Acting Principles of Nano-Scaled Matrix Additives for Composite Structures (Research Topics in Aerospace) [1st ed. 2021] 3030685225, 9783030685225

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Research Topics in Aerospace

Michael Sinapius Gerhard Ziegmann   Editors

Acting Principles of Nano-Scaled Matrix Additives for Composite Structures

Research Topics in Aerospace Series Editor Rolf Henke, Vorstand Luftfahrt, DLR, Cologne, Nordrhein-Westfalen, Germany

DLR is Germany’s national centre for space and aeronautics research. Its extensive research and development work in aeronautics, space, transportation and energy is integrated into national and international cooperative ventures. Within this series important findings in the different technical disciplines in the field of space and aeronautics, as well as interdisciplinary projects and more general topics are addressed. This demonstrates DLR’s outstanding research competences and capabilities to the worldwide scientific community and supports and motivates international research activities in exploring Earth and the universe, while also focusing on environmentally-friendly technologies and promoting mobility, communication and security.

More information about this series at http://www.springer.com/series/8625

Michael Sinapius · Gerhard Ziegmann Editors

Acting Principles of Nano-Scaled Matrix Additives for Composite Structures

Editors Michael Sinapius Institute of Mechanics and Adaptronics Technische Universität Braunschweig Braunschweig, Germany

Gerhard Ziegmann Institute of Polymer Materials and Plastics Engineering Clausthal University of Technology Clausthal-Zellerfeld, Germany

ISSN 2194-8240 ISSN 2194-8259 (electronic) Research Topics in Aerospace ISBN 978-3-030-68522-5 ISBN 978-3-030-68523-2 (eBook) https://doi.org/10.1007/978-3-030-68523-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book makes a contribution to the field of lightweight materials and in particular to research on continuous fiber reinforced polymers. The book presents the research results of a group of scientists from the North German technical universities in Braunschweig, Hannover and Clausthal-Zellerfeld in cooperation with the Federal Institute for Materials Research and Testing and the German Aerospace Center. Over the past 12, years they have been investigating the question of whether nanoscale matrix additives can significantly improve the matrix-dominated mechanical properties. The interactions of these nanoscale matrix additives have been largely elucidated at different levels. This has led to a deeper understanding of the relationships in such multiphase materials from the nano, over the micro to the macro level. The research was initially funded by the Helmholtz Association as a Virtual Institute. The successful work resulted in the Research Unit 2021 “Active Principles of Nanoscale Matrix Additives for Lightweight Fibre Composites”, which was funded by the German Research Foundation for 6 years. The results presented in this book reflect the intensive cooperation of chemists, process engineers, materials scientists and mechanical engineers. Most of the chapters of the book are condensed representations of dissertations that have been prepared in the course of several years of work. The editors would like to thank the members of the research group for their tireless and inspiring work and also for the many intensive discussions. The intensive collaboration was indeed a highlight of our many years of scientific activity. Braunschweig, Germany Clausthal-Zellerfeld, Germany January 2021

Michael Sinapius Gerhard Ziegmann

v

Acknowlegement

The financial support from the Deutsche Forschungsgemeinschaft within the Research Unit “Acting principles of nano scaled matrix additives for composite structures” (FOR2021) is gratefully acknowledged. Writing a book also involves an editor. We would like to thank Springerverlag for their interest and willingness to publish this book. We would also like to thank Prof. Rolf Henke, member of the Board of Directors of the German Center for Aeronautics, for allowing us to publish this book in the series “Research Topics in Aerospace”. January 2021

Michael Sinapius Gerhard Ziegmann

vii

Contents

Part I

Introduction

1

Motivation and Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius and Gerhard Ziegmann

2

State of Research on Fiber Reinforced Nanocomposites and Theses of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Michael Sinapius

Part II 3

3

9

Foundation

Modeling and Simulation of Nanocomposites and Their Manufacturing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behrouz Arash, Dilmurat Abliz, and Raimund Rolfes

27

4

Characterization of Polymer Nanocomposites . . . . . . . . . . . . . . . . . . . . Paulina Szymoniak, Xintong Qu, Andreas Schönhals, and Heinz Sturm

55

5

Liquid Composite Molding Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilmurat Abliz and Gerhard Ziegmann

79

Part III Particle-Matrix Interaction 6

7

Mechanical Properties of Boehmite Evaluated by Atomic Force Microscopy Experiments and Molecular Dynamic Finite Element Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johannes Fankhänel, Dorothee Silbernagl, Media Ghasem Zadeh Khorasani, Benedikt Daum, Andreas Kempe, Raimund Rolfes, and Heinz Sturm

91

Particle Surface Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Ajmal Zarinwall, Tassilo Waniek, Benedikt Finke, Reza Saadat, Heinz Sturm, and Georg Garnweitner

ix

x

Contents

8

Short- and Long-Range Particle-Matrix Interphases . . . . . . . . . . . . . . 143 Media Ghasem Zadeh Khorasani, Johannes Fankhänel, Raimund Rolfes, and Heinz Sturm

9

A Multi-scale Framework for the Prediction of the Elastic Properties of Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Atiyeh Mousavi, Johannes Fankhänel, Behrouz Arash, and Raimund Rolfes

10 Multiscale Modeling and Simulation of Polymer Nanocomposites Using Transformation Field Analysis . . . . . . . . . . . . 209 Imad Aldin Khattab and Johannes Michael Sinapius Part IV Influence of Nanoadditives on Composite Manufacturing 11 Dispersion Technology and Its Simulation . . . . . . . . . . . . . . . . . . . . . . . 241 Benedikt Finke, Arno Kwade, and Carsten Schilde 12 Cure Kinetics and Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Dilmurat Abliz, Benedikt Finke, Arno Kwade, Carsten Schilde, and Gerhard Ziegmann 13 Thermal Properties of Boehmite-Epoxy Nanocomposites . . . . . . . . . . 301 Kerstin Mandel, Dilmurat Abliz, and Gerhard Ziegmann 14 Molecular Modeling of Epoxy Resin Crosslinking Experimentally Validated by Near-Infrared Spectroscopy . . . . . . . . . 325 Robin Unger, Ulrike Braun, Johannes Fankhänel, Benedikt Daum, Behrouz Arash, and Raimund Rolfes 15 Permeability Characterization and Impregnation Strategies with Nanoparticle-Modified Resin Systems . . . . . . . . . . . . . . . . . . . . . . 351 Dilmurat Abliz and Gerhard Ziegmann Part V

Structural Mechanics of Fiber Reinforced Nanocomposites

16 Nanoscaled Boehmites’ Modes of Action in a Polymer and Its Carbon Fiber Reinforced Plastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Christine Arlt, Wibke Exner, Ulrich Riedel, Heinz Sturm, and Johannes Michael Sinapius 17 Viscoelastic Damage Behavior of Fiber Reinforced Nanoparticle-Filled Epoxy Nanocomposites: Multiscale Modeling and Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Behrouz Arash, Wibke Exner, and Raimund Rolfes 18 Effect of Particle-Surface-Modification on the Failure Behavior of Epoxy/Boehmite CFRPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Maximilian Jux and Johannes Michael Sinapius

Contents

xi

19 Surface Quality of Carbon Fibre Reinforced Nanocomposites: Investigation and Evaluation of Processing Parameters Controlling the Fibre Print-Through Effect . . . . . . . . . . . . . . . . . . . . . . 433 Thorsten Mahrholz and Johannes Michael Sinapius 20 Upscaling Effects of Carbon Fiber Reinforced Nanocomposites with Respect to Matrix-Induced Distortions and Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Wibke Exner, Maximilian Jux, Till Julian Adam, Thorsten Mahrholz, and Peter Wierach Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Part I

Introduction

Chapter 1

Motivation and Relevance Johannes Michael Sinapius and Gerhard Ziegmann

Abstract Lightweight structures play a central role in mobility. The research and development of materials with high specific strengths and stiffnesses has always been essential. Carbon fiber-reinforced plastics represent an important and still increasing part of this. However, matrix-dominated properties such as impact strength or fibreparallel compressive strength still represent a weak point in this class of materials. Nanotechnology is opening up new ways to significantly improve these properties. This chapter provides an overview of the different approaches to the use of nanotechnology in Fibre Reinforced Polymers (FRP) and introduces the book.

Carbon fibre reinforced plastics (CFRP) have a significantly higher lightweight construction potential for components with a primarily single- or biaxial stress state compared to isotropic metals. At the same time, they can contribute to a reduction in maintenance costs due to their insensitivity to corrosion and the advantageous fatigue properties. These properties make them particularly attractive in the field of mobility. Intensive research is being carried out into fibre-reinforced plastic composite (FRP) materials and cost-effective processes that nevertheless have the qualities of complex and cost-intensive prepreg components. The focus is particularly on processes with resin injection. In addition to the fibre architecture, the limits of injection materials are mainly set by the polymers used. Here, the matrix-dominated properties, such as • compressive strength parallel to the fibres, • dimensional stability through thermal and chemical shrinkage, J. M. Sinapius (B) Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] G. Ziegmann Technische Universität Clausthal, Clausthal-Zellerfeld, Germany e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_1

3

4

J. M. Sinapius and G. Ziegmann

• toughness, and • impact tolerance, which are limiting factors in the design of fibre composite components. Currently, the maximum allowable design strain is defined by these quantities, in particular by the low impact tolerance and compressive strength. Low-viscosity resin systems, as required for liquid impregnation processes, show weaknesses especially in the above mentioned properties and thus considerably limit the exploitation potential of the high-performance fibres in the laminate. If it is possible to eliminate the above-mentioned disadvantages by incorporating nanoparticles, an increase in design elongation can be expected, which naturally allows a considerable reduction in component weight and leads to the expectation of more damage-tolerant structures with greater imaging accuracy. This will create the prerequisites for achieving superior property profiles with low-cost wet impregnation processes, with additional cost reductions compared to prepreg technology. FRP materials have a micro-scale structure similar to the anisotropic lightweight construction principle of wood. Future materials will be designed on a level below this, i.e. starting at the nanoscale, thus forming a multiscale structured material. The need for this is apparent from the deficits of FRP. As a result, research into the targeted design of materials, starting at the smallest scale, is increasing. FRP materials have a micro-scale structure similar to the anisotropic lightweight construction principle of wood. Future materials will be designed on a level below this, i.e. starting at the nanoscale, thus forming a multi-scale structured material. The necessity for this is shown by the above mentioned deficits of the FRP. Research into the targeted design of materials, starting at the smallest scale, is increasingly coming to the fore [9, 11]. Nanotechnology shows a high potential for a further improvement of the properties of fibre composite materials. Three areas of the application of nanotechnology are in the focus of international research groups: 1. carbon fibers based on nanotechnology 2. influence of nanoscale additives on the interphase fibre/matrix 3. continuous fibre-reinforced nanocomposites (FRNP) by nanoscale additives to the matrix of fibre-reinforced plastics (FRP). Figure 1.1 illustrates these research areas. Several research groups [4, 8, 12, 16] carry out international research on fibers based on carbon nanotubes, so-called CNF. They aim at a significant increase in the strength and stiffness of the composites, but cannot adequately address questions of dimensional stability, compressive strength and impact strength because these properties are matrix dominated. Other international working groups are working intensively on approaches to nanotechnology in the fiber-matrix interface [6, 13]. Mäder et al. published several articles in this field [2, 3, 14, 19] as well as Akkermann et al. [7]. The Mäder working group investigated sensory functionalization via the fiber-matrix interface. The approaches also address those fiber composite properties that are determined by the interface properties between fiber and matrix which are dominant, such as

1 Motivation and Relevance

5

Fig. 1.1 Research fields of nanotechnology in Fibre Reinforced Polymers (FRP)

crack resistance or fatigue behavior. This approach can also address questions of dimensional accuracy, compressive strength and impact strength only insufficiently address the problem because they are matrix-dominated properties. Matrix dominated properties of continuous fiber reinforced polymers can be effectively improved with matrix additives. Intensive research work in the field of nanocomposites has been carried out in recent years. Nanocomposites, i.e. thermoplastic or thermoset polymer matrices filled with nanoscale particles (particle size 90%) are chosen to increase linearly from 2.5 to 4.5 Å for the esterification and from 1.5 to 3.5 Å for the etherification. For lower degrees of curing, these distances are reduced, as explained in the following subsection. The system is then relaxed by the isothermal-isobaric ensemble (NPT) and equilibrated for a period of 2.75 ns at a temperature of 300 K and a pressure of 1 atm. Subsequently, the loading simulations are performed at a temperature of 300 K with a standard strain rate of 107 1/s and a pressure of 1 atm in the transverse directions.

9.2.3 Numerical Investigation of Epoxy 9.2.3.1

Elastic Properties of the Epoxy System

An epoxy with a high degree of curing of approximately 92% is used, which is a common assumption for numerical studies in the literature, e.g. in [18, 39]. For obtaining the elastic properties, three randomly generated simulation boxes with the box size of 6 nm are loaded in the three spatial directions. Hence, the elastic properties of nine different configurations are averaged. To simplify the procedure, the behavior of the pure epoxy is assumed to be isotropic. By applying periodic

184

A. Mousavi et al.

boundary conditions (PBCs), the strains can easily be calculated from the deformation of the simulation box according to 

εi j

 V

=

ui j , h ii

(9.1)

where .V denotes the volume average and u i j the displacements on the box boundaries in j-direction perpendicular to i-direction. It has to be noted that h is defined as ⎤ ⎡ a00 h = ⎣ 0 b 0⎦ , (9.2) 00c where a, b and c are the volume element (VE) dimensions. The Virial stresses of the simulation box are obtained by [42] σ V =

 1 1  nm nm (− m n (vn − vV ).(vn − vV ) + r .f ), V 2 n,m=n n

(9.3)

where V is the volume of the simulation box, m n is the mass of atom n, vn is a velocity vector af atom n, σ V is a vector of the average velocity of all atoms in the VE, rnm is the vector pointing from atom n to atom m and f nm is the force vector between these two atoms. The first term is related to the kinetic energy which arises from thermal movements of the atoms, while the second term describes the potential energy stored in the interatomic interactions. The Young’s modulus can then be obtained based on Eq. (9.4) by fitting a straight line to the unfiltered stress-strain data in loading direction, as shown in Fig. 9.2a. σ = Eε + n

(9.4)

The blue curve shown in Fig. 9.2a is representing a filtered stress-strain curve, where each point shows the average of the unfiltered data for 50 fs. The Poisson’s ratio is calculated by εtr,1 + εtr,2 , (9.5) ν= 2εlong where εlong is the strain in loading direction and εtr,1 and εtr,2 the strains in the two transverse directions. Since the stress-strain data fluctuations are too high to obtain reasonable Young’s moduli, the strain range has to be increased from 0.0005 to 0.0025 compared to the typical experimental values. As can be seen from the comparison of Fig. 9.2a and b, this task has to be done with caution, because it can considerably influence the results. The orange straight in Fig. 9.2a is fitted in a strain range of 0.0001– 0.001, whereas the interval for the fit of the green straight in Fig. 9.2b is chosen to be 0.005–0.025. The elastic properties for these two cases are calculated with

9 A Multi-scale Framework for the Prediction …

185

Unfiltered data Averaged data Young’s modulus fit

Unfiltered data Averaged data Young’s modulus fit 200

Stress (MPa)

Stress (MPa)

200

100

0

100

0

0

0.01

0.02

0.03

0

0.01

0.02

Strain (-)

Strain (-)

(a)

(b)

0.03

Fig. 9.2 Stress-strain curve and linear fit of the reference epoxy system for a strain range of a 0.001–0.01 and b 0.005–0.025

values of E = 6070 ± 589 Mpa and ν = 0.324 ± 0.047 and E = 5236 ± 10 Mpa and ν = 0.326 ± 0.007, respectively. Based on the results, a considerable influence of the strain range on the Young’s modulus can be observed. Since a considerable nonlinearity becomes apparent for larger strains, the smaller strain range seems more realistic. However, for some of the following results, it can be noted that the smaller strain range can sometimes lead to unrealistic values caused by fluctuations of the stress-strain curves. We found out that the calculated Young’s moduli from both strain ranges overestimate the value measured in the experiment, which is reported to be 3370 MPa. The possible causes for this deviation will be addressed in the following subsections.

