in order to draw valid conclusions from simulations. On the other hand, the brittle rupture behavior of the material has to be taken into account. Fully characterizing brittle rupture behavior of orthotropic materials experimentally is a very time-consuming and costly process. A statistical characterization means at least 15, better yet, 30 samples for at least three orientation angles and testing at different impact velocities. The large sample size is required to obtain a meaningful statistical result because brittle rupture is often associated with probability distribution functions. It is relevant to study the scatter range of the stochastic rupture in order to determine the reliability of a structure. This also leads to the question if differences in the rupture behavior for different orientation angles and velocities can be observed. The possibilities and limitations to numerically reproduce this rupture behavior have to be investigated as well. A method to approach this topic is to conduct stochastic simulations that allow running a large number of parametric simulations. Minor deviations in each run emulate stochastically variations that could help to understand the origins of the scattering. These simulations could also help to identify connections between significant input parameters and their impact on the output signals.

1.2 Objective and Structure of the present Work

3

1.2 Objective and Structure of the present Work The aim of this work is to develop a method for incorporating and comparing stochastic scatter of macroscopic parameters in crash simulations. The investigations in this work are conducted on a 30 wt.% short glass fiber reinforced polypropylene. The investigations will focus on two areas, namely the fields of experimental studies and numerical simulations. The experimental rupture behavior is analyzed under quasistatic and dynamic impact velocities in the three point bending loading case. The experiments are performed at different orientation angles and strain rates. A statistical approach to the rupture behavior is chosen because this material belongs to the class of materials that break brittle. It is assumed that the experimental rupture behavior can be modeled more accurately by a Weibull distribution function than e.g. a normal or lognormal distribution function because the Weibull distribution is often associated to brittle rupture behavior. One of the central questions of this work is to what extend it is possible to model the characteristic rupture values of this material properly by using probability distribution functions. This has to be studied for different orientation angles with respect to the molding direction and for different strain rates as well. Goodness of fit tests are used as an objective measure to determine the quality of the obtained distribution functions. The second focus of this work is on stochastic simulations in the crash simulation environment. The findings of the microscopic investigations and the material characterization on a macroscopic scale help to calibrate the parameters of the utilized orthotropic material model. One of the goals of the numerical investigations is to develop an efficient process to run a large number of parametric simulations that are based on the experimental studies. The hypothesis has to be verified that it is possible to achieve a good correlation between the rupture behavior of the simulations and experiments in a statistical sense by variations of macroscopic parameters. That means that the distribution patterns of simulation and experiment are congruent to each other. The numerical approach can help to understand connections between the observed scatter in the experiments and the influences of the macroscopic parameters on this behavior. It is also investigated if certain parameters influence the outcome of the simulations more than other parameters in order to distinguish between first and second order effects. This method could have the potential of reducing the level of uncertainty in crash simulations because capabilities and limitations for structures made out of fiber reinforced polymers could be assessed while taking the stochastic scatter behavior into account. Assertions regarding the reliability and safety of a structure would increase, whereas the level of uncertainty would decrease. This result would be beneficial for safety engineers as well as all traffic participants.

4

1 Introduction

Chapter 2 focuses on the research of the current work with remarks on open challenges in related fields. The micro- and macromechanical investigations are outlined and discussed in Chapter 3. Quasistatic and dynamic three point bending tests, which are tested at 0◦ , 30◦ , 45◦ and 90◦ orientation angles, are the basis for analyzing the maximum deflections and forces at rupture. Chapter 4 focuses on the statistical analysis of the rupture values from the three point bending tests. Normal, lognormal, two and three parameter Weibull distribution functions are fitted to the data points and verified by goodness of fit tests in order to determine the best fit for each set of data points. The goal of the numerical investigations is to reproduce the stochastic rupture behavior of the experiments. In a first step, the mechanical behavior is calibrated with an orthotropic elastic plastic material model that is described in Chapter 5. Monte Carlo Simulations are performed in a second step and presented in Chapter 6. The aim of the stochastic simulations is to generate the same distribution pattern in the rupture behavior that was previously observed in the experiments. Different macroscopic parameters, like the specimen length, width and thickness, as well as other parameters like rupture strain and friction, are varied stochastically. The results from the Monte Carlo Simulations are compared with the results from the experiments in a statistical sense, which means that the obtained probability density functions are compared to each other. Another focus of the stochastic simulations is to separate significant from insignificant parameters and to quantify the influence that the significant parameters have on the rupture behavior. A summary of the central findings from this work with the outlook for future investigations is given in the final Chapter 7.

2 State of the Art The research of current work is divided into three subchapters with remarks on open challenges in each field in the last passage of each section. Firstly, an examination of the current work on experimental investigations on glass fiber reinforced polymers is outlined. Secondly, the focus will be on works that concentrate on solving numerical challenges regarding fiber reinforced polymers. A line between the two themes cannot always be drawn because simulation studies are often complemented with an experimental survey. The previous works regarding statistics and stochastic simulations are evaluated in the last section. The stated issues of these works do not exclusively refer to fiber reinforced polymers because the field of stochastic simulations is a relative modern discipline in computational engineering. However, similar prospects and application of related methods from this work make the discussed studies worth considering.

2.1 Experimental Studies A 30 wt.% short glass fiber reinforced polypropylene is investigated in the work of Gupta et al. [34]. It is investigated experimentally by means of tensile tests with coupled acoustic emission in order to identify the energy absorbing mechanisms at rupture. The embedded glass fibers are measured and the average length is determined as 410 μm. The tensile tests are performed at a quasistatic strain rate at five different orientation angles, 0◦ , 30◦ , 45◦ , 60◦ and 90◦ . It can be observed that the plateau of the stress strain curve decreases with increasing orientation angle. The authors hypothesize a process zone that is responsible for energy absorption and do not suppose that solely the plane in the rupture zone is absorbing all the energy alone. According to the authors, subcritical fibers do not break and are pulled out of the matrix at rupture instead. The left SEM image of Figure 2.1 shows that matrix plastification can be observed near and around pulled out fibers for the 0◦ orientation. Furthermore, enhanced plastic deformation of the matrix can be observed in the right image of Figure 2.1. The acoustic emission analysis, which is depict in Figure 2.2, shows that the number of events changes with increasing orientation angle. The acoustic peak amplitudes of the 90◦ orientation show a different profile than the plateau of the 0◦ orientation. Observed peaks for the analysis of the 90◦ orientation may be connected to interface debonding while peaks of the 0◦ orientation may be attributed to fiber pullout or to matrix cracks.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8_2

6

2 State of the Art

Although the mechanical behavior is studied extensively by Gupta et al., the questions of stochastically scatter range in the experiments or the use of statistical tools are not emphasized.

Figure 2.1 Scanning electron micrograph images of a 0◦ orientation sample after rupture. The left image shows pulled out fibers. The right image shows a rupture plane of the sample, where plastic deformation of the matrix can be observed, taken from [34].

Figure 2.2 Acoustic emission analysis of a 30 wt.% short glass fiber reinforced polypropylene during tensile tests. The left image shows a changing number of acoustic events with increasing orientation angle. The right image shows the peak amplitudes of the 0◦ and 90◦ orientations. The D label may be connected to fiber debonding, while the P label may be attributed to fiber pullout or to matrix cracks, taken from [34].

Spahr et al. [87] study different long and short glass fiber reinforced polypropylene compounds with various mass contents of glass fibers, ranging from 10% up to 40%. Tensile, compact tensile and Izod impact tests [1] are performed in order to study the mechanical

2.1 Experimental Studies

7

behavior and the static and dynamic fracture toughness. The microstructure is also investigated. The authors show that the volume fraction of glass fibers in the core layer is larger than in the outer layers of the specimen. They also show that with increasing fiber content, the general orientation degree of the glass fibers increases as well. The average fiber length is longer at the outer layers than in the core layer. The 40 wt.% long and short fiber reinforced polymers show fiber bundles, which does not lead to an increase in stiffness according to the authors. It is assumed that too many of these fiber bundles are causing a less good transmission of the applied force inside the specimen. A general trend can be observed that an increase in the fiber content of the specimen leads to an increase of the tensile stiffness, although the rupture strain decreases at the same time. The fracture toughness also increases with increasing fiber content. The authors assume that the ideal content of glass fibers for the polymers is between 30% and 40%. Different fiber reinforced polymers are studied by Karger-Kocsis [43] by means of notched Charpy three point bending tests in order to examine the crack propagation of the materials. 10%, 20%, 30% and 40% short and long glass fiber reinforced polypropylene compounds are investigated. The Charpy tests are performed at -40◦ C, 20◦ C and 60◦ C for two orientation angles, 0◦ and 90◦ . The testing velocity is 3.7 ms−1 and the dynamic fracture toughness, fracture energy and Young’s modulus are determined. The author also documents a three layer setup of the different oriented fibers. It is assumed that the damage zone for the short glass fiber materials is narrower for the dynamic testing than it is for static tests. This does not seem to play an important role for the long glass fiber reinforced polypropylene because the fiber length to diameter ratio is greater here. The author differentiates between stable and instable, which are synonymous to ductile and brittle, crack propagation. This is connected to the orientation angle because longitudinal aligned fibers support faster crack propagation, whereas transverse fibers prevent a fast propagation. There are only minor differences between the 0◦ and 90◦ orientation angles in terms of ductile and brittle crack propagation behavior for all three lower wt.% compounds. The 40 wt.% compound on the other hand, shows very brittle rupture behavior for the 0◦ orientation. This observation is connected to the layered structure of the composite, according to the author. The core layer that has transversal oriented fibers is not able to hold back the crack propagation solely because the outer layers support faster crack growth due to their longitudinal fibers. With help of scanning electron micrograph images it can be observed that the 0◦ orientation fibers do not prevent crack propagation, whereas 90◦ orientation fibers force the crack to propagate in a zigzag path. Furthermore, it is observed that glass fibers are debonding and pulled out at rupture, while the matrix shows crazing behavior. Schoßig et al. [79] study the mechanical behavior of a polypropylene and a polybutylene with glass fiber mass contents of 0%, 20%, 30% and 40%. Tensile tests at a strain rate range from 0.007 s−1 to 174 s−1 are performed and the Young’s modulus and tensile strength are determined. An increase of both can be observed for increasing glass fiber

8

2 State of the Art

content and with increasing strain rate. From the obtained data, a master curve is generated for the tensile strength with help of a modified G’Sell–Jonas model [26]. In another work by Schoßig et al. [81] their study is enhanced by using a third material, a high density polyethylene. In this study, the glass fiber mass content ranges from 0% to 50%. For the purpose of this work, only the results of the polypropylene compound are of interest and discussed therefore. Measurements on the microstructure yield an average fiber diameter of 13 μm and a length between 250 μm and 425 μm. Instrumented Charpy three point bending tests with and without notches are performed to measure the fracture toughness, critical crack tip opening displacement and critical J-values of the materials. Results of the study are that Young’s modulus, tensile strength and fracture toughness increase with increasing glass fiber content, while the critical crack tip opening displacement and J-values reach a maximum at 20%, respectively 30% fiber mass content. This leads to the conclusion of the authors that the ideal glass fiber content in terms of mechanical properties is 30% for polypropylene. A third publication of Schoßig et al. [82] focuses on the in situ crack detection in tensile tests with help of acoustic emission analysis. A polypropylene and a polybutylene with 20 wt.% glass fibers are studied. The tensile specimen are notched and tested at a slow velocity of 0.2 mm min−1 in order to measure the acoustic signals during the experiments. Environmental scanning electron microscopy scans are also performed during the experiments and documented accurately at different time stages in the study. An accumulation of the acoustic emission signals can be noticed with increasing force level, which is similar to the observations by Gupta et al. [34]. The authors determine the crack propagation of the polypropylene compound as stable that is contrary to the statement of Karger-Kocsis [43], who argued that the crack propagation of reinforced polypropylene is instable. The works by Schoßig et al. [79, 81, 82] focus on the rupture behavior with consideration of the strain rate dependency, without taking the orientation dependency into account. Stochastic aspects regarding the rupture behavior are not emphasized. A large experimental study is performed by Hartl et al. [36]. Puncture tests as well as tensile and three point bending tests with different notch radii are conducted on a polypropylene with 32 wt.% short glass fibers at different orientation angles for strain rates from 0.0001 s−1 to 100 s−1 at three temperatures, -30◦ C, 23◦ C and 80◦ C. μCT scans at different specimen removal positions are conducted in order to nondestructively measure the orientation degree of the glass fibers. The glass fibers are highly oriented at the edge of the plaque, while the orientation degree is lower towards the center position of the plaque. This leads to different results of the notched impact strengths. A sharp drop in strength can be observed for the 0◦ orientation specimens that are taken out from the edge of the plaque in contrast to specimens from the center position. Only a slight loss of strength can be seen for the 45◦ orientation. An increase of the strength can be seen for the 90◦ orientation towards the center removal position of the plaque.

2.2 Numerical Investigations

9

The studies on the different temperatures lead to a more ductile material behavior with increasing temperatures. The crack propagation is also investigated for the notched tensile tests. For the 90◦ orientation a planar fracture plane can be observed, while the 0◦ orientation shows a zigzag path along the fracture plane. A similar behavior had already been observed by Karger-Kocsis [43], where fibers directed in crack growth direction did not prevent the fast expansion of the crack, while transversal oriented fibers forced the crack path to dodge the interfering glass fibers. The tensile strength at room temperature is measured for 0◦ and 90◦ orientations and plotted for different strain rates, where an increasing trend can be observed. However, intermediate orientation angles are not shown. Also not investigated is if a similar or a contrary trend can be observed for the maximum strain at rupture. The research on the experimental works shows that the mechanical behavior in terms of energy absorption and crack propagation of short glass fiber reinforced polymers is extensively studied with respect to different fiber contents. Some investigations also consider the orientation dependency and the strain rate dependent material behavior as well. In terms of the rupture behavior, however, these parameters have been studied apart from each other or are not particularly examined in detail. Studying the influence of a changing orientation angle and strain rate on the rupture behavior would be of great interest therefore. A common approach to the rupture theme is by studying strength values under specific test configurations. Strain or deflection based rupture values are not explicitly emphasized. An investigation on the rupture behavior under stochastically aspects is also an open topic. Hence, it would also be interesting to research scatter range and distribution patterns of stress or force and strain or deflection based rupture values together, particularly in the context of a changing orientation angle and at different strain rates. Investigating the capability and accuracy of modeling the stochastic rupture behavior with help of distribution functions is also an open challenge.

2.2 Numerical Investigations Nutini and Vitali [65] study the mechanical behavior of a 30 wt.% short glass fiber reinforced polypropylene with help of an orthotropic elastic plastic material model. The constitutive parameters of the material model are deduced from tensile tests with optical measurements at different orientation angles. The tensile tests are prepared from the center positions of the plaques. A user defined plastic strain rupture condition is chosen. The essential point of this work is its modeling approach. The information of the process simulation is transferred to the shell element structure of the component validation tests. The mold filling process is resembled by rheological simulations that model the glass fibers, which are represented by ellipsoidal particles [42], within a viscous medium. This leads to a more physical representation of the glass fibers distribution in the component model.

10

2 State of the Art

The validation of a test box shows good accordance to the force displacement curve from the experiment. A comparison study with two isotropic material models shows a better performance for the orthotropic material model in terms of mechanical behavior and point of rupture. In a second study, the visual fracture behavior of the simulation and the experiment are compared to each other. A good correlation with help of the orthotropic material model can be achieved as well. A ribbed beam structure is tested in a three point bending loading case in a third study. The orthotropic material behavior is also able to model the force displacement curve accurately. Schöpfer et al. [77, 78] propose a layered composite-like shell structure for the simulation of reinforced polymers. They use an orthotropic elastic plastic material model to simulate the orientation dependent material behavior and an orthotropic elastic material model for orientation dependent rupture behavior. An illustration of the layer structure is shown in Figure 2.3. The layer structure is reverse engineered in order to fit the mechanical behavior of the material and does not represent the physical setup of the microstructure composition of the fiber reinforced compound in terms of layer thicknesses. Tensile and shear tests are simulated and compared to experimental tests. The simulation results show good correlation with the experimental data. Other material models, like isotropic elastic plastic material models and a micromechanical material model show less good correlation to the experiments. The correlation for the dynamic tensile and three point bending tests could not be evaluated because strain rate dependency is not supported by the material models of the layered structure. MAT_108

0.32 ȉ t

MAT_54

0.02 ȉ t

MAT_108

0.32 ȉ t

MAT_54

0.02 ȉ t

MAT_108

0.32 ȉ t

t

Figure 2.3 Composite-like shell structure for the modeling of fiber reinforced polymers. The MAT_54 is an orthotropic linear elastic material model for continuous reinforced polymers that determines the orthotropic rupture behavior, whereas the MAT_108 represents an elastic plastic material model for the orientation dependent plasticity. The arrangement of the different material models is not connected to the physical setup of the polymer compound, but is reverse engineered instead. [78]

Validation simulations of an injection molded ribbed beam component are also performed. Experiments from two injection gates are compared with the simulations. Good prediction of the inelastic behavior as well as for the rupture and post fracture behavior in terms of force displacement curve and visual comparison can be achieved.

2.2 Numerical Investigations

11

Hatt [38] uses an integrative simulation approach for the modeling of fiber reinforced polymers. The idea of an integrative simulation approach is not new because it was first presented in 2005 by Glaser and Wüst [28] and successfully applied for vehicle development in 2006 by Frik et al. [25]. By applying micromechanical mixture theory, a set of orthotropic elastic viscoplastic material models with different fiber orientation degrees is generated and the constitutive parameters are deduced from tensile tests that are performed at 0◦ , 45◦ and 90◦ orientations and strain rates from 0.001 s−1 to 100 s−1 . Mold filling simulations determine the glass fiber orientations of the components. The different orthotropic material models are stacked on each other in a composite-like shell element with 12 thickness layers. According to the actual orientation degree, a more physical based model is generated by using multiple material models with different mechanical parameters. An orientation dependent rupture condition separates the rupture behavior in a stress based criterion for the 0◦ orientation and a strain based criterion for the 90◦ orientation. Both criteria are strain rate dependent. Quasistatic and dynamic validation tests are performed on a ribbed beam structure. The beam is manufactured by two different processes, one with a single injection gate, the second with three injection gates. The simulations show good correlation for the different testing velocities and injection gates, which makes this a promising approach for the modeling of fiber reinforced polymers. A large experimental and numerical study is conducted by Lauterbach et al. [48, 49], who further developed the idea of an integrated simulation process. The orientation dependent information is now stored in a regular shell model with additional parameters, instead of a 12 thickness layer composite-like shell structure. Quasistatic and dynamic tensile and three point bending tests are used to calibrate the orthotropic elastic viscoplastic material model. Every step of the process chain is simulated and validated with help of μCT scans. The validated mold filling simulations are performed to study the dependency of the mechanical properties at different specimen removal positions and to model the physical fiber orientation of the coupon and component tests. The constitutive parameters are generated during the mapping process by means of micro mechanical homogenization methods. A bumper beam component validation test shows good correlation between simulation and experiment. The bumper tests are performed at different strain rates and impact positions to attain generally valid results for the method. Seyfarth et al. [85] use a multi scale approach to study fiber reinforced polymers. The mechanical behavior is generated by a representative volume element approach with mixed fiber and matrix properties. Schöpfer et al. [78] and Lauterbach et al. [50] also used this approach in their works. For the increase in physical characteristics of the material model one needs to accept the higher numerical costs for applying a multi scale material model because the mechanical properties are generated and calculated on the micro- and the macro-level.

12

2 State of the Art

A trend towards physically based material models with incorporation of the fiber orientation by means of mold filling simulations can be observed for numerical research studies with focus on fiber reinforced polymers. Orthotropic elastic plastic material models show good correlation for the non linear mechanical behavior of different fiber reinforced polymers and load cases. The main focus on the numerical studies is to reproduce the force displacement or stress strain curves of different experiments. It is common to use a single value rupture condition, e.g. the average plastic strain. Stochastic effects are commonly not the pivotal point of the numerical investigations. These deterministic simulations are essential because they are the starting point of building a framework for parametric variations in simulations. Combining the use of the new and advanced material models with stochastic simulations could close this gap. It would be interesting to investigate how a distribution of rupture values influences the outcome of a simulation study because scatter in terms of the rupture behavior can be observed in the experimental surveys as well. The aim of this approach would be a comparison between experimentally and numerically obtained distribution functions, instead of individual curves.

2.3 Statistical Research Pick et al. [70] studied the strength of three commercial composites in a three point bending test and a piston-on-three-balls test according to ISO 6872 [16] in order to analyze the results by means of a Weibull distribution function. A characteristic strength can be found for both test configurations. The results of both experiments are modeled by means of a Weibull distribution and a higher Weibull modulus can be observed for the piston-onthree-balls test than for the three point bending test. The authors ascribe this finding to the lower test volume of the three point bending specimens and the distribution of defects of the materials. Zeguer [100] investigates the influences of 15 input parameters for the design and analysis of a side airbag, where each design variable has a defined interval. Several target values, like bag pressure and node displacements are defined to measure the quality of the chosen parameters. Monte Carlo Simulations are used to study the input and output of the variables. Scatter plots determine the significance of a parameter and separate important from less important input variables. The results show that only 6 of 15 input variables have a significant influence on the output signal and only 2 of 10 output variables are required for further analysis of the airbag. The study shows that the design space can be reduced by 60%, respectively 80%, of the input and output variables, which reduces the simulation cost because less parameters need to be considered for the further analysis. Stochastic simulations are also applied in investigations by Özarmut et al. [67]. Orientation dependent material behavior and rupture parameters as well as friction parameters are varied in their study. Three point bending tests on a side impact beam and a hat profile are used to identify significant parameters regarding the maximum deflection and force.

2.3 Statistical Research

13

Monte Carlo Simulations with different sample sizes are conducted. The study shows that the influence of the input parameters can depend on the load case and sample size. Furthermore the authors are able to identify and allocate certain parameters to specific load cases. It can be said that the relative importance of an input variable is determined by the chosen boundary conditions for this study. This study also shows the importance of stochastic simulations as a useful tool to identify the significant input parameters and their impact on the output signals. Watson et al. [95] study aluminum castings with stochastic defects to determine their influence on the mechanical behavior of tensile tests. The tensile bars are cut out of the aluminum parts at three different locations. Two different manufacturing processes are used and 15 tensile bars from the three different positions are tested. A two parameter Weibull distribution function is used to evaluate the stress at rupture for the six different specimens. Filling simulations of the aluminum castings are run with stochastically distributed defect particles, where the defects are correlated to maximum strength values. The location of the tensile specimen in the simulation is identical to the experimental removal position of the tensile bars. The minimum strength values that lay in the gauge of the tensile bars in the simulations are evaluated and compared to the cumulative distribution function of the maximum stresses from the experiments. The comparison of the cumulative distribution functions shows that the simulation curve is located close to the experimental curve and only shows minor deviations. The field of stochastic simulations is often connected to robustness or sensitivity studies that define the limit for certain design variables of an experiment. It would be interesting to study if these processes can be adapted to the field of polymer testing. The utilization of tools from this subject area would be a new approach of comparing simulations and experiments in a statistical sense. It is therefore of great interest to investigate fiber reinforced polymers with this method and to study if congruence of simulation and experimental distribution functions can be achieved. This also means that relevant macroscopic parameters have to be identified according to the test environment. Identification and separation of significant and insignificant parameters would also be important.

