Micro-mechanical modeling of continuous glass reinforced isotactic polypropylene Influence of strain rate and temperature Senem Aktaş Çelik
MICRO-MECHANICAL MODELING OF CONTINUOUS GLASS REINFORCED ISOTACTIC POLYPROPYLENE INFLUENCE OF STRAIN RATE AND TEMPERATURE Senem Aktas¸ Çelik
MICRO-MECHANICAL MODELING OF CONTINUOUS GLASS REINFORCED ISOTACTIC POLYPROPYLENE INFLUENCE OF STRAIN RATE AND TEMPERATURE DISSERTATION to obtain the degree of doctor at the University of Twente, on the authority of the rector magnificus, Prof.dr.ir. A. Veldkamp, on account of the decision of the Doctorate Board, to be publicly defended on 10 November 2023 at 16:45 hours by Senem AKTAS¸ ÇELI˙K born on 26 August 1987 in Fatih, Turkey
This dissertation has been approved by Supervisors: Prof.dr.ir. L.E. Govaert Prof.dr.ir. R. Akkerman Co-supervisor: Dr. I. Baran TThis research was carried out under project number 15795 in the framework of the Partnership Program of the Materials innovation institute M2i (www.m2i.nl) and the Technology Foundation TTW (www.stw.nl), which is part of the Netherlands Organization for Scientific Research (www.nwo.nl). Cover: Designed by Gildeprint using two scanning electron micro-graph of glass/isotactic polypropylene composite. Original picture taken by Nick Helthuis. Printed by: Gildeprint, Enschede, The Netherlands. © 2023 by Senem Aktas¸ Çelik, Enschede, The Netherlands. ISBN:978-90-365-5902-7 DOI: 10.3990/1.9789036559034
Graduation committee: Chairman and secretary: Prof.dr.ir H.F.J.M. Koopman University of Twente Supervisors: Prof.dr.ir. L.E. Govaert Prof.dr.ir. R. Akkerman University of Twente University of Twente Co-supervisor: Dr. I. Baran University of Twente Members (in alphabetical order): Prof.dr.ir. A.H. Van Den Boogaard Dr.ir. J.A. Van Dommelen Prof.dr.ir. I.M. Gitman Prof.dr. T. Peijs University of Twente Eindhoven University of Technology University of Twente University of Warwick
Summary Thermoplastic composites have become attractive for many engineering applications due to advantages such as high specific strength, short processing cycles and low cost at high volume production. Within this class of materials, continuous fiber reinforced, unidirectional glass/isotactic polypropylene (G/iPP) is one of the strongest competitors thanks to the balance between its good mechanical properties and low manufacturing cost. To expand its applications in load bearing structures, the ability to predict the mechanical performance under anticipated conditions is essential. Structural components composed of continuous fiber reinforced G/iPP composites can be exposed to various loading and environmental conditions such as load application rates, loading angles and temperatures. As a result of load and temperature dependent nature of their polymer constituents, both the stress-strain response as well as the failure kinetics of G/iPP structures may be affected. Accurate numerical models that can simulate the materials response over a wide range of loading rates and temperatures conditions in a virtual environment would reduce the extensive experimental cost and time to understand the effects of each condition on its mechanical response and failure. Such models also enable investigation of complex, local failure phenomena that are difficult to observe in the experiments. The main goal of this thesis to provide a numerical tool that allows prediction of the mechanical response of continuous fiber reinforced, UD G/iPP composites at different temperatures, strain rates and loading angles by means of a micromechanical approach. The first step towards this objective was to obtain an accurate mathematical description of the stress-strain response of iPP over a large range of strain rates [10−5,10−1] s−1 and temperatures [5−75◦C]. This was achieved by extending the Eindhoven Glassy Polymer (EGP -) model to three processes, each with their own deformation kinetics and time-dependence. These contributions can be assigned to deformations within the amorphous and crystalline phases of iPP. A methodology was developed to determine the model parameters in a straight forward manner. Secondly, a micro-mechanical tool was developed based on the three-process EGP model to represent the mechanical response of iPP in a representative volume element (RVE) scheme. Realistic RVEs were generated from microscopic images of crosssections of the composite material. To determine the appropriate image size to be i
ii an RVE, the micro-structural irregularity of unidirectional G/iPP composite was investigated on microscopy images in terms of its averaged fiber volume fraction, local fiber volume fraction and the nearest neighbourhood distances. An appropriate size was determined as 250 μm for the material under consideration, providing a good representation of both the fiber irregularity and the mechanical response of the actual material. Next, the obtained constitutive relation was combined with realistic RVEs to perform simulations over a range of strain rates, temperatures and loading angles. As microscopic analysis revealed a good adhesion between fibers and matrix, the micromechanical simulations could be performed assuming perfect adhesion. Micromechanical simulations were subsequently employed to get insight in the possible causes of failure at various loading conditions. Simulations for different loading angles could be performed using the same RVEs by use of a novel method to apply the boundary conditions at the desired loading angle. This method was applied for the first time in large deformations. For a loading angle of 90◦, i.e. transverse to the fibers, the simulated stress-strain responses at temperatures of 23◦C, 50◦Cand 90◦C are in a good agreement with the experiments. Moreover, also the stress-strain responses under a tensile loading with angles 20◦, 30◦ and 45◦ with respect to the fiber direction are predicted well. For smaller angles such as 15◦ and 10◦, the strain hardening response was over predicted due to the significant rotation of the fibers during deformation. Finally, the micro-mechanical model was employed to study the local stress and deformation fields in the composite at the point of macroscopic failure. Two separate failure modes were observed in the uniaxial tensile experiments performed on unidirectional G/iPP composites. The first one is pre-yield failure, which is typically observed at 23◦C and 50◦C for the investigated strain rates at large loading angles. The failure mode changes to macroscopic plastic failure with decreasing strain rate, increasing temperature or decreasing loading angle. In the case of pre-yield failure, the hydrostatic stress level appeared to be the main cause. At higher temperatures, or lower loading angles, the magnitude of hydrostatic stress decreases and macroscopic plasticity is observed in the composites’ response. Failure after large macroscopic plastic deformation appears to be controlled by the local equivalent plastic strain. At a loading angle of 45◦, the hydrostatic stress still appears to have a minor role. In conclusion, the present thesis provides a micro-mechanical tool that can effectively simulate the mechanical response of continuous fiber reinforced, unidirectional G/iPP composites over a wide range of strain rates, temperatures and loading angles based on an accurate polymer constitutive model and a realistic RVE scheme with a novel method for the appropriate loading conditions.
