AutoUni – Schriftenreihe Band 99 Herausgegeben von/Edited by Volkswagen Aktiengesellschaft AutoUni

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Herausgegeben von/Edited by Volkswagen Aktiengesellschaft AutoUni Brieffach 1231 D-38436 Wolfsburg http://www.autouni.de

Steffen Ropers

Bending Behavior of Thermoplastic Composite Sheets Viscoelasticity and Temperature Dependency in the Draping Process

Steffen Ropers Wolfsburg, Germany Dissertation, Friedrich-Alexander University Erlangen Nuremberg, 2016 Any results, opinions and conclusions expressed in the AutoUni Schriftenreihe are solely those of the author(s).

AutoUni – Schriftenreihe ISBN 978-3-658-17593-1 ISBN 978-3-658-17594-8  (eBook) DOI 10.1007/978-3-658-17594-8 Library of Congress Control Number: 2017934528 © Springer Fachmedien Wiesbaden GmbH 2017 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, express 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Fachmedien Wiesbaden GmbH The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Foreword Because of their high strength- and stiffness-to-weight ratio, their processability as well as their recyclability, composite parts made of continuous fiber reinforced thermoplastic prepregs are presenting an attractive alternative to the automotive industry for structural applications. During manufacturing, these prepregs must be shaped and draped into complicated shapes, a step that is often hindered by wrinkle formation and changes in fiber orientation, which consequently affect the mechanical performance of these parts. While the shear deformation of thermoplastic prepregs is well understood, the bending behavior has remained somewhat of a mystery. In an effort to improve the draping process of thermoplastic prepregs and contribute to the understanding of the deformation behavior of such materials, Steffen Ropers presents this work which includes a series of experiments and simulations, that allow him to characterize the viscoelastic bending behavior of these prepregs as a function of deformation, time scale and temperature. Steffen Ropers outlines the general ideas and problems in a clear fashion, and presents the reader with an introduction to viscoelastic concepts as well as the technological background and state-of-the-art of thermoplastic prepregs. He pedagogically introduces viscoelasticity by using the Maxwell model to aid in the understanding of time dependent material response. He proceeds to outline the kinematic and material based, and finite element draping simulation approaches by presenting the concepts of tensile and shear deformation, as well as friction effects. Once Steffen Ropers has laid the groundwork, he shows that current approaches to characterize bending, namely the cantilever bending test, fails to introduce time dependences into the material response. Consequently, there is no direct relation between bending moment and curvature, a reason why the current approaches use complex curve fitting techniques to represent the cantilever experiments, when simulating the bending process. After clearly demonstrating the shortcomings of the current techniques, Mr. Ropers introduces time effects in the bending behavior of thermoplastic prepregs by performing dynamic mechanical analyses (DMA) as a function of temperature at three different frequencies. The frequencies of 0.1, 1 and 10 Hz are congruent with processing time scales of 10, 1 and 0.1 seconds, respectively. At this point of his thesis, Steffen Ropers has enlightened the reader on the clear impact of time and temperature in the bending behavior of these materials using small deformations and staying in the linear viscoelastic domain. To dive into the large deformation non-linear viscoelastic domain, he uses a new bending device, where temperature and curvature can be controlled, in addition to allow control of the bending speed. With this technique he is able to characterize, for the first time, the complex material behavior of thermoplastic prepregs as a function time, temperature and deformation. With single step tests at different speeds, he clearly shows stress relaxation phenomena, and with dynamic tests at different frequencies, he is able to demonstrate the inner friction or loss through classic Lissajous loops. For his dynamic tests, he introduces constant speed triangular wave experiments, as well as classic sinusoidal deformation experiments.