3 Mechanical Testing The results of all conducted experiments from this work are presented in the following chapter. Firstly, the studies on the polymer’s microstructure are shown and secondly, macroscopic observations based on mechanical testing of tensile and three point bending tests are described.

3.1 Studies on the Microstructure A polypropylene homopolymer with 30 wt.% short glass fibers is investigated in this work. The composite also consists of other additives that e.g. serve as bonding agent between the matrix and the embedded glass fibers or change the polymers color or reduce its flammability. The composite’s compound formula is only known to the supplier and therefore only the matrix and the glass fibers are taken into consideration when the microstructure is investigated and discussed, although the mentioned chemical agents could influence the mechanical behavior as well [73].

3.1.1 Characterization of the glass fibers Glass fibers that are used in fiber reinforced polymers can be separated into three sections, namely •

short fibers (characteristic length scale: μm),

•

long fibers (characteristic length scale: mm),

•

and continuous fibers (characteristic length scale: cm).

An incineration test according to DIN EN ISO 1172 [15] determines that the mass fraction of the composite is 32%. According to the technical data sheet [53] it should be 30%, see also Section A.1. A fiber volume fraction ϕ of 14.4%, instead of 13.3% is obtained, when it is calculated by

ϕ=

ψρm , ψρm + (1 − ψ)ρ f

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8_3

(3.1)

16

3 Mechanical Testing

with mass fraction of the polymer’s glass fibers ψ and the matrix density ρm = 0.91 g/cm3 and glass fiber density ρ f = 2.54 g/cm3 . Rüsch [76] studied the geometry of the glass fibers and observed that the average fiber diameter is d = 10.9 μm with a standard deviation of 0.9 μm. The glass fiber length distribution is analyzed according to ISO 22314 [40]. Here, the lengths of hundreds of glass fibers are measured in order to obtain the complete fiber spectrum of the specimen’s embedded glass fibers. The results for all measurements are listed in Table 3.1 and the fiber length distribution of the 0◦ orientation is shown in Figure 3.1. The fiber length distribution figures of the remaining orientation angles are depict in Section A.2. The average fiber length corresponds to the arithmetic mean of all measured glass fibers. The Q50 value is the 50% quantile, which means 50% of all measured glass fibers are as long as or longer than the Q50 value. Q10 and Q90 correspond to the 10% and 90% quantiles respectively. The results show that the average glass fiber of the 45◦ orientation has the smallest value, whereas the 90◦ orientation has the largest average value. The arithmetic mean of all four specimens is 486.4 μm that leads to an average fiber length to diameter ratio of l/d = 44.62. The findings on the fiber length and diameter are in the same order of magnitude as measurements by other authors [34, 81].

Table 3.1

Results of the fiber length measurements

Orientation 0◦ 30◦ 45◦ 90◦

Q10 [ μm ] 226.5 215.2 156.2 229.6

Q50 [ μm ] 443.4 440.9 423.7 468.9

Q90 [ μm ] 795.4 802.5 804.0 820.5

Average fiber length [ μm ] 487.9 485.5 463.7 508.5

The critical fiber length lc is important in terms of the micromechanical rupture mechanism [75]. It depends on the glass fiber diameter d, the ultimate glass fiber strength σ uf and the ultimate shear strength of the matrix τmu and is calculated by lc =

d · σ uf 2τmu

.

(3.2)

Fracture of the glass fibers is supposed to be the rupture mechanism if the actual fiber length in the compound is longer than the critical fiber length. If it is shorter, the likelihood that glass fibers are pulled out of the matrix during the rupture process is increased [80]. As an estimation, the shear strength can be calculated by dividing the ultimate tensile strength of the matrix by two [80], which is 30 MPa for a polypropylene homopolymer according to Dominghaus [18]. The ultimate tensile strength for the glass fibers is 1800 MPa [83]. This results in a critical fiber length of 654 μm, which is higher than the average fiber

3.1 Studies on the Microstructure

17

length of the compound. This means that a tendency towards fiber pullout can be expected as rupture mechanism as opposed to glass fiber fracture.

Figure 3.1

Results fiber length analysis 0◦ orientation

3.1.2 μCT scans Components made out of fiber reinforced polymers are manufactured via injection molding. In simplified terms, the melted polymer forms a structure with five layers stacked upon each other during the manufacturing process [56]. There are two very thin surfaces with glass fibers aligned randomly due to the instantaneous cooling of the molten polymer from the cavity walls, followed by two broader layers of fibers aligned mainly in molding direction. A core layer with fibers mainly aligned perpendicular to the molding direction is located in between. Structures with dimensions of only a few millimeters can be measured very accurately by means of micro computer tomography. Rüsch [76] analyzed the different layers of this material and evaluated the orientation degree of the embedded glass fibers by performing μCT scans. The dimension of the specimen is 3.0 x 3.0 x 3.17 mm3 . The specimen is irradiated by an X-ray source in order to nondestructively measure the microstructure. The magnitude of the X-rays is modified by the polymer when passing through the structure and then measured by a detector. The alteration of the radiation is depending on the material’s density. This distinction in densities makes the contrast between the different embedded compounds in the imaging of the specimen. The information of the data is processed by voxels, which are three dimensional pixels. The detection of the positions and

18

3 Mechanical Testing

orientations of the embedded glass fibers is determined by an algorithm that divides the test volume into small equidistant boxes and identifies connected glass fiber fractions [29]. A cross section of the reconstructed specimen can be seen in Figure 3.2. The different colors indicate the different orientations of the glass fibers that are embedded in the matrix.

Figure 3.2 Cross section of a μCT analyzed specimen, from [76]. The different colors are indicating the varying orientation angles of the glass fibers. Fibers in molding direction are shown in red, while perpendicular oriented fibers are colored in blue.

Advani and Tucker [2] proposed the use of the symmetric second order orientation tensor A to quantify the orientation degree in a cross section of the polymer. In matrix notation it is given by ⎡

Axx A = ⎣ Axy Axz

Axy Ayy Ayz

⎤ ⎡ Axx Axz (principal axes) ⎣ 0 =⇒ Ayz ⎦ Azz 0

0 Ayy 0

⎤ 0 0 ⎦. Azz

(3.3)

When the coordinate system is rotated in the directions of the principal axes, the values for Axx , Ayy and Azz describe the orientation degree of the fibers in x-, y- and z-direction respectively. However, the notation of the orientation tensor in principal axes can be deceptive as the principal direction may not be identical to the injection molding direction. An orientation tensor of e.g. Axx = 0.98, Ayy = 0.01 and Azz = 0.01 indicates a high orientation degree of glass fibers in x-direction, while an orientation tensor of Axx = 13 , Ayy = 13 and Azz = 13 would mean that all fibers are oriented equally in all three coordinate axes. The numbers of the three values always sums up to unity. A sketch of the utilized plaque is shown in Figure 3.3. Nine positions are used in order to study spatial differences of the fiber orientations across the plaque. Figure 3.4 shows the resulting through the thickness fiber orientation of position number five. The three

3.1 Studies on the Microstructure

19

orientation tensor components Axx , Ayy and Azz are plotted over the normalized thickness. It can be observed that the degree of orientation is not constant through the thickness. The orientation degree never reaches 1.0, which means that the embedded glass fibers are never oriented solely in one direction. The specimen can be separated into three different zones based on the μCT scans. Two outer areas, from −1.0 ≤ x ≤ −0.25 and 0.25 ≤ x ≤ 1.0, with a fiber orientation degree of Axx ≥ 0.6 and a core region between −0.25 < x < 0.25, where the perpendicular fiber orientation Ayy becomes dominant. The fiber orientation degree in thickness direction Azz is approximately 0.05. The results are in good accordance to the findings of other authors [49, 87]. The μCT scans of all nine positions can be found in the appendix, see Section A.3.

Figure 3.3 Sketch of the injection molded plaque with specimen removal positions. The x-axis is in molding direction.

Figure 3.4

Principal orientations of the orientation tensor from removal position five

20

3 Mechanical Testing

3.2 Macroscopic Testing The conducted tensile and three point bending tests are presented and discussed in this section. The tensile tests are conducted at a quasistatic velocity while the three point bending tests are performed at a quasistatic speed and a highly dynamic impact velocity.

3.2.1 Specimen preparation Injection molded tensile bars that are manufactured out of fiber reinforced polymers always lead to a microstructure with highly oriented fibers in filling direction. This procedure is only convenient for testing in 0◦ orientation and hence inapplicable for testing in multiple directions, e.g. 45◦ or 90◦ . The specimens are cut out of a plaque in order to fully study the orientation dependency of the material, see Figure 3.3. To minimize boundary influences of the filling process, the center position of the plaque is chosen for sample extraction. It is recommended to mill the specimen out of the plaque. A slow feed rate as well as a slow cutting speed should be used in order to achieve the best results in terms of surface quality. Increasing the parameters will reduce the surface quality that will eventually lead to a degradation of the mechanical properties. The surface quality of two different manufacturing processes, milling and laser cutting, is compared in the following part. The cutting plane of the parallel region of a tensile bar is measured with help of a surveyors’ arrow that measures the coarseness of the surface on a micro scale. The arithmetic roughness value Ra for n measurement points z1 . . . zn is determined by Ra =

1 n ∑ |zi | . n i=1

(3.4)

Rz1max is the roughness value that measures the maximum distance between two nearby peaks. Table 3.2 summarizes the results for the roughness measurements for two 0◦ and 90◦ orientation specimens that were processed by milling and laser cutting. Table 3.2

Summary of the surface roughness measurements

Manufacturing process Milling Milling Laser cutting Laser cutting

Orientation angle 0◦ 90◦ 0◦ 90◦

Ra [ μm ] 1.07 1.45 12.94 17.52

Rz1max [ μm ] 15.77 18.0 105.53 111.34

The milled specimen has a smoother surface in comparison to the laser cut specimen. For this material the laser cutting manufacturing process leads to an increase of the value for

3.2 Macroscopic Testing

21

the surface roughness Ra of more than the factor 10 in comparison to the milling process. Detailed information on the measurement path of each specimen can be found in the appendix, see Section A.4. Figure 3.5 shows a 3D surface scan of an intersection from two samples which are manufactured by the two different processes. While the milled surface on the left is very even, the laser cut surface shows high gradients locally. The peaks in the surface of the laser cut specimen in Figure 3.5 can be identified as glass fibers pointing upwards, whereas the matrix material is located below. The difference in the surface quality can be also correlated to a difference in the stress level of the specimen, which is shown in Figure 3.6. While the difference may not be significant for the 0◦ , it can be observed that the stress level of the laser cut specimen is below the stress level of the milled specimen. The difference in the stress level of the 90◦ laser cut specimen on the other hand is almost 15% less in comparison to the milled specimen. The significant reduction of the stress level can be traced back to the manufacturing process because all parameters regarding the test environment are identical. The high Rz1max values indicate that laser cutting locally induces predetermined breaking points on the surface of the specimen. The longitudinal and lateral strain over time diagram in Figure 3.7 shows that an increased straining can be observed for all laser cut specimens, except for the 90◦ orientation lateral strains. The micro cracks that are induced by the laser cutting lead to earlier elongation of the specimen for the 0◦ and 90◦ orientations likewise. At the same time, the lateral straining for the 90◦ orientation is not affected by the laser cutting process because the perpendicular aligned glass fibers resist contraction likewise for the laser cut specimen and for the milled specimen. This leads to a significant decrease of the stress level for the 90◦ orientation laser cut specimen because the elongation of the material, while withstanding lateral contraction at the same time, leads to a degradation of the internal stresses.

Figure 3.5 Comparison of surface quality via 3D microscope scan. Left: milled specimen. Right: laser cut specimen. The milled surface is very smooth and does not show any gaps on the top. The laser cut surface on the other hand shows many ”hills” and ”valleys” on top. The surface peaks can be identified as exposed glass fibers that are pointing out of the matrix.

22

3 Mechanical Testing

The investigations show that the manufacturing process can have a significant influence of the mechanical performance of the specimen. Milling at a slow rotational speed and feed rate results in the best surface quality of the specimen and is preferred to laser cutting.

Figure 3.6 Comparison of nominal stress strain curves from tensile tests prepared from two different manufacturing processes: milling machine and laser cutting.

Figure 3.7 Nominal longitudinal and lateral strain over time diagram of the experiments with different manufacturing processes. An increase for the 0◦ and 90◦ orientation longitudinal and lateral strains of the laser cut specimen can be observed in comparison to the milled specimen, except for the 90◦ orientation lateral strains. The boundary conditions for all experiments are identical.

3.2 Macroscopic Testing

23

3.2.2 Tensile tests The tensile tests are performed at a testing velocity of 0.001 ms−1 . The utilized specimen geometry is the Becker tensile bar, which is shown in Figure 3.8. The local deformation is measured with help of digital image correlation by using a GOM Aramis 4M system. Specimens are taken from the center position of the plaque at four different orientation angles, 0◦ , 30◦ , 45◦ and 90◦ , with respect to the injection molding direction, and tested until rupture.

x

y Figure 3.8

Geometry of the Becker tensile bar, from [7]

3.2.2.1 Elastic behavior In the uniaxial tensile load case the nominal stress σnom relates the applied force F to the initial cross section A0 , while the true stress σ on the other hand relates the force to the momentarily, changing cross section A (x) σnom =

σ=

F , A0

(3.5)

F . A (x)

(3.6)

For the tensile loading case the nominal strain εnom of a length l0 , e.g. the gauge length, is given by εnom =

Δl l − l0 = = λ − 1, l0 l0

λ=

l , l0

(3.7)

24

3 Mechanical Testing

with current length l and the principal stretch λ . The Hencky or logarithmic strain εlog is considering the actual change of deformation and is denoted as   l (3.8) εlog = ln = ln (λ ) = ln (1 + εnom ) . l0 Stresses and strains are linked by the Young’s modulus E in the tensile test by σ = Eε,

(3.9)

which is only defined in the linear elastic region of the stress strain curve (|ε| ≤ 0.25%). The strain rate ε˙ denotes the material’s change of strain with respect to time. The initial or nominal strain rate ε˙0 is present at the beginning of an experiment. For the tensile test the nominal strain rate is calculated by ε˙0 =

v0 , l0

(3.10)

with initial velocity v0 and gauge length l0 . The nominal strain rate for the utilized tensile tests yields ε˙0 = 0.083 s−1 . The strain rate is not constant throughout the whole test because v0 and l0 change during the experiment. The Poisson’s ratio ν is defined as the negative ratio of the lateral and longitudinal change in strains ν =−

dεx dεlat =− . dεlong dεy

(3.11)

The elastic constants of all four orientations are shown in Table 3.3. Table 3.3

Elastic constants with ±3σ errors from the quasistatic tensile tests

Orientation angle [ ◦ ] 0◦ 30◦ 45◦ 90◦

Young’s modulus E [ GPa ] 5.8 ± 0.171 4.6 ± 0.096 4.0 ± 0.135 4.1 ± 0.21

Poisson’s ratio ν [ - ] 0.38 ± 0.006 0.43 ± 0.021 0.41 ± 0.018 0.25 ± 0.009

The differences in the elastic constants can be deduced from the material’s microstructure, see also Figure 3.4. It can be observed that the Young’s modulus decreases with increasing orientation angle. For the 0◦ orientation the glass fibers in the outer layers are aligned in loading direction and therefore this orientation shows the highest force level of all four orientations. For the 90◦ orientation the outer layers are aligned perpendicular to the load direction and are not capable of picking up loads sufficiently. The core layer on the other hand is aligned in loading direction and can therefore carry at least a certain amount of

3.2 Macroscopic Testing

25

the applied forces, although not in the same range as the 0◦ orientation. The 90◦ Young’s modulus is slightly higher than the 45◦ Young’s modulus that may be attributed to the core layer orientation. It should be noted that this finding may not necessarily be observed for other material thicknesses as well. None of layers from the 30◦ and 45◦ orientations have glass fibers aligned purely in loading direction or perpendicular to it. The fibers of these specimens are aligned in a (+30◦ , −60◦ , +30◦ ) or a (+45◦ , −45◦ , +45◦ ) lay-up, which implies that the matrix material has an increasing influence on the mechanical properties. The average Young’s modulus E from the 0◦ , 45◦ and the 90◦ orientations yields E = 4.63 GPa. The average of all four Young’s moduli also results in E = 4.63 GPa. This shows that the orientation averaged mean, the so called mechanical mean, of the Young’s modulus is located at the 30◦ orientation. It could be assumed that the orientation dependent stiffness of fiber reinforced polymers is connected to trigonometric functions because the sine of 30◦ is 0.5. A proposal for calculating the orientation dependent Young’s moduli is given by E ϕ = c1 − (c1 − c2 ) |sin (ϕ)| ,

c1 , c2 > 0,

c 1 > c2 .

(3.12)

A good approximation for the 0◦ , 30◦ and the 45◦ Young’s moduli is obtained by setting c1 = 5.8 GPa and c2 = 3.4 GPa. However, the 90◦ Young’s modulus is underestimated. The deviation could be deduced from the microstructure because the 90◦ orientation tensile specimen has fibers aligned in loading direction in its core layer that means the mechanical behavior is not solely attributed to the polypropylene matrix. This aspect is also discussed in Section 3.2.2.2, where digital image correlation is used to study the localization behavior across the thickness in the tensile test. The values for the Poisson’s ratios for the 0◦ , 30◦ and 45◦ orientations lie close to the value for the Poisson’s ratio for a polypropylene homopolymer, which is given as νPP = 0.4-0.45 [9]. This observation can be correlated to the microstructure of the material. For the e.g. 0◦ most of the glass fibers of the outer layer are aligned in loading direction. That means the matrix material is responsible for the mechanical properties in lateral direction and this suggests that the observed Poisson’s ratio is within a close range to the value of the polypropylene homopolymer. The same assumptions hold for the 30◦ and 45◦ direction, where the matrix material also governs the mechanical properties in axial and lateral directions to a certain amount likewise. The 90◦ orientation on the other hand has a Poisson’s ratio of 0.25, a difference of 34-42% compared to the other orientations. A lower Poisson’s ratio indicates a higher resistance to lateral contraction, which is in good accordance with the microstructure setup of the material. The outer layers of the 90◦ orientation specimen have fibers aligned perpendicular to the loading direction. These fibers are acting against the lateral contraction resulting in a much lower value for the Poisson’s ratio in contrast to the other orientations.

26

3 Mechanical Testing

3.2.2.2 Inelastic behavior When loaded beyond the linear elastic domain in a tensile test, fiber reinforced polymers transit into a nonlinear elastic domain before the specimen shows spatially selective localization behavior prior to rupture. The physical composition of the material shows that the local microstructure is discontinuous and irregularly ordered with a heterogeneous spectral distribution of fiber lengths. Mainly two types of inelastic behavior can be identified: Firstly, pure matrix plastification and secondly, fiber-matrix debonding, which is an interaction between the two phases that eventually leads to extended matrix plastification. The microscopic image of the cross section from a 0◦ orientation tensile specimen after rupture is shown in Figure 3.9. The plastification can only be observed at one side and not across the whole area of the cross section. Additionally, the plastic deformation is not distributed homogeneously though the thickness, which is indicated by the triangle. These two observations could also be noticed for other specimen. On the one hand, there are spatially restricted plastification zones that are observed in a narrow section of the entire cross section of the tensile specimen. On the other hand, a plastification path through the thickness of the specimen can be identified. However, an a posteriori analysis of the specimen cannot reveal if the plastification path propagated from the front of the specimen to its backside, or vice versa. Enhanced plastification can be correlated to fiber-matrix debonding. This can be observed when studying the cross section of a 90◦ orientation tensile test specimen after rupture, which is shown in Figure 3.10. Glass fibers have been pulled out of the matrix of the core layer across one third of the cross section length approximately.

B) A)

Figure 3.9 (Left): Image of a 0◦ orientation tensile specimen after rupture. The rectangle indicates the zone, where enhanced plastification can be observed. (Center): Microscopic image of a part from a 0◦ orientation tensile specimen’s cross section. The perpendicular aligned glass fibers of the core layer can be identified. The triangle indicates the zone where matrix plastification occurred, recognizable from the crazing of the polymer matrix and the holes in between, indicating pulled out fibers. The rectangle indicates the zoomed area of the right picture. (Right): The holes inside the matrix can be identified either as e.g. A) one single fiber or e.g. B) multiple fibers that have been pulled out during the deformation process.

3.2 Macroscopic Testing

27

Figure 3.10 (Left): Image of a 90◦ orientation tensile specimen after rupture. The rectangle indicates the zone, where the plastification occurred, noticeable from the crazing of the matrix. Plastification can only be observed at one side of the specimen and not through the entire cross section. (Center): The perpendicular aligned glass fibers of the outer layers can be identified. The rectangle indicates the zone where fibers have been pulled out along the core layer. The smaller rectangle on the left indicates the zoomed area of the right picture. (Right): The core layer that has fibers aligned in loading direction shows enhanced matrix plastification due to pulled out fibers. A small number of holes can also be identified within a small range out of the core layer.

DIC measurements illustrate that the localized inelastic behavior is correlated to the orientation of the embedded glass fibers. This can be observed in detail for the 90◦ orientation tensile test. A difference in the strain localization between the outer layers and the core layer can be observed when the deformation is measured over the specimen’s thickness. This is shown by a picture series in Figure 3.11.

Figure 3.11 Picture series at different time steps of a DIC measurement from a 90◦ orientation tensile test. The change of deformation is measured across the thickness. A: the core layer shows enhanced localization behavior in contrast to the outer layers. For the 90◦ orientation, the process of rupture begins in the inside of the specimen, where glass fibers are oriented in loading direction. In a second step, the strain localization can be measured on the surface, when most of the damage has advanced from the inside of the specimen’s core to the outside.

28

3 Mechanical Testing

An a posteriori analysis of the 30◦ and 45◦ orientations shows a significant difference in contrast to the 0◦ and 90◦ orientations in terms of the inelastic material behavior. An improved force transmission between the different layers can be observed due to the angular rotation of the embedded glass fibers. This can be deduced from microscopic scans of the specimen’s fracture surface that shows enhanced crazing over a broader region across the cross section, which is shown in Figure 3.12.

Figure 3.12 Cross section from a 45◦ orientation tensile specimen after rupture. The plastification zones can be noticed from the whitening of the matrix across large areas of the cross section.

The nominal stress strain curves of the tensile tests are shown in Figure 3.13.

Figure 3.13 Nominal stress strain curves of all conducted tensile tests. The average curves are cut off at the end of the first ruptured specimen. The individual results are shown as slightly transparent curves.

3.2 Macroscopic Testing

29

The strain is calculated from the parallel length of the tensile bar l = 12 mm with usage of DIC. It can be observed that the 0◦ orientation has the highest force plateau, while the 90◦ orientation has the lowest. The stress level of the 90◦ orientation surpasses the 45◦ orientation’s stress level until the 2% nominal strain limit. The core layer of the 90◦ orientation supports the absorption of induced forces in contrast to the 45◦ orientation that has no fibers oriented in loading directions. The 45◦ orientation overtops the 90◦ orientation’s stress strain curve eventually again because of the improved force transmission throughout the cross section. 3.2.2.3 Rupture behavior The values of the maximum strain εmax and stress σmax at rupture in the tensile tests are listed in Table 3.4. Table 3.4

Quasistatic tensile test rupture values

0◦

μ σ σ2 CV

εmax [-] 0.038 0.004 0.0 0.105

σmax [ MPa ] 75.7 1.1 1.2 0.083

30◦ εmax [-] 0.049 0.006 0.0 0.122

σmax [ MPa ] 65.6 0.6 0.3 0.096

45◦ εmax [-] 0.054 0.005 0.0 0.026

σmax [ MPa ] 59.4 0.5 14.33 0.008

90◦ εmax [-] 0.039 0.003 0.2 0.077

σmax [ MPa ] 55.7 0.7 0.5 0.013

It could be concluded from an a posteriori observation of the microscopic images that stress concentrations located at or around the embedded glass fibers eventually break up due to loss of matrix-fiber-bonding. This event then results in a high energy release rate which cannot be absorbed by the surrounding material that leads to rupture of the specimen. This effect can vary between tests because of the spatial variations of the glass fibers that are scattered stochastically by the injection molding process. Schoßig et al. [82] use acoustic emission analysis while conducting tensile tests and observe that the signals accumulatively increase as the tests progress. Gupta et al. [34] also conducted coupled acoustic emission analysis tensile tests and observed different acoustic profiles for the different orientation angles. In situ X-ray synchrotron computed tomography measurements of tensile tests, similar to studies by Scott et al. [84] and Garcea et al. [27], could lead to further insights on the microstructure rupture process itself because they allow visual measurements of the inside of the cross sectional area.