9.2.3.2

Influence of the Strain Rate on the Elastic Properties

As mentioned above, the strain rate used for the simulations is ε˙ = 107 1/s, which is a typical strain rate for numerical applications in the literature [18, 32]. For quasi-static testing, according to the test standard DIN EN 527-4, a loading velocity of 1 mm/min is used. Specimens with 100 mm free length result in a strain rate of ε˙ = 1.67 × 10−4 1/s. This strain rate is significantly lower than the numerical investigations. In this subsection, the possible solutions and explanations to deal with this big gap will be presented. There is a disagreement in the literature regarding strain rate effects on the elastic properties of polymers. Some numerical studies like [20, 31] show a strain rate dependency of the elastic properties, while other studies like [7, 10, 18, 22, 32]

186

A. Mousavi et al. Experiment [16] Exp. 1.67 × 10−4 1/s [16] 106 1/s 8

10 1/s

107 1/s

Simulation ( = 0.001...0.01) Logarithmic fit

109 1/s

Simulation ( = 0.005...0.025) 7 Young’s modulus (GPa)

Stress (MPa)

150

100

50

0

0

0.01

0.02

Strain (-)

(a)

0.03

6

5

4

3 10−5

100

105

1010

Strain rate (1/s)

(b)

Fig. 9.3 a Time-averaged stress-strain curves of the reference epoxy system for different strain rates and b Young’s modulus for different strain rates

ignore the large gap between experiment and simulation and report a good agreement. Hence, the existence of an influence of the strain rate on the elastic behavior of polymers is not completely clear in the literature and needs to be investigated. In this part of the study, three additional strain rates of ε˙ = 106 1/s, ε˙ = 108 1/s and ε˙ = 109 1/s are simulated. Figure 9.3 shows both the time-averaged stress-strain curves and the resulting Young’s moduli. The Young’s modulus calculated for the minimum strain rate of ε˙ = 106 1/s and the maximum strain rate of ε˙ = 109 1/s differs by approximately 800–900 MPa. The main reason is that the polymer network has more time to react to the deformation and to relax the stresses at lower strain rates. It can be interpreted from Fig. 9.3b that the deviations between the experimental and numerical results can be explained by the strain rate. It has to be noted that besides the limited amount of data available in both experiments and simulations, an extrapolation over a large range is prone to errors.

9.2.3.3

Influence of the Force Field on the Elastic Properties

Another possible influencing factor can be the choice of selected force field parameters. Two options are investigated in this subsection. First, the usage of the DreidingX6 potential (Buckingham potential) instead of the Lennard-Jones potential for the nonbonding interactions can result in more realistic mechanical properties [18]. Second, in the Dreiding force field [28], it is not explicitly stated if nonbonding inter-

9 A Multi-scale Framework for the Prediction …

187

Experiment [16]

Simulation LJ, no 1-4 interactions

Sim. ( = 0.001...0.01)

Sim. Buckingham, no 1-4 interactions

Sim. ( = 0.005...0.025)

Simulation LJ, with 1-4 interactions 7 Young’s modulus (GPa)

Stress (MPa)

150

100

50

0

0

0.01

0.02

0.03

6

5

4

3

Strain (-)

Fig. 9.4 a Time-averaged stress-strain curves obtained with the different force field options and b corresponding Young’s moduli

actions between the first and fourth atoms of a dihedral should be included. In all simulations presented so far, these interactions were excluded. It is shown in Fig. 9.4 that with values of 5.33 GPa and 4.51 GPa obtained from the smaller strain range, both cases lead to lower Young’s moduli compared to the reference case. Hence, the effects of force field parameters on the elastic properties can be another reason for the gap between experiment and simulation results. It is also interesting to note that the deviation between the reference case and the case with enabled special bonds is approximately 1.6 GPa, which is even more than the maximum deviation between the different cross-linking cases.

9.3 Coarse-Grained Simulation To extend the accessible time and length scales, CG simulations are studied in this section. CG models of the epoxy resin with different levels of coarse-graining are developed. The CG force field parameters for bond and nonbond interactions are derived using the IBI method. The flow chart of the IBI method procedure is shown in Fig. 9.5. As it can be seen in the flow chart, the initial CG force fields are obtained based on probability distributions of atomistic simulation. To find the final CG force fields, the structural properties have to be updated until they converge to the reference structural data. The applicability of the CG models to predict the elastic properties is verified with the results from MD simulations. The strain rate effects on elastic properties of epoxy are also investigated and compared to the experimental data.

188

A. Mousavi et al. Initial setup of Coarse-grained (CG) potential

CG simulation

CG Probability distributions

In consistence with target

no

probability distributions?

Update CG potentials

yes Final CG potentials

Fig. 9.5 Flow chart of the IBI method procedure

9.3.1 Mapping Schemes The first step in coarse-graining is identifying possible mapping schemes. Superatoms have to be defined in such a way that the resulting CG model would be capable of keeping the identity of the underlying chemistry. Since there is no unique way to define superatoms, the mapping scheme can considerably influence the simulation results. Here, two different mapping schemes are proposed and compared to investigate the effect of different levels of coarse-graining on the elastic properties of the epoxy resin. In the first mapping scheme, the long monomer of DGEBPA (C39 O7 H44 ) is mapped into five CG beads which contains three different CG bead types named A, B and C, see Fig. 9.6. The short monomer of DGEBPA (C21 O4 H24 ) consists of two different CG bead types named A and B using the same pattern as the long one. The hardener, MTHPA (C9 O3 H10 ) is mapped into one CG bead named D (Fig. 9.6). The atomic mass of A, B, C and D beads is 194.2716, 73.0706, 90.0779 and 166.1739 amu, respectively. The center of the bead is chosen to be the center of the mass for each bead. In this CG model, compared to its full atomistic system, the degrees of freedom (DOF) decrease by 29, 10, 15 and 22 times for bead A, B, C and D, respectively. Each monomer is mapped to one bead in the second mapping scheme as illustrated in Fig. 9.7. Accordingly, the CG system consists of three beads, two for long and short monomers and one for the hardener. The atomic mass of A, B, and C beads is 624.7624, 340.4128 and 166.1739 amu, respectively. In this case, the DOF decreases by 90, 49 and 22 times for bead A, B and C, respectively. For the box size of 60 × 60 × 60 nm3 , the number of beads respectively decreases 9.86 and 17.29 times for the CG model 1 and 2, compared to the full atomistic model.

9 A Multi-scale Framework for the Prediction …

189

(a)

(b)

(d)

(c)

(e)

(f)

Fig. 9.6 a, b, c The atomistic model and d, e, f its corresponding CG illustration of a long and short chain of DGEBPA epoxy and MTHPA which is used as a hardener, respectively

(a)

(b)

(d)

(c)

(e)

(f)

Fig. 9.7 a, b, c The atomistic model and d, e, f its corresponding CG illustration of a long and short chain of DGEBPA epoxy and MTHPA, respectively

9.3.2 Coarse-Grained Force Field The CG force fields are optimized by the distribution functions obtained from atomistic simulations using the IBI method. The outstanding advantage of the IBI method is the detailed structural information which is included in the CG models. Herein, we develop two different CG models of the epoxy resin with anhydride agent. The force fields of the CG models are decomposed into bonded and non-bonded potential functions. Based on [4], the potential energy of a system is written as, E total (d, θ, r ) =

 i

E bi +

 j

Ea j +

 lm

E vdWlm + E 0 ,

(9.6)

190

A. Mousavi et al.

where E b , E a and E vdW are the terms of energy corresponding to the variation of the bonds (bond length and bond angle) and nonbond (the van der Waals (vdW)) interactions, respectively. In Eq. (9.6), E 0 is the constant free energy of the system. In the following subsections, it will be explained how the force field parameters can be driven using the IBI method. The detailed information about the atomistic model generation for the first IBI step has been previously explained explained in Sect. 9.2.2.

9.3.2.1

Coarse-Grained Stretc.hing, Bending and Non-bond Potentials

The probability distributions measured from atomistic simulations are used to derive the CG bond stretc.hing potentials via IBI [2], E bond (l) = −k B T ln(

Pbond,target (l) ), l2

(9.7)

where k B is the Boltzmann constant and T is the temperature. The term l 2 stands for the degeneracy in the position of a bead at a distance l from another bead. Ptarget (l) is a probability distribution of all bonds calculated from MD simulations. The potentials obtained in this way have a quadratic shape about their minimum. Hence, the bond-stretc.hing potentials are modeled as a harmonic spring, E bond (l) =

kb (l − l0 )2 , 2

(9.8)

where kb is the spring constant of the bond length and l0 is the equilibrium bond distance. Similarly, to derive the CG angle-bending potentials with IBI approach, the bending angle distributions have to be measured from atomistic simulations [2], E angle (θ ) = −k B T ln(

Pangle,target (θ ) ), sin θ

(9.9)

where Ptarget (θ ) is a probability distribution of all angles obtained from all atomistic models. The bending potential calculated from the above equation can be approximated as a harmonic angular spring E angle (θ ) =

kθ (θ − θ0 )2 , 2

(9.10)

where kθ and θ0 represent the spring constant and the equilibrium bond angle, respectively. To derive the CG nonbonded potentials via IBI, the initial guesses of the CG nonbonded potentials are obtained from the atomistic system [2] E nonbond (r ) = −k B T ln(gtarget (r )),

(9.11)

9 A Multi-scale Framework for the Prediction …

191

where gtarget (r ) represent the radial distribution functions (RDFs) obtained from MD simulations. The nonbond potentials are modeled by fitting the common expression of the Lennard-Jones potential E vdW (r ) = D0 [(

r0 12 r0 ) − 2( )6 ], r r

(9.12)

where D0 and r0 are associated with the equilibrium potential well depth and the equilibrium distance, respectively. Based on the differences between the two sets of RDFs calculated from CG and MD, the CG potentials can be iteratively improved by [2], n+1 (l) E bond

n+1 E angle (θ )

=

n E bond (l)

=

n E angle (θ )

+ k B T ln(

+ k B T ln(

n Pbond (l) l2

Pbond,target (l)

),

n Pangle (θ) sig(θ)

),

(9.14)

g n (r ) ), gtarget (r )

(9.15)

Pangle,target (θ )

n+1 n E nonbond (r ) = E nonbond (r ) + k B T ln(

(9.13)

where the superscript n denotes the iteration step. The iteration process continues until the target distribution from the atomistic model is reproduced by the CG simulation.

9.3.2.2

IBI Implementation

The polymer chains are placed in random, nonoverlapped configurations within a cubic simulation box. The CG models are generated by mapping the MD polymer chains based on the two mapping schemes as explained in Sect. 9.3.1. In the first step of IBI, the CG potentials are obtained from Eqs. (9.7), (9.9) and (9.11) with the probability distributions of MD simulation. The force field parameters and equilibrium values are then calculated from the fitting curve using Eqs. (9.8), (9.10) and (9.12). Each IBI iteration step starts with a 1 ns equilibration step at a temperature equal to 298 K in an isothermal–isobaric ensemble (NPT) ensemble and 1 ns to compute the RDFs. The Nosé-Hoover algorithm [29] is used for the system temperature and pressure conversions. The RDFs from CG simulation have to be compared against the atomistic RDFs to improve the estimation of CG nonbonded potentials via Eq. (9.15). The iteration steps have to be repeated until the RDFs differences between MD and CG converge to zero. All CG and MD simulations are performed using the LAMMPS and in an in-house software.

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9.3.3 Results and Discussion 9.3.3.1

Force Field Parameters

In the first CG model, six bonds and fourteen angle types have been detected based on the beads’ definition. For the second CG model, the numbers decrease to five and thirteen bond and angle types, respectively. Since all calculated bond potentials show a quadratic shape about the minimum, the bond-stretc.hing potentials are modeled as harmonic springs. Hence, the equilibrium distances and the spring constants are measured using Eq. (9.8). Figure 9.8 shows the bond length distribution for the bond between A and B, its corresponding bond stretc.hing potential and the harmonic fit as an example from the first CG model. Bending angle potentials are measured based on bending angle distributions calculated from atomistic simulation using Eq. (9.9). Like bond potentials, bending angle potentials have also a harmonic shape about the minimum. Therefore, the equilibrium bending angle and the spring constant of the bending angle are obtained using Eq. (9.10). Figure 9.9 shows one example of the bending angle distribution, its corresponding bending angle potential and the fitted curve for (B-A-C) angle from the first CG model. All RDFs corresponding to all types of nonbonded pairwise interactions exhibit a discernible peak. As it can be seen in Fig. 9.10, nonbonded potentials have a shape that can be fitted to the Lennard-Jones potentials. Hence, the Lennard-Jones parameters, i.e. the equilibrium potential well depth (D0 ) and the equilibrium distance (r0 ), can be determined by fitting a curve. The force field potentials are updated in each iteration

1 Bond potential Harmonic fit

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using Eq. (9.13), (9.14) and (9.15). The CG distributions are converted to reference values after 18 and 35 iterations for CG model number one and two, respectively.

9.3.3.2

Validation of CG Models

A simulation box size of 60 × 60 × 60 nm3 with periodic boundary conditions is initially constructed. The simulation box is composed of 181 short monomers, 32

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Table 9.1 Effect of different level of coarse graining on the Young’s modulus of the epoxy resin Young’s modulus Error (%) Wall-time simulation (GPa) (min) MD CG1 CG2

6.070 ± 0.394 6.38 ± 0.4819 4.89 ± 1.0829

– 5.21 19.56

210 11 7

long monomers are packed into the cubic lattice along with 437 agents as a hardener. The predefined mass density of the system is 1.2 cmg 3 . The simulation box consists of 1140 and 650 beads for CG model 1 and 2, indicating a 9.86 and 17.29 times decrease compared to the full atomistic model with 11241 atoms. In the MD simulation, after cross-linking with a degree of curing of 92 %, the system is equilibrated over the NPT ensemble. The equilibrated MD model is needed for an estimation of the initial CG potentials in the IBI method. A geometry optimization is initially performed to minimize the total energy of the system. After the minimization process, to capture different frames for the calculation of probability distributions, the NPT ensemble is performed for 2 ns at a room temperature of 298 K. In the final step, another NPT ensemble is performed at room temperature on the CG model after energy minimization to remove the internal stresses and prepare the system for loading. The loading simulations are performed at a temperature of 298 K with different strain rates and a pressure of 1 atm. After the preparation of the system, the method explained previously in Sect. 9.2.3.1, is used to calculate the elastic properties of the CG models. Table 9.1 illustrates the Young’s modulus for the strain rate of 107 1/s obtained from MD and CG models to verify the results. The wall time simulations obtained using 12 cores for each case, show the time reduction for CG models. The Young’s modulus is calculated in three directions using three different systems. The simulation results of the first CG model are in a good agreement with the atomistic model compared to the second CG model.

9.3.3.3

Strain Rate Effects on Elastic Properties of Epoxy Resin

The advantage of using CG models over MD models is considering the effects of a wider range of strain rates to predict the elastic modulus of the epoxy. The tensile simulations are performed with strain rates between 104 1/s to 109 1/s. As shown in Fig. 9.11, the elastic modulus of the CG method is in a good agreement with the MD results. The Young’s modulus of the epoxy decreases by decreasing the strain rates which shows the influence of the strain rate on the elastic properties. It can be concluded from Fig. 9.11 that Young’s modulus versus strain rate indicates the special trend in which the Young’s modulus, comparable to the experimental data, can be predicted.