VI

Foreword

In the second part of his work, Steffen Ropers is able to reproduce his experiments using simulation techniques, putting his experimental work on a sound theoretical foothold. Using his own cantilever test finite difference simulation, Mr. Ropers again shows that, while one can fit a single test rather well, due to the absence of time effects, the purely elastic geometrically nonlinear analysis fails to have a unified experimental-theoretical correlation over a large range of tests. Hence, Mr. Ropers concentrates on simulating his own dynamic tests. He introduces a temperature proportionality shift, which works quite well when dynamically measuring the bending stiffness of woven and unidirectional thermoplastic prepregs. He also performs a time-temperature superposition using DMA results and is able to show a time-temperature shift. Finally, he is able to methodically and successfully model the single step experiments, the triangular wave experiments and the sinusoidal experiments for different materials, thus closing the loop with his experimental work. With this work Steffen Ropers is able to demonstrate experimentally and numerically that to characterize the deformation behavior of prepregs is not only necessary to account for geometrically non-linear and complex effects, but it is also necessary to include time and temperature to have a complete analysis. Madison, U.S.A.

Prof.Dr. Prof.hon. Tim A. Osswald

Danksagung An dieser Stelle möchte ich einigen Menschen für ihre Unterstützung danken, ohne die diese Arbeit so nicht möglich gewesen wäre. Tim, Dir vielen Dank für die Betreuung über diese drei Jahre. Das freundschaftliche Verhältnis zu Dir und der lange gemeinsame akademische Weg waren einige der Gründe, überhaupt den Weg der Promotion zu gehen und waren auch ein steter Rückhalt. Deine Ratschläge und, dass Du immer für ein Gespräch Zeit hattest, hat mir ebenfalls sehr geholfen. Prof. Dirk Schubert, Ihnen danke ich für die Diskussion meiner Arbeit und Ihre Bereitschaft das Zweitgutachten zu übernehmen. Danke an meine Kollegen der Volkswagen Konzernforschung für das freundschaftliche Umfeld und die vielen Freiheiten, um an meinem Thema zu arbeiten. Ich hoffe, die eine oder andere Freundschaft bleibt auch nach meiner Zeit hier erhalten. Vielen Dank Anne für deinen Rat und Tat, dein offenes Ohr, deine ehrliche Meinung und die Betreuung. Die schriftliche Ausarbeitung wäre sicherlich noch voller Fehler ohne dein beherztes Korrektur-Lesen, Katja, und die Mechanik Nachhilfe war immer eine Bereicherung. Olaf, Dir vielen Dank, dass Du mir die Promotion ermöglicht hast und für die angeregten Diskussionen. Besonderer Dank gilt Dir, Marton, für deine Unterstützung bei den zahlreichen Versuchen und deiner anhaltenden Begeisterung. Ich wünsche Dir weiterhin viel Erfolg bei deiner Promotion. Mama und Papa, danke, dass Ihr meinen akademischen Werdegang überhaupt ermöglicht habt und für eure Unterstützung und Zuversicht. Dirk und Beni, euch ebenfalls danke für eure Unterstützung, das Lesen und die Diskussion. Berta, Hansi, Phili und Rom, danke für jede Gelegenheit dem Doktoranden-Alltag zu entkommen. Caro, danke, dass Du stets an meiner Seite gestanden hast und mich immer wieder bestärkt hast. Igelsdorf, Germany

Steffen Ropers

Bending Behavior of Thermoplastic Composite Sheets Viscoelasticity and Temperature Dependency in the Draping Process Biegeverhalten von Organoblechen Viskoelastizität und Temperatur-Abhängigkeit im Drapier-Prozess Der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Doktor der Ingenieurswissenschaften (Dr.-Ing.) vorgelegt von Steffen Ropers aus Forchheim Als Dissertation genehmigt von der Technischen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der mündlichen Prüfung: 16.12.2016 Vorsitzender des Promotionsorgans: Prof. Dr. Reinhard Lerch Gutachter: Prof. Dr. Tim A. Osswald Prof. Dr. Dirk W. Schubert

Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XV

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XVII

Kurzfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XXI

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XXIII

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4

2

Thermoplastic Prepregs . . . . . . . . . . . . . . . . . 2.1 Thermoplastic Polymers . . . . . . . . . . . . . . 2.2 Viscoelasticity of Polymers . . . . . . . . . . . . . 2.2.1 Boltzmann Superposition Principle . . . . 2.2.2 Maxwell Model . . . . . . . . . . . . . . . 2.3 Structure of Continuous Fiber Reinforced Polymers 2.4 Manufacturing of Thermoplastic Prepregs . . . . . 2.5 Processing: Thermoforming Process . . . . . . . .