Samenvatting Thermoplastische composieten zijn aantrekkelijk voor veel technische toepassingen vanwege hun voordelen zoals hoge specifieke sterkte, korte verwerkingscycli en lage kosten bij productie in grote volumes. Binnen deze klasse van materialen is unidirectioneel glasvezelversterkt isotactisch polypropyleen (G/iPP) een van de sterkste kandidaten vanwege de balans tussen de goede mechanische eigenschappen en lage productiekosten. Om hun toepassingen in dragende constructies verder uit te kunnen breiden, is het essentieel om hun mechanische respons onder bedrijfsomstandigheden te kunnen voorspellen. Structurele, lastdragende componenten kunnen worden blootgesteld aan verschillende belastings- en omgevingsomstandigheden, zoals verschillende reksnelheden, belastingrichtingen en temperaturen. Als gevolg van de tijd- en temperatuurafhankelijke aard van de polymere matrix, kunnen hierbij zowel de spannings-rekrespons als de faalkinetiek van G/iPP-structuren worden beïnvloed. Nauwkeurige numerieke modellen die de materiaalrespons kunnen simuleren over een groot bereik van reksnelheid en temperatuur maken het mogelijk om de mechanische respons in complexe belastingsomstandigheden te bestuderen. Dergelijke modellen maken het ook mogelijk om complexe, lokale faalverschijnselen te onderzoeken die experimenteel moeilijk waarneembaar zijn. Het doel van dit proefschrift is om een micro-mechanisch model te ontwikkelen waarmee de mechanische respons van uni-directionele, glasvezelversterkte G/iPPcomposieten bij verschillende temperaturen, reksnelheden en belastingrichtingen kan worden voorspeld. De eerste stap in deze richting was het verkrijgen van een nauwkeurige wiskundige beschrijving van de spanning-rekrespons van iPP over een groot bereik van reksnelheid [10−5 . . .10−1] s−1 en temperatuur [5 −75◦C]. Dit werd bereikt door het bestaande Eindhoven Glassy Polymer (EGP-) model uit te breiden naar een model met drie afzonderlijke processen, elk met hun eigen deformatiekinetiek en tijdsafhankelijkheid. Deze individuele bijdragen kunnen ieder worden toegewezen aan specifieke moleculaire deformatie processen binnen de amorfe en kristallijne fasen van iPP. Er is een methodologie ontwikkeld om de modelparameters op een eenvoudige manier te bepalen aan de hand van uniaxiale trek en compressie metingen bij verschillende reksnelheden en temperaturen. Met deze uitbreiding van het EGP-model is het voor het eerst mogelijk om het gedrag van een polymeer over zo’n groot temperatuurbereik met een enkele set parameters te beschrijven. iii
iv Het ontwikkelde EGP-model is vervolgens gebruikt in een micromechanische benadering om de mechanische respons van de iPP-matrix te simuleren tijdens belasting van een uni-directioneel (UD) G/iPP composiet. Hierbij wordt gebruik gemaakt van een representatief volume-element (RVE). Realistische RVE’s werden gegenereerd uit microscopische beelden van doorsneden van het composietmateriaal. Om de juiste RVE-grootte te bepalen, werd de micro-structurele onregelmatigheid van unidirectionele G/iPP composiet onderzocht aan microscopische opnamen in termen van de gemiddelde vezelvolumefractie, de lokale vezelvolumefractie en de dichtstbijzijnde buurtafstanden. Een geschikte grootte werd bepaald als 250 μm, wat een goede weergave geeft van zowel de vezelonregelmatigheid als de mechanische respons van het onderhavige materiaal. Het realistische RVE is vervolgens gecombineerd met het verder ontwikkelde EGPmodel om simulaties uit te voeren over een breed spectrum van reksnelheden, temperaturen en belastingrichtingen om inzicht te krijgen in de mogelijke oorzaken van het falen van het composiet onder verschillende belasting omstandigheden. Aangezien microscopische-analyse van breukvlakken een goede hechting tussen vezels en matrix aantoonde, werd in de micro- mechanische simulaties een perfecte hechting tussen vezel en matrix toegepast. Simulaties voor verschillende belastingrichtingen werden uitgevoerd met behulp van dezelfde 3D-RVE met behulp van een nieuwe methode om de randvoorwaarden bij de gewenste belastingrichting toe te passen. Deze methode is hier voor het eerst toegepast bij grote vervormingen. Voor een belastingrichting van 90◦ zijn de gesimuleerde spannings-rekresponsies bij temperaturen van 23◦C, 50◦C en 90◦C in goede overeenstemming met de experimenten. Bovendien wordt de spanningsrekrespons onder een trekbelasting met hoeken 20◦, 30◦ en 45◦ ten opzichte van de vezelrichting goed voorspeld. Voor kleinere hoeken zoals 15◦ en 10◦ werd een te sterke rekverstevigingsrespons voorspeld. Dit verschijnsel wordt veroorzaakt door de aanzienlijke rotatie van de vezels die bij deze belastingrichtingen optreden tijdens vervorming van de RVE. Ten slotte is het micro-mechanische model gebruikt om de lokale spannings- en vervormingsvelden in het composiet te bestuderen op het punt van macroscopisch falen. In de uitgevoerde uniaxiale trekexperimenten werden twee afzonderlijke faalmodi waargenomen in unidirectionele G/iPP-composieten. De eerste faalmodus is brosse breuk voordat de macroscopische vloeispanning bereikt is (pre-yield failure). Deze werd waargenomen bij 23◦C en 50◦C voor de onderzochte reksnelheden bij grote belastingrichtingen. De faalmodus verandert in macroscopisch plastisch falen wanneer de reksnelheid afneemt, de temperatuur stijgt of de belastinghoek kleiner wordt. In het geval van pre-yield failure bleek de hydrostatische spanning de belangrijkste oorzaak te zijn. Bij hogere temperaturen of lagere belastinghoeken neemt de omvang van de hydrostatische spanning af en wordt macroscopische plasticiteit
v waargenomen in de reactie van de composieten. Falen na grote macroscopische plastische vervormingen lijkt te worden gecontroleerd door de lokale equivalente plastische rek. Bij een belastinghoek van 45◦ lijkt de hydrostatische spanning nog steeds een ondergeschikte rol te spelen. Samengevat biedt het huidige proefschrift een micro-mechanisch hulpmiddel dat de mechanische respons van continue vezelversterkte, unidirectionele G/iPP-composieten effectief kan simuleren over een groot bereik van reksnelheden, temperaturen en belastingrichtingen op basis van een nauwkeurig constitutief model en een realistisch RVE-schema met een nieuwe methode om de juiste mechanische randvoorwaarden op te leggen.