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Fig. 9.11 Strain rate effects on Young’s modulus of the epoxy resin using CG simulation. The values are averaged from 9 different configurations

9.4 Micro-Scale Simulation The transition from the atomistic level to the continuum-based micro-scale is accomplished in this chapter. The FE models are constructed based on the elastic properties of the constituents and geometrical aspects, such as the interphase thickness. Two cases in two different levels are considered as shown in Fig. 9.12. In the first case, the FE model of agglomerate UCs, the influence of the reduction of the interphase volume due to agglomeration can be captured. It can also be addressed whether agglomerates are filled with polymer or not. The second case, the agglomerate RVEs, contains a representative number of agglomerates according to the experimental agglomerate size distribution. These models utilize the properties obtained from the agglomerate UCs.

9.4.1 Agglomerate UCs 9.4.1.1

Modeling and Simulation Aspects

The agglomerates are generated with a hardcore algorithm, as illustrated in Fig. 9.13. The first particle is always placed in the middle of the simulation box with a random orientation. To place the next particle, a sphere around the center of mass (CM) of

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Fig. 9.12 Exemplary FE model of a an agglomerate UC and b an agglomerate RVE

(a)

(b)

the first particle is introduced. The algorithm tries to randomly rotate and place the second particle in such a way that its center of mass (CMi) is inside of the sphere. The particle can only be placed, if there is no intersection with the already existing particle. Otherwise, the new particle is removed and the process is repeated until the particle can be placed or a maximum number of iterations, n max , is reached. If the particle cannot be placed in n max iterations, the radius r of the sphere is gradually increased by r , until it is possible to add the particle to the system. After placing the particle, the agglomerate is shifted, so that the center of mass of the agglomerate coincides with the center of the simulation box. Then, the whole process is repeated, until the desired number of primary particles i max is reached. After placing the particles, there are two possibilities for the further model generation, which are invoked depending on whether the model shall contain RFAs or not. In case an RFA area is present, a DEM-like approach is introduced to describe the interparticulate interactions inside the agglomerate. The particles are cut out of the simulation box and a reference point is introduced for each primary particle. Additionally, spring elements are introduced between each pair of particles (each pair of reference points). By replacing each particle with a reference point, the particles are treated as rigid bodies. Subsequently, the interphases are created. The following generation of the RFAs is not trivial, since, there is neither experimental data about their existence nor infor-

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vious particles been detected?

i=i+1

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(b)

Fig. 9.13 a Schematic illustration and b flow chart of the agglomerate generation

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nvert = 4

Particles Convex hull Resin free area Agglomerate center of mass

Fig. 9.14 Schematic illustration of the modeling of the RFAs. The colored dots mark the n ver t vertices of each particle, which are closest to the agglomerate center of mass. The small colored dots are unused, since they are located inside of the convex hull

mation about their possible geometry. Here, the RFAs are modeled as the convex hull of the n ver t vertices of each particle, that are closest to the agglomerate center of mass. This idea is schematically illustrated in Fig. 9.14. For each vertex of each primary particle, the distance to the agglomerate center of mass is determined. Then the convex hull is calculated, considering only the n ver t (e.g. three) vertices of each particle that are closest to the agglomerate center of mass. The resulting convex hull is cut from the models, leading to an empty, non-meshed region representing the RFAs.

9.4.1.2

Simulation Results

The n ver t parameter, which determines the size of the RFA, is addressed in this part. As shown in Fig. 9.15a, there is a linear dependency of the RFA volume fraction on the n ver t parameter. Moreover, the volume fraction and its slope increases with increasing number of primary particles per agglomerate. The influence of the RFA volume fraction on the Young’s modulus is illustrated in Fig. 9.15b indicating a linear relation. The blue curve represents a small agglomerate consisting of three primary particles. A low influence on the Young’s modulus up to differences of 3% can be observed due to a small RFA volume fraction. With increasing agglomerate size and the RFA volume fraction, the Young’s moduli are significantly influenced. For n ver t = 5, which is the largest investigated value, the reduction of the Young’s modulus is 600 MPa compared to n ver t = 3. This means the Young’s modulus decreases by approximately 10%. Due to the lack of experimental data, the choice of the n ver t parameter is ambiguous. In the following, a conservative value of n ver t = 3 will be utilized even though

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(a)

(b)

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Fig. 9.15 a RFA volume and b Young’s modulus in dependency of the n ver t parameter and the number of primary particles per agglomerate

this choice tends to underestimate the experimental Young’s moduli and, hence, overestimate the RFA volume. The three main factors, which influence the elastic properties of the agglomerate UCs, are the interfacial bonding, the number of primary particles per agglomerate and the presence of a resin-free area. Since the mass fraction of the agglomerates is fixed with a value of 10%, the size of the agglomerate UCs increases with an increasing number of primary particles. Figure 9.16 illustrates the influence of the interfacial bonding, the RFA and the agglomerate size on the Young’s modulus of the agglomerate UC. To compare the results, experimental values for the pure epoxy and the NC from [17] are also shown. The average agglomerate size from the experimental study (around 300 primary particles) can be found in the diagram. In principle, two major effects can be observed in Fig. 9.15. Firstly, ignoring the RFAs, the influence of the reduction of the interphase volume with an increasing number of primary particles can be observed. The green curve (Interphase V1, no chemical bonding) is almost independent of the number of primary particles per agglomerate because the interphase stiffness is close to the bulk epoxy stiffness. In contrast, the red case (Interphase V2, strong chemical bonding) shows a significant decrease in the UC stiffness with an increasing number of primary particles per agglomerate. Since the interphase is much stiffer than the bulk epoxy, the reduction of the interphase volume has a significant influence on the results. The Young’s moduli in the red curve converge because of the percentage reduction of the interphase volume with increasing agglomerate size. Secondly, comparing the blue and the orange curves, an overlayed effect can be observed by the presence of the RFA. As explained before, with increasing agglomerate size, the volume fraction of the

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RFA increases. This means the amount of weakly interacting inner surface increases where the particles are not bordering with polymer. The reason is a large reduction of the Young’s modulus of the agglomerate UCs for both interphase cases. So far, the chemically unbonded case with filled agglomerates provides the best approximation of the experimental NC response. However, a comparison between simulations and experiments is not possible here, since the agglomerate size distribution is not included. Still, there are some recommendations to optimize the elastic properties. A finer dispersion of the agglomerates as well as the enhanced interfacial bonding between the reinforcement and the polymer leads to a higher stiffness. It is also beneficial to ensure that agglomerates are filled with polymer to obtain the best elastic NC properties.

9.4.2 Agglomerate RVEs In this subsection, agglomerate RVEs are discussed as the outcome and the highest level of the micro-scale simulations in this chapter.

Pure epoxy

NC from Jux et al. [16] Homogeneous RVE Interphase V2 Interphase V2 and RFA Interphase V2 and filled with epoxy

Homogeneous RVE Interphase V1 Interphase V1 and RFA Interphase V1 and filled with epoxy

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Fig. 9.16 Dependency of the Young’s modulus of the agglomerate UCs on the number of primary particles per agglomerate, the interfacial bonding and the existence of a resin free area

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Modeling and Simulation Aspects

The agglomerate RVEs can be explained as RVEs consisting of the discussed agglomerate UCs. The first step in the model generation is the determination of the agglomerate sizes based on the experimental agglomerate size distribution as shown in Fig. 9.17a. The experimental agglomerate size distribution is a probability distribution of agglomerate sizes. Three examples for RVEs containing 20 UCs are shown in Fig. 9.17b. The material parameters of each UC are extracted from the results shown in Fig. 9.16 using a logarithmic function. Since it is impossible to fill the complete RVE volume with UCs and satisfy the agglomerate size distribution at the same time, the RVEs presented here demand a lower particle mass fraction. Hence, a mass fraction of the agglomerate UCs has to be as high as possible. For the generation of the agglomerate RVEs, a simple hardcore algorithm is used for randomly placing the UCs into the RVE starting with the largest UC. To compare the results of the agglomerate RVEs to experiments, the mass fraction of the RVEs is chosen to be 2.5%.

9.4.2.2

Representativeness of the Agglomerate RVEs

It is essential to investigate the representative agglomerate RVEs. Hence, agglomerate RVEs with an increasing number (i.e. 1, 3, 5, 10 and 20) of agglomerate UCs are simulated. To observe the convergence of the standard deviation, a statistically representative number of random samples is simulated for each case. The dependence of the Young’s modulus on the number of UCs per RVE is shown in Fig. 9.18a. The usage of one UC per RVE, leads to a large standard deviation and underestimation of the Young’s modulus. When using 5 UCs per RVE, Young’s modulus is converged and the difference to the RVE containing 10 UCs is only 0.02%. To reduce the computational effort for the simulations, RVEs containing 10 UCs will be used. From Fig. 9.18b, a sufficient convergence of the standard deviation of the Young’s modulus can be observed for 20 random realizations.

9.4.2.3

Influence of the Interphase and the RFA on the Elastic Properties of Boehmite/Epoxy NCs

The two interfacial bonding cases (Interphase V1: chemically unbonded and Interphase V2: chemically bonded) and the two RFA cases for each interphase (resinfree area and filled with polymer) are investigated in the following. The normalized Young’s moduli of all four cases are presented in Fig. 9.19. To make a comparison, the experimental results for pure epoxy and the NC with a BNP weight fraction of 2.5% are also shown. The Young’s modulus of two interphase cases without an RFA show an increase of 1.5 and 5.9% for Interphase V1 and Interphase V2, respectively, compared to the pure epoxy. The best prediction of the real NC properties is the chemically unbonded case without an RFA (Interphase V1) due to the low devia-

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Normalized probability (-)

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(b) Fig. 9.17 a Normalized agglomerate size distribution and b three example distributions for agglomerate RVEs consisting of 20 UCs

tion from experiments (approximately 0.6%). Since the experimental results suggest a slightly higher Young’s modulus compared to the simulations with reduced primary particle size, a certain chemical bond between the BNPs and the epoxy can be expected in the real material. To obtain more realistic elastic properties, it is particularly important to ensure that agglomerates are filled with polymer. Moreover, by improving the chemical interactions between the BNPs and the epoxy, an increase of the elastic properties of up to 6% can be reached, which is almost three times higher than the experimentally measured values. The effect of the agglomerate size distribution on the Young’s modulus of the two cases without the RFA for a constant weight fraction of 2.5% is shown in Fig. 9.20a. The experimental agglomerate size distribution (shown in Fig. 9.17a) was scaled to smaller or larger mean agglomerate sizes. The chemically unbonded case (Interphase V1) shows an almost constant Young’s modulus independent of the agglomerate size.

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Fig. 9.18 a Dependency of the Young’s modulus on the number of agglomerate UCs per RVE and b convergence behavior of the standard deviation for the RVE containing 10 UCs Pure epoxy Without resin free area With resin free area

Nanocomposite Jux et al. [16] Interphase V1 Interphase V2

to the pure epoxy (-)

Young’s modulus relative

1.1

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Fig. 9.19 Dependency of the Young’s modulus of the agglomerate RVEs on the interphase case and whether the agglomerates are filled with polymer or not

For the chemically bonded case, in contrast, the Young’s modulus increases with decreasing agglomerate size. For agglomerate sizes above 100 nm, the influence is small and the simulations predict an almost converged Young’s modulus. For agglomerates smaller than 100 nm, a significant increase of the Young’s modulus is observed. Using the primary particle size of 14nm shows a percentage increase of Young’s modulus of 12.3% compared to the bulk epoxy which is more than twice the increase of the 105 nm mean agglomerate size with the same weight fraction. It

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Fig. 9.20 Numerical investigation of the influence of a the mean agglomerate size on the Young’s modulus for a mass fraction of 2.5% and b the BNP mass fraction on the Young’s modulus for a mean agglomerate size of 105 nm

has to be noted that this is in the same range as the experimental result of the 105 nm agglomerate size with a weight fraction of 10%. The influence of the BNP weight fraction on the Young’s modulus is shown in Fig. 9.20b. An approximately linear dependence is observed for both interphase cases. The Interphase V1 case provides a better approximation compared to experimental measurements. The extrapolation of the curves is done to a weight fraction of 10% to achieve more realistic results for real technical applications. This results in a percentage increase of the Young’s moduli of 8.9 and 27.6% for the chemically unbonded and bonded cases, respectively. Interphase V1 provides a closer approximation compared to experiments. Moreover, the percentage increase of the elastic properties can potentially be doubled by improving interfacial bonding.

9.5 Conclusions In this chapter, a multi-scale framework for the determination of the elastic properties of NCs was proposed. The elastic properties of the epoxy were calculated and homogenized on the nanoscale level and passed to a higher scale. On the microlevel, the input from atomistic simulations was assembled and continuum models that contain agglomerates were developed. The epoxy was characterized through numerical tensile tests. The influence of the strain rate, the network structure and the chosen force field parameters on the elastic properties were studied. In view of a considerable influence on the elastic properties of the polymer, a proper calibration of the epoxy models with the given experimental data seems unfeasible. For the work presented here, the elastic properties of the reference system (E = 6.07G Pa, /nu = 0.324) were utilized despite

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the large deviations from the experiments. To extend the accessible time and length scales, two different CG models were developed. The effect of different levels of coarse-graining on the elastic properties was studied. The IBI method was applied to find bond potentials parameters. The models enabled a decrease in the number of DOF of the epoxy resin up to 9.22 and 18.5 times compared to MD simulations. The Young’s modulus of the epoxy predicted by the CG models was verified with atomistic simulation results. The CG results demonstrate that the Young’s modulus of the first CG model shows a good agreement between the CG model and MD simulations. The second CG model, on the other hand, could not reveal the expected accuracy and thus needed to be modified. The wall time simulation was also obtained using 12 cores to show the reduction of computational cost compared to MD. After the verification process, the first CG model was selected to study the strain rate effects on elastic modulus. The Young’s modulus decreases by decreasing the strain rate and shows a reasonable trend to predict the data comparable to the experimental data. Ultimately, the previous results were assembled to continuum models. Agglomeration reduces the stiffness of the NC. Stiffness of the NC is reduced in the case of agglomeration, because the overall interphase volume decreases, if primary particles are closely located inside of the agglomerates. The best prediction of the experimentally measured Young’s modulus is observed for the chemically unbonded case without an RFA. Since the experiments are slightly stiffer, a certain chemical bonding can be expected in the real material. Copyright Notice This chapter is based on Fankhänel’s Ph.D. thesis [14] and the paper by Mousavi et al. [1].