30

3 Mechanical Testing

3.2.3 Three point bending tests The three point bending tests for the statistical analysis are performed at two testing velocities, a quasistatic speed of 0.001 ms−1 and a high dynamic impact velocity of 4.0 ms−1 . The nominal geometry of the samples is 40.0 x 10.0 x 3.17 mm3 . The exact geometries of all specimens are listed in the appendix, see Tables A.1-A.8. Details on the testing plan and the setup can be found in Table 3.5 and Figure 3.14 respectively. Table 3.5

Statistical testing plan for three point bending tests

Velocity 0.001 m/s 4.0 m/s

Nom. strain rate ε˙0 0.021 1/s 85.0 1/s

Support 30.0 mm 30.0 mm

Orientations 0◦ , 30◦ , 45◦ , 90◦ 0◦ , 30◦ , 45◦ , 90◦

Sample size 30, 30, 30, 50 30, 30, 30, 30

The quasistatic tests are conducted at a universal testing machine while the dynamic tests are performed at an instrumented pendulum. The test setup of the quasistatic and dynamic tests is identical in order to compare the force deflection curves to each other. All samples are conditioned at least 72 hours at 23 ◦ C and 50 %rh before testing.

R2

R2

R2

T = 3.17 mm 40 mm 30 mm Figure 3.14

Three point bending test setup with nominal specimen geometry

3.2.3.1 Quasistatic three point bending tests All four orientations are tested at a constant forward speed of 0.001 ms−1 until the specimen ruptures. The 0◦ , 30◦ and the 45◦ orientations are tested with 30 repetitions and the 90◦ orientation is tested with 50 repetitions, resulting in 140 total samples for the quasistatic tests. A 2.0 kN load cell measures the force signal during the test and the deflection is measured via traverse translation. The testing time is ranging from 2.6 seconds for the 0◦ orientation to 4.0 seconds for the 90◦ orientation. The recording frequency is 100 Hz which means 260 or 400 data points are obtained for the 0◦ orientation and the 90◦ orientation respectively.

3.2 Macroscopic Testing

31

A summary of the quasistatic results is shown in Table 3.6. Detailed information on every specimens’ geometry and rupture values can also be found in the appendix, see Tables A.1-A.4. Table 3.6

Quasistatic three point bending results

0◦

μ σ σ2 CV

wmax [ mm ] 1.95 0.061 0.004 0.031

30◦ Fmax [N] 349.2 6.94 48.17 0.02

wmax [ mm ] 2.34 0.072 0.005 0.031

Fmax [N] 293.32 4.18 17.48 0.01

45◦ wmax [ mm ] 2.7 0.07 0.005 0.026

Fmax [N] 255.58 3.78 14.33 0.01

90◦ wmax [ mm ] 2.97 0.148 0.022 0.05

Fmax [N] 225.11 3.46 12.0 0.02

The force deflection curves of all conducted quasistatic three point bending tests are shown in Figure 3.15. All displayed curves are plotted until peak force. The complete curves can be found in the appendix, see Figure A.19.

Figure 3.15 Quasistatic three point bending force deflection curves. The curves are plotted until the peak force. The sample size for the 0◦ , 30◦ and 45◦ orientations is 30 and 50 for the 90◦ orientation.

With increasing orientation angle the three point bending tests show a decreasing force and an increasing deflection level. A similar material behavior could also be observed by other authors [34, 79]. The difference in the onset of the 45◦ and the 90◦ orientation is

32

3 Mechanical Testing

more pronounced in the three point bending test than in the tensile test. The applied load in the three point bending test is not acting across the whole cross section but at the edge region of the specimen instead. Therefore, the absorbed force level of the 90◦ orientation is lower than the 45◦ orientation because it has less glass fibers aligned in stress direction at the outer layer. The Figures 3.16–3.19 show the characteristic fracture behavior for the four investigated orientation angles. Every specimen of the conducted experiments fractures, yet only the specimens 11 and 35 of the 90◦ orientation entirely break apart into two pieces. The remaining specimens break at the bottom, where the tensile stress acts, while the top, where the compression stress acts, is still connected. The rupture of the 0◦ orientation shows a discontinuous but straight fracture pattern. The 30◦ and 45◦ orientations on the other hand, break under an angle. This fracture behavior appears to follow the orientation degree of the embedded glass fibers from the outer layer. Hartl et al. [36] could observe a similar fracture behavior for their specimen as well. The 90◦ orientation specimen ruptures under a straight line, with a more pronounced crack opening than the 0◦ orientation.

Figure 3.16

Specimen 26 of the 0◦ orientation after the quasistatic three point bending test [76]

Figure 3.17

Specimen 9 of the 30◦ orientation after the quasistatic three point bending test [76]

3.2 Macroscopic Testing

33

Figure 3.18 Specimen 23 of the 45◦ orientation after the quasistatic three point bending test [76]

Figure 3.19 Specimen 17 of the 90◦ orientation after the quasistatic three point bending test [76]

3.2.3.2 Dynamic three point bending tests The dynamic three point bending tests are performed on an instrumented pendulum testing machine, which is shown in Figure 3.20. Support- and fin radii are equal to the quasistatic setup.

Figure 3.20 Details on the instrumented pendulum testing machine Impetus

34

3 Mechanical Testing

The velocity at impact is 4.0 ms−1 and the pendulum has a mass of 0.43 kg which leads to a total potential impact energy of 3.44 J. 30 repetitions are performed for every orientation angle, so a total number of 120 dynamic tests are conducted. A 400g uniaxial acceleration sensor that is fixed to the pendulum captures the acceleration signal with a recording frequency of 225 kHz. This means that 135 data points are obtained for a test duration of 0.6 ms for a 0◦ orientation test. The deflection is recorded by an angular sensor that measures the movement of the pendulum on its circular path. The summarized results of the dynamic three point bending tests are shown in Table 3.7. The filtered force deflection curves from the experiments are shown in Figure 3.21. The unfiltered curves can be seen in Figure A.20. Table 3.7

Dynamic three point bending results

0◦

μ σ σ2 CV

wmax [ mm ] 2.33 0.074 0.005 0.032

30◦ Fmax [N] 552.92 23.19 537.94 0.04

wmax [ mm ] 2.52 0.14 0.02 0.056

Fmax [N] 444.73 18.38 337.78 0.04

45◦ wmax [ mm ] 2.63 0.167 0.028 0.064

Fmax [N] 384.86 24.47 598.67 0.06

90◦ wmax [ mm ] 2.31 0.176 0.031 0.076

Fmax [N] 320.67 26.56 705.47 0.08

Figure 3.21 Filtered 4 ms−1 impact velocity three point bending force deflection curves. The curves are plotted until the peak force. The sample size for the 0◦ , 30◦ , 45◦ and 90◦ orientations is 30.

3.2 Macroscopic Testing

35

The high oscillations in the curves particularly stand out. They are induced vibrations on the specimen by the impact. It can be observed that the force signals of the unfiltered curves show a negative algebraic sign during the experiments. This is due to the piezoelectric effect and the interaction between the physical excitation of the sensor and its electrical response to oscillations. Due to the oscillations, the piezoelectric acceleration sensor measures a positive force signal on impact while a negative force signal is registered when the amplitude decays. The overall force level is higher for the dynamic curves than for the quasistatic curves that could be attributed to the viscoelastic viscoplastic material behavior of the matrix. The yield stress of polypropylene increases when loaded at higher strain rates [66]. The force level decreases with increasing orientation angle. This is similar to the quasistatic tests. From a visually analyzing point of view, a distinct separation of the maximum deflections between different orientation angles is not possible because the relative distances between the maximum deflections seem to vanish. This can also be seen in the tabulated rupture values, see Table 3.7. The mean values for maximum deflections for all four orientations are within a range of 0.32 mm. Overlaps in the maximum deflections for the different orientation angles will be studied more profound when the rupture behavior is statistically analyzed in Chapter 4. It is remarkable that the average maximum deflection of the 90◦ orientation has the lowest value of all four orientations for the dynamic testing velocity, while it has the highest value for the quasistatic tests. This observation can be attributed to the viscoelastic viscoplastic material behavior of the polypropylene matrix that eventually leads to brittle rupture of the matrix [44] and this severe drop of the maximum deflection. In this case the average value for the maximum deflection for the 90◦ orientation is lower than for the 0◦ orientation that formerly showed the least maximum deflection in the quasistatic experiments. The Figures 3.22–3.25 show the fracture pattern of the specimens of all four orientation angles after the dynamic three point bending test. All specimens fracture into two parts during the experiment. The 0◦ orientation specimen fractures at a straight line from the top to the bottom. Similar to the quasistatic tests, the 30◦ and 45◦ orientations show a rupture behavior under a specific angle. Yet, this behavior can only be observed on the bottom of the specimen, where the tensile stress acts. The forming of a fracture step can be observed in the middle of the specimen that may indicate a transition zone in the core layer. The top side shows a straight fracture pattern. A straight rupture line on the top and the bottom with a fracture step in the middle can be observed for the 90◦ orientation. The step may be connected to the core layer that has fibers oriented in 0◦ direction.

36

3 Mechanical Testing

Figure 3.22

Specimen 14 of the 0◦ orientation after the dynamic three point bending test [76]

Figure 3.23

Specimen 20 of the 30◦ orientation after the dynamic three point bending test [76]

Figure 3.24

Specimen 22 of the 45◦ orientation after the dynamic three point bending test [76]

Figure 3.25

Specimen 24 of the 90◦ orientation after the dynamic three point bending test [76]

3.2 Macroscopic Testing

37

In addition to the photographs of the fracture patterns, the fracture planes of the dynamic tests are investigated by means of SEM. Figure 3.26 shows a SEM image from a fracture surface of a 0◦ orientation specimen. Glass fibers are pointing out of the matrix in the lower part of the image. Ends of glass fibers, which are shorter than the fibers in the lower part, are also pointing out of the matrix in the upper part of the image. These glass fiber ends are partially embedded in the matrix. Holes in the matrix are labeled as a and indicate locations where glass fibers are pulled out during the deformation process. Matrix deformation can also be observed in the lower left corner of the image, which is labeled with b. Holes from pulled out glass fibers can also be found for the 30◦ orientation in Figure 3.27 at the a labeled circles. The b labeled circle shows a glass fiber that is still partially embedded in the matrix. Partially embedded glass fibers can also be determined for a 45◦ orientation specimen, which is depicted in Figure 3.28. The holes from pulled out fibers are indicated by the a label in Figure 3.28. A SEM image of a 90◦ orientation specimen is illustrated in Figure 3.29. A hole from a pulled out fiber is shown at the a label. Matrix material at the edge of a glass fiber can be found at the b labeled circle. One similarity between the images of the dynamic tests at different orientation angles is that pieces of matrix do not remain on the glass fibers after the detachment, which could also be observed by other authors [34, 82].

Figure 3.26 SEM image from specimen 14 of the 0◦ orientation [76]

38

3 Mechanical Testing

Figure 3.27

SEM image from specimen 20 of the 30◦ orientation [76]

Figure 3.28

SEM image from specimen 22 of the 45◦ orientation [76]

3.2 Macroscopic Testing

Figure 3.29 SEM image from specimen 24 of the 90◦ orientation [76]

39

4 Statistical Analysis 4.1 Mathematical Framework Using the tools of statistical mathematics is essential in order to evaluate a large number of data points. Tools from the field of empirical statistics are presented and applied on results of the three point bending tests in the following section.

4.1.1 General definitions The maximum force Fmax and maximum deflection wmax from the conducted three point bending experiments are analyzed statistically in this work. The formulae for the arithmetic mean μ, empirical variance σ 2 , standard deviation σ and coefficient of variation CV for n measurement points xi , . . . , xn are given as μ = x¯ =

σ2 =

σ=

1 n ∑ (xi − x)¯ 2 , n − 1 i=1

√

CV =

1 n ∑ xi , n i=1

(4.1)

(4.2)

σ 2,

(4.3)

σ . x¯

(4.4)

Variance and standard deviation are measures of scatter within a series. The coefficient of variation is a tool to compare scatter between different series [37]. The confidence interval assumes that the real mean value of a data set is within a specific interval with an e.g. 95% confidence. If the confidence intervals of two series are not intersecting, a significant © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8_4

42

4 Statistical Analysis

difference between both series exists. The upper and lower bounds of a 95% confidence interval are defined as σ cu = x¯ + t(1− α ,k−1) √ , 2 n (4.5) σ cl = x¯ − t(1− α ,k−1) √ , 2 n  with the 1 − α2 quantile of the Student’s t-distribution [88] with (k − 1) degrees of freedom and α = 5%.

4.1.2 Distribution functions The four utilized probability distribution functions of this work are from the family of bell shaped distribution functions and extreme value distribution functions, namely, the normal distribution, lognormal distribution, two parameter Weibull distribution and three parameter Weibull distribution. The normal distribution, also known as the Gaussian distribution, is widely used in different disciplines, e.g. economic or social sciences. The random variables of a normal distribution are spread around the mean in such a way that approximately 68.3% of the data is within a range of μ ± σ , approximately 95.5% of the data is within a range of μ ± 2σ and approximately 99.7% of the data is in a range of μ ± 3σ . The probability density function of the normal distribution is given as

   1 x−μ 2 1 , σ > 0. (4.6) fN (x) = √ exp − 2 σ σ 2π The normal distribution is a mirror-symmetrical function with axis of symmetry at x = μ and inflection points at μ ±σ . With the z-score z = (x − μ) /σ , any normal distribution can be transformed into the standard normal distribution. With μ = 0 and σ = 1, the standard normal distribution is given as 1 2 1 ϕ (x) = √ e− 2 x . 2π

(4.7)

An important characteristic of distribution functions is that the sum of all probabilities adds up to one, which means that all possible outcomes of the probability function yield 100% of the results. This is mathematically expressed as 1 √ 2π

∞ −∞

1 2

e− 2 x dx = 1,

(4.8)

4.1 Mathematical Framework

43

for the standard normal distribution. The probability space in Eq. (4.8) ranges from −∞ to +∞ that from a technical perspective is not realistic for all possible experimental observations. It would be more realistic to use a truncated normal distribution with a probability space (0, +∞) instead because e.g. the maximum force at rupture in a tensile test cannot be negative. However, this is not required from a practical point of view. The probability that a random variable falls below zero is beyond the 6σ limit for these type of problems, which means that less than 2 parts per billion lie outside this limit. While this means that it is still possible for values to fall below the 6σ bounds, it is very unlikely to happen. According to the central limit theorem, the sum of a large number of independent random variables approximately yields the normal distribution [68]. On the other hand, the lognormal distribution results from the product of a large number of positive independent random variables in the lognormal domain. The lognormal probability density function is defined as

   1 ln (x) − μ 2 1 √ exp − , x, σ > 0. (4.9) fLN (x) = 2 σ xσ 2π It is only defined for positive x and often follows the form of a skewed bell shape, depending on the chosen parameters. μ and σ for the lognormal distribution are not identical to the arithmetic mean and standard deviation from the normal distribution. These parameters correspond to the geometric mean μg and multiplicative standard deviation σg [52] that are also denoted as location and scale parameter and given as

n √ (4.10) μg = x¯g = n x1 · x2 · . . . · xn = n ∏ xi , i=1

 σg = exp

 2  xi 1 n . ln ∑ n − 1 i=1 x¯g

(4.11)

The Weibull distribution [96, 97] belongs to the class of generalized extreme value distributions [22] and is a type III of those. Other extreme value distributions are the Fréchet distribution [23], which is a type II GEV distribution and the Gumbel distribution [32], which is a type I GEV distribution. The three classes of generalized extreme value distributions are categorized by the existence or non existence of a finite limit for their upper or lower bounds. Type I distribution functions are unbounded, type II distribution functions have a lower threshold and type III distribution functions have an upper limit value. The interested reader is referred to Gumbel [33] for more information and further details on the topic of generalized extreme value distributions. Only a Weibull distribution function is suitable for analyzing rupture of materials from a technical point of view because it can be expected that a specimen ultimately ruptures un-

44

4 Statistical Analysis

der a persistent load. In other words, the probability density function does have an upper limit. A defined minimum threshold value is unlikely because the presence of flaws and defects in a material cannot completely be excluded. A completely unbounded probability density function is also unlikely because all materials eventually rupture under accumulative loads. The Weibull distribution is associated to rupture of materials because it is affiliated to the weakest link theory. According to the weakest link theory, a body is allegorized as a chain. The entire strength of the body is irrecoverably lost as soon as the first element of the chain ruptures. Only the classical two and three parameter Weibull distributions are considered in this work, although a large number of different Weibull distributions exist. The two parameter Weibull probability density function is given as fW2 (x) =

  x c  c  x c−1 , exp − b b b

x, b, c > 0,

(4.12)

with scale and shape parameter b and c. The shape parameter c is also called Weibull modulus. The Weibull modulus corresponds to the scatter of the distribution, e.g. for larger moduli the scatter decreases and vice versa. The three parameter Weibull distribution is obtained by adding a location parameter a to Eq. (4.12), which yields c fW3 (x) = b



x−a b

c−1

   x−a c exp − , b

x > a,

b, c > 0.

(4.13)

The parameter a is also denoted as the threshold parameter. The underlying assumption of this parameter is that the potential risk of rupture only exists for values greater than a. However, this has to be scrutinized from a physical point of view because flaws and defects in the material cannot be completely excluded. It may still be possible for structures to rupture earlier than predicted by the three parameter Weibull distribution in reality. On the other hand, adding an extra parameter to the distribution function increases the variability and is beneficial to the degree of fitting. On the downside, the three parameters can only be determined by numerical optimization that adds complexity to the solution finding process. The data is plotted in a double logarithmic probability plot in order to fit the parameters of the Weibull distribution. According to Rinne [74], the procedure for creating a probability plot is given in the following sequence: •

Sort values xi in ascending order; x{i:n} = x1 ≤ x2 ≤ . . . ≤ xn ,

•

take the natural logarithm of all values; xi∗ = ln (xi ),

•

calculate the plotting positions Pi for every value xi ; Pi = i/ (n + 1),

•

calculate the reduced values ui = ln[−ln(1 − Pi )],

•

∗ graph the probability plot with abscissa x{i:n} versus ordinate ui ,

4.1 Mathematical Framework

45

•

if the data points do not form a straight line a location parameter a may exist; else,  continue fitting the data with help of the function y (x) = cxi∗ + b by means of linear regression,

•

the parameterc of Eq.  (4.12) is equal to the slope of y (x), while b is calculated  with b = exp −b /c ,

•

compute the coefficient of determination R2 for y (x).

The coefficient of determination is given by 2

R2 = 1 −

∑ni=1 (xi − yi ) . ¯2 ∑ni=1 (xi − x)

(4.14)

In order to create probability plots for the three parameter Weibull distribution, y (x) is   replaced by y∗ (x) = c [ln (xi∗ − a)] + b and solved for a, b , c by means of numerical opti mization. The parameter b can be computed with help of b and c in the same manner as for the two parameter Weibull distribution. It is required to pre-process the data because several results of e.g. the maximum deflections have the same value, see Tables A.1-A.8. This would lead to different probabilities for identical values because of the use of plotting positions Pi , which would be unrealistic. Instead, according to Wilker [99], the data is sorted in k number of bins by k=

√

n,

(4.15)

with bin width b=

xmax − xmin , k−1

(4.16)

that depends on the maximum value xmax and the minimum value xmin of each data set. Furthermore, the average values xi,m of each bin xi,m =

bui − bli , 2

(4.17)

with the upper bound bui and lower bound bli of each bin, are used in the probability plot process instead of the individual xi values. Additionally, the summed up relative frequencies fi of the i-th bin replace the i-th number of each sample in the plotting positions formula Pi = i/ (n + 1). Currently, there is still an open debate over the correct choice for the plotting position Pi , see [12, 54, 55]. Different proposals for expressing the plotting position can be found in [54]. The plotting position proposed by Weibull [96], Pi = i/ (n + 1) is used in the present work.

46

4 Statistical Analysis

4.1.3 Goodness of fit tests Goodness of fit tests can be useful mathematical tools because they compute a scalar quality criterion for the obtained fit of a probability function. Goodness of fit tests do not prove a hypothetical distribution function. The underlying idea is to falsify that the data is actually distributed in a hypothetical manner. The goodness of fit tests therefore give an answer whether or not the null hypothesis H0 should be rejected. Three quality estimators are given in the following section and their advantages and disadvantages are discussed.

4.1.3.1 Kolmogorov-Smirnov test The Kolmogorov-Smirnov test [46, 86] checks for the maximum distance between an assumed cumulative distribution function F (x) and the empirical distribution function F0 (x) by calculating pKS = sup |F0 (x) − F (x)|.

(4.18)

The pKS value is compared to a tabulated reference value that is given for a specific sample size N and significance level α. The null hypothesis is non rejected, if pKS is smaller than the reference value. Otherwise, the null hypothesis is declined. On the one hand the Kolmogorov-Smirnov test is a simple and easy to implement test, on the other hand it only compares one sample of the whole sample size and therefore lacks convincing significance.

4.1.3.2 Anderson-Darling test The Anderson-Darling test [3] takes the whole sample size into account in order to determine the Anderson-Darling value pAD . The computation for pAD is given by 2i − 1 [ln (F (Yi )) + ln (1 − F (Yn+1−i ))] , n i=1 n

pAD = −N − ∑

(4.19)

with the cumulative distribution function F(Y ) and data points Y1 , . . . ,Yn . pAD has to be lower than a critical tabulated value in order not to reject the null hypothesis. The Anderson-Darling test can be seen as superior in contradistinction to the KolmogorovSmirnov test. On the one hand, the computational costs for the Anderson-Darling test are slightly higher. In return, an increased significance can be achieved because the whole distribution function is taken into consideration, including the critical tails. The outer tails of a probability function can be seen as crucial under certain circumstances. This can be essential for the evaluation of the fracture strength of materials in safety-related environments like pedestrian protection load cases.