Contents Summary i Samenvatting iii 1 Introduction 1 1.1 BackgroundandMotivation.......................... 1 1.2 ProblemStatement ............................... 2 1.3 Objective,ApproachandOutline . . . . . . . . . . . . . . . . . . . . . . . 4 References ....................................... 6 2 Characterization rate and temperature dependent stress-strain response of isotactic polypropylene using a multi-process approach 9 2.1 Introduction ................................... 10 2.2 Phenomenology................................. 12 2.3 NumericalModelling.............................. 15 2.3.1 ConstitutiveModel........................... 15 2.3.2 Methods for Parameter Identification . . . . . . . . . . . . . . . . 18 2.4 Results&Discussion.............................. 26 2.4.1 Process I Spectrum Identification . . . . . . . . . . . . . . . . . . . 26 2.4.2 Process II Spectrum Identification . . . . . . . . . . . . . . . . . . 32 2.4.3 Process III Spectrum Identification . . . . . . . . . . . . . . . . . . 33 2.4.4 Discussion................................ 35 2.5 Conclusion.................................... 36 2.6 Appendix .................................... 38 2.6.1 Relaxation Spectrum Determination . . . . . . . . . . . . . . . . . 38 2.6.2 Relaxation Spectrum Parameters . . . . . . . . . . . . . . . . . . . 41 2.6.3 Updated Ree-Eyring Equation Parameters . . . . . . . . . . . . . 42 References ....................................... 43 vii
viii Contents 3 Determination A Representative Volume Element For Continuous Fiber Reinforced Glass/IPP Composites 47 3.1 Introduction ................................... 48 3.2 ViscoelasticMaterialModel .......................... 50 3.3 Micro-mechanicalFramework. . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.1 DeterminationofRVESize . . . . . . . . . . . . . . . . . . . . . . 52 3.3.2 RVEConstruction ........................... 55 3.3.3 FiniteElementModel ......................... 55 3.4 ResultsandDiscussion............................. 58 3.4.1 The Effect Of Fiber Distribution . . . . . . . . . . . . . . . . . . . 58 3.4.2 Morphological Validation of Image Size . . . . . . . . . . . . . . . 64 3.4.3 Mechanical Validation of Image Size . . . . . . . . . . . . . . . . . 66 3.5 Conclusion.................................... 70 3.6 Appendix .................................... 71 References ....................................... 73 4 Micro-mechanical Analysis Of Continuous Glass Fiber Reinforced iPP Composites At Different Loading Conditions 77 4.1 Introduction ................................... 78 4.2 Methodology .................................. 81 4.2.1 Experimental .............................. 81 4.2.2 NumericalFramework......................... 82 4.3 Results&Discussion.............................. 97 4.4 Conclusion....................................101 4.5 Appendix ....................................103 4.5.1 Identification of Relaxation Spectrum Parameters At Different Temperatures ..............................105 References .......................................106 5 Micro-mechanical Analysis Of Changes In Failure Mode Of UD Glass/iPP Composites Due To The Changes in Strain Rate, Temperature and Loading Angle 113 5.1 Introduction ...................................115 5.2 Experiments ...................................118 5.2.1 Laminate Preparation and Optical Microscopy . . . . . . . . . . . 118 5.2.2 Transverse Tensile Experiments and Scanning Electron Microscopy(SEM).............................119 5.3 NumericalModelling..............................124 5.3.1 ConstitutiveModel...........................124
Contents ix 5.3.2 RVEConstruction ...........................126 5.3.3 FiniteElementModel .........................126 5.3.4 Processing the Results of the Micro-mechanical Analysis . . . . . 129 5.4 Results&Discussion..............................131 5.4.1 Transversetensileloading . . . . . . . . . . . . . . . . . . . . . . . 131 5.4.2 Off-axisloading.............................140 5.5 Conclusion....................................144 5.6 Appendix ....................................146 5.6.1 Off-axis Loading with Periodic Boundary Conditions . . . . . . . 148 References .......................................150 6 Conclusions and Recommendations 157 6.1 SummaryandConclusions ..........................157 6.2 Recommendations ...............................159 References .......................................160 Acknowledgments 162 Cirriculum Vitæ 166 Publications 168
Chapter 1 Introduction 1.1 Background and Motivation The climate crisis is reported among the top risks of the next decade [1]. To fight against the climate change, European carbon emission regulations require reduction of carbon dioxide CO2 emission by 7.6% each year from 2020 to 2030 [2], forcing automotive industry to set CO2 targets and increase their fuel efficiency by 2030 [3]. For instance, automotive groups such as BMW and Volkswagen plan to decrease their CO2 emissions per vehicle by 40% until 2030 [4, 5]. One possible solution to the emission problem is manufacturing lighter cars in order to reduce fuel consumption and hence CO2 emission. New material strategies and innovative materials are necessary to achieve this goal. Thermoplastic composite materials can significantly contribute to the weight reduction, up to 30% and 70% compared with aluminum and steel, respectively. Moreover, they also meet the material recycling standard mandated by the European Union. Thermoplastic composites have been recognized by automotive industry thanks to their high specific strength as well as short processing cycles and low cost at high volume production [6]. Today, many components in cars such as front-end modules (Volkswagen’s Atlas and Teramont SUVs [7]), door (Ford Fiesta [8]) and roof modules, underbody panels (BMW 7 series and Z4 [9]), instrumental panels (Mercedes-Benz S-Class [10]), etc. are made from thermoplastics. The growth in the thermoplastic composite market is expected to increase as the demand for environmentally friendly electric vehicles (EVs) increases [11]. The increasing demand of these materials makes understanding the mechanical behavior of thermoplastic composites essential for safe design and production of thermoplastic composite components in structural load bearing applications. 1