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31. Odegard GM, Jensen BD, Gowtham S, Wu J, He J, Zhang Z (2014) Predicting mechanical response of crosslinked epoxy using ReaxFF. Chem Phys Lett 591:175–178 32. Okabe T, Oya Y, Tanabe K, Kikugawa G, Yoshioka K (2016) Molecular dynamics simulation of crosslinked epoxy resins: curing and mechanical properties. Eur Polym J 80:78–88 33. Orrite SD, Stoll S, Schurtenberger P (2005) Off-lattice monte carlo simulations of irreversible and reversible aggregation processes. Soft Matter 1(5):364–371 34. Peter C, Kremer K (2009) Multiscale simulation of soft matter systems-from the atomistic to the coarse-grained level and back. Soft Matter 5(22):4357–4366 35. Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19 36. Pontefisso A, Zappalorto M, Quaresimin M (2015) An efficient RVE formulation for the analysis of the elastic properties of spherical nanoparticle reinforced polymers. Comput Mater Sci 96:319–326 37. Prasad A, Grover T, Basu S (2010) Coarse-grained molecular dynamics simulation of crosslinking of DGEBA epoxy resin and estimation of the adhesive strength. Int J Eng Sci Technol 2(4):17–30 38. Rzepiela AJ, Louhivuori M, Peter C, Marrink SJ (2011) Hybrid simulations: combining atomistic and coarse-grained force fields using virtual sites. Phys Chem Chem Phys 13(22):10437– 10448 39. Shenogina NB, Tsige M, Patnaik SS, Mukhopadhyay SM (2013) Molecular modeling of elastic properties of thermosetting polymers using a dynamic deformation approach. Polymer 54(13):3370–3376 40. Shi DL, Feng XQ, Huang YY, Hwang KC, Gao H (2004) The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites. J Eng Mater Technol 126(3):250–257 41. Shokrieh MM, Rafiee R (2010) Stochastic multi-scale modeling of CNT/polymer composites. Comput Mater Sci 50(2):437–446 42. Subramaniyan AK, Sun C (2008) Continuum interpretation of virial stress in molecular simulations. Int J Solids Struct 45(14–15):4340–4346 43. Theodorou DN, Suter UW (1986) Atomistic modeling of mechanical properties of polymeric glasses. Macromolecules 19(1):139–154 44. Varshney V, Patnaik SS, Roy AK, Farmer BL (2008) A molecular dynamics study of epoxybased networks: cross-linking procedure and prediction of molecular and material properties. Macromolecules 41(18):6837–6842 45. Wongsto A, Li S (2005) Micromechanical FE analysis of UD fibre-reinforced composites with fibres distributed at random over the transverse cross-section. Compos Part A Appl Sci Manuf 36(9):1246–1266 46. Wu C, Xu W (2006) Atomistic molecular modelling of crosslinked epoxy resin. Polymer 47(16):6004–6009 47. Xie Z, Chai D, Wang Y, Tan H (2016) Directly modifying the nonbonded potential based on the standard iterative boltzmann inversion method for coarse-grained force fields. J Phys Chem B 120(45):11834–11844 48. Yagyu H (2015) Coarse-grained molecular dynamics simulation of the effects of strain rate on tensile stress of cross-linked rubber. Soft Mater 13(4):263–270 49. Yang S, Cui Z, Qu J (2014) A coarse-grained model for epoxy molding compound. J Phys Chem B 118(6):1660–1669 50. Yang S, Qu J (2012) Computing thermomechanical properties of crosslinked epoxy by molecular dynamic simulations. Polymer 53(21):4806–4817 51. Yarovsky I, Evans E (2002) Computer simulation of structure and properties of crosslinked polymers: application to epoxy resins. Polymer 43(3):963–969

Chapter 10

Multiscale Modeling and Simulation of Polymer Nanocomposites Using Transformation Field Analysis Imad Aldin Khattab and Johannes Michael Sinapius

Abstract This chapter presents the adaptation of the Transformation Field Analysis (TFA) proposed by Dvorak et al. for predicting the properties of a representative volume of inelastic heterogeneous materials consisting of polymers and nanoparticles. The TFA method is implemented at the nanoscale as a user routine integrated into Representative Volume Elements (RVE) micro scale model, where the homogenously modelled modified matrix is reinforced by a certain volume fraction of aligned fibres. This yields to save computational resources and time compared to todays established methods with comparable accuracy. Finally, a single nano level application based TFA approach and numerical multi-scale (nano/TFA-micro/FEA) application are implemented to verify the behaviour of elastic-plastic epoxy nanocomposite consisting of agglomerated fillers in nano-scale and reinforced by aligned fibres in micro-scale under different external loads. A multiscale approach based TFA micro scale and TFA nano scale is presented and evaluated.

10.1 Introduction Fibre reinforced nanocomposites are currently the subject of international research and show remarkable improvements in mechanical and thermal characteristics compared to unmodified resins. The reinforced nanocomposite structure in a macro scale consists of fibres and a modified matrix. Subsequently, the modeling has to involve three scales in order to predict the influence of nanoparticles on the macro structure behaviour under external loads as shown in Fig. 10.1:

I. A. Khattab · J. M. Sinapius (B) Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_10

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Fig. 10.1 Triple-scales representing the nanocomposites to predict the influence of the nanoparticles on the mechanical behaviour of the composite structures

Fig. 10.2 Multi scale approach nano-micro (TFA-FEA) scale

• Nano scale consisting of possibly agglomerated nanoparticles and a bulk matrix • Micro scale consisting of a homogenously modelled modified bulk matrix and fibres • Macro scale nanocomposite structures, e.g., a wing-box or a fuselage structure As a result, the process requires two stages, nano-micro scale and micro-macro scale. A multiscale computational approach covering all methods (from atomistic/molecular through mesoscale to macroscale) for each length and time scale can play an ever-increasing role in predicting and designing material properties, and guiding such experimental work as synthesis and characterization. This chapter presents such a novel multi-scale approach. The mixture of nanoparticles and bulk matrix is homogenized to provide effective properties of the modified matrix by the Transformation Field Analysis (TFA) approach. The microscopic structure is thereafter modelled as unit cell subjected to periodic boundary conditions and the overall transformation strains, or stresses, are computed through a multiscale model as shown in Fig. 10.2.

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The Transformation Field Analysis (TFA) is a method for incremental solution of loading problems in inelastic heterogeneous media and composite materials but has never been investigated for nanoscale problems before. It is hypothesized that the TFA proposed by Dvorak [15], Dvorak et al. [17, 18], Bahei-El-Din and Dvorak [5] can be seen as an elegant way of reducing the number of microscopic internal variables by assuming the nanoscopic fields of internal variables to be piecewise uniform regarding their local distribution. Additionally, the TFA method provides the same information about the inelastic behaviour of a unit cell as does a standard finite element program for elastic-plastic or other inelastic materials, however, usually at much less expense. Oskay and Fish [31], for example, emphasize the large savings in computational cost in comparison to standard finite element programs for inelastic analysis. Scientific publications that illustrate the applications of this method are, among others, by Procházka [32] on slope optimization, by Baweja et al. [6] for creep of concrete, by Benveniste [7, 8], Dvorak and Benveniste [14] on piezoelectric composites, by Dvorak [12] on damage modeling, by Franciosi and Berberinni [21] on polycrystal plasticity, by Sacco [34] on periodic masonry, by Bahei-El-Din et al. [1–3] on electromechanical coupling in woven composites exhibiting damage and on multiscale damage mechanics and by El-Etriby et al. [19] and Bahei-El-Din [4] on a multi-scale based model for composite materials with embedded PZT filaments for energy harvesting. A refined procedure, so called non-uniform TFA (NTFA), considering a locally non-uniform distribution of the inelastic strain in the material, has been proposed by Michel and Suquet [29, 30]. The NTFA approach has been adopted to model different composite materials, e.g., it has been used to study the polycrystal plasticity by Franciosi and Berbenni [21, 22] the elastic-viscoplastic response of composite materials by Roussette et al. [33] and the three dimensional behaviour of metal matrix composites by Fritzen and Böhlke [23, 24]. Although many efforts have been made in order to improve the non-uniform TFA approach, two key aspects of TFA are still object of research; they concern the approximation of the inelastic field and the solution of the evolution problem that are directly related to the number of history variables, Covezzi et al. [11]. Clearly, all the mentioned studies and researches using the TFA approach have been developed in the microscopic or mesoscopic domains. Consequently, the effects of nanocomposites on the stiffness and strength of real structures through the TFA and NTFA methods have not been investigated. This chapter describes the principle of the TFA method containing various definitions of the local and overall properties in Sect. 10.2. The details of the TFA method and formulas for composites with elasticplastic phases are specified in Sect. 10.3. Section 10.4 comprehensively describes a two-scale micro-nano-behavioural model based on the so-called FE2 approach, where the TFA approach is integrated at nano scale. Additionally, the step by step of the two pre-processing and implementing TFA stages are presented. Finally, Sect. 10.5 demonstrates the implementation of single scale and multi-scale applications to verify

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the behaviour of elastic-plastic epoxy nanocomposite consisting of agglomerating fillers in nano-scale and reinforced by aligned fibres in micro-scale under different external loads.

10.2 Transformation Field Analysis (TFA) In the current work, the RVE of the equivalent continuum mechanics model of nanocomposites based on TFA method is utilized to predict the elastic-plastic behaviour of a matrix with nanoparticles or agglomerates under different external loads. The TFA provides piecewise uniform approximations of the instantaneous local strain and stress fields in the phases, and estimates of the overall instantaneous thermo-mechanical properties of a representative volume of the heterogeneous solid.

10.2.1 Local Fields As depicted in Fig. 10.3, the actual stress and strain fields in the RVE, which vary pointwise, are averaged over a finite number M of subvolumes Vη , η = 1, 2, ..., M, such  that each subvolume belongs either to the matrix or the nanoparticles, where Vη = V is the volume of the RVE. Hence, the overall fields are the weighted volume sum of the local average fields, σ¯ =

M  η=1

Fig. 10.3 RVE with subvolumes

cη ση, ε¯ =

M  η=1

cη εη,

(10.1)

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where σ¯ and ε¯ are the global stress and strain. Vectors ση and εη represent the average stress and strain in subvolume Vη , and cη = Vη/V is the volume fraction such M cη = 1. that η=1 The representative volume V of nanocomposites, i.e. nanoparticle imbedded into a matrix, is divided into subvolumes represented by finite elements, Vη , η = 1, 2, ..., M such that each subvolume belongs either to the matrix or the nanoparticles. A uniform overall strain load ε0 or uniform overall stress load σ 0 and uniform thermal load θ are acting which cause the elastic-plastic behaviour of the unit cell. It shall be described in nanoscopic level by means of the TFA method. The subvolumes of the RVE are assumed to undergo elastic deformations due to the local stress averaged over the subvolume. In addition, uniform transformation stresses, λη , or strains, μη , can be introduced in the subvolumes to represent local fields that are not reversed by removing the mechanical loads. For example, the transformation fields can be due to temperature changes or inelastic deformations as in the present work. Consequently, rate dependent load sets are applied, denoted by {˙ε0 , θ˙ } and {σ˙ 0 , θ˙ }, where ε˙ 0 and σ˙ 0 are overall strain and stress rates, resp., whereas θ˙ denotes the overall thermal load increments. The total local strain or stress field increment in each subvolume (element) generated by such application can be additively decomposed as: ε˙ η = Mη σ˙ ηe + μ˙ η σ˙ η = L η ε˙ ηe + λ˙ η

(10.2a) (10.2b)

where ε˙ ηe and σ˙ ηe are elastic local strain and stress, respectively. The eigenstrains μ˙ η includes both, inelastic strain and total thermal strains, μ˙ η → ε˙ ηin + ε˙ ηθ , the eigenstresses λ˙ η consists of relaxation stress and thermal stress λ˙ η → σ˙ ηr e + σ˙ ηθ . L η and Mη = L −1 η are elastic stiffness and compliance, respectively. Subsequently, at each increment, the relation between the local strain or stress and overall strain or stress loads is described by the following equations ε˙ η = Aη ε˙ 0 +

M 

  Dηρ ε˙ ρin + ε˙ ρθ

(10.3a)

  Fηρ σ˙ ρr e + σ˙ ρθ

(10.3b)

ρ=1

σ˙ η = Bη σ˙ 0 +

M  ρ=1

where Aη and Bη are the elastic strain and stress concentration factors of the volume elements Vη . Dηρ and Fηρ are the eigenstrain and eigenstress concentration factor matrices (the transformation influence functions). Preparing a TFA solution, the elastic strain and stress concentration factors Aη and Bη of the elements Vη and the eigenstrain and eigenstress concentration factor matrices, Dηρ and Fηρ have to be computed for the M subvolumes (elements) of the

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RVE (ρ, η = 1, 2, ..., M). The concentration factors Aη and Bη are computed under nodal displacements and forces. For example, the kth column (k = 1, 2, ..., 6) of the stress concentration factor Bη is given by the stresses caused in Vη by setting one of the overall stress component σ¯ k = 1 and all others to zero under periodic boundary conditions, as suggested by the first term of Eq. (10.3b). Evaluation of the eigenstrain and eigenstress concentration factor matrices follows a similar procedure that is inferred from Eq. (10.3b). For example, the kth column (k = 1, 2, ..., 6) of eigenstress concentration factor matrix Fρη is given by the average stresses computed in subvolume Vη , due to transformation stress λk = 1 applied to Vρ , where the boundary condition of the RVE is ε0 = 0. A total of 6M similar problems are solved to compute all six columns of the transformation influence functions. However, since the load vector is the only variable among these problems and the stiffness matrix is constant, the finite element solution of the RVE for evaluating the transformation influence functions generally does not require extensive computing time. The overall effective elastic properties of the RVE, i.e. stiffness and compliance matrices, are computed by using a homogenisation method. The homogenisation method is performed by standard FE tools such as Abaqus, where periodic boundary conditions are imposed on the RVE.

10.2.2 Total Response The response rate of the RVE under uniform strain rate ε˙ , stress rate σ˙ , uniform transformation strain rate μ, ˙ and stress rate λ˙ can be written as ε˙ = M σ˙ + μ˙ σ˙ = L ε˙ + λ˙ .

(10.4a) (10.4b)

The total rate of the transformation fields are related via λ˙ = −L μ. ˙ In case of elastic behaviour and absence of transformation fields, the overall stiffness and compliance are predicted by Hill [26]: L=

M 

cη L η Aη

(10.5a)

cη Mη Bη .

(10.5b)

η=1

M=

M  η=1

If the rate of the local transformation stresses and strains are known, their overall equivalent values can be calculated from generalized Levin’s formula, Levin [28] and Dvorak and Benveniste [16], as follow

10 Multiscale Modeling and Simulation of Polymer Nanocomposites . . .

λ˙ =

M 

215

cη AηT λ˙ η

(10.6a)

cη BηT μ˙ η .

(10.6b)

η=1

μ˙ =

M  η=1

Consequently, the instantaneous tangent compliance and stiffness is computed as it is described in the next section and under inelastic behaviour of matrix.