4.2 Statistical Evaluation

47

4.1.3.3 Chi-squared test (χ 2 test) The procedure of the chi-squared test [69] differs from the other goodness of fit tests. While the Kolmogorov-Smirnov and the Anderson-Darling test measure distances in the cumulative distribution space, the chi-squared test compares deviations between the observed relative bin frequencies oi = o1 , . . . , ok and the expected bin frequencies ei = e1 , . . . , ek (oi − ei )2 . ei i=1 k

pχ 2 = ∑

(4.20)

2 , with the sigThe calculated pχ 2 value is then compared to a tabulated value χ1−α,k−1 nificance level α and k − 1 degrees of freedom. The null hypothesis is not rejected if 2 holds. The chi-squared test is a goodness of fit test for large data sets with pχ 2 < χ1−α,k−1 n > 50 because each bin should contain at least five samples [68]. Another drawback of the chi-squared test is that the chi-squared distribution function can be deduced from the normal distribution function. The chi-squared distribution function with k degrees of freedom corresponds to the sum of k independent, squared standard normal random variables Z12 + . . . + Zk2 . Hence, the chi-squared test gives preference to the normal distribution in contradiction to the other distribution functions because the normal and the chi-squared distribution functions are similar to each other.

4.2 Statistical Evaluation Figure 4.1 summarizes the results of the quasistatic and dynamic three point bending tests. The mean values with corresponding 95% confidence intervals are plotted. An orientation dependent material behavior can be observed for the quasistatic and dynamic three point bending tests in terms of the maximum forces. The force level decreases with increasing orientation angle likewise for the quasistatic and the dynamic tests. The orientation dependency in terms of the maximum deflections is particularly observed for the quasistatic velocity, whereas the relative differences are very small for the dynamic tests.

48

4 Statistical Analysis

Figure 4.1 Mean values with 95% confidence intervals of the maximum deflections and forces for all four orientations from the quasistatic and dynamic three point bending tests [91]

4.2.1 Quasistatic three point bending tests The Anderson-Darling test is chosen as goodness of fit test for this work because it can be seen as superior to the Kolmogorov-Smirnov and chi-squared test. The Anderson-Darling test sets an upper limit, the threshold limit value pT LV , for a given significance level α and a number of n samples. The calculated pAD value of the probability function fit needs to fall below this TLV in order not to reject the null hypothesis. A significance level of α = 5% and sample number of n = 30 results in a critical value of pT LV = 0.741, according to Dodson [17]. If the Anderson-Darling test returns a pAD value less than 0.741 for a given probability function, the fit will most likely represent the distribution of the data. However, if several probability functions fall below the critical value, the probability function with the lowest pAD value is most probable the candidate that represents the correct distribution of the data. A fit for a probability function fails the test if the Anderson-Darling test returns a pAD value higher than 0.741. The normal distribution, lognormal distribution as well as the two and three parameter Weibull distributions are fitted to each data set and tested against the Anderson-Darling test. The data sets consist of maximum deflections wmax and maximum forces Fmax for four orientation angles, 0◦ , 30◦ , 45◦ and 90◦ and two velocities, quasistatic 0.001 ms−1 and dynamic 4.0 ms−1 .

4.2 Statistical Evaluation

49

The statistical evaluations of the maximum deflections of the quasistatic tests can be seen in Table 4.1 and Figure 4.2. For the 0◦ orientation the normal and lognormal distributions pass the Anderson-Darling test with values of 0.398 and 0.361 respectively. The null hypothesis for the two and three parameter Weibull distributions is refused because their values yield 1.175 and 3.585. From a visual point of view, a clear separation between the normal and lognormal distribution is not possible, although the difference between their pAD values is approximately 10%. The normal, lognormal and three parameter Weibull distribution functions pass the Anderson-Darling test for the 30◦ orientation. The lowest value of 0.368 belongs to the lognormal distribution, which therefore offers the best fit for this orientation angle. The difference between the normal and the lognormal distribution is marginal again. Solely the two parameter Weibull distribution function does not pass the Anderson-Darling test. The statistical evaluation for the 45◦ orientation is similar to the 30◦ . Only the two parameter Weibull distribution exceeds the Anderson-Darling threshold limit of 0.741. With 0.231 and 0.226 the pAD values for the normal and lognormal distribution are very close to each other. The shape of both distribution functions is also very similar. The maximum deflections of the 90◦ orientation are fitted best by a normal or a lognormal distribution. Like before, the difference of their shapes is only marginal. Neither the two nor the three parameter Weibull distributions reach a value below 0.741 and are therefore refused by the Anderson-Darling test. The stochastically scatter of the maximum deflections of the quasistatic three point bending tests are most likely represented by a lognormal distribution function that showed the least pAD values for all orientation angles. However, normal distributions could be used for all orientation angles as well because the observed differences between the corresponding lognormal distributions are not significant. The presence of a large number of outliers could not be observed. The results of the maximum forces from the quasistatic three point bending tests are given in Table 4.2 and Figure 4.3. All PDFs pass the Anderson-Darling test for the 0◦ orientation. The normal distribution has the lowest value of 0.295 and therefore offers the best fit for the data. The value of 0.306 from the lognormal distribution is slightly higher. The appearance of the normal and the lognormal distributions is very similar again. The two parameter Weibull distribution never passed the Anderson-Darling test for any of the orientation angles for the maximum deflections. For the maximum forces on the other hand, the null hypothesis is accepted solely for the 0◦ orientation, although the calculated pAD value of the two parameter Weibull distribution is almost twice as large as the normal distribution’s pAD value. This finding for the 0◦ orientation might indicate a connection between the Weibull distribution and the rupture of glass fibers because the glass fibers are the mechanically dominant part of the material for the 0◦ orientation angle. Increasing the sample size could help to clarify this point. The results of the 30◦ and 45◦ orientations can be summarized in one part. The null

50

4 Statistical Analysis

hypothesis is accepted for the normal and lognormal distributions. The lognormal distributions have the lower values but the curves of the normal and the lognormal distributions look alike. The two and three parameter Weibull distributions do not pass the AndersonDarling test. The pAD values for the 90◦ orientation range from 0.229 for the lognormal distribution to 0.236 for the normal distribution to 0.271 for the three parameter Weibull distribution. The two parameter Weibull distribution does not pass the test for this orientation angle. The differences of the normal and the lognormal probability density functions are marginal for the maximum deflections and forces. Their calculated pAD values are also very close to each other, although the lognormal’s values are slightly lower for the different configurations except for the maximum forces of the 0◦ orientation. It should be noted that negative probabilities exist for the normal distribution because its domain of definition is (−∞, ∞), which is not realistic. The lognormal distribution is only defined for positive x and its application is therefore recommended from a practical point of view. The Weibull probability plots for the two and three parameter Weibull distributions of the maximum deflections and forces are shown in the appendix, see Section A.5. The best fit PDFs for the maximum deflections and forces are depict in Figure 4.4 with corresponding parameters and pAD values shown in Table 4.3 and Table 4.4 respectively. The probability functions of the maximum deflections in Figure 4.4 show minor overlaps at the tails except for the 45◦ and 90◦ orientations, which show major overlaps. It could be concluded from the confidence interval plots that a significant difference between these two orientation angles might exist. However, these overlaps can only be observed when plotting the probability density functions next to each other. On the other hand, the differences for the maximum forces with respect to the orientation angle are significant and a specific force spectrum correlates to a specific orientation angle. The orientation dependency in terms of maximum forces is very pronounced therefore. The 0◦ orientation shows the largest scatter and the level of scatter decreases with increasing orientation angle. This can be observed for the values of the standard deviations that follow a decreasing pattern from the 0◦ to the 90◦ orientation, see also Table 4.2. This could be interpreted as an influence of the glass fibers on the mechanical properties of the material. The more glass fibers are aligned in stress direction, the larger the scatter of the probability distribution function.

4.2 Statistical Evaluation

51

Table 4.1 Normal (N), lognormal (LN), two parameter Weibull (W2P) and three parameter Weibull (W3P) goodness of fit tests for the maximum deflection of the quasistatic three point bending tests

Orientation PDF μ/μg σ /σg AD value AD TLV Reject H0 PDF a b c R2 AD value AD TLV Reject H0

0◦ N 1.953 0.062 0.398 0.741 no W2P 0 1.977 27.509 0.886 1.175 0.741 yes

30◦ LN 1.952 1.032 0.361 0.741 no W3P 1.798 0.143 1.799 0.986 3.585 0.741 yes

N 2.337 0.072 0.376 0.741 no W2P 0 2.354 34.832 0.929 1.22 0.741 yes

45◦ LN 2.337 1.032 0.368 0.741 no W3P 2.122 0.203 2.351 0.992 0.43 0.741 no

N 2.703 0.07 0.231 0.741 no W2P 0 2.715 31.245 0.978 0.916 0.741 yes

90◦ LN 2.702 1.027 0.226 0.741 no W3P 2.409 0.292 3.264 0.994 0.414 0.741 no

N 2.969 0.148 0.309 0.741 no W2P 0 3.043 20.809 0.92 0.872 0.741 yes

LN 2.967 1.052 0.274 0.741 no W3P 2.492 0.489 2.787 0.99 0.82 0.741 yes

Figure 4.2 Comparison of all PDFs for the maximum deflections of the quasistatic three point bending tests

52

4 Statistical Analysis

Table 4.2 Normal (N), lognormal (LN), two parameter Weibull (W2P) and three parameter Weibull (W3P) goodness of fit tests for the maximum force of the quasistatic three point bending tests

Orientation PDF μ/μg σ /σg AD value AD TLV Reject H0 PDF a b c R2 AD value AD TLV Reject H0

0◦ N 349.24 6.945 0.295 0.741 no W2P 0 350.11 49.29 0.959 0.586 0.741 no

30◦ LN 349.16 1.015 0.306 0.741 no W3P 324.77 23.28 2.85 0.996 0.424 0.741 no

N 293.31 4.186 0.534 0.741 no W2P 0 294.73 73.22 0.908 1.15 0.741 yes

LN 293.29 1.027 0.515 0.741 no W3P 281.71 10.95 2.11 0.999 3.26 0.741 yes

45◦ N 255.57 3.788 0.456 0.741 no W2P 0 257.95 63.36 0.816 1.853 0.741 yes

LN 255.55 1.015 0.427 0.741 no W3P 247.38 7.42 1.46 0.988 3.653 0.741 yes

90◦ N 225.11 3.462 0.236 0.741 no W2P 0 226.33 75.14 0.938 1.027 0.741 yes

LN 225.09 1.016 0.229 0.741 no W3P 212.72 12.51 3.4 0.992 0.271 0.741 no

Figure 4.3 Comparison of all PDFs for the maximum deflections of the quasistatic three point bending tests

4.2 Statistical Evaluation

Table 4.3

53

Best fit PDFs for the maximum deflection wmax of the quasistatic three point bending tests

Orientation PDF Parameters AD value

Table 4.4

0◦ Lognormal μg = 1.952 σg = 1.032 0.361

30◦ Lognormal μg = 2.337 σg = 1.032 0.368

45◦ Lognormal μg = 2.702 σg = 1.027 0.226

90◦ Lognormal μg = 2.967 σg = 1.052 0.274

Best fit PDFs for the maximum force Fmax of the quasistatic three point bending tests

Orientation PDF Parameters AD value

Figure 4.4 ing tests

0◦ Normal μ = 349.24 σ = 6.945 0.295

30◦ Lognormal μg = 293.29 σg = 1.015 0.515

45◦ Lognormal μg = 255.55 σg = 1.015 0.427

90◦ Lognormal μg = 225.09 σg = 1.016 0.229

Best fit PDFs for the maximum deflections and forces of the quasistatic three point bend-

54

4 Statistical Analysis

4.2.2 Dynamic three point bending tests The results of the maximum deflections from the dynamic three point bending tests are shown in Table 4.5 and Figure 4.5. None of the chosen PDFs passes the Anderson-Darling test for the 0◦ orientation. It is noteworthy that approximately 2/3 of the values concentrate in the first two bars of the histogram. None of the applied PDFs is able to represent this behavior sufficiently. The lognormal distribution reaches the lowest pAD value with 0.997 followed by the normal distribution with 1.079. The maximum deflections of the 30◦ orientation can be represented well by the normal distribution with a value of 0.398, followed by the lognormal and three parameter Weibull distributions, with values of 0.453 and 0.514 respectively. The data is distributed over a broader spectrum in contrast to the 0◦ orientation. The difference between the normal and the lognormal distribution is not significant, which is similar to the observations of the quasistatic tests. The 45◦ and 90◦ orientations are similar to the 0◦ orientation, and therefore discussed in one part. The data mostly concentrates in a narrow section of the whole spectrum. The PDFs are not able to adequately represent the distributional pattern of the data. The best fit for the 45◦ orientation is the normal distribution with a value of 1.101 while the lognormal distribution fits best for the 90◦ orientation, with a value of 1.642. However, the null hypothesis for both PDFs are rejected by the Anderson-Darling test because their values are not below the threshold limit value of 0.741. Details of the statistical evaluations of the maximum forces of the dynamic tests are shown in Table 4.6, with diagrams of the PDFs depict in Figure 4.6. All corresponding pAD values from the 0◦ orientation PDFs exceed the threshold limit value by a factor of two to three. The PDFs look similar to each other, except for minor deviation. It appears that there is a gap of approximately 10 N in the list of force values. A separation can be observed between the majority of the results on the larger force scale, with Fmax > 540 N and six data points that fall below the 530 N mark. The data cannot be labeled as a cluster of outliers because this cluster represents 1/5 of all samples. The skewed bell shape of the Weibull distribution is able to outline the lower cluster of data points appropriately. The resulting values of the two and three parameter Weibull distributions point in that direction because they are lower than the values of the normal and lognormal distribution. However, the null hypotheses are rejected by the Anderson-Darling test technically. It may be possible that the data could be fitted better by a different type of Weibull distribution, e.g. a multimodal Weibull distribution. Increasing the sample size could also help to clarify this point. A formation of clusters cannot be observed for the 30◦ orientation. The normal distribution achieves the best pAD value of 0.264. The lognormal and the three parameter Weibull distribution reach values of 0.311 and 0.739 and are also accepted by the Anderson-Darling test. The null hypothesis of the two parameter Weibull distribution is rejected because its value is 1.244. The maximum force values of the 45◦ orientation are also broadly distributed over the

4.2 Statistical Evaluation

55

spectrum. Clustering of the data cannot be observed. The value of the normal distribution shows the lowest value with 0.559, followed by the lognormal distribution function, with a value of 0.728. The two and three parameter Weibull distributions do not pass the Anderson-Darling test. The 90◦ orientation shows similarities to the 0◦ orientation. Two clusters of data points can be observed. The first, smaller cluster consists of seven samples that reach a maximum force of 300 N. The second cluster contains samples with a maximum force above 315 N. A total number of 22 data points are in this second cluster. The remaining data point is located between both clusters at a maximum force of 305 N. Approximately 80% of the data is located in the upper cluster and 20% of the data is in the lower cluster at the 0◦ and 90◦ orientation angles. The upper cluster of the 0◦ orientation consists of 24 and the lower of 6 values. A similar relationship can also be observed for the 90◦ orientation, if the single data point between the two clusters is counted to the lower cluster. It is possible that a correlation between the forming of clusters at certain orientation angles at high strain rates exists. The orientation angle alone is not significant for the cluster forming. The strain rate must play a role too because the formation of clusters could not be observed for the quasistatic tests. The forming of clusters could be connected to brittle rupture because the glass fibers, which are the mechanically dominant part for the 0◦ orientation, typically show a brittle type of rupture and the strain rate dependent matrix behavior leads to brittle rupture at the 90◦ orientation as well. Additional three point bending tests could resolve if the clusters remain or disappear with an increasing sample size. The Weibull probability plots for the two and three parameter Weibull distributions of the dynamic three point bending tests are shown in the appendix, see Section A.5. The best fit PDFs for the dynamic maximum deflections and forces are shown in Figure 4.7 with corresponding parameters and pAD values given in Table 4.7 and Table 4.8 respectively. The normal and lognormal distribution functions are similar to each other, which could also be observed for the quasistatic evaluations. As a pragmatic approach, solely the lognormal distribution could be used to model the data of the dynamic tests. The relative differences between the maximum deflections for the dynamic tests diminish and an assigning of specific maximum deflections to individual orientation angles is not reliably possible. The degree of orientation dependent rupture behavior in terms of the maximum deflections minimizes while it is retained for the maximum forces. A separation between the maximum forces with respect to the orientation angle is still possible, although minor overlaps can be observed.

56

4 Statistical Analysis

Table 4.5 Normal (N), lognormal (LN), two parameter Weibull (W2P) and three parameter Weibull (W3P) goodness of fit tests for the maximum deflection of the dynamic three point bending tests

Orientation PDF μ/μg σ /σg AD value AD TLV Reject H0 PDF a b c R2 AD value AD TLV Reject H0

Figure 4.5 tests

0◦ N 2.331 0.075 1.079 0.741 yes W2P 0 2.359 25.232 0.872 1.796 0.741 yes

30◦ LN 2.331 1.032 0.997 0.741 yes W3P 2.213 0.099 1.092 0.999 2.824 0.741 yes

N 2.516 0.141 0.398 0.741 no W2P 0 2.541 19.735 0.972 1.209 0.741 yes

45◦ LN 2.512 1.059 0.453 0.741 no W3P 1.945 0.57 3.693 0.995 0.514 0.741 no

N 2.634 0.166 1.101 0.741 yes W2P 0 2.653 14.188 0.964 1.674 0.741 yes

LN 2.629 1.069 1.314 0.741 yes W3P 0.326 2.324 12.349 0.964 1.626 0.741 yes

90◦ N 2.308 0.177 1.776 0.741 yes W2P 0 2.387 13.831 0.792 2.207 0.741 yes

LN 2.304 1.078 1.642 0.741 yes W3P 2.003 0.216 0.99 0.995 6.929 0.741 yes

Comparison of all PDFs for the maximum deflections of the dynamic three point bending

4.2 Statistical Evaluation

57

Table 4.6 Normal (N) and lognormal (LN), two parameter Weibull (W2P) and three parameter Weibull (W3P) goodness of fit tests for the maximum force of the dynamic three point bending tests

Orientation PDF μ/μg σ /σg AD value AD TLV Reject H0 PDF a b c R2 AD value AD TLV Reject H0

Figure 4.6

0◦ N 552.93 23.195 1.868 0.741 yes W2P 0 555.28 23.49 0.963 1.491 0.741 yes

30◦ LN 552.42 1.045 2.006 0.741 yes W3P 152.18 403.01 16.67 0.96 1.535 0.741 yes

N 444.71 18.379 0.264 0.741 no W2P 0 448.71 21.84 0.959 1.244 0.741 yes

LN 444.34 1.043 0.311 0.741 no W3P 335.59 110.59 5.102 0.968 0.739 0.741 no

45◦ N 384.85 24.466 0.559 0.741 no W2P 0 390.75 16.18 0.983 1.102 0.741 yes

LN 384.05 1.069 0.728 0.741 no W3P 172.61 216.79 8.523 0.985 1.485 0.741 yes

90◦ N 320.67 26.56 2.144 0.741 yes W2P 0 322.02 12.38 0.954 1.416 0.741 yes

LN 319.5 1.093 2.358 0.741 yes W3P 0 322.02 12.38 0.954 1.416 0.741 yes

Comparison of all PDFs for the maximum forces of the dynamic three point bending tests

58

Table 4.7

4 Statistical Analysis

Best fit PDFs the for maximum deflection wmax of the dynamic three point bending tests

Orientation PDF Parameters AD value

Table 4.8

30◦ Normal μ = 2.516 σ = 0.141 0.398

45◦ Normal μ = 2.634 σ = 0.167 1.1

90◦ Lognormal μg = 2.304 σg = 1.078 1.642

Best fit PDFs for the maximum force Fmax of the dynamic three point bending tests

Orientation PDF Parameters AD value

Figure 4.7 tests

0◦ Lognormal μg = 2.331 σg = 1.032 0.997

0◦ Weibull 2P b = 555.28 c = 23.497 1.491

30◦ Normal μ = 444.71 σ = 18.379 0.264

45◦ Normal μ = 384.85 σ = 24.466 0.559

90◦ Weibull 2P b = 322.02 c = 12.38 1.416

Best fit PDFs for the maximum deflections and forces of the dynamic three point bending

5 Material Modeling The finite element method is introduced in this section. Further, the constitutive equations for the orientation dependent elastic plastic material model are outlined. The contact algorithms of the numerical simulations are also described in this part because the three point bending test involves a specimen that is penetrated by an impactor, which requires a contact definition in the simulation.

5.1 Explicit Finite Element Method The finite element method is a numerical tool to solve partial differential equations. Structural mechanics is one field of application. The geometry of a body is discretized by a finite number of elements that are connected with nodal points. Specific features, like density and constitutive properties, are assigned to the elements and boundary conditions, like external forces or bearings, are attached. The structural problem is then solved by means of numerical algorithms, which solve the set of differential equations by means of explicit or implicit solution procedures. Explicit time integration can be used for short simulation durations on a milliseconds scale. The computational cost for each time step is small and the simulation runs stable until the end. On the other hand, the explicit time integration does not check if the system is in a state of equilibrium during the calculation. Quasistatic simulations are commonly simulated implicitly. The equilibrium of the system is ensured at every time step during the simulation. The computational cost for every time step is high. In practice, quasistatic loading cases, like the RCAR insurance classification 10◦ structure test, are also run with explicit time integration, due to the lower computational costs. The load case simulation is therefore run with time scaling and without viscous effects like damping or strain rate dependent material behavior. Time scaling can be applied when the mechanical response in the simulation refers to a time-independent material model. The quasistatic simulations of this work are also run with explicit time integration and time scaling. This way it is ensured that the method can be applied to explicitly run quasistatic load cases as well. Further details on the finite element method can be found in e.g. [6]. The numerical simulations in this work are performed with the commercial FE solver LS-DYNA. Among others, LS-DYNA is used for crash simulations for its explicit time integration solver and its built-in contact-impact algorithm. © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8_5

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5 Material Modeling

5.2 Constitutive Equations The constitutive equations describe how stresses and strains in a continuous body are linked to each other. The one dimensional equations from Section 3.2.2.1 are now expanded to three dimensional Euclidean space, which implies an orthonormal set of uniform basis vectors. All tensors are given in matrix notation if not stated otherwise.

5.2.1 Material theory The second order Cauchy stress tensor is given as ⎡

σxx σ = ⎣ σxy σxz

σxy σyy σyz

⎤ ⎡ σx σxz σyz ⎦ = ⎣ σxy σzz σxz

σxy σy σyz

⎤ σxz σyz ⎦ , σz

(5.1)

with normal stresses σxx , σyy and σzz and shear angles σxy , σxz and σyz . The second order infinitesimal strain tensor is given as ⎡

εxx ε = ⎣ 12 γxy 1 2 γxz

1 2 γxy εyy 1 2 γyz

1 2 γxz 1 2 γyz

εzz

⎤

⎡

εx ⎦ = ⎣ 1 γxy 2 1 2 γxz

1 2 γxy εy 1 2 γyz

1 2 γxz 1 2 γyz

⎤ ⎦,

(5.2)

εz

with normal strains εxx , εyy and εzz and shear strains γxy , γxz and γyz .