2 Chapter 1. Introduction 1.2 Problem Statement Continuous fiber reinforced, unidirectional (UD) glass/isotactic polypropylene (G/iPP) is one of the strongest competitors among other thermoplastic composites due to the balance between its good mechanical properties and low manufacturing cost [12]. Today, it has relatively limited application fields in automotive industry, which are door modules and instrumental panels [12–14]. The mechanical response of continuous fiber reinforced unidirectional G/iPP composites is of interest to expand their applications further for automotive structures. During their life time, continuous fiber reinforced, UD G/iPP composites might undergo different loading and harsh environmental conditions involving high or low loading speeds, different loading angles and elevated temperatures. Load and temperature dependent behavior of the iPP polymer ultimately contributes to changes not only in the stress-strain response, but also in failure kinetics of these materials. Addressing the effects of each condition on the mechanical response and failure of the material requires extensive experiments, which are costly and time consuming. In addition, investigating the fine details of failure phenomena requires advanced equipment and techniques in the experiments. These challenges can be overcome by developing high-fidelity numerical models that can replicate extreme load and temperature conditions in a virtual environment. These models also allow insight into failure phenomena that are not yet characterized or very complicated to observe in the experiments. UD G/iPP Laminate Polymer T°C , , A schematic overview of the micro-structure Intrinsic response of iPP Homogenization of RVEs Deformation response and failure of UD G/iPP Fibers Figure 1.1 Schematic overview of the micro-mechanical analysis for UD G/iPP composites. Micro-mechanical models have become attractive since they can provide an estimation of the macro-mechanical properties from the known behavior of composite
1.2. Problem Statement 3 constituents [15, 16]. Moreover, they enable assessment of stress, strain, deformation and damage at a fundamental scale for multiple load scenarios [17]. Figure 1.1 displays a schematic overview of the micro-mechanical analysis for UD G/iPP composites. Mechanical behaviors of fibers and matrix are modeled separately by using representative volume elements (RVEs). Then, the volume average of the local fields are computed for the corresponding effective properties of the material, a technique referred to homogenization. The fiber distribution in composites is generally non-uniform and depends on the constituents, the chosen manufacturing techniques as well as the chosen process parameters. The fibers in UD G/iPP composites are clustered around the layers by forming a layer-like structure, as seen in Figure 1.1. It is important to represent the irregular fiber patterns of the material in the RVEs to obtain reliable predictions of local stress-strain fields, deformation and damage in micro-structure [18–20]. The polymer and fiber responses can be described by using different constitutive equations. The fibers are usually assumed as linear elastic materials in the literature [21–23]. The constitutive model selection for the polymer component is an important aspect in the micro-mechanics. The mechanical response of a polymer is described by its intrinsic behavior, as seen in Figure 1.1, which can be measured under homogeneous deformation. To predict the intrinsic behavior of the polymers, a great amount of effort has been devoted to develop suitable constitutive models. Some of them are the Haward and Thackray model [24], the Boyce-Parks-Argon (BPA) model from MIT [25], the Oxford-Glass-Rubber (OGR) model [26, 27] and the Eindhoven Glassy Polymer (EGP) model [28–32]. The intrinsic response of thermoplastic polymers is strongly strain rate and temperature dependent [32–34]. It was shown for many polymers such as polycarbonate (PC) in [35, 36], polymethylmethacrylate (PMMA) [37], isotactic polypropylene (iPP) in [37], and for polyamide-6 (PA6) in [38] that an increase in strain rate increases the yield stress of the polymers. Furthermore, temperature increase has a profound decreasing effect on yield stress. Therefore, it is important to establish a constitutive model that can capture rate and temperature dependent yield kinetics of the polymers. It is worth to mention that there are multiple grades of iPP as a result of different processing conditions or molecular weight distribution. The iPP used in G/iPP UD tapes and the pure iPP used in this study from Caeler’s thesis [39] have different molecular weights and lamellar thicknesses due to the differences in their production techniques. The molecular weight of iPP matrix used in the G/iPP composites are 150.000 g mol−1 with a crystallinity of 63 wt% (±4.6%) while the pure polymer iPP used in this thesis has a molecular weight of 320.000 g mol−1 with a 63.3 of crystallinity. Moreover, the iPP in G/iPP composites has a lower lamellar thickness than the pure iPP used for this thesis. It was studied by van Erp et al. [40] that the different morphologies do not change the strain rate and the temperature dependence but the pronounced changes might be observed in yield stress. Therefore, it is necessary to modify some material properties of pure iPP polymer for G/iPP composites.