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Draping Simulation of Thermoplastic Prepregs . . . . . 3.1 Overview of Draping Simulation Techniques . . . . . 3.1.1 Kinematic Approach . . . . . . . . . . . . . 3.1.2 Finite Element Approach . . . . . . . . . . . 3.2 Material Characterization of Thermoplastic Prepregs 3.2.1 Tensile Characterization . . . . . . . . . . . 3.2.2 Shear Characterization . . . . . . . . . . . . 3.2.3 Friction Characterization . . . . . . . . . . .

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4

Bending Characterization of Textile Composites . . . 4.1 Materials . . . . . . . . . . . . . . . . . . . . . . 4.2 Cantilever Test . . . . . . . . . . . . . . . . . . . 4.2.1 Cantilever Experiments on GF/PA6 Fabric . 4.3 Dynamic Mechanical Analysis . . . . . . . . . . . 4.3.1 DMA Experimental Set-Up . . . . . . . . 4.3.2 DMA Results GF/PA6 Fabric . . . . . . . 4.3.3 DMA Results GF/PA6 Fabric, conditioned

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31 31 33 36 40 41 43 43

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Contents . . . . . . .

45 45 47 48 51 54 58

Simulation of Bending Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Cantilever Results as Input for Draping Simulation . . . . . . . . . . . . . . . 5.2 Temperature Proportionality Method . . . . . . . . . . . . . . . . . . . . . . . 5.3 Linear Viscoelastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Material Model subjected to Single Deformation in Rheometer Bending Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Material Model subjected to Oscillatory Constant Deformation Rate . . 5.4 Non-Linear Viscoelastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Material Model subjected to Single Deformation and Relaxation . . . . 5.4.2 Temperature Dependency . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Material Model subjected to Oscillatory Constant Deformation Rate . . 5.4.4 Material Model subjected to Sinusoidal Deformation . . . . . . . . . .

61 61 63 66

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Bending Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Simulation of Bending Behavior . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 85

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.4

4.5 4.6 5

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4.3.4 DMA Results CF/PA6 UD Tape . . . . . . . . . . Dynamic, High Curvature Bending Tests . . . . . . . . . . 4.4.1 Test Procedure for GF/PA6 and CF/PA6 . . . . . . 4.4.2 Results of Bending Experiments CF/PA6 UD Tape 4.4.3 Results of Bending Experiments GF/PA6 Fabric . Oscillatory Constant Deformation Rate . . . . . . . . . . . Sinusoidal Deformation . . . . . . . . . . . . . . . . . . .

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70 72 74 77 78 80 81

List of Figures 1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 3.1

3.2 3.3 3.4 3.5

Dependency of the tensile strength of composites on the angle between fiber and load direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . μCT scan of an exemplary part made of thermoplastic pre-impregnated fabric material with shearing and wrinkling . . . . . . . . . . . . . . . . . . . . . . . Molecular morphology of amorphous (left) and semi-crystalline (right) thermoplastics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dependency of the shear modulus of amorphous and semi-crystalline polymers Stress response to sinusoidal strain for (a) elastic, (b) viscous and (c) viscoelastic material behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic representation of the complex modulus E ∗ with its real and imaginary part, E  and E  respectively, and the phase lag δ . . . . . . . . . . . . . . . . . Stress response to sinusoidal strain for non-linear viscoelastic material behavior Schematic demonstration of Boltzmann’s superposition principle . . . . . . . . Schematic representation of the Maxwell model . . . . . . . . . . . . . . . . . Storage and loss modulus of a Maxwell model as a function of angular frequency Normalized Lissajous figures of a Maxwell model at various frequencies for the parameters E = 0.5 GPa and τ = 3 s . . . . . . . . . . . . . . . . . . . . . . . Schematic structure of unidirectional tapes (left), fabrics (middle) and non-crimp fabrics (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Most common fabric types: plain (left), twill (middle) and satin weave (right) . Schematic of polymer solution impregnation process . . . . . . . . . . . . . . Schematic of powder impregnation process . . . . . . . . . . . . . . . . . . . Schematic of commingling process along with a cross section of a commingled yarn; schematic of hybrid textile . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of a monomer impregnation process . . . . . . . . . . . . . . . . . Basic steps of the thermoforming process . . . . . . . . . . . . . . . . . . . . Deformation mechanisms of textile materials . . . . . . . . . . . . . . . . . . Schematic illustration of the kinematic approach: (a) component surface with starting point and initial orthogonal fiber directions (b) spheres with radius equal to the defined mesh size on fiber paths (c) new mesh node in section point of spheres (d) new mesh element generated by connection of new node with nearest neighbor nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of the material stiffnesses: (a) only tensile stiffness, (b) tensile and shear stiffness, (c) tensile, shear and bending stiffness . . . . . . . . . . . . . . Schematic of uniaxial (left) and biaxial (right) tensile testing . . . . . . . . . . Schematic of shear characterization testing with shear force Fs and shear angle γ Schematic of the picture frame test setup . . . . . . . . . . . . . . . . . . . . .