5.2.1.1 Elasticity In terms of linear elastic theory, stresses and strains are delineated by Hooke’s law, which is given by σ = Cε,

(5.3)

with the fourth order stiffness tensor C. C has 81 independent constants in its generalized 3D form. The number of independent constants reduces to 21 by making use of symmetry characteristics and under the assumption that the elastic deformation of a body can be

5.2 Constitutive Equations

deduced from given as ⎡ σx ⎢ σ ⎢ y ⎢ ⎢ σz ⎢ ⎢ σyz ⎢ ⎣ σxz σxy

61

a specific elastic potential [8]. In matrix notation by Voigt, Eq. (5.3) is ⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

C11 C12 C13 C14 C15 C16

C12 C22 C23 C24 C25 C26

C13 C23 C33 C34 C35 C36

C14 C24 C34 C44 C45 C46

C15 C25 C35 C45 C55 C56

C16 C26 C36 C46 C56 C66

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

εx εy εz 2εyz 2εxz 2εxy

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(5.4)

Further simplifications of C can be achieved when studying orthotropic materials, which have three orthogonal planes of symmetry. The number of elastic constants can then be reduced to 9 and Eq. (5.4) yields ⎤ ⎡ ⎤⎡ ⎤ ⎡ 0 0 C11 C12 C13 0 εx σx ⎢ ⎥ ⎢ σ ⎥ ⎢ C 0 0 ⎥ ⎥ ⎢ εy ⎥ ⎢ y ⎥ ⎢ 12 C22 C23 0 ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 0 ⎥ ⎢ εz ⎥ ⎢ σz ⎥ ⎢ C13 C23 C33 0 (5.5) ⎥=⎢ ⎥⎢ ⎥. ⎢ ⎢ σyz ⎥ ⎢ 0 0 ⎥ ⎢ 2εyz ⎥ 0 0 C44 0 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎣ σxz ⎦ ⎣ 0 0 0 0 C55 0 ⎦ ⎣ 2εxz ⎦ σxy 2εxy 0 0 0 0 0 C66 For isotropic linear elastic material behavior the number of elastic constants creases to two and Eq. (5.5) can be expressed as ⎡ ⎤ ⎡ 0 0 0 σx C11 C12 C12 ⎢ σ ⎥ ⎢ C 0 0 0 ⎢ y ⎥ ⎢ 12 C11 C12 ⎢ ⎥ ⎢ 0 0 0 ⎢ σz ⎥ ⎢ C12 C12 C11 ⎢ ⎥=⎢ 1 ⎢ σyz ⎥ ⎢ 0 (C −C ) 0 0 0 0 11 12 2 ⎢ ⎥ ⎢ 1 ⎣ σxz ⎦ ⎣ 0 (C −C ) 0 0 0 0 11 12 2 1 σxy 0 0 0 0 0 2 (C11 −C12 )

further de⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

⎤ εx εy ⎥ ⎥ ⎥ εz ⎥ ⎥ 2εyz ⎥ ⎥ 2εxz ⎦ 2εxy (5.6)

with the two material parameters C11 and C12 given as C11 =

E (1 − ν) , (1 + ν) (1 − 2ν)

C12 =

Eν . (1 + ν) (1 − 2ν)

(5.7)

Amorphous substances like glass, bitumen or wax show isotropic material behavior [98]. Glass fiber reinforced polypropylene is a candidate for the orthotropic material behavior.

62

5 Material Modeling

5.2.1.2 Plasticity Permanent deformations ε pl = ε − εel can be observed after unloading a tensile specimen that was previously loaded beyond its yield stress σy . The concept of the additive decomposing of the total strain ε into an elastic part εel and a plastic part ε pl is shown in Figure 5.1.

Figure 5.1 Schematic stress strain diagram of a tensile test. Plastic deformation ε pl can be observed when the tensile bar is unloaded after it had been loaded beyond its yield stress σy , from [30].

This relation can be transferred to the general 3D case for small deformations |ε| < 5%, where the second order infinitesimal strain tensor ε is decomposed into an elastic strain tensor εel and a plastic strain tensor ε pl ε = εel + ε pl ,

|ε| ≤ 5%.

(5.8)

Plastic yielding can be observed for a material when the following equation is satisfied F (σ) − σy ≤ 0,

(5.9)

with the yield function F (σ) that is given as a function of the Cauchy stress tensor. Two yield functions are discussed in detail. The von Mises yield criterion [57] for isotropic materials and the Hill yield criterion [39] for orthotropic materials. Both yield criteria are phenomenological plasticity models and do not take processes into account that happen on an atomic scale and are not considering thermal effects.

5.2 Constitutive Equations

63

The von Mises criterion describes yielding of isotropic materials on a macroscopic scale. This yield function expresses a multi axial state of stress as a scalar value. The underlying assumption of this criterion is that the body deforms plastically while maintaining a constant volume. This yield criterion is associated to metals but is also used for polymers, although polymers do not show isochoric plastic behavior [90]. There are different ways to mathematically express the yield function. It can be given as a function of the Cauchy stress tensor components in the general case   2 +σ2 +σ2 , (5.10) σvM = σx2 + σy2 + σz2 − σx σy − σx σz − σy σz + σxy xz yz or in principal stress tensor components    1 σvM = (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 . 2 On the assumption of plane stress conditions Eq. (5.11) yields  σvM = σ12 + σ22 − σ1 σ2 .

(5.11)

(5.12)

Substituting Eq. (5.12) into Eq. (5.9) leads to an expression that defines if a material is currently in an elastic or plastic state of deformation  σ12 + σ22 − σ1 σ2 − σy ≤ 0. (5.13) The material undergoes plastic deformation as soon as Eq. (5.13) is satisfied. σy is depending on the material and does exceed values of 60-100 MPa for fiber reinforced polymers, depending on the glass fiber content and the orientation angle. For mild steel σy can reach 235 MPa. The von Mises model is usually associated to metals that do not show a significant difference between tension and compression loading. However, this is not the case for polymers [7, 89]. A drawback of this method is the lack of flexibility because of its fixed yield function. A generalized yield function as proposed in [45] would be more feasible when studying polymer materials numerically. On the downside, the increasing flexibility of the yield function requires an increase of the experimental effort in order to calibrate the new material model. ∗ M AT _024 and ∗ M AT _123, which are studied by Sygusch and Lauterbach [90] and Sygusch et al. [92], are two materials models in L S -DYNA that have the von Mises yield function implemented. The correlation of the material models for each orientation angle is good but a drawback of this method is that one needs four material cards in order to model the four orientation angles. The Hill yield criterion can be used to model orthotropic plastic material behavior. Physically this yield function is closer to the actual mechanical behavior of fiber reinforced

64

5 Material Modeling

polymers than the von Mises yield function. The Hill criterion for coinciding orthotropic and principal material directions is given by 2 2 2 + 2Mσ13 + 2Nσ12 = 1, (5.14) F (σ22 − σ33 )2 + G (σ33 − σ11 )2 + H (σ11 − σ22 )2 + 2Lσ23

with constants F, G, H, L, M, N that depend on the tensile yield stresses X,Y, Z and the shear yield stresses R, S, T . The parameters have to be determined by mechanical tests or by reverse engineering 1 1 1 1 = G + H, 2F = 2 + 2 − 2 , 2 X Y Z X 1 1 1 1 = H + F, 2G = 2 + 2 − 2 , 2 Y Z X Y 1 1 1 1 = F + G, 2H = 2 + 2 − 2 , Z2 X Y Z 1 1 1 2M = 2 , 2N = 2 . 2H = 2 , R S T

(5.15)

Under plane stress conditions Eq. (5.14) yields 2 2 2 (G + H) σ11 − 2Hσ11 σ22 + (H + F) σ22 + 2Nσ12 = 1.

(5.16)

Another way of expressing the Hill yield criterion is by using the Lankford parameter r [47]. It is defined as the ratio of the transversal and the thickness strains in a tensile test r=

ε22 , ε33

r=

ln ww0 ln tt0

,

with the principal stretches λi = λ1 =

l , l0

λ2 =

w , w0

r= ai a0i

ln λ2 , ln λ3

(5.17)

given as λ3 =

t . t0

(5.18)

The mechanism is illustrated in Figure 5.2. If r is greater than one the resistance toward transversal straining is greater than against thinning straining. If it is below one the transversal strains increase faster than the thinning strains. The mechanical material behavior is isotropic if the Lankford parameter is equal to unity. On the assumption of isochoric plastic deformation Eq. (5.17) can also be rewritten in the form of r=−

ε22 , ε11 + ε22

r=

− ln ww0 ln ll0 + ln ww0

,

r=

− ln λ2 . ln λ1 + ln λ2

(5.19)

5.2 Constitutive Equations

One can also express Eq. (5.16) as   1 1 1 r0 + r90 1 2 1 2 σ11 − + 2− 2 σ11 σ22 + 2 σ22 2 2 X X Y X r90 (1 + r0 ) r90 1 r0 + r90 2 + 2 = 1, (2r45 + 1) σ12 X r90 (1 + r0 )

65

(5.20)

by defining r-values r0 , r45 and r90 for respectively 0◦ , 45◦ and 90◦ orientation angles [14].

Figure 5.2

Geometry of a section from a tensile specimen: (a) before and (b) after deformation [5]

5.2.2 Orthotropic modeling Eq. (5.20) is implemented in L S -DYNA as material model ∗ M AT _157. The orthotropic linear elastic material behavior, as shown in Eq. (5.5), is also incorporated for this material card. This material model has been successfully used by other authors as well [38, 48]. The determination of the constitutive parameters is performed iteratively. Firstly, the elastic properties are generated by applying micro mechanical homogenization methods. The findings of Section 3.1 help to generate the stiffness tensor that is required to model the orientation dependent elastic behavior. Secondly, the inelastic behavior is determined by a reverse engineering process [90, 93]. Different micromechanically motivated approaches exist to generate the stiffness tensor parameters, see also [35, 58, 59]. Mori Tanaka homogenization [58] with orientation averaging according to Advani and Tucker [2] is used for the computation of the stiffness tensor in this work. The matrix’s density ρm , Young’s modulus Em and Poisson’s ratio νm and likewise the glass fiber’s density ρ f , Young’s modulus E f and Poisson’s ratio ν f are used as input parameters. The volume fractions of the matrix and glass fibers serve as

66

5 Material Modeling

weighting factors in the model. The volume fraction is determined as ϕ = 14.4% in Section 3.1.1. The obtained length to diameter ratio l/d of 44.62 also plays an important role because the embedded glass fibers are represented as ellipsoidal inclusions, as proposed by Eshelby [20]. The fourth order Eshelby tensor E, with entries according to Mura [60], is computed in the first step. Secondly, the fourth order strain concentration tensor A(SC) for a dilute fiber concentration is calculated by −1   , A(SC) = I + E : C−1 m : C f − Cm

(5.21)

with the fourth order identity tensor I [11] and the isotropic stiffness tensors of the matrix Cm and the glass fibers C f , according to Eq. (5.6). The : operator in Eq. (5.22) is called double contraction [41] and maps a fourth order tensor on another fourth order tensor. The fourth order Mori Tanaka strain concentration tensor A(MT ) is computed in the next step  −1 A(MT ) = A(SC) : (1 − ϕ) I + A(SC) .

(5.22)

The unidirectional homogeneous composite stiffness tensor C(UD) is computed with help of the Mori Tanaka strain concentration tensor A(MT ) by  C(UD) = Cm + ϕ C f − Cm : A(SC) ,

(5.23)

and is the basis for the orientation averaged stiffness tensor C(OA) of the fiber reinforced polymer. C(OA) is calculated according to [2] by combining the fiber orientation tensor A from Eq. (3.3) and the unidirectional homogeneous composite stiffness tensor C(UD) in the form of  (OA) Ci jkl = B1 Ai jkl + B2 (Ai j δkl ) + B3 Aik δ jl + Ail δ jk + A jl δik + A jk δil (5.24)  + B4 (δi j δkl ) + B5 δik δ jl + δil δ jk , (OA)

with the stiffness tensor Ci jkl in index notation, the Kronecker delta δi j , the second order orientation tensor Ai j and fourth order orientation tensor Ai jkl [2] in index notation. All

5.2 Constitutive Equations

67

second and fourth order tensors are given with an orthonormal set of uniform basis vectors ( e#»i ⊗ e#»j ), respectively ( e#»i ⊗ e#»j ⊗ e#»k ⊗ e#»l ). The Bi constants are calculated by (UD)

(UD)

(UD)

(UD)

(UD)

(UD)

B1 = C1111 +C2222 − 2C1122 − 4C1212 , B2 = C1122 −C2233 ,  1  (UD) (UD) (UD) C2233 −C2222 , B3 = C1212 + 2 (UD) B4 = C2233 ,  1  (UD) (UD) B5 = C2222 −C2233 , 2

(5.25)

according to [2]. The fiber orientation distribution is idealized as one homogeneous layer across the thickness. This may be suitable for coupon sized geometries but could be an issue for larger sized structures. The principal components Axx , Ayy and Azz of the orientation tensor are set to correspond average values across the thickness. This results in the values Axx = 0.64, Ayy = 0.31 and Azz = 0.05 for the removal position five, where all samples of the conducted experiments are taken from. The micromechanically assembled components of the orientation averaged stiffness tensor C(OA) are incorporated into the material model as ⎡ ⎤ 8.78 3.59 3.2 0 0 0 ⎢ 3.59 5.63 3.2 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 ⎥ ⎢ 3.2 3.2 4.91 (OA) =⎢ C (5.26) ⎥ (GPa) . ⎢ 0 0 0 1.39 0 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 0 0.91 0 ⎦ 0 0 0 0 0 0.98 The r values are determined by a sequential reverse engineering process. The 0◦ , 45◦ and 90◦ orientation angles are used to calibrate the inelastic material behavior. The 0◦ orientation is modeled without consideration of the other orientations in a first step because it serves as a basis function for the other orientations. The 45◦ and 90◦ orientations are modified from this basis function by setting the r45 and r90 values in a simultaneous optimization process. The r values are optimized iteratively in order to fit the nonlinear curvature of the experiments that lead to r values of r0 = 1.0, r45 = 0.5 and r90 = 0.21. The material model is validated by testing against the experimental curve of the 30◦ orientation. An advantage of this method is that orientation dependent inelastic material behavior can be incorporated by only calibrating a few parameters. This approach requires the parameters of the analytical function plus the three r values. However, the incorporated curves are self-similar. This could be an issue if the inelastic behavior shows significant differences in the curve shapes for the different orientations. Yet, this is not the case for this material. #» The local element orientation vector b is determined with help of the global orientation

68

5 Material Modeling

#» vector V = (V1 ,V2 ,V3 ) and the orientation angle β [62], which is illustrated in Figure 5.3. ՜ 

՜  ՜ ՜ ՜ „ ൌ š

Ⱦ

Figure 5.3 Concept of the local shell element orientation. The local shell element orientation vector #» #» #» b = n × V is the starting rotation point of the element orientation angle β .

This determination of the element orientations is suitable for analyzing 2D problems, where the specimen’s geometry is flat and has no complex curvatures. For larger structures, however, the local fiber orientation can vary, which studies on the utilized plaque have shown in Section 3.1.2. The element orientations need to be mapped locally for three dimensional structures on the other hand, as carried out by Lauterbach et al. [48]. Internal variables are used to store the orientation of each integration point from each element that enables to incorporate the physical fiber orientations of a mold filling simulations in a three dimensional structure. Every integration point of an element has its individual constitutive properties and its own orientation in that case. These properties are dependent on the corresponding glass fiber #» content and orientation angle from the process simulation, see also [51]. V and β are obsolete in that case.

5.2.3 Rupture condition The phenomenological maximum principal strain criterion is used to model simulation rupture in this work. This criterion takes the total straining of a body into account and does not differentiate between tension  compression load cases. The rupture surface  and  represents a cube with edge lengths 2 ε f  in principal strain space. In principal directions, the rupture condition is given by ε1 = ε f

if

ε1 > 0

and

ε1 = −ε f

if

ε1 < 0,

ε2 = ε f

if

ε2 > 0

and

ε2 = −ε f

if

ε2 < 0,

ε3 = ε f

if

ε3 > 0

and

ε3 = −ε f

if

ε3 < 0.

(5.27)

This rupture condition can be activated in L S -DYNA by using a ∗ M AT _A DD _E ROSION material card. It is not a standalone material model but has to

5.3 Contact Modeling

69

be used in combination with an existing material card like the orthotropic material model from Section 5.2.2. A new rupture model is calibrated for every orientation angle for the conducted simulations in this work. Another approach to the rupture behavior could be the use of a stress based criterion like the maximum principal stress condition. Stress based criteria are usually associated to rupture of brittle materials [31]. However, minor oscillations cannot be excluded in the conducted simulations because the simulations are run with explicit time integration. These oscillations are a drawback for the application of a stress based rupture condition. Presimulations have shown that the oscillation’s amplitude serves as an offset to the current stress value in the FE model that leads to inaccurate rupture parameter estimation. This is not an issue for a strain based rupture condition, which increases the robustness of the rupture estimation and is preferred therefore.

5.3 Contact Modeling The contact algorithms of L S -DYNA can be separated into groups by their formalism or mechanism. Formalism means that the intersection search between e.g. two contact partners happens either solely from one side or from both sides at the same time. Only these one-way or two-way contacts are investigated in this work, although other formalisms are implemented as well. The full list of contact algorithms can be found in [61]. The contact mechanism determines the way internal contact forces are computed and applied to the corresponding parts. From the implemented kinematic constrained method, the distributed parameter method and the penalty method, only the latter is further investigated. The two contact algorithms used in this work are both from the class of penalty method algorithms. In the penalty method, a force that is proportional to the depth of penetration is applied between the contact partners on intersection. These forces act as an additional barrier and can be interpreted as linear springs. The calculation of these springs is depending on the chosen contact algorithm. The nodes-to-surface contact belongs to the group of one-way contacts. Search for intersection between parts is only performed from one side, the so called slave part of the contact definition. A penetration check on behalf of the master’s part is not performed. A benefit of searching from one side only is the reduced calculation time. A drawback of this method is that it may be possible for master sections to move into the slave part without detection by the algorithm for contact parts with different mesh densities. However, this is not the case for the conducted three point bending simulations of this work.

70

5 Material Modeling

A penetration check is performed by computing the orthogonal distance between the slave #» r . The node ns at location t and the point of contact on the master segment at location #» principle is shown in Figure 5.4.

#» Figure 5.4 Schematic depiction of a slave node ns with location vector t in front of a master segment. #» The point of contact on the master segment is at location vector r . The master segment is shown with parameterized coordinates ξ , η, from [63].

The contact point on the master segment is parameterized with local coordinates ξ , η  ∂ #» r (ξ , η)  #» #» · t − r (ξ , η) = 0, ∂ξ  ∂ #» r (ξ , η)  #» #» · t − r (ξ , η) = 0, ∂η

(5.28)

and the depth of penetration is estimated the inner product of the master  #» by calculating  r (ξ , η) segment’s normal #» n m with the vector t − #»  #»  r (ξ , η) . l = #» n m · t − #»

(5.29)

Nodes of the slave part are penetrating the surface of the master part if l < 0. The calculation routine of the contact forces is invoked subsequently and forces are placed between the slave nodes and the contact point on the master surface. The contact force on the slave node is then calculated by #» n m, F c = −lk #»

if l < 0

(5.30)

with proportional factor k k=

fs AKm , dmin

(5.31)

5.3 Contact Modeling

71

which is depending on the scale parameter fs , the master’s element area A, the compression modulus Km of the master part material and the diagonal of the smallest contact shell element dmin . The adjacent forces of the master nodes are calculated by #» F 1c = (1 − ξ ) k #» g, #» #» F = ξk g ,

(5.32)

2c

as depict in Figure 5.5. ՜

F1c

՜

F2c

n՜ m ՜

Fc

1

2

g՜

Ɍ Figure 5.5 Force calculation for slave and master nodes in the penalty method contact algorithm of LS-DYNA [4]

The mortar contact is a contact algorithm that focuses on accuracy and robustness [71, 72]. However, the increased accuracy requires a higher CPU time. The mortar contact is a two-way contact algorithm and was initially implemented for implicit calculations but can be used for explicit simulations as well. The mortar contact calculates normal and tangential stresses and deduces the nodal forces from the stresses instead of calculating the forces directly. The contact normal stress σn is given by   d σn = αβs βm εEs f , (5.33) εdcs with scaling factors α and ε, stiffness scale factors βs and βm for the slave segment and the master segment respectively, Young’s modulus of the slave segment Es , characteristic length of the slave segment dcs and function f (x) that is defined as ⎧ dmax 1 2 ⎪ x < 2εd s ⎨ 4x c f (x) = (5.34) cubic function, which depends ⎪ dmax ⎩ on the scaling parameter IGAP 2εd s ≤ x, c

with maximum contact penetration dmax . The tangential stresses are then derived from the normal stresses by Coulomb’s friction law [63]. The mechanical response of the mortar contact is softer in contrast to the nodes-to-surface contact because it uses the Young’s modulus of the slave part for its calculations, whereas

72

5 Material Modeling

the nodes-to-surface contact uses the bulk modulus of the master part. The contact force in the mortar contact increases smoothly because the underlying function contributes to a continuous increment. The scale factors are set to default values in this work.

6 Numerical Results This chapter covers the results of the conducted simulations of the three point bending tests. The chapter is divided into two sections, namely the deterministic and the stochastic simulations. The calibration of the material model is realized with the deterministic simulations, which are the basis for the stochastic simulations. The material parameters and the parameters of the test setup are fixed for the deterministic simulations. The geometry parameters of the specimen, as well as material and test setup parameters are varied in the stochastic simulations. The objective of the stochastic simulations is to model the experimental rupture behavior numerically in terms of its distribution function and to gain knowledge of the influence and significance of the input parameters and their impact on the system response. The FE models in this study are built up with shell elements and the orientation of the glass fibers is homogeneously distributed over the specimen’s thickness. Furthermore, only quasistatic simulations of the 0◦ orientation are conducted and compared to their experimental results. All simulations are run with the orthotropic elastic plastic material model of Section 5.2.2 and explicit time integration with time scaling. The quasistatic three point bending tests are simulated explicitly to verify that the evaluation of the method is working correctly without the consideration of dynamic effects. A time scaling factor of 1000 is used, so that one second in the experiment is equivalent to one millisecond in the simulation. An advantage of this method is that quasistatic simulations can be simulated explicitly, which is less computational costly than running simulations with implicit time integration. Convergence problems can also become an issue with implicit simulations for large deformations. A disadvantage of the applied method is that oscillations, which are not observed in the physical experiment, can be observed in the response of the simulation.

6.1 Deterministic Three Point Bending Simulations The elastic and inelastic material behavior is calibrated with the orthotropic material model for the 0◦ , 45◦ and 90◦ orientations of the quasistatic three point bending tests. The validation is performed with the 30◦ orientation. The utilized material parameters of the model are shown in Table 6.1. The specimen of the three point bending test is modeled with fully integrated shell elements with nine integration points over the thickness and an element size of 0.5 mm. The impactor and the support of the test setup are modeled with an elastic material model with steel material properties, which means a density of 7.85e-06 kg/mm3 © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8_6

74

6 Numerical Results

and a Young’s modulus of 210.0 GPa. The nodes-to-surface contact algorithm is searching for intersection during the simulation. The FE model of the three point bending test is shown in Figure 6.1. Table 6.1 Constitutive parameters of the orthotropic stiffness tensor and Lankford coefficients of the utilized material model

C11 8.78 GPa C44 1.39 GPa

specimen

C12 3.59 GPa C55 0.91 GPa

C13 3.2 GPa C66 0.98 GPa

C22 5.63 GPa r0 1.0

C23 3.2 GPa r45 0.5

C33 4.91 GPa r90 0.21

impactor

left & right support without visible shell thicknesses

ई

उ ऊ

with visible shell thicknesses

Figure 6.1 FE model of the three point bending test. The shell element size of the specimen is 0.5 mm. Left: without visible shell thicknesses. Right: with visible shell thicknesses.