4 Chapter 1. Introduction An accurate numerical tool is still needed in industry to predict the behavior of continuous fiber reinforced G/iPP composites at varying loading and environmental conditions which in turn helps engineers to optimize the design and production parameters of these materials to be used in load carrying structures. 1.3 Objective, Approach and Outline This thesis aims to provide an accurate numerical tool that enables predicting the mechanical response of continuous fiber reinforced, UD G/iPP composites at different temperatures, strain rates and loading angles by following a micromechanical approach. Consequently, the thesis is structured in a way that the following sub-goals are achieved: • Characterization of iPP polymer over a wide strain rate and temperature intervals • Understanding the effect of RVE selection on the mechanical response • Development of a micro-mechanical framework that enables one to determine the appropriate RVEs from micro-graphs • Development of proper boundary conditions to apply off-axis loads • Prediction of the stress-strain responses at varying loading conditions • Investigation of the mechanisms leading to different failure modes in the microstructure The schematic outline of the thesis described in Figure 1.2.
1.3. Objective, Approach and Outline 5 Characterization of rate and temperature dependent iPP behavior Chapter 2: A rate and temperature dependent constitutive model Determination appropriate RVEs for continuous fiber reinforced G/iPP Chapter 3: A micro-mechanical framework to determine the RVEs from microscopy images Chapter 4: Development of boundary conditions to apply off-axis loading. Prediction of the deformation response at different strain rates, temperatures, loading angles Micro-mechanical prediction the deformation response of continuous fiber reinforced G/iPP at different loading conditions Chapter 5: Understanding failure mode at different strain rates, temperatures, loading angles. Investigation of local stress-strain fields that could contribute to the failure at the corresponding conditions. Micro-mechanical investigation of changing failure modes due to the changes in loading conditions , , T , , T Pre-yield failure plastic failure , T Figure 1.2 Schematic description of the outline of this thesis. The chapters of this thesis were written as separate articles for the publication so that each chapter is self-contained with this format, allowing the reader to study them independently. As a consequence, some of the essential details are repeated in several chapters. The author apologizes for any inconvenience that may be caused by the repetition. In Chapter 2, iPP polymer matrix behavior was characterized over a wide range of strain rates and temperatures. In this regard, a rate and temperature dependent viscoplastic model was improved to capture the deformation response of iPP accurately. In Chapter 3, a micro-mechanical approach was established to determine the appropriate RVEs for continuous fiber reinforced G/iPP composites. First, fiber distribution of the material was investigated by optical microscopy. The morphological properties of differently generated RVEs were compared to determine the RVE
6 Chapter 1. Introduction generation method. Then, morphological and mechanical characterizations through the microscopy images were carried out to determine size and location independent RVEs from micro-graphs. In Chapter 4, a micro-mechanical model was implemented to predict the stress-strain response of continuous fiber reinforced G/iPP composites. A new approach to apply off-axis loading to the continuous fiber reinforced RVEs was established for the first time in this thesis. The elaboration of simulation results were made for different load and temperature conditions. In Chapter 5, an experimental examination was made to investigate the fracture surfaces. Then, the changes in failure modes with varying temperature, strain rate and loading angles were investigated by using micro-mechanics. To this end, the stress and strain fields in micro-structure were explored to determine their roles in failure of these materials and how these fields change with changing conditions were also elaborated. Finally in Chapter 6, the results in the previous chapters were summarized and future directions for research are recommended. References [1] World Economic Forum, The Global Risks Report 2021: 16th Edition, 2021. [2] United Nations Environment Programme, “Emissions Gap Report 21,” Tech. rep., 2021. [3] European Commission, “Fit for 55 - Delivering the EU’s 2030 climate target on the way to climate neutrality,” European Commission, , No. 550 final, 2021, pp. 15. [4] Group, B., “Renewed success in meeting CO2 targets,” Tech. rep., 2021. [5] Volkswagen, “Volkswagen again significantly exceeds European CO2 fleet targets for 2021,” , No. 6, 2022, pp. 2021–2022. [6] Baran, I., Cinar, K., Ersoy, N., Akkerman, R., and Hattel, J. H., “A Review on the Mechanical Modeling of Composite Manufacturing Processes,” Archives of Computational Methods in Engineering, Vol. 24, No. 2, apr 2017, pp. 365–395. [7] Moore, S., “Continuous-fiber-reinforced thermoplastic composite displays potential as tool for localized reinforcement,” Plastics Today, 2017. [8] Ciliberti, M. W. S., “Advances in Plastic Components,” Library, 2003. [9] Jacob, A., “Thermoplastic composite specified for BMW Z4 underbody shielding,” 2009. [10] Group, J., “Lightweight bracket for firm clamping,” . [11] Doris, W., “Automotive Thermoplastic Resin Composites Market Analysis 20212028,” Tech. rep., 2022. [12] Stewart, R., “Rebounding automotive industry welcome news for FRP,” Reinforced Plastics, Vol. 55, No. 1, 2011, pp. 38–44. [13] Stewart, R., “Thermoplastic composites - Recyclable and fast to process,” Reinforced Plastics, Vol. 55, No. 3, 2011, pp. 22–28.