1 3 6 6 7 8 9 9 11 12 13 13 14 16 16 17 18 19 20

22 23 24 25 26

XII

List of Figures 3.6 3.7

Schematic of a specimen under bias extension test . . . . . . . . . . . . . . . . Schematic friction test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 29

4.1 4.2 4.3 4.4 4.5

Microscope picture of glass fiber yarn (left) and single glass fiber (right) . . . . Twill weave of Tepex fabric material . . . . . . . . . . . . . . . . . . . . . . . Microscope picture of carbon fiber yarn (left) and single carbon fiber (right) . . Samples of fabric and unidirectional material; matrix was removed at one end . Cantilever setup with 1 area of support, 2 slide with ruler, 3 stop, 4 side frame (transparent) and 5 specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of behavior of the bending moment over curvature at the embedded point for different bending lengths and load cases . . . . . . . . . . . . . . . . . . . Fixture for bending experiments with thermoplastic prepregs above the matrix melting point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of a cantilever beam under the distributed load of its own weight . . Bending lines for 90° orientation show highly non-linear behavior from bending lengths of 90 mm on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moment over curvature for 90° orientation along the bending lines of different lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DMA: dual cantilever fixture (left) and three-point bending fixture (right) . . . Tepex fabric material: Storage modulus, loss modulus and damping over temperature at 0.1, 1 and 10 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditioned Tepex fabric material: Storage modulus, loss modulus and damping over temperature at 0.1, 1 and 10 Hz . . . . . . . . . . . . . . . . . . . . . . . Unidirectional material: Storage modulus, loss modulus and damping over temperature at 0.1, 1 and 10 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . Bending setup in a rheometer with one fixed and one rotating shaft . . . . . . . Bending moment for a CF/PA6 UD during deformation and relaxation at 60 ◦C and with various deformation times . . . . . . . . . . . . . . . . . . . . . . . . Bending moment for a CF/PA6 UD during deformation and relaxation in the solid regime and with a deformation time of 11.8 s . . . . . . . . . . . . . . . Buckled Tepex specimen at a deformation time of 118 s and a temperature of 220 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending moment for a CF/PA6 UD during deformation and relaxation in the molten regime and with a deformation time of 1.18 s . . . . . . . . . . . . . . Bending moment for a CF/PA6 UD during deformation and relaxation at 260 ◦C and various deformation times . . . . . . . . . . . . . . . . . . . . . . . . . . Bending moment for a GF/PA6 organo sheet during deformation in the solid regime with a deformation time of 1.18 s; Relaxation was not measured, thus not depicted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending moment for a GF/PA6 organo sheet during deformation and relaxation in the molten regime with a deformation time of 11.8 s . . . . . . . . . . . . . Bending moment for a GF/PA6 organo sheet during deformation and relaxation in the molten regime with a deformation time of 1.18 s . . . . . . . . . . . . . Bending moment for a GF/PA6 organo sheet during deformation and relaxation at 260 ◦C and various deformation times . . . . . . . . . . . . . . . . . . . . .