10.3 TFA of Inelastic Deformation Two different load rate sets are applied, {˙ε0 , θ˙ } and {σ˙ 0 , θ˙ }. The eigenstrains and eigenstresses contributions caused by one of the applied load sets are expressed as a function of total stress or strain, Eq. (10.2). p Using the local instantaneous compliances and stiffnesses Mη = Mηe + Mη and p e Lη = L η + Lη , and introducing them in Eq. (10.2), the total local increments can be decomposed into elastic (superscript e), inelastic (superscript p), and thermal parts (superscript θ ) ε˙ ηe + ε˙ ηin + ε˙ ηθ Mη σ˙ η + mη θ˙   e Mη + Mηp σ˙ η + mη θ˙

=

ε˙ η

= = ε˙ ηin + ε˙ ηθ

applying = Mηp σ˙ η + mη θ˙

(10.7) (10.8)

and σ˙ η

σ˙ ηe + σ˙ ηr e + σ˙ ηθ Lη ε˙ η + lη θ˙  e  L η + Lηp ε˙ η + lη θ˙

= = =

σ˙ ηr e p

+

σ˙ ηθ

applying = Lηp σ˙ η + m η θ˙ p

(10.9) (10.10)

where mη = m eη + mη and lη = lηe + lη are instantaneous strain and stress thermal vectors, respectively. Thus, the eigenstresses are related to the stress relaxation. The total thermal stress and the eigenstrains are included in the inelastic strain and the total thermal strains

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  σ˙ ηr e + σ˙ ηθ = −L η ε˙ ηin + ε˙ ηθ   = −L eη Mηp σ˙ η + m η θ˙   ε˙ ηin + ε˙ ηθ = −Mη σ˙ ηr e + σ˙ ηθ   = −Mηe Lηp ε˙ η + lη θ˙

(10.11) (10.12)

where Mηe and L eη are the elastic compliance and stiffness of each subvolume, and p p Mη and Lη are the instantaneous plastic compliance and stiffness of each subvolume. The plastic compliance and stiffness for each element can be calculated when the element reaches the plastic deformation at initial yield surface or subsequent loading surface in the six-dimensional stress space. In particular, the Mises loading surface frequently is adopted for the elastic-plastic resin at each increment. Isotropic hardening and flow rules are utilized to predict the plastic compliance and stiffness at each increment, see the appendix section and Dvorak et al. [18]. All the constitutive relations for materials that exhibit incremental elastic-plastic deformation in response to an applied loading path which extends beyond their initial yield surface are provided in detail by Dvorak [13, Chapter 11]. Merging Eqs. (10.3) and (10.11), resp. Eq. (10.12), provides one system of equations for each of the two loading sets {˙ε0 , θ˙ } and {σ˙ 0 , θ˙ } that can be solved for the respective rates of local strain ε˙ η and local stress σ˙ η , respectively: ε˙ η −

M 

Dηρ Mρe Lρp ε˙ ρ = Aη ε˙ 0 +

ρ=1

σ˙ η −

M 

M 

Dηρ Mρe lρ θ˙

(10.13a)

Fηρ L eρ mρ θ˙

(10.13b)

ρ=1

Fηρ L eρ Mρp σ˙ ρ

ρ=1

= Bη σ˙ + 0

M  ρ=1

For numerical solutions such as those carried out in this work under uniform total speed of the applied strain and temperature for the load set {˙ε0 , θ˙ }, the array of strain increments in the volume element Vη is ⎡ {˙εη } = ⎣diag(I ) +

M  ρ=1

⎫ ⎤−1 ⎧ M ⎬ ⎨  Dηρ Mρe Lρp ⎦ Dηρ Mρe lρ θ˙ , Aη ε˙ 0 + ⎭ ⎩

(10.14)

ρ=1

reduced to a matrix notation −1   0      {˙εη } = diag(I ) + Dηρ Mρe Lρp Aη ε˙ + Dηρ Mρe lρ θ˙ .

(10.15)

The solutions of the local strain vectors of all subdivisions are sought in the form     {˙εη } = Aη ε˙ 0 + aη θ˙

(10.16)

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  where the instantaneous  strain concentration factors Aη and the thermal strain consentration factors aη are computed by  Aη =   aη =



 −1   diag(I ) + Dηρ Mρe Lρp Aη     e p −1 diag(I ) + Dηρ Mρ Lρ Dηρ Mρe lρ .

(10.17) (10.18)

The overall instantaneous stiffness matrix and thermal strain vector for the RVE can be found by substituting Eqs. (10.17) and (10.18) in Eq. (10.5a) L=

M 

cη Lη Aη

(10.19)

cη lη aη

(10.20)

η=1

l=

M  η=1

where the local instantaneous stiffness matrix Lη and the thermal strain vector lη are calculated from Lη = L eη + Lηp lη = lηe + lηp

(10.21) (10.22)

Finally, the overall stress rate can be found ˙ σ˙ = L˙ε0 + lθ,

(10.23)

σ˙ = L e ε˙ 0 + λ˙   = L e ε˙ 0 − μ˙

(10.24)

or from Eq. (10.2)

where the overall inelastic strain rate is computed from Eq. (10.6a) μ˙ =

M 

cη BηT μ˙ η

η=1

=

M 

  cη BηT ε˙ ηp + ε˙ ηθ

(10.25)

η=1

after substituting the local plastic strain rate ε˙ η and the thermal strain rate ε˙ ηθ from Eq. (10.7). For the load set {σ˙ 0 , θ˙ } similar sequences of equations are applied. p

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An iterative procedure and additional details that are employed to find the local strain rates in agreement with instantaneous constitutive relations and yield conditions of the elements are described in the next section.

10.4 Finite Element-Based Implementation The TFA method is used to determine the mechanical properties of the polymer nanocomposite consisting of agglomerated nanoparticles with interphase area. The yielded mechanical properties are then used in the equivalent unit cell of a FE model. Cho et al. [10] show that agglomerating fillers with interphase area can be simulated by FEA, as shown in Fig. 10.4. adpoted from Shin et al. [35]. In this chapter, the focus lies on the nano-micro scale level of the process without considering the micro-macro scale level. By the two-scale nano-micro, a multiscale behaviour model based on a multilevel finite element (FE2 )-approach is used to take into account heterogeneities in the behaviour between the matrix and nanoparticles, Carrère et al. [9]. The major benefit of the (FE2 )-method is the ability to analyse complex mechanical problems with heterogeneous phases that present a variety of behaviour at different scales. Consequently, this new proposition will be beneficial to define an accelerated multiscale investigation of a highly heterogeneous nanocomposite materials. In the (FE2 )-method, each micro level integration point of a structural FE simulation is associated with a full resolution finite element discretization of the underlying nano level. The micro level strain ε¯ at the integration point defines the Dirichlet boundary conditions [25] for the nano level problem which is then solved numerically. After the equilibrium of the nonlinear nano level simulation is attained, the effective stress σ¯ is computed by means of volume averaging and it is returned to the micro level simulation. Note that the micro level algorithmic stiffness operator D¯ is also an important output of the nano level RVE problem which can account for a substantial amount of additional algebraic operations further increasing the overall computational complexity.

Fig. 10.4 A multi particulate nanocomposite systems that includes four agglomerated nanoparticles a Transmission electron microscopy (TEM) image of silica nanocomposites b equivalent finite element model, after Shin et al. [35]

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The two-scale process is performed by a commercial finite element program, where the RVE of the micro level is modelled by FE tools such as Abaqus and the RVE model of the nano scale is applied by the TFA method and integrated in the UMAT subroutine. Regarding the (FE2 )-method, the TFA process of the nano level will be under strain set loads. The TFA approach is divided into two stages: offline or pre-processing stage and the online or implementing phase. Offline phase: 1. The finite element RVE of the nano scale consisting of bulk matrix and agglomerated nanoparticles is modelled under periodic boundary conditions. 2. The global effective stiffness and compliance are calculated. 3. In case of incremental strain load, the linear elastic data of the elastic strain concentration factors of elements Vη , Aη and the eigenstrain concentration factor matrices Dηρ are computed for the M subvolumes (elements) of the RVE, where η, ρ = 1, 2, ..., M. Online phase: At each Gauss point of the micro level, the solution algorithm for elastic-plastic composites of the nano level under strain incremental load set Δε will be constructed as follows 1. For the given interval t1 ≤ t ≤ tn+1 of the prescribed histories Δε = εi+1 − εi the number of sub-increments n are selected, e.g., n ≤ 10, and compute the time increment h = tn+1 −t1/n where i is increment index of the micro level. Here, the sub-increment value n should not be too big, since the increment of micro level is already small. 2. The RVE of the nano level will be subjected to the overall sub incremental strain εt+1 = εt + Δε/n. This incremental strain values are different for each micro level Gauss point. 3. Calculate the initial stress σρ (t1 ) = σˆ ρ and initial strain ερ (t1 ) = εˆ ρ of each nano-element according to Eq. 10.2, and the tensile yield stress Yρ (t1 ) = Yˆρ (see appendix). 4. For k = 1, 2, ..., n, repeat the following steps 4–9.   5. For each nano-element ρ, compute the yield function gρ(k) σρ(k) , as described in the appendix section. 6. If gρ(k) < 0, i.e. the element is elastic, go to step 9; if gρ(k) = 0, compute {dερ }k from Eqs. (10.15) and (10.16). The calculation of ε˙ 0 = Δε and the instantaneous p plastic stiffness Lρ is explained in the appendix section. 7. The differential equation Eq. (10.16) is solved by second order Runge-Kutta method and the strain field at time t (k + 1) compute as follows:      h dερ + dερ∗ k  ερ k+1 = ερ k + 2

(10.26)

220

where

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        ∗ dερ k = dερ tk+1 , ερ k + h dερ k

(10.27)

  8. The corresponding stress σρ k+1 is calculated by the constitutive equations given in the appendix section. 9. The overall tangent operator at each micro level increment and for each Gauss point can be computed by Eq. (10.19). At the end of the process at each micro level, the overall stress and overall tangent operator are obtained from Eqs. (10.24) and (10.20). Please note that there are no thermal loads in the current application, i.e. θ˙ = 0. The convergence of the solution is verified by changing the increment step of micro and nano models. The solution depends on the selected magnitude of the nano time increments h; an error of order h 2 is expected in the applied Range-Kutta method. In case of micro finite element model, the time increment of the geometrical linear and static implicit analyses should be selected carefully and controlled to achieve the convergence by taking into account the nonlinearity of the nano level model which is connected to each Gauss point. In the next section, a single nano level application is presented to predict the inelastic mechanical behaviour of resin modified by nano particles. Additionally, a numerical multi-scale (nano/TFA -micro/FEA) application is implemented to inspect the behaviour of elastic-plastic epoxy nanocomposite in nano scale and reinforced by aligned fibres in micro-scale under different external loads.

10.5 Numerical Applications The current work presents two numerical applications: one application at nano-scale and two applications at nano-micro level. The aim at the nano-scale for a couple of agglomerated nanoparticles is to analyze the ability of the TFA method to predict the inelastic behavior of the matrix containing nanoparticles. The second application is to investigate the influence of agglomerated nanoparticles within the matrix phase in nano-scale by the TFA method on the aligned fibre reinforced composites RVE model in micro-scale under different external loads. Both applications follow the steps of the process described in Sect. 10.4. However, in the case of a single nano level, these steps are applied once, so that the time increment is h = (tn+1 −t1 )/n , where n should be big enough for overall external loads ε0/n and σ 0/n to achieve the required accuracy. In case of multiscale approach the steps of process are repeated at each micro level increment and for each micro Gauss point. In this case, the tangent operator is important to attain the convergence of the solution.

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10.5.1 Single Nano-Scale Model A typical Representative Volume Element (RVE) consists of 90% weight fraction (wt) resin epoxy and 5 agglomerated Boehmite nanoparticles having a 10% wt, is used for this task as shown in Fig. 10.5. The TFA method described by Dvorak et al. [13, 18] is employed to simulate the elastic-plastic response of RVE on a nanoscale level. The RVE of volume V is divided into subvolumes represented by finite elements, Vη , η = 1, 2, ..., M, such that each subvolume belongs either to the matrix or the particles. The nano RVE is subjected to external strain and stress loads. The results are compared with the full FE method, where the RVE is modelled by a standard FE tool such as Abaqus using 659 general purpose tetrahedral elements (type C3D4). All the geometrical and mechanical properties input data are defined and given by Fankhänel et al. [20] in order to make the results comparable. The length edge of RVE is 13.3 nm, the size of the orthorhombic Boehmite particle is 3.1 × 3 × 3.1 nm as depicted in Fig. 10.5. The anisotropic Young’s moduli of the individual boehmite nanoparticles yield values of 232, 136 and 267 GPa, as determined by MDFEM simulations in the directions [100], [010] and [001], respectively. Poisson’s Ratio is ν = 0.3. Young’s modulus of the epoxy resin is 3370.0 MPa, it’s Poisson’s ratio ν = 0.39. While the nanoparticles are assumed to remain elastic, the matrix (LY556 Epoxy) is elasticplastic with initial tensile yield stress 34.59 MPa. The hardening characteristic is plotted in Fig. 10.6. In preparation for the TFA analysis, the elastic strain and stress concentration factors of elements Vη , Aη and Bη and the eigenstrain and eigenstress concentration factor matrices, Dηρ and Fηρ , η, ρ = 1, 2, ..., M, have to be computed for the M subvolumes (elements) of the RVE. The overall effective elastic properties of the RVE, stiffness and compliance matrices, are computed by using a homogenisation method. The homogenisation method is performed by Abaqus, where periodic boundary conditions are imposed on the RVE.

Fig. 10.5 Left: Representative Volume Element (RVE) consisting of 90% (weight fraction) resin epoxy and 10% agglomerate boehmite nanoparticles. Right: The orthorhombic shape of a single boehmite nanoparticle

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Fig. 10.6 Specified data points of the plastic behavior of the LY556 epoxy

A strain controlled load case ε0 and a stress controlled load case σ 0 are applied to determine the elastic-plastic behaviour of the unit cell in nanoscopic level by using the TFA method. Consequently, the rate sets are applied, denoted by {˙ε0 } and {σ˙ 0 }, where ε˙ 0 and σ˙ 0 are overall strain or stress increments, respectivly.

10.5.2 Results of the Single Nano-Scale Model The RVE is subjected mainly to two load cases. The first load case is given by a stress load in the x-direction and a shear stress load in the xy-plane, the second load case is determined by strain loads in the x-direction and the xy-plane. The computations for these load cases are performed for different number of load increments in order to investigate the convergence. The elastic-plastic stress-strain response of the RVE is computed under five cases: a tension stress of σ11 = 100 × 10−14 MPa, a shear stress of σ12 = 50 × 10−14 MPa; strain loads of εx = 0.05 and εx y = 0.05, for case 3 and 4 respectively; combined strain load εx = εx y = 0.05 for case 5. All the results are plotted and compared with full FE analysis as shown in Figs. 10.7, 10.8, 10.9, 10.10 and 10.11. The influence of the time increments h and the number of increments n on the final result is substantiated for case number five, where different numbers of increment are selected n = 5, 10, 20, 40, 100. The overall strain response under tension and shear stress loads calculated by the TFA method coincides with the solutions found by direct evaluation of the overall strains using the standard elastic-plastic FE method as shown in Figs. 10.7 and 10.8. The percentage difference of overall equivalent strain between TFA and FE solutions err or = 3.2% and for shear stresses in the xy-plane are for tensile stresses Δεequivalent

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Fig. 10.7 The equivalent strain response of the RVE nanocomposite under tensile stress in x-direction

Fig. 10.8 The equivalent strain response of the RVE nanocomposite under shear stress in xy-plane

Fig. 10.9 von Mises stress response of the RVE nanocomposite under strain in x-direction

err or Δεequivalent = 4.3%. The percentage difference of the total von Mises stress response err or = between TFA and FE solutions are in case of a strain load in x-direction Δσ Mises err or 0.21% and in case of shear strain load in xy plane Δσ Mises = 0.46%, as shwon in Figs. 10.9 and 10.10.

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Fig. 10.10 von Mises stress response of the RVE nanocomposite under strain in xy-plane

Fig. 10.11 von Mises stress response of the RVE nanocomposite under equivalent strain composed of maximum strain in x-direction and xy-plane of 0.05 each

Table 10.1 The effective von Mises stress responses of nano RVE model under combined strain loads εx = 0.05 and εx y = 0.05, and for various increment numbers number of increments

100

40

20

10

5

FE σ Mises (10−14 MPa) T F A (10−14 MPa) σ Mises

97.836 95.106

97.834 95.224

97.830 95.424

97.822 95.714

97.808 96.406

The behaviour of the RVE nanocomposites under the composed strains εx = εx y = 0.05 loads is plotted in Fig. 10.9, where the percentage difference of von Mises err or = 2.8%. The effective stresses between the FEA and TFA solution is Δσ Mises von Mises stress responses of nano RVE model under the later load case for varied increment numbers are recorded in Table 10.1, where the percentage tolerance of von Mises stress between the lower n = 5 and higher increment value n = 100 for err or = 0.02%, and for the TFA the full finite element solutions is not noticeable Δσ Mises err or solutions is Δσ Mises = 1.3%. The TFA method in comparison with the finite-element (FE) solution appears to be more efficient in terms of the CPU time required. Within the initial phase of this work the first milestone has been achieved being the successful implementation of the

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TFA for a model of a nano-scale heterogeneous materials. Moreover, the comparison with FEM showed a convincing agreement which is required for the next step which is the integration of the TFA in a multi-scale approach. Certainly, the cost of both the TFA and the FE solution depends on many other factors, such as the specified tolerance and the number of iterations required for achieving such tolerance.