References 7 [14] Mike and Brady, P., “The road to lightweight performance,” Reinforced Plastics, Vol. 52, No. 10, 2008, pp. 32–36. [15] Xia, Z., Chen, Y., and Ellyin, F., “A meso/micro-mechanical model for damage progression in glass-fiber/epoxy cross-ply laminates by finite-element analysis,” Composites Science and Technology, Vol. 60, No. 8, 2000, pp. 1171–1179. [16] Sun, C. T. and Vaidya, R. S., “Prediction of composite properties from a representative volume element,” Composites Science and Technology, Vol. 56, No. 2, 1996, pp. 171–179. [17] Aboudi, J., Arnold, S. M., and Bednarcyk, B. A., Fundamentals of the Mechanics of Multiphase Materials, 2013. [18] Pyrz, R., “Correlation of microstructure variability and local stress field in twophase materials,” Materials Science and Engineering A, Vol. 177, No. 1-2, apr 1994, pp. 253–259. [19] Hojo, M., Mizuno, M., Hobbiebrunken, T., Adachi, T., Tanaka, M., and Ha, S. K., “Effect of fiber array irregularities on microscopic interfacial normal stress states of transversely loaded UD-CFRP from viewpoint of failure initiation,”Composites Science and Technology, Vol. 69, No. 11-12, 2009, pp. 1726–1734. [20] Bulsara, V. N., Talreja, R., and Qu, J., “Damage initiation under transverse loading of unidirectional composites with arbitrarily distributed fibers,” Composites Science and Technology, Vol. 59, No. 5, apr 1999, pp. 673–682. [21] Romanov, V., Lomov, S. V., Swolfs, Y., Orlova, S., Gorbatikh, L., and Verpoest, I., “Statistical analysis of real and simulated fibre arrangements in unidirectional composites,” Composites Science and Technology, Vol. 87, oct 2013, pp. 126–134. [22] Pulungan, D., Lubineau, G., Yudhanto, A., Yaldiz, R., and Schijve, W., “Identifying design parameters controlling damage behaviors of continuous fiber-reinforced thermoplastic composites using micromechanics as a virtual testing tool,” International Journal of Solids and Structures, Vol. 117, 2017, pp. 177– 190. [23] Maragoni, L., Carraro, P. A., and Quaresimin, M., “Development, validation and analysis of an efficient micro-scale representative volume element for unidirectional composites,” Composites Part A: Applied Science and Manufacturing, Vol. 110, No. April, 2018, pp. 268–283. [24] Haward, R. N. and Thackray, G., “The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 302, No. 1471, 1968, pp. 453–472. [25] Boyce, M. C., Parks, D. M., and Argon, A. S., “Large inelastic deformation of glassy polymers. part I: rate dependent constitutive model,” Mechanics of Materials, Vol. 7, No. 1, sep 1988, pp. 15–33. [26] Buckley, C. P. and Jones, D. C., “Glass-rubber constitutive model for amorphous polymers near the glass transition,”Polymer, Vol. 36, No. 17, 1995, pp. 3301–3312. [27] Buckley, C. P., Dooling, P. J., Harding, J., and Ruiz, C., “Deformation of thermosetting resins at impact rates of strain. Part 2: Constitutive model with rejuvenation,” Journal of the Mechanics and Physics of Solids, Vol. 52, No. 10, 2004,
8 Chapter 1. Introduction pp. 2355–2377. [28] Tervoort, T. A., Smit, R. J., Brekelmans, W. A., and Govaert, L. E., “A Constitutive Equation for the Elasto-Viscoplastic Deformation of Glassy Polymers,”Mechanics Time-Dependent Materials, Vol. 1, No. 3, 1998, pp. 269–291. [29] Baaijens, F. P., “Calculation of residual stresses in injection molded products,” Rheologica Acta, Vol. 30, No. 3, 1991, pp. 284–299. [30] Klompen, E. T., Engels, T. A., Govaert, L. E., and Meijer, H. E., “Modeling of the postyield response of glassy polymers: Influence of thermomechanical history,” Macromolecules, Vol. 38, No. 16, 2005, pp. 6997–7008. [31] Van Breemen, L. C., Klompen, E. T., Govaert, L. E., and Meijer, H. E., “Extending the EGP constitutive model for polymer glasses to multiple relaxation times,” Journal of the Mechanics and Physics of Solids, Vol. 59, No. 10, 2011, pp. 2191–2207. [32] Van Breemen, L. C., Engels, T. A., Klompen, E. T., Senden, D. J., and Govaert, L. E., “Rate- and temperature-dependent strain softening in solid polymers,” Journal of Polymer Science, Part B: Polymer Physics, Vol. 50, No. 24, 2012, pp. 1757– 1771. [33] Govaert, L. E., Timmermans, P. H., and Brekelmans, W. A., “The influence of intrinsic strain softening on strain localization in polycarbonate: Modeling and experimental validation,” Journal of Engineering Materials and Technology, Transactions of the ASME, Vol. 122, No. 2, 2000, pp. 177–185. [34] Erp, T. B. V., Cavallo, D., Peters, G. W. M., and Govaert, L. E., “Rate-, Temperature-, and Structure-Dependent Yield Kinetics of Isotactic Polypropylene,” 2012, pp. 1438–1451. [35] Bauwens-Crowet, C., Bauwens, J. C., and Homès, G., “Tensile yield-stress behavior of glassy polymers,” Journal of Polymer Science Part A-2: Polymer Physics, Vol. 7, No. 4, 1969, pp. 735–742. [36] Bauwens-Crowet, C., Bauwens, J. C., and Homes, G., “The temperature dependence of yield of polycarbonate in uniaxial compression and tensile tests,” Journal of Materials Science, Vol. 7, No. 2, 1972, pp. 176–183. [37] Roetling, J. A., “Yield stress behaviour of isotactic polypropylene,” Polymer, Vol. 7, No. 7, 1966, pp. 303–306. [38] Parodi, E., Peters, G. W. M., and Govaert, L. E., “Prediction of plasticitycontrolled failure in polyamide 6 : Influence of temperature and relative humidity,” Vol. 45942, 2018, pp. 15–17. [39] Caelers, H. J., Parodi, E., Cavallo, D., Peters, G. W., and Govaert, L. E., “Deformation and failure kinetics of iPP polymorphs,” Journal of Polymer Science, Part B: Polymer Physics, Vol. 55, No. 9, 2017, pp. 729–747. [40] Van Erp, T. B., Cavallo, D., Peters, G. W., and Govaert, L. E., “Rate-, temperature- , and structure-dependent yield kinetics of isotactic polypropylene,” Journal of Polymer Science, Part B: Polymer Physics, Vol. 50, No. 20, 2012, pp. 1438–1451.