31 32 33 33

4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21

4.22 4.23 4.24

34 35 37 38 39 39 42 44 44 45 46 48 49 49 50 50

51 52 53 53

List of Figures 4.25 Quasi-static deformation in 118 s at 260 ◦C and cantilever results of 105 mm at 250 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.26 Course of the rotation angle over time during triangular wave experiments . . . 4.27 Bending moment for the Tepex GF/PA6 fabric material at 260 ◦C under triangular wave deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 Bending moment for the Tepex GF/PA6 fabric material at 220 ◦C under triangular wave deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.29 Bending moment for the Tepex GF/PA6 fabric material under triangular wave deformation with a deformation time of 1.18 s . . . . . . . . . . . . . . . . . . 4.30 Bending moment for the Tepex GF/PA6 fabric material under triangular wave deformation with a deformation time of 11.8 s . . . . . . . . . . . . . . . . . . 4.31 Course of the rotation angle over time during sinusoidal wave experiments . . . 4.32 Bending moment for the Tepex GF/PA6 fabric material under sine wave deformation with a frequency of 0.1 Hz . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17

Moment with respect to curvature of 105 mm bending length cantilever experiment, input for FDM simulation and simulation result . . . . . . . . . . . . . . Geometries of simulated and experimental bending lines; the bending stiffness for all simulations was derived from the 105 mm experiment . . . . . . . . . . Predicting the bending stiffness of Tepex at various temperatures by transferring the temperature dependency of the storage modulus . . . . . . . . . . . . . . . Predicting the bending stiffness of Ultratape at various temperatures by transferring the temperature dependency of the storage modulus . . . . . . . . . . . . Isotherms for the Tepex fabric material from DMA measurements . . . . . . . Mastercurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shift factors and WLF representation for Tepex material . . . . . . . . . . . . Schematic of a viscoelastic network with elastic element 0 and n viscoelastic elements 1 − n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless Prony coefficients ei with respect to the corresponding relaxation times τi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rheometer bending experiments: Tepex fabric material at a deformation time of 11.8 s simulated with viscoelastic Prony material . . . . . . . . . . . . . . . . Rheometer bending experiments: Tepex fabric material at a deformation time of 1.18 s simulated with viscoelastic Prony materia . . . . . . . . . . . . . . . . . Rheometer bending experiments: Ultratape UD material at a deformation time of 1.18 s simulated with viscoelastic Prony material . . . . . . . . . . . . . . . Comparison of material response in experiment and simulation with linear viscoelastic material model with a deformation time of 1.18 s . . . . . . . . . . Comparison of material response in experiment and simulation with linear viscoelastic material model with a deformation time of 11.8 s . . . . . . . . . . Schematic of a viscoelastic network with hyperelastic element 0 and three viscoelastic elements 1 − 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depiction of iterative process, experimental and simulated bending moment of the Tepex fabric material at 220 ◦C and a deformation time of 1.18 s . . . . . . Experimental and simulated bending moment of the Tepex fabric material at 220 ◦C and a deformation time of 11.8 s . . . . . . . . . . . . . . . . . . . . .

XIII

54 55 56 56 57 57 58 59 62 63 65 66 67 67 68 69 70 71 72 72 73 74 75 76 77

XIV

List of Figures

5.18 Course of the stress over strain in each power-law strain hardening element for two different strain rates and temperatures . . . . . . . . . . . . . . . . . . . . 5.19 Experimental and simulated bending moment of Tepex fabric material at 210 ◦C and a deformation time of 1.18 s . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Experimental and simulated bending moment of Tepex fabric material at 210 ◦C and a deformation time of 11.8 s . . . . . . . . . . . . . . . . . . . . . . . . . 5.21 Comparison of the Tepex material response in experiment and simulation with non-linear viscoelastic material model with a deformation time of 1.18 s . . . . 5.22 Comparison of the Tepex material response in experiment and simulation with non-linear viscoelastic material model with a deformation time of 11.8 s . . . . 5.23 Comparison of the Tepex material response in experiment and simulation with non-linear viscoelastic material model at 0.1 Hz sine deformation . . . . . . . .