10.5.3 Nano-Micro Scale The micro RVE model consists of 55% volume fraction glass fibres aligned in hexagonal array and embedded in 45% modified epoxy resin where the x-axis aligned with fibre direction. The RVE model subjected to periodic boundary conditions is modelled to study the influence of the nanoparticles on the mechanical behaviour of the fibre reinforced composites in micro scale under external mechanical loads. The Young’s modulus of the glass fibre is 77.6 GPa and Poisson’s ratio is ν = 0.25. The anisotropic Young’s modulus values of modified epoxy resin which calculated from RVE nano scale model through the off-line stage are 4086.6, 4168.3 and 4158.6 MPa in x-, y-, and z-direction, respectively. Poisson’s ratios are ν = 0.36, 0.37, and 0.368. The linear analysis of the micro RVE model is implemented by Abaqus and the TFA process is integrated to the UMAT subroutine for each Gauss point of the modified epoxy resin element. The same nano RVE model of the first application, described in the previous section, is utilized to simulate the modified resin by using TFA approach as shown in Fig. 10.12. The fibre behaviour is considered elastic modelled by the matrial model provided by Abaqus, so there is not need to call the UMAT at the fibre element Gauss points. The nano scale RVE model is subjected to incremental strain load set {˙ε0 } and only the elastic strain concentration matrices Aη ,

Fig. 10.12 Schematic of the FE2 multiscale analysis between nano-micro (TFA-FEA) scale

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the eigenstrain concentration factor matrices Dρη , and the volume fraction cη of each element η, ρ = 1, 2, ..., M are called by the TFA process in additional to the overall effective stiffness. To save computational cost, only the incremental strain {˙ε0 } at each Gauss point at increment i + 1 is passed to the TFA process. This technical step requires to save the initial values of effective stress and strain in additional to the overall strains and stresses results of the nano RVE elements which is equal to the final results of the previous increment i. The NSTATV state variable for each Gauss point in UMAT subroutine is defined to save and update these values at the end of the current increment and pass them to the next increment.

10.5.4 Extended Nano-Micro (TFA-TFA) Scale Approach To validate the multiscale approach based on a TFA nano scale and a full FEA micro scale, a multiscale approach based TFA micro scale and TFA nano scale is processed in the following steps: Offline stage: 1. The entire offline of the nano TFA model steps are considered. 2. The micro scale RVE model is based on periodic boundary conditions. Thus, the offline steps of the nano TFA model are repeated for the micro TFA model, where in case of incremental strain load, the linear elastic data of the elastic strain cono of each element volume Vηmicr o and the overall effective centration factors Amicr η e(micr o) stiffness L are computed for the M subvolumes (elements) of the micro RVE, where ηmicr o = 1, 2, ..., M micr o . The effective stiffness and compliance of the modified resin is already calculated from the overall effective stiffness and compliance of the nano-scale model. 3. As an approximate procedure, the interactions of inelastic strain and stress in modified resin is neglected. Since the current multiscale process is full a complete TFA-TFA analysis, the tangent operator calculated from Eq. (10.19) is not micr o is not inquired. Accordingly, the eigenstrain concentration factor matrices Dηρ micr o micr o micr o ,ρ = 1, 2, ..., M . calculated, where η Online stage: At each Gauss point of the micro level, the solution algorithm for elastic-plastic composites of the nano level under strain set load {˙ε0(micr o) } will be constructed exactly as the steps of the online stage in Sect. 10.4. (steps 1–8) and without calculating the tangent operator neither for nano scale nor for micro scale. The different last step of multiscale FE analysis consists here in calculating the total response stress of microscale model from Eq. (10.24) as follows   micr o = σimicr o + L e(micr o) Δε0(micr o) + Δλmicr o σi+1   = σimicr o + L e(micr o) Δε0(micr o) − Δμmicr o ,

(10.28)

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where the overall inelastic strain Δμmicr o is calculated from the Eq. (10.6b) as follows Δμ

micr o

=

micr o M

o cηmicr o BηT (micr o) Δμmicr η

(10.29)

ηmicr o =1

Here, the inelastic strain at each micro Gauss point equal to the overall inelastic strain of nano RVE model and can be calculated by Eq. (10.25) as well o = Δμmicr η

M 

cη BηT Δμη

(10.30)

η=1

where the inelastic strain at each nano-Gauss point Δμη is calculated by the process steps 1–8 (Sect. 10.4). The accuracy of the solution is sensitive to the increment number of micro scale n micr o and nano scale n. In case of the TFA-TFA multiscale method, the computational cost is a function of the number of Gauss points (internal variables) and time increment number of the micro and nano scale.

10.5.5 Results of Two Micro-Nano Scale Approaches The micro RVE under periodic boundary conditions is meshed by 552 eight-node brick elements with reduced integration (type C3D8R) divided into 318 modified epoxy resin elements and 234 glass fibres elements. This micro model is utilized for both multiscale approaches TFA-FEA and TFA-TFA, where for later approach, it is used to subtract the linear parameters mentioned in offline TFA process, Sect. 10.4. For both multiscale approaches, the micro RVE is subjected to three strain loads in y-direction ε y = 0.02, xy-plane εx y = 0.02 individually and simultaneously. The non-combined load cases are also investigated by a full finite element analysis. The single-level implementation on the same micro RVE model consists of unmodified epoxy resin. The results are considered as reference to investigate the influence of the nanoparticle on the mechanical behaviour of the nanocomposites. For each strain load case, the effective von Mises stress responses of the two multiscale approaches and the reference RVE model of pure resin are plotted in Figs. 10.13, 10.14, and 10.15. The overall von Mises stress response under strain in transverse y-direction ε y = 0.02 is plotted as shown in Fig. 10.13. Although of the different behaviour between the two multi-scale approaches the figure shows a good agreement at the end of strain load where the percentage difference is negligible and equal to 0.01%. The maximum percentage difference is Δσ Mises = 6.34% at ε y = 0.0076. It can be perceived an improvement in the strength and stiffness of the micro RVE with modified resin compared with unmodified resin, where the stress of micro model at the end of load

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Fig. 10.13 The von Mises stress response of the micro RVE model under strain load in y direction

with modified resin is 125.5 MPa and with pure resin is just 110.2 MPa. The 5% wt of the agglomerated nanoparticles increase the stiffness E y from 10746 to 12504 MPa. The computational time under the same hardware conditions of the TFA-TFA method is less than six times, 40 min, compared to the TFA-FEA method with corresponding 260 min. Correspondingly, the von Mises stress response under strain load in xy-plane εx y = 0.02 of the two multiscale approaches and unmodified resin are plotted in Fig. 10.14. As the previous case, the percentage difference of von Mises stress at the end of load is insignificant and equal to 0.97%, whereas the effective von Mises stress at εx y = 0.02 is 92.44 MPa and 93.39 MPa for TFA-FEA and TFA-TFA respectively. The maximum percentage difference is Δσ Mises = 5.28% at εx y = 0.008. The shear stress in xyplane increases by 20.9 % compared to the micro RVE model with unmodified resin. Obviously, the shear Modulus G x y increases because of nanoparticles contents from 1212 to 1456 MPa. The computational cost in case of TFA-TFA is six times less than TFA-FEA. The von Mises response of the micro RVE model under combined strain ε y = 0.02 and εx y = 0.02 are plotted in Fig. 10.15. Although it is a complex strain load case, the percentage difference of von Mises stress for TFA and FEA at the end of load path is still negligible and is 0.7%, and overall behaviour for both multiscale approaches are similar. However, the computational time with TFA-TFA method is less than one third, namely 96 min, compared with TFA-FEA method which corresponds to 300 min, whereby an automatic incrementation option is selected for TFA-FEA RVE model, as well as a fixed increment number, n micr o = 40 in case of TFA-TFA RVE model. The nano increment number for both cases is n = 5. A further investigation of the influence of the increment number (n nano = 2, 3, 5 and n micro (auto ≈ 46), f i xed (40) and f i xed (100) ) on the accuracy and the solution convergence for micro and nano scale models is carried out, again for both multiscale approaches as shown in Fig. 10.16. The von Mises stress response under strain load in y-direction ε y = 0.02 (Fig. 10.15) shows the accuracy and stability of the both multiscale models TFA-FEA and TFA-TFA, with all the results being almost identical for different micro and nano increment numbers. From this it can be

10 Multiscale Modeling and Simulation of Polymer Nanocomposites . . . Fig. 10.14 von Mises stress response of the micro RVE model under strain load in y-direction

Fig. 10.15 von Mises stress response of the micro RVE model under equivalent strain composed of strain in y-direction and xy-plane

Fig. 10.16 On the influence of the increment number n = 2, 3, 5 and n micr o = auto(≈ 46), n micr o = f i xed (40), and n micr o = f i xed (100) on the accuracy and the solution convergence for micro and nano scale models

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concluded that for relative acceptable accuracy the lower the increment number, the lower the computational effort and that the TFA-TFA multiscale approach is faster than TFA-FEA approach. The convergence of solution of TFA-TFA approach can be achieved for smaller increment numbers of TFA micro model than the micro model of FEA, due to the lack of the iteration by the autoincrement, which requires more iterations to achieve the convergence.

10.5.6 Multiscale Modeling of Nanoparticle-Modified Epoxy Tension Specimens Using (TFA) The impact of the nanoparticles on the mechanical properties (tensile modulus, and tensile strength) of nanoparticle reinforced epoxy resins is investigated using experimental and numerical methods. Tension specimens with a thickness of 2 mm are produced and tested according to testing standard DIN EN ISO 527-1 and DIN EN ISO 527-2, sample type 1B as shown in Fig. 10.17. The stress/strain curves are recorded with a test speed of 1 mm/min [27]. A typical Representative Volume Element (RVE) consists of 90% (weight fraction) resin epoxy and 10% of 5 agglomerated Boehmite nanoparticles is used to simulate nano particle-modified epoxy tension specimens (Sect. 10.5.1). The TFA method described in Sects. 10.4 and 10.5.3 is employed to simulate the elastic-plastic response of tension specimens. The RVE of volume V is modelled by Abaqus using 16536 ten-node tetrahedral elements (type C3D10) and divided into subvolumes represented by finite elements, Vη , η = 1, 2, ..., M, such that each subvolume belongs either to the matrix or the particles. All the input data are defined and given by Fankhänel et al. [20] and Jux et al. [27] in order to make the results comparable. The length edge of RVE is 13.3 nm, the size of Boehmite particle is 3.1 × 3 × 3.1 nm3 (orthorhombic shape) as shown in Fig. 10.5 (Sect. 10.5.1). The anisotropic Young’s moduli of the single boehmite nanoparticles are determined by means of MDFEM simulations, yielding values of 232, 136 and 267 GPa, in the directions [100], [010] and [001], respectively. Poisson’s ratio is ν = 0.3, The elstic properties of the epoxy resin are E = 3370.0 MPa and Poisson’s ratio ν = 0.39. While the nanoparticles are assumed to remain elastic, the matrix

Fig. 10.17 FE model of the modified resin tension test specimen discretized by 2700 four-node tetrahedral elements (type C3D4)

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Fig. 10.18 Specified data points of the plastic behavior of the LY556 epoxy

Fig. 10.19 The resin RVE modified with nanoparticles, discretized to 64 subvolumes of resin plus 5 of nano-particles

(LY556 epoxy) is elastic-plastic with initial tensile yield stress Y = 27.0 MPa. The hardening curve is given in the Fig. 10.18. The hardening curve is derived from the stress-strain curve of the tension test of pure resin specimens. In preparation for a TFA solution, the elastic strain and stress concentration factors of elements Vη , Aη and Bη and the eigenstrain and eigenstress concentration factor matrices, Dηρ and Fηρ , η, ρ = 1, 2, ..., M, have to be computed for the M subvolumes (elements) of the RVE. The overall effective elastic properties of the RVE, stiffness and compliance matrices, are computed by using a homogenisation method. The homogenisation method is performed by Abaqus, where periodic boundary conditions are imposed on the RVE. To avoid the computational cost because of the high intensive mesh, the RVE model is divided to 4 x 4 x 4 = 64 subvolumes in additional to 5 subvolumes representing the five boehmite particles as shown in Fig. 10.19, where the TFA parameters and the mechanical properties in each sub volume is uniform. The modified resin tension test specimen is modelled by FE Abaqus and 2700 of C3D4 element type is used, 1800 elements for the tested area and 900 elements for grip parts.

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Fig. 10.20 Schematic of the FE2 multiscale analysis between tension specimen model (FEA) scale and nano (RVE-TFA) scale

The linear analysis of the specimen model is implemented by Abaqus and the TFA process is integrated to the UMAT subroutine for each Gauss point of the modified epoxy resin element as shown in Fig. 10.20. The grips elements are considered elastic to reduce the computational cost that it means no needs to call the UMAT at each Gauss point of grips elements. The specimen is subjected to displacement load equal to 5 mm exactly as the experimental tension test. Since the specimen is consisted of agglomerated nano particles and pure resin, the unit of the nano scale RVE is considered in mm to adapt between the real scale of the specimen model and the RVE-TFA model of modified resin at each integration point.

10.5.7 Results of Multiscale Modeling of Modified Epoxy Tension Specimens The grips of the specimen are connected to two reference points. The full constrains are applied at the one reference point and a displacement equal to U y = 5 mm is applied at the opposite reference point. The time load is divided to 100 increments. The strain at each increment is calculated exactly as the experimental process and given by ΔL ab (10.31) εi = L ab where L ab is the instantaneous distance between two middle points a and b at each increment i, and ΔL ab = 25 mm is the constant distance. The stress is calculated by Fi (10.32) σi = A

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Fig. 10.21 Stress–strain curves of tension test for reference pure resin and nanoparticle modified resin specimens

where Fi is the instantaneous reaction force at the reference point and A = 10 × 2 = 20 mm2 is the area of the middle section. The result of the stress-strain of multiscale analysis for the tension modified resin specimens and tension reference pure resin specimens in additional to the stress– strain curve of FE pure resin specimen model are plotted in Fig. 10.21. It can be observed from the results displayed in Fig. 10.21: • The stiffness and strength of nano particle-modified epoxy results increase comparing with the results of the tension pure resin tension specimens. • The result of the stress–strain of the FE pure resin model very matching the experimental results where the model is implemented by taking in account the hardening curve of pure resin which is derived from the experimental tension test. • The result of the stress–strain of the FE2 multiscale modified resin model very matching the experimental results until the point where the ultimate stress point is reached, where the behavior is stiffer. • The stiffer behavior at this stage of stress–strain curve relates to the following points: – The FE2 multiscale model does not take in account any kind of failure criteria. – The approximation of sub volume procedure makes the nano RVE model is stiffer. – The nano RVE model does not take in account the interface area between the nano particles and the pure resin in additional to the free resin area inside the agglomerate.