Chapter 2 Characterization rate and temperature dependent stress-strain response of isotactic polypropylene using a multi-process approach Abstract The objective of this study is to develop a mathematical description of the 3D stress-strain response of isotactic polypropylene over a large range of strain rates (10−5-10−1 s−1) and temperature range (5 −75◦C). The constitutive relation is based on the Eindhoven Glassy Polymer (EGP-)model, which is extended to a version with three deformation processes acting in parallel. The parameter identification was performed using experimental data from Caelers et al. [1] on uniaxial compression and extension loading. The obtained model gives an accurate presentation of the pre- and post yield response of isotactic polypropylene over the entire rate and temperature range experimentally covered. 9
10 Chapter 2. Characterization rate and temperature dependent stress-strain response of isotactic polypropylene using a multi-process approach 2.1 Introduction Polymers have a wide range of application fields thanks to their production easiness, versatility and low cost. The increasing demand of these materials in load bearing applications [2] requires the understanding of their mechanical behavior. The mechanical response of a polymer is described by its intrinsic stress-strain response under homogeneous deformation. In intrinsic behavior, after an initial elastic part, a nonlinear viscoelastic response is followed by the yield point where the deformation becomes irreversible. Then, strain softening takes place due to plastic deformation. In order to obtain reliable polymer-based components, it is necessary to understand and predict this intrinsic response since it controls the mechanical performance which is very important in the design and optimization of these structures. The intrinsic response of thermoplastic polymers strongly depends on the loading conditions such as strain rate and temperature. For instance, this has been shown [3, 4] for the yield behavior of polycarbonate (PC). Similar trends were also observed for the yield stress kinetics of polymethylmethacrylate (PMMA) [5], isotactic polypropylene (iPP) [6], and for polyamide-6 (PA6) [7] over a wide temperature and strain rate interval. iPP is one of the most commonly used semi-crystalline polymers due to the balance between cost and mechanical performance [8–10]. As a semi-crystalline polymer, iPP contains both crystalline and amorphous domains in its molecular structure and both will contribute to the mechanical performance. The yield kinetics of iPP over a wide temperature and strain rate interval were studied in [6]. The effect of cooling rate on the lamellar thickness of iPP was discussed by [11] to reveal the relation between the structural properties and the mechanical performance over a wide strain rate and temperature range. The deformation response and failure kinetics of different iPP-polymorphs α−, β−and γ-iPP were investigated experimentally in [12]. In this study, three different slopes were observed in the rate-dependent yield kinetics of iPP-polymorphs. These changes in the slopes were attributed to the deformations of these amorphous and crystalline phases that are triggered at different temperature or strain rates. In addition to experiments, a considerable effort has been dedicated to the development of suitable constitutive models to describe the large deformation response of the polymers under varying strain rate and temperature conditions. The first attempt was done by [13] by developing a one-dimensional constitutive model in which the post yield response of the polymer was split additively into viscous and hardening parts. This model was extended to three-dimensional models such as the BoyceParks-Argon (BPA) model [14] with 3-chain, 8-chain or full-chain strain hardening approaches [15–17], the Oxford-Glass-Rubber (OGR) model [18, 19] with EdwardsVilgis hardening expression [20] and the Eindhoven Glassy Polymer (EGP) model [21–25] employing a neo-Hookean model [26] for strain hardening. The EGP model was improved by introducing the pressure sensitivity and the softening kinetics [27], and the physical ageing [28]. For a more accurate pre-yield description, the EGP
2.1. Introduction 11 model was extended to a multi-mode constitutive relation in [24] with a method to determine the spectrum of multiple relaxation times. However, these models were not adequate to predict the thermorheologically complex polymer responses such as iPP. In the case of iPP, there are different molecular deformation processes contributing to the yield and each of them has its own relaxation kinetics. To capture such a stress-strain response, the multi-process EGP model was developed [25], in which the stress contribution of each process is described individually. This model was employed successfully to describe the compressive stress-strain response of iPP over a temperature range of 5 −40◦C using one deformation proces at 40◦C and two processes that are acting in parallel at temperatures 5◦C and 20◦C. A later study by Caelers et al. [1], however, demonstrated that at 40◦C, already stress contributions of two processes should be considered. So, the contributions at 5◦Cand 20◦C originate from a third process. Figure 2.1 shows the active processes at certain temperatures and strain rates. The grey area shows the range of rates and temperatures at which the existing literature studied numerically. 10-5 10-4 10-3 10-2 10-1 0 5 20 40 75 100 Process I Process I+II Process I+II+III Temperature [°C] Strain rate [1/s] Existing literature Figure 2.1 Active processes at certain temperatures and strain rates, and the temperatures and strain rates at which the intrinsic response was analyzed in the literature. The characterisation of each process is needed for expanding the application areas of iPP. Fiber reinforced iPP composite structures as examples of the application fields of iPP show rate and temperature dependent behavior due to their polymer constituents. The composite structures may be exposed to high temperature or creep loading conditions during their lifetime and only process I determines the behaviour of the polymer under these conditions. The components of these structure may also be subjected to impact loading, a situation where all three processes contribute to the deformation behaviour of the polymer. Hence, three deformation processes should
12 Chapter 2. Characterization rate and temperature dependent stress-strain response of isotactic polypropylene using a multi-process approach be employed in the model to get an accurate description. In the current study, we, therefore, use the multi-process EGP model in a threeprocess approach to capture the pressure dependent (compressive-tensile) stressstrain response of iPP over a larger temperature range (5−75◦C). The parameter identification was performed on the experimental data covered by Caelers et al. [1] onα-iPP. A method was described explicitly to determine the required parameters of each process such as the relaxation spectrum and the ageing parameters which were not trivial for iPP. Such a precise model would enable to understand and analyze the effect of strain rate and temperature on iPP based composites which could be difficult and expensive to characterize their behavior experimentally. Moreover, the model enables to analyze local strains, variation of localization due to the fiber distribution and their effects on failure specifically when the shear bands are present. 2.2 Phenomenology For iPP, multiple crystallographic structures can be present depending on the applied pressure and cooling rate conditions [29, 30], which leads to variations in the physical and mechanical properties of the polymer. Different iPP-polymorphs were investigated by Caelers et al. [1] in terms of deformation and the yield kinetics. These are α-crystals, β-crystals and γ-crystals that can be formed depending on the isotacticity level, applied conditions such as pressure and cooling rate and the additives such as nucleating agents. In this study, α-iPP with a molecular weight of 320.000 g mol−1 and a crystallinity of 63 wt%, which is the result of a sufficiently high isotacticity, atmospheric pressure and moderate cooling rate conditions has been addressed to characterize its mechanical response and further referred to as iPP in thiswork.