78 79 80 81 81 82

List of Tables 4.1 4.2 4.3 4.4

5.1 5.2 5.3

Main material properties,* information is not disclaimed, ** warp/ weft direction Cantilever bending parameters . . . . . . . . . . . . . . . . . . . . . . . . . . Test procedure for temperatures above melting point . . . . . . . . . . . . . . Rheometer bending test matrix; Temperatures that are marked with ↓ are approached from above, meaning the chamber was heated up to 255 ◦C and subsequently cooled to set-point temperature . . . . . . . . . . . . . . . . . . . .

32 38 47

Material parameters for Prony series and WLF equation . . . . . . . . . . . . . Parameters of Yeoh and Power-law strain hardening model for Tepex material at 210 ◦C and 220 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of Yeoh and Power-law strain hardening model for the Tepex fabric material at 210 ◦C and 220 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

47

77 79

Nomenclature Symbol

Unit

Description

aT

[–]

Shift factor

B

[N mm2 ]

Bending stiffness

BV F

[◦C],

Vogel-Fulcher constant

C1

[–]

WLF model parameter 1

C2

[◦C]

WLF model parameter 2

d

[mm]

Displacement

E

[N mm−2 ], [MPa]

Young’s modulus

e

[–]

Euler number

E∗

[N mm−2 ],

[MPa]

Complex modulus

E

[N mm−2 ],

[MPa]

Storage modulus

E 

[N mm−2 ],

[MPa]

Ea

[J mol−1 ]

Arrhenius model activation energy

F

[N]

Force

Fs

[N mm−1 ]

Shear force, normalized

g

[mm s−2 ]

Gravity

I

[mm4 ]

Moment of inertia

I¯

[–]

Strain invariant

i

[–]

Imaginary number

L

[mm]

Length, constant

l

[mm]

Length, variable

Ltex

[mm]

Textile specimen edge length

M

[N mm]

Moment

m

[kg]

Mass

Mn

[N]

Moment, normalized

P

[–]

Proportionality factor

[K]

Loss modulus

XVIII

Nomenclature

Symbol

Unit

Description

R

[J mol−1 K−1 ]

Universal gas constant

s

[mm]

Curvilinear coordinate

t

[mm]

Thickness

t

[s]

Time

T0

[◦C], [K]

Reference temperature

Tg

[◦C],

[K]

Glass transition temperature

Tm

[◦C],

[K]

Melting temperature

TV F

[◦C],

[K]

Vogel-Fulcher temperature

Ts

[◦C],

[K]

Softening temperature

Tz

[◦C], [K]

Absolute zero temperature

U

[–]

Strain energy density

u

[mm]

Frenet coordinate

v

[mm s−1 ]

Velocity

w

[N mm−1 ]

Weight per unit length

x

[mm]

Cartesian coordinate

y

[mm]

Cartesian coordinate

z

[mm]

Cartesian coordinate

α

[°], [rad]

Rotation angle

δ

[°], [rad]

Phase lag

tan δ

[–]

Damping

ε

[–]

Strain

η

[Pa s]

Viscosity

γ

[°], [rad]

Shear angle

κ

[mm−1 ]

Curvature

ω

[s−1 ]

Angular frequency

φ

[°], [rad]

Tangent angle

σ

[N mm−2 ], [MPa]

Stress

τ

[N mm−2 ],

Shear stress

τi

[s]

[MPa]

Relaxation times

Nomenclature

XIX

Symbol

Unit

Description

θ

[°], [rad]