The contact force of the simulation is evaluated between the impactor and the specimen. The translation of the traverse is resembled by evaluating the translation from one node of the impactor. All nodes of the impactor experience the same translation as the impactor is moving on a directional motion. The comparison of the simulation results with the experimental three point bending curves is shown in Figure 6.2. The following values are determined for the rupture strain criterion: 6.2% for the 0◦ orientation, 9.3% for the 30◦ orientation, 12.0% for the 45◦ orientation and 14.0% for the 90◦ orientation. The rupture strains are reverse engineered to fit the average deflection values of each orientation angle. The observed oscillations in the simulation curves come from the explicit time integration and the dynamic response of the specimen due to the impact. This results in an increase of the uncertainty of the force prediction because the oscillation’s amplitude shifts the average force level. The oscillations could

6.1 Deterministic Three Point Bending Simulations

75

reduce if the simulations are run with the actual time duration of the experiment, which is further investigated in Section 6.2.3. Another option could be to run the simulations with implicit time integration. It can be observed that the amplitude of the oscillating force signal is not constant over the time with the nodes-to-surface contact. This further increases the complexity of estimating the average force level correctly. The effect is connected to the chosen contact algorithm because the nodes-to-surface contact calculates the contact forces by a nodes to segment penalty formulation [94]. Studies have shown that the segment to segment penalty formulation of the mortar contact shows a more consistent oscillating response. This option would be favorable in terms of analyzing the force signal with help of a fast Fourier transform. The force evaluation at rupture with a FFT algorithm is described in Section 6.2.1.2. An overall good fit for all four orientations of the quasistatic three point bending tests can be obtained with the nodes-to-surface contact algorithm. The orthotropic material model is able to simulate the orientation dependent elastic and inelastic material behavior accurately. The parameters of the material model are identical for all four orientations, except for the orientation angle parameter β , see Section 5.2.2. The simulations are run with the segment to segment mortar contact algorithm in a next step. The same rupture values are used in order to compare differences in the calculation between the two contact algorithms. The results are shown in Figure 6.3. Several differences between the contact algorithms can be observed. Foremost different is that the amplitudes of the oscillation reduce with the mortar contact. This simplifies the evaluation of the maximum forces in the simulations. Secondly, an increase of the force level towards the end of the simulation can be observed for the 45◦ and the 90◦ orientation angles. This is due to an issue of a tolerance parameter in the calculation of the contact search routine. Studies have shown that this is not an issue anymore with a more recent version of the FE solver. However, this tolerance issue does not impair the 0◦ orientation simulations. The sudden increase of the curvature is only observable above a certain level of penetration, which is not reached in the 0◦ orientation tests. The same rupture values from the simulations with the nodes-to-surface contact result in a higher maximum deflection with the mortar contact. The point of rupture is now closer to the upper limit of the maximum deflections with the mortar contact, which shows that the choice of contact algorithm affects the point of rupture as well. The higher oscillating amplitudes of the nodes-to-surface contact might lead to an increased plastic strain accumulation in the material, which then results in an earlier point of rupture. Studies with significant higher oscillating amplitudes undermine this hypothesis. A tendency towards earlier rupture of the specimen can be observed with increasing contact stiffness, which leads to an increase of the oscillating amplitudes. However, an overall good fit can be achieved with the mortar contact besides minor deviations.

76

6 Numerical Results

Figure 6.2 Comparison between simulations and experiments of the quasistatic three point bending tests with the nodes-to-surface contact algorithm

Figure 6.3 Comparison between simulations and experiments of the quasistatic three point bending tests with the mortar contact algorithm

6.2 Stochastic Three Point Bending Simulations

77

6.2 Stochastic Three Point Bending Simulations The object of the stochastic simulations is to accurately reproduce the stochastic rupture behavior of the experiments. The investigations will focus on the quasistatic three point bending tests of the 0◦ orientation.

6.2.1 Monte Carlo Simulation The idea behind a Monte Carlo Simulation is to run a large number of simulations with small variations in specific parameters, like e.g. element thickness. The simulation results are statistically evaluated. The aim of this study is to emulate the real physical scatter of the rupture behavior of the experiment. The resulting probability function of the Monte Carlo Simulation is compared to the experimentally obtained probability function in terms of overall position and scatter range. The three point bending FE model from Section 6.1 is used with the mortar contact. Seven design variables are identified as macroscopic uncertainties and varied in the simulations. Three geometry parameters: length, width and thickness of the specimen; three test setup parameters: x-translation and rotation around the z-axis of the specimen and variation of the friction; one material parameter, the variation of the maximum principal strain at rupture. The input of the geometric variations comes directly from measurements of the specimens. Mean and standard deviation are calculated from the dimensions shown in Table A.1 and used as parameters for normal distributions. The use of a normal distribution is suitable for this type of input data because outliers would be neglected from the experiments beforehand. Using a general extreme value distribution is not recommended because it would result in non physical deviations from the norm. A standard normal distribution with mean zero and standard deviation one is assumed for the uncertainty in the placement of the specimen in terms of translational and rotational discrepancies from the norm. The variation in the friction is also assumed to be normally distributed. The friction in the three point bending test is calibrated at 0.3 with help of digital image correlation by Sygusch et al. [92]. A standard deviation of 0.05 is assumed to study the magnitude of an influence from an uncertainty in the friction calibration. The distribution of the rupture strain is computed from the maximum deflections in a separate step and discussed in the following sequence. 6.2.1.1 Determining the rupture distribution function The information of the specimen’s local deformation is missing because the obtained data from the three point bending tests are global force deflection curves. The local information can be obtained in two ways. It can be either directly calculated by means of linear elastic Bernoulli beam theory or indirectly, in an additional reverse engineering process. The

78

6 Numerical Results

flexural strain ε f lex in the three point bending load case for a given deflection w, thickness t and support span L is calculated by ε f lex =

6wt . L2

(6.1)

However, this relation is only valid in the linear elastic region and does not hold for large deformations and non linear material behavior. In a first step, the maximum strains are calculated from the maximum deflections of Table A.1 from the quasistatic three point bending tests of the 0◦ orientation angle. The mean and standard deviation are calculated from the 30 strain values in a next step. These values could be used an input for a normal distribution with mean μ = 0.04128 and standard deviation σ = 0.00132. It would be more suitable to determine the rupture distribution function by reverse engineering because the linear theory can only be seen as a first estimation for elastic plastic problems. Only then it is ensured that the rupture criterion and mesh density are in accordance to each other and that the non linear material and geometric behavior is considered. These uncertainties cannot be eliminated when the rupture strains are solely calculated from linear beam theory. In order to reverse engineer a rupture distribution function the FE model is run with six arbitrary rupture strain values that are in the same magnitude as the expected average rupture strain. The values of this study are 0.051, 0.053, 0.055, 0.057, 0.06 and 0.062. The corresponding maximum deflections in the simulations are 1.882 mm, 1.922 mm, 1.956 mm, 1.986 mm, 2.036 mm and 2.071 mm. The following equation is obtained with a coefficient of determination R2 of 0.998 when the rupture strains ε f and maximum deflections wmax are cross plotted and linear regression is applied wmax = 16.89 mm · ε f + 1.0241 mm.

(6.2)

Solving Eq. (6.2) for ε f results in the desired equation for reverse engineering the rupture strains εf =

wmax − 1.0241 mm , 16.89 mm

wmax > 1.0241 mm.

(6.3)

A normal distribution with mean μ = 0.055 and standard deviation σ = 0.00363 is obtained for all 30 maximum deflections of Table A.1 that are used as input for Eq. (6.3). This reverse engineering method can be seen as advantageous in contrast to the calculation of the rupture strains from linear elastic Bernoulli beam theory because the mesh density and the non linear geometric and non linear material behavior is considered.

6.2 Stochastic Three Point Bending Simulations

79

6.2.1.2 Determining the maximum force at rupture

Vibrations are induced on the three point bending specimen after the initial contact from the impactor. This can be an issue when the maximum force at rupture has to be identified. It can be a challenge to measure the average peak force because the harmonic vibrations act as an offset to the numerical force signal. The response frequency of the oscillation can change by a small portion, especially in a Monte Carlo Simulation, where geometric or material parameters are varied. Additionally, it is not known beforehand at which point of the oscillation the curve ends, which is due to the fact that the rupture strain is a parameter of the simulation study as well. These points have to be considered for an accurately evaluation of the maximum force. A way to measure the maximum force at rupture is to average over the last period of the oscillating curve. This can be done by analyzing the curve in the Fourier space. A generalized discrete Fourier transform algorithm that identifies the primarily vibration frequency f of the specimen in the Fourier space is introduced and applied. This frequency information is used to calculate the period T = 1/ f of the high frequent sinusoidal wave. The data points of the last period of the force signal are automatically stored and their mean value is then calculated and determined as the maximum force at rupture. The Fourier analysis of the force time curve decomposes the signal into its periodic component functions. The Blackman window function [10] is applied to the signal before the Fourier algorithm is utilized. It transforms the finite force signal into a continuous function and is given as  n  n + 0.08 cos 4π , n = 0, . . . , M − 1, (6.4) wB (M) = 0.42 − 0.5 cos 2π M M with window width M that corresponds to the number of data points of the force time signal. A generic example for the application of the Blackman window function to a force time curve is shown in Figure 6.4. The DFT algorithm after Cooley and Tukey [13] is used in this work. It maps a discrete time series of points xM=0 . . . xn−1 to the frequency domain Yk=0 . . .Yn−1 and is given as n−1



kM Y [k] = ∑ x (M) exp −2πi n M=0

 ,

k = 0, . . . , n − 1.

(6.5)

The frequency spectrum of the signal and its corresponding amplitudes can be analyzed by applying a DFT to the curve. The corresponding frequency of a periodic oscillating curve will stand out in the frequency domain. This information can then used to define the number of discrete evaluation points of the last period of the curve prior to rupture. All discrete force values are averaged over this section to obtain the maximum force at rupture for the specific simulation curve. The resulting spectrum of the force time signal

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in the frequency domain is shown in Figure 6.5. The peak at 40 kHz is determined as the interfering frequency in the signal.

Figure 6.4 The upper image shows the finite force time curve of a three point bending test simulation. The lower image shows the transformed, periodic force time curve. The transformation is a result from the application of the Blackman window function.

Figure 6.5 The image shows the frequency spectrum of the force signal upon 50 kHz. The oscillations in the signal come from the peak around 40 kHz. The frequency spectrum curve is truncated because amplitudes above 50 kHz are zero.

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6.2.1.3 Monte Carlo study with variations of all parameters The aim of this work is to investigate if the stochastic rupture behavior of the experiments can be modeled accurately by using Monte Carlo Simulations and if certain input parameters influence the simulation output more than other. A Monte Carlo Simulation with 100 samples is run with variations in the seven parameters that are given in Table 6.2. All input parameters are varied by the formalism of a normal distribution. The resulting maximum deflection and force values are evaluated and compared with the experimental data in a statistical sense by comparison of their probability density functions. Table 6.2

Normal distribution input parameters for the Monte Carlo study

Parameter Specimen length Specimen width Specimen thickness Translation specimen Rotation specimen Friction Rupture strain

Expected value μ 40.09 mm 10.02 mm 3.17 mm 0.0 mm 0.0◦ 0.3 0.055

Standard deviation σ 0.093 mm 0.15 mm 0.007 mm 1.0 mm 1.0◦ 0.05 0.00363

The variations of the geometric parameters are taken from Table A.1. Translational and rotational variations of the specimen placement on the support are assumed to be within an 3σ error of ±3 mm respectively ±3◦ . The variation of the friction coefficient is presumed to vary in a 3σ range of ±50%, to capture possible influences of an improper calibration. The rupture strain distribution is deduced from the maximum deflections at rupture and is described in Section 6.2.1.1. The results of the Monte Carlo Simulation are shown in Figure 6.6. The maximum deflection probability density function of the simulation is compared to the experimental probability density function in the upper image. The accuracy of the stochastically rupd = 1.95 mm is identical to the ture behavior is very high. The experimental mean of μexp d simulation mean μmcs = 1.95 mm. The variation between the standard deviations of the experiment and the simulation is less than 5%. The standard deviation of the experiment is d = 0.061 while it is σ d = 0.064 for the simulation. The red plus signs indicate the loσexp mcs cation of the maximum deflection values. A strong connection between the rupture strains and maximum deflections can be observed in the scatter plot, which is shown in Figure 6.7. A low respectively high rupture strain correlates to a low respectively high maximum deflection at rupture. The images in Section A.6.2 show that correlations according to the other parameters of the study could be not observed for the maximum deflection at rupture. This indicates that the maximum deflection in the three point bending tests is strongly dependent on the material behavior and less dependent on the test setup or small geometric variations.

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The result from the force evaluation on the other hand show minor deviations. A good F = 349.2 N is achieved by a simulation mean accordance of the experimental mean of μexp F of μmcs = 347.93 N, which is a deviation of less than 1%. The discrepancy in the standard deviations is almost 14%. The scatter range of the maximum forces in the simulations is larger than in the experiments. The larger scatter could be deduced from the amount of parameters that are varied. A tendency towards an increase of the scatter range with increasing use of parameters can be observed. Less parameters are varied in the validation study in Section 6.2.2 and a narrower scatter range of the force signal can be observed. A correlation between the width and the force is recognizable from the scatter plot, which is shown in Figure 6.8. This behavior could be expected because the width and thickness are part of the area moment of inertia formula. The impact of the thickness on the maximum force at rupture is not clear at hand from its scatter plot, although the thickness enters the equation of the second moment of area to the third power. The standard deviation of the thickness is small compared to the standard deviation of the width and it does not preponderate therefore, although the thickness influences the force to a certain degree. Coherence can be observed between the rupture strain and the maximum force. This behavior could be expected as well because a higher strain at rupture means that the specimen is able to sustain further loading. The level of influence from the width and the rupture strain on the maximum force and has yet to be quantified. Further details on the Monte Carlo study can be found in the appendix, see Section A.6. The histograms of the input parameters and the measured outputs are shown. The deviations for a given mean and standard deviation are less than 1% for every variable. The full documentation of all scatter plots from this study are also located in Section A.6.

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Figure 6.6 Comparison between the normal distributions of the Monte Carlo Simulation and the experiments. All seven design variables, specimen length, width and thickness, translation and rotation of the specimen, friction and rupture strain are varied according to Table 6.2. The upper image shows the results of the maximum deflection. The evaluation of the maximum force is shown in the lower image. The red plus signs indicate the location of the individual values from the samples.

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Figure 6.7 Scatter plot of the rupture strain and the maximum deflection at rupture. A strong correlation can be observed because a low respectively high rupture strain corresponds to a low respectively high deflection.

Figure 6.8 Scatter plot of the width and the maximum force at rupture. A correlation between both parameters can be observed.

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6.2.2 Evaluation and validation The ANOVA [21], short for analysis of variance, is a useful tool for analyzing statistical data with multiple input and output signals. The ANOVA decomposes the input of the Monte Carlo study and evaluates the importance of each design variable variance on the output variances individually. The seven input parameters are specimen length, width and thickness; translation and rotation of the specimen on the support; friction and rupture strain. Output signals are the maximum deflection and force at rupture. The ANOVA also checks for non linear coherences by analyzing dependencies of multiplications from two input variances on the output variance. Higher order terms are not taken into consideration. 2 is given by the expected value μ of all input parameters and the sum The output signal σout of first and second order variance terms  2 2 2 σ1 , . . . , σn2 = μ + σ12 + σ22 + . . . + σn2 + σ12 σ22 + σ12 σ32 + . . . + σn−1 σn2 . σout

(6.6)

The results of the ANOVA for the maximum deflections can be seen in Figure 6.9 and Figure 6.10. Figure 6.9 shows the cumulative contributions of each input parameter on the value of the maximum deflection. The rupture strain is contributing the most to the deflection output with 97.6%. The influence of all other parameters is less than 2.5%. Figure 6.10 shows the effect that a change in value of an input variable has on the value of the maximum deflection. The main effect plot identifies the rupture strain as the most important parameter because the curve has the highest slope of all curves in the Figure 6.10. The degree of impact by other parameters is not significant in comparison, although they influence the maximum deflection partially. The oscillating pattern of the translation specimen parameter curve could be related to numerical issues from the contact algorithm because different number of nodes and elements are in contact with the support when the specimen is shifted. This behavior is not directly obtained from the scatter plots alone because multiple effects are overlapping each other. The evaluations of the maximum force at rupture are depict in the cumulative contribution plot in Figure 6.11 and the main effect plot in Figure 6.12. The results of the cumulative contribution plot can be separated into three parts. The major input parameters are the specimen width and the rupture strain with 55.57% respectively 32.14% contribution. Minor parameters are the friction and the specimen’s thickness with 7.3% and 4.15% contribution. Parameters with less than 1% contribution are the specimen’s rotation and translation and the length of the probe. It could be assumed that the width and thickness influence the force level to a certain degree because both parameters enter the formula of the area moment of inertia. The contribution of the thickness is 4.15% while it is 55.57% for the width, although the thickness enters the formula of the second moment of area to the third power. It can be assumed that the structure of the data is related to this observation because the mean value of the width is 10.02 mm with a standard deviation of 0.15 mm and the mean value of the thickness is 3.17 mm with a standard deviation of 0.007 mm.

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Deviations of the thickness do not strongly influence the force level because of the narrow scatter range. The quantity of the contribution of the rupture strain is remarkable because the rupture strain alone makes 32.14% of the total contribution. A combined contribution of 87.17% is achieved when the values of the width and the rupture strain are added. Figure 6.12 shows the effect of each parameter on the maximum force. Width and rupture strain can be identified as major parameters due to the highest curve slopes in the graph. Friction and thickness are the minor influential parameters of this study. An oscillating pattern of the translation specimen parameter can be observed as well. This is again connected to the contact algorithm. The rotation specimen parameter shows a parabolic curvature. This behavior is related to the orthotropic material model. The 0◦ orientation, which is investigated in this Monte Carlo study, is the orientation angle with the highest stiffness. A rotational shift in any direction results in a less rigid material behavior, which causes the forming of a parabolic curve shape. Studies on other orientation angles have shown that e.g. for a 45◦ orientation the material behavior is stiffer when the orientation degree decreases while the mechanical behavior is less rigid when the probe is rotationally shifted towards a higher orientation angle. This behavior results in a S-shaped curve. However, the translational and rotational variations have a marginal effect on the value of the maximum force at rupture. No influence on the maximum force can be observed for the change in length of the specimen.

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Figure 6.9 Cumulative contributions of the input parameters on the maximum deflection value. The bar chart shows that the rupture strain is the most important parameter, while the other parameters do not have a significant influence on the maximum deflection.

Figure 6.10 Main effect plot of the ANOVA. This image shows the influence that each input variable has on the maximum deflection. The rupture strain is the most important parameter in this study due to the highest curve slope. The oscillating pattern from the translation specimen curve is related to the contact algorithm.

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Figure 6.11 Cumulative contributions of the input parameters on the maximum force at rupture. The width and rupture strain are the two most important input parameters here. Friction and thickness do also influence the output signal, but their impact does not preponderate in comparison to the major parameters.

Figure 6.12 Main effect plot of the ANOVA. Width and rupture strain are the major parameters for the maximum force, whereas friction and thickness can be seen as minor parameters. The other parameters do not preponderate. The oscillating pattern of the translation specimen parameter is related to the contact algorithm.

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The ANOVA of the Monte Carlo study with variations in all parameters reveals that the rupture strain is the main influential parameter of the maximum deflections, whereas a combination of the width and rupture strain are the major decisive parameters for the maximum force at rupture. The rupture strain is contributing up to 97.6% to the outcome of the maximum deflection result. 87.17% is the contribution of the combination from the width and the rupture strain on the maximum force. A validation study will investigate the trustworthiness of these findings. Only two of seven parameters, namely the width and the rupture strain, are varied in the following Monte Carlo validation study. The parameter variations of the width and the rupture strain are shown in Table 6.3. The variations are identical to the Monte Carlo study of Section 6.2.1.3. The remaining parameters are constant. The validation study is performed with 100 samples and a shell element size of the specimen of 0.5 mm, which is identical to the previous Monte Carlo study as well. Table 6.3 Normal distribution input parameters for the Monte Carlo validation study. Specimen length and thickness, translation and rotation of the specimen and friction are constant.

Parameter Specimen width Rupture strain

Expected value μ 10.02 mm 0.055

Standard deviation σ 0.15 mm 0.00363

The evaluations of the Monte Carlo validation study are shown in Figure 6.13. The upper image shows the comparison of the maximum deflection probability density functions. The dashed line from the Monte Carlo Simulation is in good accordance with the experimental curve. The variations are not significantly large because the difference in the mean is less than 1%, whereas it is smaller than 5% for the standard deviation. A similar result had already been obtained by the previous Monte Carlo study. This shows that the rupture strain is the most important parameter for the maximum deflections. The variation of the specimen width does not significantly alter the outcome of the maximum deflection result. The concordance of the probability density functions between the Monte Carlo validation study and the experiment is very good in terms of the maximum force as well. The difference is less than 1% for the mean value and 6.27% for the standard deviation. This shows that the width and rupture strain alone are capable to emulate the maximum force at rupture precisely. This also proves that only the most important parameters need to be scattered stochastically in order to model the stochastic rupture behavior accurately.

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Figure 6.13 Comparison between the normal distributions of the Monte Carlo validation study and the experiments. Two design variables, namely specimen width and rupture strain are varied according to Table 6.3 in this study. The upper image shows the evaluation of the maximum deflections, whereas the comparison of the maximum forces is shown in the lower image. Both simulation curves are in good accordance with the experimental curves. The red plus signs indicate the location of the individual sample values.

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6.2.3 Influence of the explicit time integration The previous studies are run with explicit time integration with time scaling because of the long experiment time. Explicit time integration can lead to convergence problems when the number of time step cycles increases too large. The physical time is sped up by factor 1000 to solve this time step issue. Time scaling can be genuinely used to reduce the number of cycles as long as viscous effects are neglected in the simulation. Time scaling is applicable here because the simulation is run without numerical damping and without viscoplastic material behavior. There are two ways to run the simulation with physical time. The simulation can be run with explicit time integration and deactivated time scaling or with implicit time integration. Running simulations implicitly can be cost intensive and is therefore uncommon in the crash simulation environment. Hence, the simulations are run with explicit time integration and deactivated time scaling. The setup of the validation study from Section 6.2.2 is used as a starting point. Furthermore, two additional studies are performed, one with 50% of the initial time scaling and the second with 10%. The sample number of 100 and the mesh density of 0.5 mm remains the same. The results in Table 6.4 and Figure 6.14 show that it is not possible to simulate the experiment with the actual physical time with explicit time integration. The validation and the 50% time scale studies show good accordance to the experimental probability density function. However, this does not hold for the 10% time scale study. The number of cycles is too large with almost 577500. This leads to erroneous results due to the accumulation of the numerical round off errors. These round off errors come from the approximation of the numerical time integration. The simulation runs stable until the end, but the solution diverges. There is a trade-off because the number of cycles is connected to the time step of the simulation. Especially the curvature of the supports requires a fine mesh density in order to model the support radius precisely. This detailed mesh density consequently causes the small time step in the simulation. The small time step leads to a large amount of cycles due to the long experiment time, which causes these numerical issues. A further time scaling reduction is not possible because the solution continuously diverges. Table 6.4 Evaluation of the mean and standard deviation for different time scaling studies. The validation study corresponds to time scale factor of 1.0 that is equivalent to an explicit time integration simulation factor of 1000. Physical time is reached with a simulation factor of 1.