10.6 Conclusions The Transformation Field Analysis (TFA) proposed by Dvorak et al. is utilized to predict the properties of a representative volume of inelastic heterogeneous materials consisting of polymer and nanoparticles. Consequently, the TFA method at the

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nanoscale can be implemented as a user routine integrated into RVE-micro scale model, where the modified matrix is reinforced by a certain volume fraction of aligned fibres. The results of the first application in Sect. 10.5.1 reveal that the TFA is a proper method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities on a nanoscale level. The method easily accommodates any uniform overall loading path and inelastic constitutive equation. The model geometries are all incorporated in a similar manner, through the mechanical transformation influence functions or concentration factor tensors derived from elastic solutions for the chosen geometry and the elastic moduli. In comparison with the finite-element method in unit cell model solutions, the present method is more efficient. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. The results of the multiscale approaches TFA-FEA and TFA-TFA are reasonable. The TFA approach is attested to be the more efficient and stable approach as mathematical tool in the nanocomposite multiscale simulation. The results of this work motivate to an enhanced research on the nanocomposites under mechanical and thermal loads to examine several and different aspects of nanocomposite materials consisting of elastic-plastic, viscoelastic, and viscoplastic phases. The milestones of the next step of this work is to predict the effect of the mechanical properties of a polymer nanocomposite consisting of agglomerating fillers on the stiffness and strength of real macro scale structures, e.g. wing-box and fuselage structures, through triple scale nano-micro-macro scales under mechanical and thermal loads by employing the TFA approach to model the micro and nano RVE models. In addition to estimate the material behaviour, there is a possibility to utilize the TFA approach to predict the response the RVE of the nanocomposites involving damage progression model.

10.7 Appendix The constitutive equations for elastic-plastic homogeneous materials subjected to uniform stress or strain changes are determined with the help of a yield surface g(σ ) = 0, which contains the stress states that cause purely elastic deformations. Assuming isotropic hardening, the von Mises form of the yield surface is given by g(σ ) =

3 (s) : (s) − Y 2 2

(10.33)

where s is the deviatoric stress and Y is the yield stress in simple tension. The product (s) : (s) is the inner product of second order tensor si j skl . The elastic behaviour is dominant if g < 0 or g = 0 and (∂g/∂σ ) : dσ ≤ 0. Otherwise the elastic-plastic behaviour is dominant if g = 0 and (∂g/∂σ ) : dσ ≥ 0. In this case, the time derivative

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of the equivalent stress/plastic strain curve defines  p  the plastic tangent modulus or the hardening rule at the current stress σeq = Y εeq :  p  σ˙ eq = p H εeq ε˙ eq

(10.34)

where 

3 s˙ : s˙ 2  2 p p ε˙ : ε˙ = 3

σ˙ eq =

(10.35)

p ε˙ eq

(10.36)

Under traction boundary conditions, the total strain increment caused by application of a stress increment σ˙ is ε˙ = ε˙ e + ε˙ p . (10.37) and the instantaneous elastic-plastic compliance matrix is given by the following equation: (10.38) M = Me + Mp. where Me is the elastic compliance matrix and M p is the instantaneous plastic compliance matrix and given by  M = p

  3  n : nT . 2H

(10.39)

Under displacement boundary conditions, the stress increment caused by an applied total strain increment ε˙ , which includes plastic parts, is   σ˙ = L e ε˙ − ε˙ p

(10.40)

and the instantaneous elastic-plastic stiffness matrix is given by the following equation: (10.41) L = Le + Lp, where L e is the elastic stiffness matrix and L p is the instantaneous plastic stiffness matrix given by     2G p L = (10.42) n : nT H 1 + 2G   G is elastic shear modulus. The product n : nT in Eqs. (10.39) and (10.42) is a tensor product n i j n kl , where

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1 n =

2 Y 3



s11 s22 s33 2s32 2s31 2s12

T

.

(10.43)

10.8 Nomenclature The Voigt notation is adopted; (6 × 1) vectors are denoted by lightface lower case Roman or Greek letters, (6 × 6) matrices by lightface uppercase Roman letters, and A A−1 = A−1 A = I , if the inverse exists. Scalars are denoted by italic lower and upper letters. Bold characters are reserved for material properties that change during deformation. Local field: Vη ση , εη cη = Vη/V ε˙ ηe , σ˙ ηe ε˙ ηθ , σ˙ ηθ ε˙ ηin , σ˙ ηr e p ε˙ η = ε˙ ηin μ˙ η λ˙ η Mη = L −1 η Aη , Bη Aη , Bη aη , bη Dηρ , Fηρ Mηe , L eη Mη , Lη p p Mη , Lη m eη , lηe p p mη , lη mη , lη

Subvolume , η = 1, 2, . . . , M is the finite number of local field Average stress and strain in subvolume Vη , η = 1, 2, . . . , M Volume fraction, such that cη = 1 Elastic local strain and stress Thermal local strain and stress Inelastic local strain and relaxation stress plastic local strain Eigenstrain consisting of inelastic strain and thermal strains, μ˙ η → p ε˙ η + ε˙ ηθ Eigenstress (residual stress) consisting of relaxation stress and thermal stress, λ˙ η → σ˙ ηr e + σ˙ ηθ Elastic compliance and stiffness matrix of subvolume Vη Local elastic strain and stress concentration factors of element Vη Local instantaneous strain and stress concentration factors of element Vη Local instantaneous thermal strain and stress concentration factors of element Vη Local eigenstrain and eigenstress concentration factor matrices η, ρ = 1, 2, ..., M Local elastic stiffness and compliance matrix of subvolume Vη Local instantaneous compliance and stiffness matrix of subvolume Vη Local instantaneous inelastic compliance and stiffness matrix of subvolume Vη Local elastic strain and stress thermal vectors of subvolume Vη Local instantaneous inelastic strain and stress thermal vectors of subvolume Vη Local instantaneous strain and stress thermal vectors of subvolume Vη

Global fields: ε0 , σ 0 and θ

Overall uniform applied strain, stress and thermal load

10 Multiscale Modeling and Simulation of Polymer Nanocomposites . . .

M = L −1 M = L μ, λ ε, σ ε¯ , σ¯ ε˙ , σ˙ μ, ˙ λ˙ g(σ ) Y s (s) : (s) p  H εeq σ˙ eq p ε˙ eq (n) : (n) G

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Overall elastic compliance and stiffness matrix Overall instantaneous compliance and stiffness matrix Overall uniform transformation strain and stress Overall response strain and stress Overall elastic strain and stress Overall strain and stress rate Overall uniform transformation strain and stress rate Yield surface Yield stress Deviatoric stress The inner product of second order tensor si j skl plastic tangent modulus or the hardening rule Time derivative of the instantaneous equivalent stress Time derivative of the instantaneous plastic strain tensor product n i j n kl Elastic shear modulus.

References 1. Bahei-El-Din Y (2009) Modelling electromechanical coupling in woven composites exhibiting damage. J Aerosp Eng 223(5):485–495 2. Bahei-El-Din Y, Khire R, Hajela P (2010) Multiscale transformation field analysis of progressive damage in fibrous laminates. Int J Multiscale Comput Eng 8(1):69–80 3. Bahei-El-Din Y, Micheal A (2013) Multiscale analysis of multifunctional composite structures. In: Proceeding of the ASME international mechanical engineering congress and exposition, vol 9 4. Bahei-El-Din YA (2013) Multiscale modelling and multifunctional composites. Tech. Rep. FA9550-11-1-0076, The Britisch University in Egypt (2013) 5. Bahei-El-Din YA, Dvorak GJ (2000) Comprehensive composite materials, 1 edn, chap. Micromechanics of inelastic composite materials. Elsevier, pp 403–430 6. Baweja S, Dvorak GJ, Bazant ZP (1998) Triaxial composite model for basic creep of concrete. J Eng Mech 124(9):959–965 7. Benveniste Y (1987) A differential effective medium theory with a composite sphere embedding. J Appl Mech 54(2):466–468 8. Benveniste Y (1987) A new approach to the application of mori-tanaka’s theory in composite materials. Mech Mater 6(2):147–157 9. Carrere N, Feyel F, Kanoute P (2004) A comparison between an embedded fe2 approach and a tfa-like model. Int J Multiscale Comput Eng 2(4):20–38 10. Cho M, Cho M, Chang S, Yu S (2010) A study on the prediction of the mechanical properties of nanoparticulate composites using the homogenization method with the effective interface concept. Int’l J Numer Methods Eng 85(12):1564–1583 11. Covezzi F, de Miranda S, Marfia S, Sacco E (2017) Homogenization of elastic-viscoplastic composites by the mixed TFA. Computer Methods in Applied Mechanics and Engineering 318:701–723 12. Dvorak G (2000) Damage analysis and prevention in composite materials. In: Aref H, Philips J (eds) Mechanics for a new mellennium, pp 197–210 13. Dvorak G (2013) Micromechanics of composite materials. Springer, Solid Mechanics and Its Applications

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14. Dvorak G, Benveniste Y (1996) Theoretical and applied mechanics 1996, chap. On micromechanics of inelastic and piezoelectric composites. Elsevier, pp 217–237 15. Dvorak GJ (1993) Micromechanics of inelastic composite materials: theory and experiment. ASME J Eng Mater Technol 115:327–338 16. Dvorak GJ, Benveniste Y (1992) On transformation strains and uniform fields in multiphase elastic media. In: Proceeding of the Royal Society, vol 437 17. Dvorak GJ, Lagoudas DC, Huang CM (1994) Fatigue damage and shakedown in metal matrix composite laminates. Mech Compos Mater Struct 1:171–202 18. Dvorak GJ, Wafa AM, Bahei-El-Din YA (1994) Implementation of the transformation field analysis for inelastic composite materials. Comput Mech 14:201–228 19. El-Etriby AE, Abdel-Meguid ME, Shalan KM, Hatem TM, Bahei-El-Din YA (2015) A multiscale based model for composite materials with embedded pzt filaments for energy harvesting. In: Proceedings of the TMS middle east–mediterranean materials congress on energy and infrastructure systems, pp 361–379 20. Fankhänel J, Silbernagl D, Ghasem Zadeh Khorasani M, Daum B, Kempe A, Sturm H, Rolfes R (2016) Mechanical properties of boehmite evaluated by atomic force microscopy experiments and molecular dynamic finite element simulations. J Nanomater (Article ID 5017213) 21. Franciosi P, Berbenni S (2007) Heterogeneous crystal and poly-crystal plasticity modeling from a transformation field analysis within a regularized schmid law. J Mech Phys Solids 55(11):2265–2299 22. Franciosi P, Berbenni S (2008) Multi-laminate plastic-strain organization for non-uniform TFA modeling of poly-crystal regularized plastic flow. Int J Plast 24(9):1549–1580 23. Fritzen F, Böhlke T (2010) Threeÿdimensional finite element implementation of the nonuniform transformation field analysis. J Numer Methods Eng 84(7):803–829 24. Fritzen F, Böhlke T (2011) Nonuniform transformation field analysis of materials with morphological anisotropy. Compos Sci Technol 71(4):433–442 25. Gilbarg D, Trudinger N (1998) Partial differential equations of second order. Springer 26. Hill RJ (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357–372 27. Jux M, Fankhänel J, Daum B, Mahrholz T, Sinapius M, Rolfes R (2018) Mechanical properties of epoxy/boehmite nanocomposites in dependency of mass fraction and surface modification-an experimental and numerical approach. Polymer 141:34–45. https://doi.org/10.1016/j.polymer. 2018.02.059 28. Levin VM (1967) Thermal expansion coefficients of heterogeneous materials. Mekhanika Tverdogo Tela (translated: Mechanics of Solids) 2:88–94 29. Michel J, Suquet P (2003) Nonuniform transformation field analysis. Int J Solids Struct 40(25):6937–6955 30. Michel J, Suquet P (2004) Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Comput Methods Appl Mech Eng 193(48–51):5477– 5502 31. Oskay C, Fish J (2007) Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials. Comput Methods Appl Mech Eng 196(7):1216–1243 32. Procházka P (1997) Slope optimization using transformation field analysis. Eng Anal Bound Elem 20(3):179–184 33. Roussette S, Michel J, Suquet P (2009) Nonuniform transformation field analysis of elasticviscoplastic composites. Compos Sci Technol 69(1):22–27 34. Sacco E (2009) A nonlinear homogenization procedure for periodic masonry. Eur J Mech— A/Solids 28(2):209–222 35. Shin H, Yang S, Choi J, Chang S, Cho M (2015) Effect of interphase percolation on mechanical behavior of nanoparticle-reinforced polymer nanocomposite with filler agglomeration: A multiscale approach. Chem Phys Lett 635:80–85

Part IV

Influence of Nanoadditives on Composite Manufacturing

Chapter 11

Dispersion Technology and Its Simulation Benedikt Finke, Arno Kwade, and Carsten Schilde

Abstract Dispersion quality plays a key role in the processing and product quality of fiber reinforced composites. Process control on the dispersing process in terms of a predictable product quality and required run times as well as their scalability are crucial for the production of fiber reinforced composites. In this chapter, an overview on available dispersing methods is given for the production of fiber reinforced composites. Advantages and weaknesses of the respective dispersing methods are discussed and related to their stress mechanism. The simulation and modeling of such dispersing processes is discussed and strategies are outlined to obtain the data necessary to describe them entirely. Examples on how to parameterize such simulations based on simulations on a lower size and time scale as well as based on experiments and material parameters are given.

11.1 Dispersing Technology for the Production of Nanoparticle Reinforced Composites The unit operation of dispersing combines the wetting and mixing of particles in a fluid with the reduction of the size of the particles. In this field, the term “particles” also refers to assemblies of primary particles clustered in agglomerates and aggregates. Agglomerates are considered to be multiple primary particles, held together by strong surface interaction forces. In contrast, aggregates refer to clusters of particles jointed by solid bridges or partially sintered connections. Dispersing refers to the B. Finke (B) · A. Kwade · C. Schilde Technische Universität Braunschweig - Institute for Particle Technology, Braunschweig, Germany e-mail: [email protected] A. Kwade e-mail: [email protected] C. Schilde e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_11

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destruction of such clusters by overcoming inter-particle forces and possibly solid bonds formed between the primary particles. Dispersing is therefore demarked from the unit operation of real grinding in which breakage of primary particles is induced. The disruption of aggregates and agglomerates typically requires energies orders of magnitude lower than the breakage of primary particles and therefore follows other rules than real grinding in terms of process design [12]. Good dispersion quality, in the sense of a well-defined and narrow particle size distribution without traces of coarse particles, has proven vital for the production and properties of fiber reinforced nanocomposites. As laid out in detail in Chap. 15 a filtering of nanoparticles can occur during the infusion process of fiber reinforced composites. This effect can be used as a design tool, where desired. It can also be prevented by adjusting design parameters. However, even small amounts of coarse material (in the scale of micron sized particles) impose serious difficulties on the infusion process, as the particles filter out in front of the fiber layers, causing a steep pressure drop along with a loss in particle content and ultimately block the resin flow [1]. To great extent, the disperse properties (solids content, particle size distribution and particle morphology) of the nanocomposite also define the resultant mechanical properties of the particle-matrix material [14, 16]. Coarse particles are known to act as imperfections in the matrix material and provoke premature failure of the material. In the same way well-controlled disperse properties can counteract an embrittlement of the material, a side effect often found in particle filled polymers [17].

11.1.1 Stress Mechanisms and Dispersing Methods During dispersing processes as well as grinding processes, particle breakage is induced by applying stress to a particle, which exceeds the particle strength and causes enough elastic energy to be stored in the material to allow the initiated crack to run through the particle completely. Particle strength increases over-proportionately with the particle fineness (see Fig. 11.5), as the number of imperfections on which cracks can be initiated is reduced proportionally with the particle volume [8]. The stress acting on the particles can be applied by various means. Rumpf [41] categorized these stress mechanisms by the characteristic of stress introduction (see Fig. 11.1) into: 1. 2. 3. 4.