2.2. Phenomenology 13 10-6 10-5 10-4 10-3 10-2 10-1 100 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 0 20 40 60 80 100 True stress [MPa] True strain [-] 5°C 23°C 50°C 75°C Strain rate[1/s] Compressive yield [MPa] Strain rate[1/s] 10-6 10-5 10-4 10-3 10-2 10-1 100 0 20 40 60 80 100 a) b) 50°C 5°C 23°C 50°C 75°C c) Compressive yield [MPa] I II I+II Figure 2.2 a) Stress-strain response of iPP under compression loading at 5◦C, 23◦C, 50◦C and 75◦C at different strain rates; b) The compressive yield kinetics of iPP; c) The yield kinetics of iPP at 50◦C. The slopes representing the different deformation processes are shown with I and II. (Data taken from [1] was reproduced.) The intrinsic stress-strain data of iPP under compression loading was generated by [1]. Figure 2.2a shows the stress-strain response at 5◦C in a strain rate interval [10−5,10−2] s−1, at 23◦C in a strain rate interval [10−4,10−2] s−1, at 50◦C in a strain rate interval [10−5,10−2] s−1 and at 75◦C in a strain rate interval [10−4,10−3] and at 10−1 s−1. The maximum stress value in the stress-strain response is described as the yield stress where plastic deformation balances the applied strain rate due to stress-induced molecular mobility. The yield stress increases with an increasing strain rate and also with a decrease in temperature. In case the yield kinetics are represented for all strain rates and temperatures, a linear relation between the yield stress and the logarithm of the strain rate can be noticed at first glance at Figure 2.2b. Another observation is that the rate dependency gets stronger at lower temperatures. Moreover, distinct changes in the slopes of the yield kinetics are noticeable in the
14 Chapter 2. Characterization rate and temperature dependent stress-strain response of isotactic polypropylene using a multi-process approach figure. For instance, the slope of the curve at 75◦C increases after a strain rate 10−2 s−1 and the slope of 50◦C increases after 10−4 s−1. The curve sections with steeper slopes at 50◦C and 75◦C are parallel to the curve at 23◦C between the strain rates [10−5,10−2] s−1. A third increase is subtle but the slope of the curve at 23◦C increases after 10−2 s−1 and becomes parallel to the curve at 5◦C. These changes in the slopes are attributed to the different deformation mechanisms. As mentioned in [1], process I is observed at high temperatures or lower strain rates and it is linked to the intralamellar deformation or crystal slip. At moderate strain rates and temperatures, process II is activated and it originates from the inter-lamellar deformation or crystal slip. The third process activated at low temperatures or high strain rates is associated with the glass transition of the amorphous phase. The deformation processes act in parallel so they contribute to the yield response additively. As seen in Figure 2.2c, the second process activity begins after the strain rate 10−4 s−1 and the total contribution of the first and the second processes is observed in the yield stress response. In a similar way, the onset of the third process also increases the yield stress since it is an additive contribution to process I and process II. Then, it can be concluded that for iPP, three deformation processes are active at 5◦C and at 23◦C for the strain rates higher than 10−2 s−1. This behavior can be described by the modified Ree-Eyring equation by taking the sum of the three processes: σyield = ∑ x=1,2,3 kT V∗x sinh−1( ˙ ˙ 0,x exp(ΔUx/RT) ) (2.1) where k is the Boltzmann constant, T is the temperature in [K], Ris the gas constant and ˙ is the applied strain rate. ΔUx is the activation energy, which is related to temperature dependency and V∗x is the activation volume, which determines the rate-dependency. The rate constant ˙ 0,x is the state parameter that depends on the morphological properties of the rigid amorphous. The Ree-Eyring fitting parameters for these three processes are listed in Table 2.1. Table 2.1 Ree-Eyring equation parameters. Process I Process II Process III ˙ 0,I VI ΔUI ˙ 0,II VII ΔUII ˙ 0,III VIII ΔUIII [s−1] [nm3] [kJmol−1] [s−1] [nm3] [kJmol−1] [s−1] [nm3] [kJmol−1] 1x1032 10.39 274.0 6x1028 4.33 197.0 1x1064 7.79 340.0 The resulting Ree-Eyring fit can be seen in Figure 2.2b with solid lines passing through the experimental yield stresses. After the characterization of Ree-Eyring parameters describing the rate and temperature dependent yield kinetics, the
2.3. Numerical Modelling 15 EGP model requires the parameters listed in Appendix to describe the intrinsic deformation response of the iPP in Figure 2.2a. 2.3 Numerical Modelling 2.3.1 Constitutive Model In this study, we extended the two-process EGP model [25] to three processes to capture the stress-strain response of iPP over a wide strain rate and temperature interval. Kinematics The deformation tensor is defined as the multiplicative decomposition of the elastic, Fe, and plastic, Fp, contributions of the total deformation: F =Fe.Fp (2.2) This definition has two implications. The first one is that the velocity gradient tensor L can be split additively into its elastic, Le, and plastic, Lp, parts as follows: L = ˙F.F−1 = ˙Fe.F− 1 e +Fe. ˙Fp.F− 1 p .F− 1 e =Le +Lp (2.3) Secondly, the decomposition in eq. (2.2) is not unique because it does not specify the amount of rotation associated with the elastic and the plastic parts of the deformation. The solution is to assume that the plastic deformation is spin-free [14]: Lp =Dp +Ωp =Dp (2.4) where Dp is the plastic rate of deformation tensor andΩp is the plastic spin tensor. Furthermore, the plastic deformation is assumed as isochoric so that the volume ratio J only depends on the elastic deformation gradient tensor: J =det(F)=det(Fe) (2.5)
www.gildeprint.nlRkJQdWJsaXNoZXIy MjY0ODMw