Study μ σ cycles

Maximum deflection Validation 50% 10% 1.94 mm 1.95 mm 1.96 mm 0.064 mm 0.065 mm 0.085 mm 57800 115500 577500

Maximum force Validation 50% 10% 349.54 N 349.83 N 328.74 N 6.505 N 7.003 N 13.947 N 57800 115500 577500

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Figure 6.14 Comparison between the normal distributions of the three time scaling studies and the experimental results. Specimen width and rupture strain are varied according to Table 6.3. The Monte Carlo Simulations are the validation study, a 50% time scaling study and a 10% time scaling study. The evaluation for the maximum deflections is shown in the upper image, whereas the maximum forces are shown in the lower image. It can be seen that with decreasing time scaling the results differ from the original validation study. Especially the standard deviation increases significantly for the 10% time scaling study for the maximum deflection and force. The validation study simulation requires approximately 57800 cycles, while the 50% study needs approximately 115500 cycles and the 10% time scaling study requires approximately 577500 cycles to finish the simulation.

7 Summary and Outlook In the context of increasing regulatory standards in the vehicle development, a method for incorporating and comparing stochastic scatter of macroscopic parameters in crash simulations is developed in the present work. The method is applied on a 30 wt.% short glass fiber reinforced polypropylene with the objectives to statistically characterize the stochastic rupture behavior of the experiments by means of probability functions and to reproduce the stochastic rupture behavior of the experiments by means of Monte Carlo Simulations. The experiments focus on three point bending tests that are performed at four different orientation angles, 0◦ , 30◦ , 45◦ and 90◦ and two impact velocities 0.001 ms−1 and 4.0 ms−1 . The orientation angles are with respect to the molding direction and the impact velocities correspond to strain rates of 0.021 s−1 and 85 s−1 respectively. Tensile tests are also taken into consideration for the material tests. However, the point of rupture almost entirely concentrates on the transition from the parallel region towards the radius of the tensile specimen. The three point bending test is preferred to the tensile test because of this challenge. Investigations on the manufacturing process of the specimens show that milling at a slow rotational speed and feed rate generate the best results in terms of the surface quality and is preferred to laser cutting. A statistical testing plan with a minimum of 30 repetitions for each configuration is performed because the material shows brittle rupture behavior. A distinct orientation dependent rupture behavior can be observed for the quasistatic tests in terms of the maximum force and deflection at rupture. It is therefore suggested to increase the application of orientation dependent rupture criteria for this material in the simulation because the orientation dependency is relevant in terms of the quasistatic rupture behavior. This could increase the numerical computation costs in contrast to the use of isotropic rupture criteria but can also be seen as an opportunity for more predictive statements regarding the fracture behavior of structures made out of fiber reinforced polymers. The orientation dependency in terms of the maximum deflections at rupture is not pronounced in the dynamic tests. This could simplify the application of strain based rupture criteria in crash simulations with high velocities. It could be expected to achieve sufficient results by utilization of an average rupture strain value because the relative error margin between the different orientations is narrower. However, the separation regarding the maximum forces at rupture remains for the dynamic tests. Hence, a suggestion of a simplified rupture approach does not hold for stress based rupture criteria. The statistical analysis of the experimental survey also indicates that the stochastic rupture © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8_7

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behavior of the maximum deflections and forces from the quasistatic three point bending tests can be modeled accurately by means of the normal and lognormal distributions. The evaluations also show that this observation does not hold for all dynamic tests where the Weibull distribution can achieve the better results for some of the test configurations. Suggestions cannot be projected from the three point bending test to the tensile test because the rupture behavior in a tensile test is not subjected to a statistical analysis in this work and because of the different boundary conditions and specimen geometry. The numerical investigations are built on the basis of the experiments, which are also used for the material characterization. The investigations of the glass fibers and the microstructure help to calibrate the orthotropic elastic plastic material model. The material model is calibrated with a homogeneous fiber distribution across the thickness, which is a simplified assumption of the actual fiber orientation in the material. An overall good correlation between the quasistatic simulations and the experimental force deflection curves in the three point bending test can be achieved. However, the contact modeling can be a challenge for large deformations at the 45◦ and 90◦ orientations and requires further investigations. Monte Carlo Simulations with variations in macroscopic parameters are run to emulate the stochastic scatter behavior of the experimentally obtained maximum deflections and forces. The results show that the stochastic rupture behavior in terms of the probability density functions can be reproduced very well for the 0◦ orientation. Data analysis leads to the conclusion that the maximum principal strain and specimen width are the decisive parameters of the stochastic simulations. This result is also verified in a validation study. The simulations show that the stochastic rupture behavior in the experiments can be deduced very well from variations of macroscopic parameters for the 0◦ orientation. Deviations of the nominal specimen geometry should be reduced to a minimum in order to reduce their influence on the rupture behavior. The incorrect placement of the specimen is only of secondary importance for the rupture behavior, at least for the selected boundary conditions in this work. Further experiments need to resolve the general validity of the obtained probability distribution functions because this work focuses on investigations of the center position of an injection molded plaque. It has to be investigated to what extend these findings can be transferred to other removal positions as well. This could clarify correlations between the different microstructure setups and their influence on the stochastic rupture behavior. Larger three point bending specimens could also be the subject of further experimental studies in order to study size or scaling effects. These larger specimens would have to be taken from the center position of the plaque in order to compare the resulting probability distribution functions with the findings of this work. Further investigations could also study if similarities between statistical evaluations of the three point bending tests from this work and experiments on component parts can be found. It can be assumed that the microstructure setup and the fiber orientation in the component part varies significantly from the analyzed coupon geometry of this work.

7 Summary and Outlook

95

For some configurations, the Weibull distribution does not fall below the critical threshold value in order to be generally valid in the context of the applied goodness of fit test, although it performed significantly better than the normal and lognormal distributions. Hence, it could to be investigated if another type of distribution function, like e.g. a multimodal Weibull distribution function, can lead to additional insight into this issue. Increasing the sample size could also help to resolve this challenge. The present method has proven to work very well for the specific test configuration of the 0◦ orientation of the quasistatic three point bending test, whereas it has to be applied and validated for the other orientation angles in the quasistatic and dynamic three point bending test as well. However, the contact related discrepancies at large deformations for the quasistatic 45◦ and 90◦ orientations need to be investigated beforehand by e.g. applying different contact algorithms. A convergence study with different mesh densities is also required before applying this method to component parts because coarser mesh densities than the utilized mesh density of this work are commonly used in crash simulations. The use of an orthotropic material model for fiber reinforced polymers is one step towards a physically motivated setup in the simulation model. However, the assumption of a single orientation layer over the thickness simplifies the real setup of the specimen that has different oriented layers that are stacked upon and interconnected with each other. It is possible that the incorporation of local fiber orientations could influence the obtained results of the Monte Carlo Simulations. This would also enable additional parameters for the stochastic simulations, like local fiber orientation and fiber volume content. The variation of these local properties could be studied in future works as well. This could be a first step towards investigations on the influences of the injection molding process. The orthotropic material model of this work allows the integration of local fiber orientation properties and is therefore suitable for further investigations in that direction. The introduced method has shown to function very well for the investigated load case in this work. The discussion shows that there is a versatile spectrum for different applications as well. Future researchers could concentrate on the emphasized challenges in order to improve the significance of the method, so that engineers can build confidence in dimensioning structures from the perspective of statistical certainty.

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List of symbols A a A0 A (x) A Ai j Ai jkl A(MT ) A(SC) Axx Ayy Azz α

master shell element area threshold parameter initial cross section of the tensile specimen momentary cross section of the tensile specimen second order orientation tensor second order orientation tensor in index notation, orientation tensor components fourth order orientation tensor in index notation fourth order Mori Tanaka strain concentration tensor fourth order strain concentration tensor for dilute fiber concentrations orientation degree in x-direction orientation degree in y-direction orientation degree in z-direction significance level, scale factor

b  b Bi bli bui b β βm βs

bin width, scale parameter temporary scale parameter constants lower bound of the i-th bin upper bound of the i-th bin local shell element orientation vector shell element orientation angle stiffness scale factor master segment stiffness scale factor slave segment

c c1 c2 cl cu Ci j (OA) Ci jkl C Cf

shape parameter, Weibull modulus constant constant lower bound of the 95% confidence interval upper bound of the 95% confidence interval stiffness tensor components fourth order orientation averaged stiffness tensor in index notation fourth order stiffness tensor fourth order isotropic stiffness tensor glass fibers

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8

106

List of symbols

Cm C(OA) C(UD) (UD) Ci jkl χ2 CV

fourth order isotropic stiffness tensor matrix fourth order orientation averaged stiffness tensor fourth order unidirectional homogenized composite stiffness tensor unidirectional homogenized composite stiffness tensor components chi-squared distribution coefficient of variation

d dcs dmax dmin δi j

Glass fiber diameter, contact penetration characteristic length slave segment maximum contact penetration smallest contact element diagonal Kronecker delta in index notation

E Ef ei ei Em Eϕ Es ε ε ε1 ε2 ε3 ε22 ε33 ε˙0 εel εel εf ε f lex εi j εlat εlog εlong εmax εnom ε pl ε pl εx

Young’s modulus Young’s modulus glass fibers expected relative bin frequencies uniform basis vector Young’s modulus matrix orientation dependent Young’s modulus Young’s modulus slave segment material total strain, scale factor infinitesimal strain tensor principal strain 1-direction principal strain 2-direction principal strain 3-direction transversal strain thickness strain nominal strain rate infinitesimal elastic strain tensor elastic strain maximum principal strain criterion flexural strain infinitesimal strain tensor components lateral strain Hencky or logarithmic strain longitudinal strain maximum strain nominal strain infinitesimal plastic strain tensor plastic strain lateral strain

List of symbols

107

εy E η

longitudinal strain fourth order Eshelby tensor local shell coordinate

F f F0 (x) F1c F2c c F fi fLN (x) Fmax fN (x) ϕ (x) fs F (σ) f W2 f W3 F (x) F (Y )

force, Hill criterion constant frequency empirical distribution function contact force on master node 1 contact force on master node 2 contact force on slave node summed up relative frequencies of the i-th bin lognormal distribution maximum force normal distribution standard normal distribution scale parameter yield function two parameter Weibull distribution three parameter Weibull distribution cumulative distribution function cumulative distribution function

G g

Hill criterion constant depth of penetration

H0 H

null hypothesis Hill criterion constant

I

fourth order identity tensor

k Km

number of bins, degrees of freedom, proportional factor, frequencies compression modulus of master material

l l0 l/d Δl L lc λ λ1 λ2 λ3

momentary gauge length of the tensile specimen, length, depth of penetration initial gauge length of the tensile specimen, initial length length to diameter ratio change of gauge length of the tensile specimen Hill criterion constant, support span critical fiber length principal stretch of the tensile specimen principal length stretch principal width stretch principal thickness stretch

M

Hill criterion constant, window width

108

List of symbols

μ d μexp d μmcs F μexp F μmcs μg

arithmetic mean value experimental mean value deflections simulation mean value deflections experimental mean value forces simulation mean value forces geometric mean value

N n nm ns n ν νf νm

sample size, Hill criterion constant number of measurement points master segment’s normal vector slave node shell element normal vector Poisson’s ratio Poisson’s ratio glass fibers Poisson’s ratio matrix

oi

observed relative bin frequencies

Pi pAD pχ 2 pKS pT LV ϕ ψ

plotting positions Anderson-Darling value chi-squared value Kolmogorov-Smirnov value critical value, threshold limit value fiber volume fraction, orientation angle matrix volume fraction

Q10 Q50 Q90

10% quantile 50% quantile 90% quantile

r r0 r45 r90 R2 Ra R r ρf ρm Rz1max

Lankford parameter, r-value r-value 0◦ orientation r-value 45◦ orientation r-value 90◦ orientation coefficient of determination arithmetic roughness shear yield stress master segment location vector density glass fibers density matrix maximum roughness between to nearby peaks

S σ

shear yield stress Cauchy stress tensor

List of symbols

σi2 2 σout d σexp d σmcs σ uf σg σi j σmax σn σnom σ σ2 σvM σy

input signal variances output signal variance experimental standard deviation deflections simulation standard deviation deflections ultimate strength glass fibers multiplicative standard deviation Cauchy stress tensor components maximum stress contact normal stress nominal stress true stress, empirical standard deviation empirical variance von Mises yield criterion yield stress

t t0 T t τmu

Student’s t-distribution, thickness initial thickness shear yield stress, frequency period slave node location vector ultimate shear strength matrix

ui

reduced values

v0 V

initial velocity global orientation vector

w w0 wB (M) wmax

width initial width Blackman window function maximum deflection

x¯ x¯g X xi xi,m xi∗ x{i:n} ∗ x{i:n} x (M) xmax xmin ξ

arithmetic mean value geometric mean value tensile yield stress x-direction measurement points average value of the i-th bin natural logarithm of xi xi in ascending order natural logarithm of xi in ascending order discrete points of a time series maximum value of a data set minimum value of a data set local shell coordinate

109

110

List of symbols

Y Yi Y [k] y (x) y∗ (x)

tensile yield stress y-direction data points discrete Fourier transformation fitting function two parameter Weibull distribution fitting function three parameter Weibull distribution

z Z Zi2 zi

z-score tensile yield stress z-direction squared standard normal random variables measurement points

A Appendix Contents A.1 A.2 A.3 A.4 A.5 A.6

Technical data sheet Hostacom G3 R05 105555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fiber length analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . μCT scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface roughness measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three point bending test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6.1 Histograms of all design variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6.2 Scatter plots of all design variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2020 N. Sygusch, Stochastic Approach to Rupture Probability of Short Glass Fiber Reinforced Polypropylene based on Three-Point-Bending Tests, Mechanik, Werkstoffe und Konstruktion im Bauwesen 52, https://doi.org/10.1007/978-3-658-27113-8

112 114 116 121 123 135 135 139

112

A Appendix

A.1 Technical data sheet Hostacom G3 R05 105555

Hostacom G3 R05 105555 Compounded Polyolefin Product Description Hostacom G3 R05 105555 is a 30% glass fiber reinforced PP homopolymer, with excellent stiffness, good dimensional stability and low creep under load at elevated temperatures. The product is available in black color, pellet form. This grade is delivered in 105555 color version. This grade is not intended for medical, pharmaceutical, food and drinking water applications.

Product Characteristics Status

Commercial

Availability

Europe

Processing Method

Injection molding

Features

Stiffness, dimensional stability, creep under load at elevated temperatures.

Typical Customer Applications

Used for structural parts.

Typical Properties Physical Melt Flow Rate (230 °C, 2.16 kg) Melt Volume Rate (230 °C, 2.16 kg) Density (23 °C) Mechanical Tensile Modulus (23 °C) Tensile Stress at Break (23 °C) Tens.Strain at Break Flexural Modulus (23 °C) Tech. A Flexural Strength (23 °C) Tech. A Impact Charpy Impact Strength, unnotched (23 °C) Charpy Impact Strength, notched (23 °C) Charpy Impact Strength, notched (-30 °C) Thermal Vicat Softening Temperature B (50 N) Heat Deflection Temperature A (1.8 MPa) Heat Deflection Temperature B (0.45 MPa)

(1)

Method

Value

Unit

ISO 1133 ISO 1133 ISO 1183-1/A

4 4 1.14

g/10 min cm3/10 min g/cm3

ISO 527-1, -2 ISO 527-1, -2 ISO 527-1, -2 ISO 178/A1 ISO 178/A1

7200 100 3 7000 140

MPa MPa % MPa MPa

ISO 179-1/1eU ISO 179-1/1eA ISO 179-1/1eA ISO 306 ISO 75-1, -2 ISO 75-1, -2

50 11 8 138 150 160

kJ/m2 kJ/m2 kJ/m2 °C °C °C

Product Storage and Handling • Product should be stored in dry conditions at temperatures below 50°C and protected from UV-light. • Improper storage may bring damage to the packaging and can negatively affects on the quality of this product • Keep material completely dry for good processing.

Notes Typical properties; not to be construed as specifications. (1) : Here is indicated the region where the material is produced. For importation or demand of a local equivalent grade, please contact our Sales Representatives.

A.1 Technical data sheet Hostacom G3 R05 105555

© LyondellBasell Industries Holdings, B.V. 2014 LyondellBasell markets this product through the following entities: • Equistar Chemicals, LP • Basell Sales & Marketing Company B.V. • Basell Asia Pacific Limited • Basell International Trading FZE • LyondellBasell Australia Pty Ltd For the contact details of the LyondellBasell company selling this product in your country, please visit http://www.lyondellbasell.com/. Before using a product sold by a company of the LyondellBasell family of companies, users should make their own independent determination that the product is suitable for the intended use and can be used safely and legally. SELLER MAKES NO WARRANTY; EXPRESS OR IMPLIED (INCLUDING ANY WARRANTY OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE OR ANY WARRANTY) OTHER THAN AS SEPARATELY AGREED TO BY THE PARTIES IN A CONTRACT. This product(s) may not be used in: (i) any U.S. FDA Class I, Health Canada Class I, and/or European Union Class I Medical Devices, without prior notification to Seller for each specific product and application; or (ii) the manufacture of any of the following, without prior written approval by Seller for each specific product and application: (1) U.S. FDA Class II, Health Canada Class II or Class III, and/or European Union Class II Medical Devices; (2) film, overwrap and/or product packaging that is considered a part or component of one of the aforementioned Medical Devices; (3) packaging in direct contact with a pharmaceutical active ingredient and/or dosage form that is intended for inhalation, injection, intravenous, nasal, ophthalmic (eye), digestive, or topical (skin) administration; (4) tobacco related products and applications; (5) electronic cigarettes and similar devices; and (6) pressure pipe or fittings that are considered a part or component of a nuclear reactor. (iii) Additionally, the product(s) may not be used in: (1) U.S. FDA Class III, Health Canada Class IV, and/or European Class III Medical Devices; (2) applications involving permanent implantation into the body; (3) life-sustaining medical applications; and (4) lead, asbestos or MTBE related applications. All references to U.S. FDA, Health Canada, and European Union regulations include another country’s equivalent regulatory classification.

Users should review the applicable Material Safety Data Sheet before handling the product. Addhere, Adflex, Adstif, Adsyl, Akoafloor, Akoalit, Alastian, Alathon, Alkylate, Amazing Chemistry, Aquamarine, Aquathene, Arctic Plus, Arctic Shield, Avant, Catalloy, Clyrell, CRP, Crystex, Dexflex, Duopac, Duoprime, Explore & Experiment, Filmex, Flexathene, Fueling the power to win, Get in touch with, Glacido, Hifax, Histif, Hostacom, Hostalen PP, Hostalen ACP, Ideal, Indure, Integrate, Koattro, LIPP, Lucalen, Luflexen, Lupolen, Lupolex, Luposim, Lupostress, Lupotech, Metocene, Microthene, Moplen, MPDIOL, Nerolex, Nexprene, Petrothene, Plexar, Polymeg, Pristene, Prodflex, Pro-fax, Punctilious, Purell, Refax, SAA100, SAA101, Sequel, Softell, Spherilene, Spheripol, Spherizone, Starflex, Stretchene, Superflex, TBAc , Tebol, T-Hydro, Toppyl, Trans4m, Tufflo, Ultrathene, Vacido and Valtec, are trademarks owned and/or used by the LyondellBasell family of companies. Adsyl, Akoafloor, Akoalit, Alastian, Alathon, Aquamarine, Arctic Plus, Arctic Shield, Avant, CRP, Crystex, Dexflex, Duopac, Duoprime, Explore & Experiment, Filmex, Flexathene, Hifax, Hostacom, Hostalen, Ideal, Integrate, Koattro, Lucalen, Lupolen, Metocene, Microthene, Moplen, MPDIOL, Nexprene, Petrothene, Plexar, Polymeg, Pristene, Pro-fax, Punctilious, Purell, Sequel, Softell, Spheripol, Spherizone, Starflex, Tebol, T-Hydro, Toppyl, Tufflo, Ultrathene are registered in the U.S. Patent and Trademark Office. Release date: 12/01/2015

113

114

A.2 Fiber length analysis

Figure A.1 Results fiber length analysis 0◦ orientation

Figure A.2 Results fiber length analysis 30◦ orientation

A Appendix

A.2 Fiber length analysis

Figure A.3 Results fiber length analysis 45◦ orientation

Figure A.4 Results fiber length analysis 90◦ orientation

115

116

A Appendix

A.3 μCT scans

Figure A.5 Sketch of the utilized injection molded plaque with specimen removal positions. The x-axis is in molding direction.

Figure A.6

Principal orientations of the orientation tensor from removal position 1

A.3 μCT scans

Figure A.7 Principal orientations of the orientation tensor from removal position 2

Figure A.8 Principal orientations of the orientation tensor from removal position 3

117

118

A Appendix

Figure A.9 Principal orientations of the orientation tensor from removal position 4

Figure A.10

Principal orientations of the orientation tensor from removal position 5

A.3 μCT scans

Figure A.11

Principal orientations of the orientation tensor from removal position 6

Figure A.12

Principal orientations of the orientation tensor from removal position 7

119

120

A Appendix

Figure A.13

Principal orientations of the orientation tensor from removal position 8

Figure A.14

Principal orientations of the orientation tensor from removal position 9

A.4 Surface roughness measurements

121

A.4 Surface roughness measurements

10

5

5

-0

-0

-5

-5

-10

-10

-15

-15

-20

-20 -2

-1

0

1

Ra

2

3

4

1,0700

μm

Rz1max

15,7675

μm

Rz

12,0419

μm

Figure A.15

5

6

7 8 9 Millimeter

10

11

12

13

14

15

16

17

Surface measurement for a 0◦ orientation specimen. Manufactured by milling.

15

Mikrometer

Mikrometer

15

10

15

10

10

5

5

-0

-0

-5

-5

-10

-10

-15

-15

-20

Mikrometer

Mikrometer

15

-20 -21

Ra

-20

-19

-18

-17

-16

-15

1,4460

μm

Rz1max

18,0026

μm

Rz

13,4026

μm

Figure A.16

-14

-13

-12 -11 -10 Millimeter

-9

-8

-7

-6

-5

-4

-3

-2

Surface measurement for a 90◦ orientation specimen. Manufactured by milling.

60

60

40

40

20

20

-0

-0

-20

-20

-40

-40

-60

-60

-80

-80 -1

0

Ra Rz1max

2

3

4

12,9351

μm

105,5254

μm

85,1874

μm

Rz

Figure A.17

1

5

6

7 8 9 Millimeter

10

11

12

13

14

15

16

17

Surface measurement for a 0◦ orientation specimen. Manufactured by laser cutting.

60

60

40

40

20

20

-0

-0

-20

-20

-40

-40

-60

-60

-80

Mikrometer

-2

Mikrometer

Mikrometer

A Appendix

Mikrometer

122

-80 -2

Ra Rz1max Rz

Figure A.18

-1

0

1

2

3

4

16,2626

μm

111,3365

μm

97,5717

μm

5

6

7 8 9 Millimeter

10

11

12

13

14

15

16

17

Surface measurement for a 90◦ orientation specimen. Manufactured by laser cutting.

A.5 Three point bending test results

123

A.5 Three point bending test results In this section the quasistatic and the non filtered 4 ms−1 dynamic force deflection curves of the three point bending tests, the detailed results of three point bending tests and the obtained probability plots are shown.