Half fiber angle

Index

Description

0

Long-term

b

Bending

f

Picture frame, Fiber

E

Elastic

m

Matrix

N

Normal

R

Friction

s

Shear

t

Tensile

tex

Textile

u

Overhang

η

Viscous

Kurzfassung Faser-Verbund-Kunststoffe (FVK), wie thermoplastische Prepregs, bieten ein breites Spektrum an hervorragenden mechanischen Eigenschaften bezogen auf ihre Dichte. Der Verbund wird dabei aus einer Kunststoffmatrix und Verstärkungsfasern aus z.B. Glas oder Kohlenstoff gebildet. Durch diesen Aufbau ist das Material in seinen Eigenschaften anisotrop, d.h. abhängig von der Orientierung der Fasern. Thermoplastische Matrix-Systeme sind, im Vergleich zu duroplastischen, wiederaufschmelzbar. So können kurze Zykluszeiten und damit großserienfähige Prozesse realisiert werden. Um eine Großserie robust zu gestalten, ist eine Absicherung durch die Prozesssimulation unerlässlich. Dafür stehen bereits kommerzielle CAE Methoden zur Verfügung, die Ergebnisgenauigkeit, jedoch, ist noch zu verbessern. Hierzu ist ein Verständnis über das komplexe Materialverhalten, geeignete Charakterisierungsmethoden sowie Modelle vonnöten, die das Materialverhalten simulativ abbilden können. Bei der Verarbeitung von thermoplastischen Prepregs wird die Matrix vor der Umformung aufgeschmolzen, ist also im Verbund bereits enthalten und beeinflusst damit auch deutlich das Materialverhalten während der Umformung. Insbesondere das Verhalten quer zur Faserrichtung und damit auch die Biegesteifigkeit wird durch die Matrix dominiert. Hinsichtlich der Umformsimulation von endlos-faserverstärkten Kunststoffen, der Drapiersimulation, ist eine realitätsnahe Abbildung der Biegesteifigkeit essenziell, da sie entscheidend die Faltenbildung beeinflusst. Im Rahmen dieser Arbeit wurden unterschiedliche Charakterisierungsversuche durchgeführt, um den Einfluss der Temperatur auf das Biegeverhalten und, inwiefern dieses durch Viskoelastizität geprägt ist, zu untersuchen. Cantilever-Versuche ließen dabei nur eine Untersuchung der Temperaturabhängigkeit zu, während dynamisch mechanische Analyse und Rheometer-basierte Biegeversuche zusätzlich auch die Möglichkeit boten viskoelastisches Materialverhalten zu identifizieren. Aufbauend auf den Versuchsergebnissen wurden elastische und verschiedenartige viskoelastische Materialmodelle hinsichtlich ihrer Fähigkeit, das experimentell bestimmte Verhalten abzubilden, analysiert. Eine umfassende Beschreibung des Biegeverhaltens von thermoplastischen Prepregs ist letztlich nur durch ein temperatur-abhängiges, nicht-linear viskoelastisches Materialmodell möglich.

Abstract Fiber reinforced plastics (FRP) such as thermoplastic prepregs feature a wide range of outstanding mechanical properties with respect to their density. The composite material is composed of a polymer matrix system and glass or carbon reinforcement fibers. This composite structure comes with anisotropic material behavior, i.e., a dependency on the fiber orientations. Thermoplastic matrix systems, in contrast to thermoset resins, can be remelted, enabling short cycle times and, therefore, large-scale production processes. For a robust large-scale production, a validation of the process by process simulations is indispensable. Therefore, commercial Computer-Aided-Engineering (CAE) tools are available, yet, the accuracy of their results is to be improved. An enhancement of those tools can solely be achieved by understanding the complex material behavior and finding appropriate testing methods as well as models which are capable of representing the actual material behavior. For processing thermoplastic prepregs, the material is heated beyond the matrix’ melting temperature to allow it to be shaped. However, the polymer melt has a significant impact on the forming behavior for the sheets. Especially the behavior transverse to the fiber directions and, thereby, also the bending stiffness are dominated by the matrix. With respect to forming simulations of continuous-fiber reinforced polymers, i.e., draping simulations, it is essential to realistically model the bending stiffness, since it largely determines the formation of wrinkles. In the scope of this work, various characterization experiments were performed, in order to account for the temperature dependency and the viscoelastic nature of the bending stiffness. Cantilever experiments only allowed for a determination of the temperature dependency, while dynamic mechanical analysis and bending experiments, carried out in a rheometer additionally allowed for the identification of viscoelastic material behavior. Based on the experimental results, various elastic and viscoelastic material models were assessed with respect to their capability of representing the experimentally determined material behavior. It was concluded that only a temperature-dependent, non-linear viscoelastic material model is capable of representing all aspects of the bending behavior of thermoplastic prepregs.