Figure A.19

Quasistatic three point bending force deflection curves

Figure A.20

Non filtered 4 ms−1 impact velocity three point bending force deflection curves

124

A Appendix

Table A.1

# 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 μ σ σ2 CV

Details on the quasistatic 0◦ orientation three point bending tests

Length L [ mm ] 40.08 40.17 40.04 40.05 40.03 40.04 40.00 40.08 40.13 40.06 40.10 40.34 40.11 40.16 40.16 40.09 40.00 40.15 40.11 40.14 40.21 40.03 40.15 40.04 39.82 40.08 40.27 40.12 40.00 40.02 40.09 0.093 0.009 0.002

Width W [ mm ] 9.98 10.02 9.90 9.98 9.99 10.28 10.01 10.16 10.08 10.02 10.12 10.15 10.16 9.86 9.94 10.17 10.16 9.81 10.18 9.75 9.83 10.27 10.02 9.86 10.07 10.08 9.77 10.16 10.05 9.72 10.02 0.15 0.023 0.015

Thickness T [ mm ] 3.17 3.17 3.17 3.16 3.17 3.17 3.17 3.17 3.17 3.17 3.18 3.17 3.18 3.18 3.18 3.16 3.18 3.17 3.19 3.16 3.17 3.17 3.16 3.18 3.16 3.17 3.18 3.17 3.17 3.16 3.17 0.007 0.0 0.002

Mass m[g] 1.41 1.42 1.41 1.42 1.42 1.47 1.42 1.45 1.45 1.43 1.45 1.46 1.45 1.41 1.43 1.45 1.44 1.40 1.45 1.39 1.41 1.46 1.42 1.40 1.41 1.43 1.39 1.45 1.43 1.37 1.43 0.024 0.001 0.017

Density ρ [ g/ccm ] 1.11 1.11 1.12 1.12 1.12 1.12 1.12 1.13 1.13 1.12 1.13 1.12 1.12 1.12 1.12 1.13 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.11 1.12 1.12 1.12 1.12 0.004 0.0 0.004

Max. w [ mm ] 1.95 2.07 1.91 1.96 1.97 1.95 1.92 2.11 1.94 2.06 1.93 1.92 1.95 1.99 1.84 1.88 1.94 1.96 1.83 1.92 1.89 1.89 1.99 2.00 1.90 1.96 2.04 1.99 1.99 1.95 1.95 0.061 0.004 0.031

Max. f [N] 343.9 354.3 345.3 348.0 343.7 361.5 351.4 359.1 358.0 357.2 355.5 351.1 356.6 347.4 346.7 354.5 353.0 343.8 347.2 333.8 342.8 349.8 343.4 343.1 351.1 348.8 337.7 358.0 354.7 336.0 349.2 6.94 48.17 0.02

A.5 Three point bending test results

Table A.2

# 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 μ σ σ2 CV

125

Details on the quasistatic 30◦ orientation three point bending tests

Length L [ mm ] 39.98 39.86 39.99 40.01 39.93 39.80 40.00 40.03 39.98 40.02 40.00 40.07 40.02 39.99 40.01 39.91 40.08 40.00 40.05 39.98 40.04 39.90 39.99 40.01 40.02 40.00 40.04 39.96 40.01 40.01 39.99 0.058 0.003 0.001

Width W [ mm ] 10.05 9.99 10.05 10.05 10.06 10.10 10.06 10.07 10.06 10.01 9.97 10.02 10.00 10.01 10.00 10.04 9.95 10.00 9.97 9.89 10.06 9.98 9.97 10.02 9.96 10.06 10.08 9.87 10.06 9.93 10.01 0.055 0.003 0.005

Thickness T [ mm ] 3.16 3.17 3.15 3.19 3.18 3.19 3.15 3.15 3.18 3.16 3.18 3.19 3.16 3.15 3.19 3.19 3.18 3.16 3.18 3.17 3.17 3.18 3.17 3.19 3.16 3.19 3.18 3.18 3.16 3.18 3.17 0.014 0.000 0.004

Mass m[g] 1.43 1.42 1.43 1.44 1.44 1.45 1.43 1.43 1.44 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.42 1.42 1.41 1.43 1.42 1.43 1.44 1.42 1.44 1.44 1.41 1.43 1.41 1.43 0.009 0.000 0.006

Density ρ [ g/ccm ] 1.13 1.12 1.13 1.13 1.13 1.13 1.13 1.13 1.12 1.13 1.13 1.12 1.13 1.13 1.12 1.12 1.13 1.12 1.12 1.12 1.12 1.12 1.13 1.12 1.13 1.12 1.12 1.13 1.13 1.12 1.12 0.003 0.000 0.003

Max. w [ mm ] 2.28 2.29 2.28 2.42 2.17 2.27 2.43 2.36 2.39 2.37 2.38 2.33 2.26 2.47 2.47 2.42 2.39 2.29 2.30 2.24 2.32 2.36 2.26 2.36 2.40 2.43 2.32 2.33 2.27 2.27 2.34 0.072 0.005 0.031

Max. f [N] 292.40 289.58 290.51 298.89 283.89 295.44 294.13 292.39 297.75 295.19 294.23 291.40 289.62 295.44 301.86 299.36 293.78 289.73 291.13 287.62 289.08 298.78 290.61 300.96 291.52 298.92 293.08 291.91 289.51 290.98 293.32 4.18 17.48 0.01

126

A Appendix

Table A.3

# 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 μ σ σ2 CV

Details on the quasistatic 45◦ orientation three point bending tests

Length L [ mm ] 39.92 40.05 39.97 40.02 40.07 40.03 39.91 40.05 40.10 40.04 40.04 39.96 40.06 40.10 40.17 40.02 39.87 39.61 39.99 40.01 40.08 39.94 39.98 39.99 40.03 40.02 39.99 40.08 40.05 40.02 40.01 0.096 0.009 0.002

Width W [ mm ] 9.92 10.16 10.01 9.94 10.00 9.93 9.89 10.00 9.92 9.90 9.96 9.83 9.92 9.94 9.98 10.01 9.92 9.84 10.00 9.81 9.91 9.84 9.91 9.85 10.03 9.81 10.05 9.91 10.20 9.81 9.94 0.093 0.009 0.009

Thickness T [ mm ] 3.19 3.19 3.18 3.17 3.18 3.15 3.19 3.15 3.17 3.19 3.16 3.16 3.16 3.15 3.16 3.15 3.18 3.19 3.19 3.18 3.18 3.18 3.18 3.19 3.19 3.18 3.17 3.19 3.17 3.19 3.18 0.014 0.000 0.004

Mass m[g] 1.41 1.45 1.42 1.41 1.42 1.41 1.41 1.42 1.41 1.41 1.41 1.39 1.41 1.41 1.42 1.42 1.41 1.38 1.42 1.39 1.41 1.39 1.41 1.39 1.43 1.40 1.43 1.40 1.45 1.39 1.41 0.016 0.000 0.012

Density ρ [ g/ccm ] 1.12 1.12 1.12 1.12 1.11 1.13 1.12 1.13 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.11 1.11 1.11 1.12 1.12 1.12 1.10 1.12 1.12 1.13 1.11 1.12 1.11 1.12 0.005 0.000 0.005

Max. w [ mm ] 2.60 2.81 2.66 2.75 2.72 2.68 2.74 2.67 2.79 2.73 2.66 2.84 2.59 2.74 2.69 2.61 2.70 2.64 2.66 2.80 2.79 2.75 2.73 2.71 2.63 2.68 2.61 2.58 2.82 2.71 2.70 0.070 0.005 0.026

Max. f [N] 254.99 264.91 255.39 256.88 260.31 251.79 257.46 257.89 256.12 254.84 251.57 255.12 251.24 255.92 254.04 255.02 253.76 251.32 257.86 254.43 256.51 254.48 257.23 250.59 259.12 253.29 259.04 252.46 265.75 248.12 255.58 3.78 14.33 0.01

A.5 Three point bending test results

Table A.4

# 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 36 37 38 39 40

127

Details on the quasistatic 90◦ orientation three point bending tests

Length L [ mm ] 39.97 39.95 40.04 39.98 40.05 40.12 40.02 40.09 40.03 40.18 40.23 40.01 40.21 40.07 40.00 40.10 40.01 39.98 40.05 40.08 39.97 40.01 40.01 40.00 39.98 40.17 40.04 40.05 39.98 40.08 40.18 40.07 40.07 39.97 39.95 39.20 39.98 40.01 40.13 40.05

Width W [ mm ] 9.95 9.93 9.92 9.96 9.95 9.96 10.05 9.94 9.94 9.95 9.88 9.93 9.99 9.96 9.94 9.97 9.97 9.98 9.92 10.01 9.97 9.93 9.94 9.88 9.94 9.95 9.99 9.95 9.91 9.97 10.04 10.04 9.98 10.00 9.99 10.01 9.98 9.95 9.99 9.97

Thickness T [ mm ] 3.19 3.18 3.17 3.19 3.18 3.18 3.17 3.17 3.18 3.20 3.18 3.18 3.18 3.17 3.19 3.18 3.17 3.21 3.18 3.17 3.18 3.18 3.20 3.18 3.19 3.18 3.18 3.21 3.17 3.18 3.19 3.17 3.19 3.16 3.18 3.16 3.17 3.17 3.17 3.17

Mass m[g] 1.43 1.42 1.42 1.43 1.42 1.43 1.44 1.43 1.43 1.43 1.43 1.43 1.44 1.43 1.43 1.43 1.43 1.44 1.42 1.44 1.43 1.43 1.43 1.42 1.43 1.43 1.43 1.43 1.42 1.43 1.45 1.43 1.44 1.43 1.43 1.40 1.43 1.43 1.44 1.44

Density ρ [ g/ccm ] 1.12 1.13 1.13 1.13 1.12 1.13 1.13 1.13 1.13 1.12 1.13 1.13 1.12 1.13 1.12 1.12 1.13 1.13 1.12 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.12 1.13 1.13 1.13 1.12 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13

Max. w [ mm ] 3.17 2.61 2.87 3.14 3.07 2.93 2.86 2.98 3.13 2.91 3.10 3.09 3.16 3.16 2.77 2.87 3.05 2.74 2.82 2.96 2.96 2.82 3.08 3.03 2.87 2.89 2.87 2.89 2.97 3.36 3.02 2.73 2.98 2.83 3.03 3.02 3.18 2.86 2.91 3.11

Max. f [N] 224.96 216.63 223.43 230.46 222.11 223.98 222.29 226.19 227.09 224.71 224.79 227.26 226.15 227.99 220.16 219.29 222.60 226.47 221.89 222.61 225.45 222.05 228.09 224.95 223.15 222.65 226.45 221.09 221.86 229.01 230.99 218.88 224.82 220.69 228.55 228.38 230.01 222.43 225.53 230.52

128

A Appendix

Continuation of details on the quasistatic 90◦ orientation three point bending tests

# 41 42 43 44 45 46 47 48 49 50 μ σ σ2 CV

Length L [ mm ] 40.00 40.03 39.97 40.05 40.00 40.01 40.02 40.01 40.09 40.12 40.03 0.136 0.018 0.003

Width W [ mm ] 9.99 10.02 9.98 9.97 9.98 9.99 9.97 9.99 10.00 9.98 9.97 0.036 0.001 0.004

Thickness T [ mm ] 3.17 3.17 3.17 3.18 3.18 3.18 3.18 3.18 3.18 3.17 3.18 0.011 0.000 0.003

Mass m[g] 1.44 1.44 1.43 1.43 1.43 1.44 1.43 1.44 1.44 1.43 1.43 0.008 0.000 0.006

Density ρ [ g/ccm ] 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 1.13 0.004 0.000 0.003

Max. w [ mm ] 3.18 2.96 2.85 3.25 2.90 2.87 3.06 2.94 2.73 2.93 2.97 0.148 0.022 0.050

Max. f [N] 232.41 230.94 224.60 229.29 223.22 225.78 222.81 227.07 224.71 228.17 225.11 3.46 12.00 0.02

A.5 Three point bending test results

Table A.5

# 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 μ σ σ2 CV

129

Details on the dynamic 0◦ orientation three point bending tests

Length L [ mm ] 40.14 40.01 40.22 40.20 40.09 40.13 39.99 40.15 40.07 40.19 40.13 39.97 40.09 40.22 40.04 40.16 40.05 40.03 40.01 40.11 40.11 40.09 40.16 40.08 40.08 40.19 40.18 40.12 40.09 40.09 40.11 0.066 0.004 0.002

Width W [ mm ] 9.97 9.90 9.70 9.96 10.01 9.97 10.00 9.96 10.00 9.97 9.97 9.98 9.99 9.98 9.99 9.98 9.97 9.97 10.00 9.95 9.94 9.99 9.95 9.90 10.00 9.93 9.93 9.96 9.94 9.95 9.96 0.055 0.003 0.006

Thickness T [ mm ] 3.18 3.21 3.19 3.18 3.20 3.18 3.17 3.16 3.17 3.17 3.17 3.15 3.16 3.16 3.17 3.19 3.18 3.17 3.16 3.16 3.18 3.16 3.18 3.21 3.17 3.19 3.17 3.19 3.18 3.15 3.18 0.015 0.000 0.005

Mass m[g] 1.42 1.41 1.39 1.42 1.43 1.42 1.42 1.42 1.42 1.41 1.42 1.41 1.41 1.42 1.42 1.42 1.42 1.41 1.42 1.41 1.42 1.42 1.42 1.40 1.42 1.42 1.41 1.41 1.41 1.40 1.41 0.008 0.000 0.005

Density ρ [ g/ccm ] 1.12 1.11 1.11 1.12 1.11 1.12 1.12 1.12 1.12 1.11 1.12 1.12 1.12 1.12 1.12 1.11 1.12 1.11 1.12 1.12 1.12 1.12 1.12 1.10 1.12 1.11 1.12 1.11 1.11 1.12 1.12 0.004 0.000 0.004

Max. w [ mm ] 2.47 2.23 2.48 2.46 2.40 2.32 2.23 2.51 2.28 2.23 2.35 2.29 2.34 2.33 2.30 2.29 2.27 2.35 2.37 2.44 2.35 2.28 2.26 2.27 2.31 2.30 2.35 2.30 2.28 2.29 2.33 0.074 0.005 0.032

Max. f [N] 552.97 516.51 513.49 523.92 499.04 568.18 554.41 555.09 551.02 524.90 563.34 553.75 574.99 562.05 568.29 571.83 546.65 572.88 571.94 489.27 572.11 559.50 572.63 561.67 581.35 568.12 555.86 565.63 547.67 568.66 552.92 23.19 537.94 0.04

130

A Appendix

Table A.6

# 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 μ σ σ2 CV

Details on the dynamic 30◦ orientation three point bending tests

Length L [ mm ] 40.02 39.89 40.09 40.10 40.10 40.18 40.14 40.11 40.19 39.81 40.01 40.10 40.08 40.01 40.16 40.12 39.90 39.92 40.08 40.10 40.13 40.08 40.06 40.06 40.03 40.05 40.06 40.04 40.11 40.08 40.06 0.085 0.007 0.002

Width W [ mm ] 9.89 9.96 10.01 10.00 9.98 9.93 9.97 9.99 9.96 10.01 9.97 9.99 9.97 9.98 9.97 9.91 9.92 10.01 9.86 10.00 9.94 10.02 9.90 10.01 9.97 9.94 9.97 9.97 9.98 10.01 9.97 0.039 0.002 0.004

Thickness T [ mm ] 3.18 3.19 3.23 3.17 3.18 3.17 3.16 3.17 3.17 3.21 3.17 3.17 3.17 3.18 3.20 3.17 3.18 3.18 3.16 3.18 3.16 3.18 3.17 3.17 3.20 3.18 3.17 3.18 3.19 3.18 3.18 0.015 0.000 0.005

Mass m[g] 1.41 1.42 1.43 1.42 1.42 1.42 1.42 1.42 1.42 1.43 1.42 1.42 1.42 1.43 1.43 1.41 1.41 1.43 1.40 1.43 1.41 1.43 1.41 1.43 1.42 1.42 1.42 1.44 1.44 1.42 1.42 0.008 0.000 0.006

Density ρ [ g/ccm ] 1.12 1.12 1.10 1.12 1.12 1.12 1.12 1.11 1.12 1.11 1.12 1.12 1.12 1.13 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.11 1.12 1.12 1.13 1.12 1.12 1.12 0.005 0.000 0.005

Max. w [ mm ] 2.33 2.48 2.49 2.41 2.74 2.48 2.57 2.57 2.73 2.52 2.59 2.30 2.72 2.67 2.54 2.55 2.47 2.69 2.73 2.51 2.28 2.50 2.61 2.52 2.47 2.51 2.42 2.31 2.17 2.62 2.52 0.14 0.02 0.056

Max. f [N] 457.78 443.75 456.54 425.35 466.93 413.51 454.44 444.39 481.19 448.64 460.79 418.29 452.03 435.31 450.88 444.20 409.36 469.28 475.66 436.44 428.34 447.16 447.34 441.35 404.67 434.53 447.58 434.44 464.84 446.81 444.73 18.38 337.78 0.04

A.5 Three point bending test results

Table A.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 26 27 28 29 30 μ σ σ2 CV

131

Details on the dynamic 45◦ orientation three point bending tests

Length L [ mm ] 39.99 39.83 40.06 39.94 39.94 40.05 40.10 40.17 39.92 40.07 40.08 40.09 40.16 40.12 40.07 40.08 39.96 40.07 40.12 40.10 40.09 40.12 40.04 40.04 39.89 40.11 40.06 40.14 40.10 40.10 40.05 0.081 0.007 0.002

Width W [ mm ] 9.98 9.86 9.97 9.95 9.86 9.95 9.93 9.92 9.99 9.87 9.87 9.89 9.89 9.90 9.89 9.88 9.92 9.97 9.88 9.91 9.96 9.97 9.87 9.90 9.95 9.90 10.00 9.96 9.90 9.97 9.92 0.042 0.002 0.004

Thickness T [ mm ] 3.16 3.15 3.16 3.15 3.17 3.17 3.18 3.17 3.21 3.17 3.17 3.16 3.16 3.15 3.17 3.17 3.17 3.17 3.17 3.18 3.16 3.16 3.16 3.18 3.17 3.17 3.19 3.16 3.17 3.18 3.17 0.012 0.0 0.004

Mass m[g] 1.41 1.39 1.42 1.41 1.40 1.41 1.42 1.42 1.44 1.41 1.41 1.41 1.41 1.41 1.40 1.41 1.41 1.42 1.41 1.42 1.42 1.42 1.40 1.42 1.41 1.41 1.42 1.41 1.41 1.43 1.41 0.009 0.0 0.007

Density ρ [ g/ccm ] 1.12 1.13 1.13 1.13 1.12 1.12 1.12 1.12 1.13 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.13 1.12 1.12 1.12 1.12 1.12 1.11 1.12 1.12 1.13 1.12 0.003 0.0 0.003

Max. w [ mm ] 2.73 2.73 2.57 2.72 2.58 2.75 2.56 2.61 2.75 2.59 2.26 2.79 2.65 2.17 2.38 2.81 2.73 2.66 2.51 2.76 2.67 2.67 2.75 2.61 2.95 2.78 2.62 2.57 2.76 2.33 2.63 0.167 0.028 0.064

Max. f [N] 428.83 396.77 388.37 416.12 381.22 390.65 359.36 394.33 403.40 377.61 387.43 377.19 391.12 360.04 347.87 370.53 387.47 383.78 316.01 396.56 390.04 385.88 419.70 395.81 331.02 408.91 407.94 379.64 409.93 362.34 384.86 24.47 598.67 0.06

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A Appendix

Table A.8

# 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 μ σ σ2 CV

Details on the dynamic 90◦ orientation three point bending tests

Length L [ mm ] 40.06 40.02 39.97 40.08 40.00 40.05 40.04 39.96 39.96 39.97 40.14 39.93 40.03 40.08 40.09 40.04 40.01 40.04 39.97 40.11 40.06 39.98 39.97 40.01 39.96 39.97 40.05 40.00 40.05 39.99 40.02 0.051 0.003 0.001

Width W [ mm ] 10.03 10.01 9.98 10.03 9.99 9.98 9.99 10.00 9.99 10.01 9.97 9.97 10.00 9.87 9.97 9.98 10.00 10.01 10.05 10.01 10.00 9.98 10.01 9.99 9.97 9.99 9.98 10.00 10.07 10.05 10.00 0.034 0.001 0.003

Thickness T [ mm ] 3.16 3.17 3.15 3.18 3.18 3.17 3.16 3.18 3.17 3.17 3.17 3.16 3.17 3.16 3.17 3.17 3.18 3.17 3.17 3.18 3.16 3.17 3.17 3.17 3.19 3.18 3.16 3.16 3.18 3.17 3.17 0.009 0.0 0.003

Mass m[g] 1.43 1.43 1.42 1.43 1.43 1.43 1.43 1.42 1.43 1.43 1.43 1.42 1.43 1.41 1.43 1.43 1.44 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.44 1.44 1.43 0.005 0.0 0.004

Density ρ [ g/ccm ] 1.13 1.13 1.13 1.12 1.12 1.13 1.13 1.12 1.13 1.12 1.13 1.13 1.13 1.13 1.13 1.12 1.13 1.12 1.13 1.12 1.13 1.13 1.13 1.13 1.12 1.13 1.13 1.13 1.12 1.13 1.13 0.003 0.0 0.003

Max. w [ mm ] 2.62 2.20 2.01 2.47 2.18 2.58 2.53 2.38 2.58 2.21 2.13 2.20 2.42 2.17 2.21 2.26 2.73 2.21 2.16 2.21 2.24 2.59 2.21 2.34 2.23 2.12 2.20 2.20 2.23 2.44 2.31 0.176 0.031 0.076

Max. f [N] 347.46 332.46 271.56 275.04 348.52 328.18 304.93 283.82 339.93 339.94 258.51 342.28 267.65 339.83 332.88 333.45 325.89 334.27 287.62 343.93 346.21 329.16 346.04 318.82 327.94 335.75 324.57 339.55 323.81 290.12 320.67 26.56 705.47 0.08

A.5 Three point bending test results

Figure A.21

Maximum deflection probability plots of the quasistatic three point bending tests

Figure A.22

Maximum force probability plots of the quasistatic three point bending tests

133

134

A Appendix

Figure A.23

Maximum deflection probability plots of the dynamic three point bending tests

Figure A.24

Maximum force probability plots of the dynamic three point bending tests

A.6 Results Monte Carlo Simulation

A.6 Results Monte Carlo Simulation A.6.1 Histograms of all design variables

Figure A.25

Histogram length

Figure A.26

Histogram width

135

136

A Appendix

Figure A.27

Histogram thickness

Figure A.28

Histogram translation specimen

A.6 Results Monte Carlo Simulation

Figure A.29

Histogram rotation specimen

Figure A.30

Histogram friction

137

138

Figure A.31

A Appendix

Histogram rupture strain

A.6 Results Monte Carlo Simulation

A.6.2 Scatter plots of all design variables

Figure A.32

Scatter plot length - deflection

Figure A.33

Scatter plot length - force

139

140

A Appendix

Figure A.34

Scatter plot width - deflection

Figure A.35

Scatter plot width - force

A.6 Results Monte Carlo Simulation

Figure A.36

Scatter plot thickness - deflection

Figure A.37

Scatter plot thickness - force

141

142

A Appendix

Figure A.38

Scatter plot translation specimen - deflection

Figure A.39

Scatter plot translation specimen - force

A.6 Results Monte Carlo Simulation

Figure A.40

Scatter plot rotation specimen - deflection

Figure A.41

Scatter plot rotation specimen - force

143

144

A Appendix

Figure A.42

Scatter plot friction - deflection

Figure A.43

Scatter plot friction - force

A.6 Results Monte Carlo Simulation

Figure A.44

Scatter plot rupture strain - deflection

Figure A.45

Scatter plot rupture strain - force

145