Stress applied by two surfaces Stress applied by a single surface Stress applied by surrounding fluid Stress applied by other physical means

Stressing between surfaces (Fig. 11.1a) is the stress mechanism dominating in dispersing and grinding devices such as stirred media mills (Fig. 11.2c, d), where a particle is stressed when captured between colliding grinding beads. A product related stress model [23] exists, which enables a mathematical description of the stressing conditions in a dispersing process in a stirred media mill. It is

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Fig. 11.1 Stress mechanisms according to Rumpf [41]. a Stress applied by two surfaces. b Stress applied by a single surface c Stress applied by surrounding fluid d Stress applied by other physical means

based on the model assumption, that the stress energy S E which is utilized to stress a particle when it is caught between colliding grinding media is proportional to the kinetic energy in the contact. The proportionality is given in Eq. 11.1, with ρG B being the grinding media density, dG B the grinding media diameter and vt the tip speed of the stirred media mill rotor. The model assumes the relative velocity between the grinding beads to be proportional to the tangential velocity vt of the mills rotor. S E ∝ E kin,G B ∝ ρG B · dG3 B · vt2

(11.1)

For questions related to the efficiency of milling processes, the stress energy does not suffice to describe the stress state resulting in the stressed particle. The stress intensity S I is defined to relate the stress energy to the particle volume (Eq. (11.2)). SI ∝

SE x3

(11.2)

When two grinding beads approach, the fluid is squeezed out of the gap between them. At high viscosities, this fluid displacement takes considerable energy, decelerates the particles before contact and hence reduces kinetic energy. Knieke et al. [19] applied Davis [40] work on this phenomenon to grinding bead contacts and developed the energy transfer factor rη . It depends on the initial distance s0 between the colliding spheres, which is taken to be the size of one grinding bead diameter, as well as the

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final distance between the colliding spheres s. The final distance is considered to be the diameter x of the captured product particle (Eq. (11.3)).  rη = 1 +

 s 2   x 2 1 1 ln = 1+ ln StG B s0 StG B dG B

(11.3)

The Stokes number St is given in Eq. (11.4) as: StG B =

vG B · dG B · ρG B 9·η

(11.4)

The maximum stress energy left to stress the particle is therefore expressed according to Eq. (11.5). (11.5) S E = rη · ρG B · dG3 B · vt2 High viscosities, as featured by e.g. epoxy resins, therefore considerably dampen the grinding media collisions and significantly reduce the available stress energy. This effect is especially pronounced with small grinding media diameters, which is why for dispersing in stirred media mills at high viscosities, large grinding media diameters must be chosen [12]. Stressing can also be achieved by collision of a particle with a single surface (Fig. 11.1b), either of another particle or a surface of the grinding machine. The energy available at each stress event is proportional to the kinetic energy of the particle. For nano-scale particles, this translates to very small kinetic energies available to stress the particle, which are reduced further, as nanoparticles are known to closely follow the fluids flow and hence, be decelerated before impact. This is why this stress mechanism is not utilized for the dispersing of nanoparticles. There are several ways, by which stressing in fluid flow (Fig. 11.1c) can be achieved for dispersing nanoparticulate suspensions. One possibility is to produce turbulent flow by means of strong agitation in a stirred tank (Fig. 11.2a) or rotorstator devices (Fig. 11.2b). Here, the applicable stress on the particle is governed by the energy dissipation rate, the viscosity of the fluid and the size of the particle. The dependency on these parameters varies with the ratio of particle size to the size of the Kolmogorov-Eddy (also referred to as Kolmogorov’s length scale) [31]. Due to this, the stress on the particles cannot be described continuously. Reaching turbulent flow conditions in a suspension with high viscosity requires powerful drives and adjusted agitator geometries. At the same time, a high viscosity yields high stresses on the particle making it a potential method for the production of nanocomposites. Practical constrains limit the applicability of stirred tanks. The rheological behavior of nanoparticulate suspensions often exhibits yield points, which leads to the formation of dead zones in areas of low shear rates. Secondary agitators close to the wall allow material to be transported back into the high shear regime, which can increase the range of applicability. In rotor-stator systems, material is transported to the stress zone by intrinsic suction of the dispersing unit or by pumping. In continuous or pas-

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b)

c)

d)

e)

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f)

g)

Fig. 11.2 Various dispersing devices with different stress mechanisms. a Stirred tank (dissolver) b rotor stator device c basket mill d stirred media mill e kneader f planetary mixer g three roller mill

sage operation mode, dead zone effects can be avoided as long as the viscosity and the related pressure loss allows material transport. Ultrasonic homogenizers induce rapid and high amplitude oscillations in local pressure within the suspension. When the pressure falls below the boiling pressure of the fluid, bubbles form and collapse as the next sound wave increases the pressure. Microjets form as the fluid fills the volume previously occupied by the gaseous phase. Velocities close to the speed of sound were detected in these areas. As particle surfaces serve as nucleation points during bubble formation, these microjets occur in close proximity to the particles and micro jet flow is directed towards the particle surface, as the presence of the particle surface itself prevents fluid flow in the opposite direction. Fluid forces therefore stress the particles heavily as cavitation arises and

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local energy input can be considered high [9]. Practical limitations arise as/because the fluid needs to exhibit a high boiling pressure for cavitation to arise at acceptable ultrasonic amplitudes. This is not necessarily the case for monomeric resins and rarely the case for oligo- and polymers. High fluid viscosities also impede mixing of the material in the dispersing device, causing local temperature maxima and degradation of the material. When subjected to laminar shear flow, particles encounter a shear stress along their equatorial plane (see Fig. 11.9a) due to the differing drag forces acting on both sides of the particle. The velocity gradient also induces a rotating motion onto the particle. Raasch [39] derived the energy W to be applied per unit of time t onto a single particle to be 2  x 3 W = · · π · γ˙ 2 · η (11.6) P= t 3 2 With x being the particle diameter and γ˙ the shear rate. According to Krekel [20], the time required for encountering one stress event (either tensile or compression force) can be expressed as 4·π (11.7) tstress event = γ˙ The number of stress events S N is therefore SN =

γ˙ · t 4·π

(11.8)

The energy dissipated per stress event is considered to serve as a characteristic stress energy S E [13], which can therefore be expressed as S E = 2 · π · η · γ˙ · Vparticle

(11.9)

While the stress intensity S I is proportional to the shear stress τ acting in the equatorial plane of the particle (derived in Raasch [39]) given by S I ∝ τ = 2.5 · η · γ˙

(11.10)

The maximum available stress intensity is known to determine the minimum reachable particle size. Typical dispersing machines which utilize laminar shear flow as the source of stressing are kneaders (Fig. 11.2e), extruders and three roller mills (Fig. 11.2g). Planetary mixers (Fig. 11.2f) are known to also operate in a laminar regime, when high viscous suspensions are processed, despite the high speed of the agitators. All these devices have in common, that they force the high viscous fluid through small gaps between surfaces of differing speed to induce shear flow. Their agitators are designed to ensure flow even at high viscosities. From Eq. (11.10) is evident that unlike any other dispersing mechanism, stressing in laminar shear flow profits immensely from high viscosities which makes it an ideal method for the

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production of nanoparticulate resin-based suspensions, when only the power of the machines drive limits applicability.

11.1.2 Comparison of Dispersing Methods for the Production of Highly Viscous Nanocomposites From the effects discussed in Sect. 11.1.1 it is clear, that each dispersing device has its individual fields of applicability. While ultrasonic homogenizers exhibit excellent performance in terms of minimum reachable particle size and dispersing efficiency in low viscous nanoparticulate systems [47], a thermal degradation due to the dissipated energy cannot be prevented despite intense cooling and additional mixing by a stirrer. In similar manner, the abovementioned limiting effects of dead zones and immense pressure loss during pumping prevent the utilization of dissolvers or rotor stator devices if solids contents of practical use shall be produced with the materials covered in this book. They can therefore not serve as machines for the production of high quality nanocomposites. As dispersing in laminar shear flow gains power with rising viscosity it seems like an obvious choice for the production of highly viscous nanocomposite materials. Equations (11.6)–(11.10) prove the methods dependency on the viscosity. As the viscosity depends on the particle size and the particle size is changed during the process depending on the stressing conditions, a trilateral dependency between the parameters stress level, viscosity and particle size distribution arises. The final particle size of a dispersing process depends on the shear rate applied to the suspension and the viscosity reached during the process [34]. Especially by raising the solids content, the viscosity and hence the stress intensity can be increased by orders of magnitude [17]. This allows close control over the final particle size based on a few dispersing experiments. Figure 11.3 exemplarily depicts the dispersing progress of a kneading dispersing process run in fed batch operation mode to determine the final particle size at each solids content. The kneading hook velocity and hence the maximum shear rate is kept constant throughout the experiment. When a final particle size is reached, new particle material is added to the suspension, which leads to an intermediate increase in particle size until a lower limit to the final particle size is reached. Despite the modest increases in viscosity at low solids contents (see Fig. 12.18), large differences in final particle size can be observed. At larger solids contents, the decrease in final particle size ceases. This effect originates from the non-linear steep increase of particle strength with increasing fineness (see Fig. 11.5 for a visualization of this relationship). The final particle sizes were found to be very reproducible for a given solids content [12]. The method proved applicable for the material system focused in this book as well and yielded variations in the x50,3 -value at the scale of the particle size measurements accuracy [17]. Despite the good reproducibility of dispersing processes in a kneader, further processing of the material along the process chain of fiber-reinforced nanocomposites revealed the residue of small amounts of coarse particles in the suspension after the

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Fig. 11.3 Dispersing experiment in a lab scale kneader (IKA HD -T 0.6) operated in fed batch mode for alumina nanoparticles Aeroxide AluC 100 (Evonik) and epoxy resin Hexflow RTM6-2

kneading process. As visible from Fig. 11.4 resp. Figure 15.5, despite most of the particle material being dispersed to nanometer size range, some particles remain in the micrometer scale. This coarse material introduces strong undesired effects such as high pressure loss during injection and a filtering effect, which decreases the amount of particles in the composite [1]. As the same processing route was used successfully to produce nanocomposites before [34], this effect can be considered to be material dependent. Particles which originate from a pyrogenic synthesis do not show such behavior, as their highly fractal structure allows reliable dispersing at the given stress states. However, sol-gel synthesized particles, as the ones considered here, form more compact structures during synthesis and subsequent drying step, which can endure larger stresses. The coarse material can be removed successfully by a subsequent dispersing step in a three roller mill. In a kneader, only small fractions of the grinding chamber ( Tg or T < Tg . It seems the stiffer the system is, depending on the phase, the lower the heat capacity probably due to the decreased mobility of the molecules.

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Chapter 14

Molecular Modeling of Epoxy Resin Crosslinking Experimentally Validated by Near-Infrared Spectroscopy Robin Unger, Ulrike Braun, Johannes Fankhänel, Benedikt Daum, Behrouz Arash, and Raimund Rolfes Abstract Reliable simulation of polymers on an atomistic length scale requires a realistic representation of the cured material. A molecular modeling method for the curing of epoxy systems is presented, which is developed with respect to efficiency while maintaining a well equilibrated system. The main criterion for bond formation is the distance between reactive groups and no specific reaction probability is prescribed. The molecular modeling is studied for three different mixing ratios with respect to the curing evolution of reactive groups and the final curing stage. For the first time, the evolution of reactive groups during the curing process predicted by the molecular modeling is validated with near-infrared spectroscopy data, showing a good agreement between simulation results and experimental measurements. With the proposed method, deeper insights into the curing mechanism of epoxy systems can be gained and it allows us to provide reliable input data for molecular dynamics simulations of material properties.

R. Unger (B) · J. Fankhänel · B. Daum · B. Arash · R. Rolfes Leibniz Universität Hannover, Hannover, Germany e-mail: [email protected] J. Fankhänel e-mail: [email protected] B. Daum e-mail: [email protected] B. Arash e-mail: [email protected] R. Rolfes e-mail: [email protected] U. Braun Umweltbundesamt (UBA), Berlin, Germany e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_14

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14.1 Introduction Thermosetting polymers belong to a class of polymers that are characterised by their irreversible curing process, changing the material consistency from a liquid soluble prepolymer to a solid material. The physical properties of the highly crosslinked network formation depend very much on the underlying molecule network structure, which is primarily influenced by the constituent monomers and the polymerisation/polyaddition reaction [14, 36]. As there is a large number of constituent monomers, that allow a huge variety of combinations, computer-aided testing and design of new polymers is desirable in order to reduce the needed time and costs. With molecular dynamics (MD), a suitable method is available for simulating nanoscale effects and predicting material properties, however, it requires the associated simulation model to be an accurate representation of the real material system. This necessity of an appropriate molecular modeling led to research in this field, resulting in a variety of molecular modeling methods for different polymer materials that can be found in literature. One initial method for a MD crosslinking simulation, focussing on the polymerisation of poly(methacrylate) networks, was presented by Doherty et al. [7]. The crosslinking was realised by forming covalent bonds between reactive groups with the smallest distance (reaction radius) in between, succeeded by a short relaxation to let the system react to the changes of the network structure. Other publications suggested molecular modeling methods for various polymer systems focussing on different aspects of the curing mechanism. However, all of them followed the same fundamental principle of a dynamic covalent bond formation, with the main criterion being the reactive distance between two possible reaction partners. These molecular modeling methods share the following procedure: (1) generation of the unlinked model, (2) initialisation of the MD simulation, (3) MD crosslinking simulation, (4) final equilibration of the system. The first step involves the initial generation of a random unlinked model from separate monomer and curing agent molecules. An energy optimisation is applied to the initial model to minimise internal energy as a preparation for the following MD simulation. The crosslinking simulation is initialised by heating up the simulation box to the desired curing temperature and an equilibration before the crosslinking is started. In the actual crosslinking procedure, a number of bonds are formed in every crosslinking step, followed by a relaxation time to let the system react to the changes of the network structure. Depending on the specific method, only the bond with the shortest reactive distance [7, 46, 47] or all bonds within a certain reaction cut-off are formed [13, 43, 49]. The crosslinking is either continued until a desired degree of crosslinking is achieved or until no possible bonds are found any more. The final cured system is equilibrated and charges are updated, if not done dynamically. Approaches focussing on amine-cured epoxy systems differ especially in their assessment of reactive amine group reactivity. Wu and Xu [46, 47] and Bandyopadhyay et al. [4] assumed an equal reactivity of primary and secondary amines. Varshney et al. [43] continued on this research by adopting the methodology of Wu et al. [46, 47] and comparing it with a newly developed method, which finally

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leads to better equilibrated structures with lower energies and reduced computational time. Extensive investigations on the influence of different reaction probabilities of curing agents of an amine-cured epoxy system were done by Estridge [10]. The study showed, that significantly changed reactivities of primary and secondary amines lead to different network structures of the cured epoxy system. Furthermore, Varshney et al. [43], Estridge [10], as well as Yang and Qu [48] investigated the molecular weight of the largest and second largest molecular cluster as a function of the degree of curing. The authors were able to predict a theoretical gel point and the investigation of the effect of the network structure showed an influence on the physical properties. These results underline the need for an accurate molecular modeling in order to reliably predict material properties. For a general review on the evolution of polymer networks with respect to the structure evolution and the gel point the reader is referred to Li and Strachan [21]. Okabe et al. [32, 33] studied different mixtures of epoxy resin and amine curing agents with MD simulations and experimental differential scanning calorimetry (DSC). The molecular modeling results showed an acceptable agreement to experimental results in the general behaviour of the curing curves and reactivity of the epoxy groups. However, specific characterisation of the different crosslinking reactions during the curing process was not possible with DSC. Jang et al. [16] presented a relative reactivity volume criterion for the crosslinking of vinyl ester resins that takes the regioselectivity (head-to-tail chain propagation) and the corresponding monomer reactivity ratios into account. Li and Strachan [19, 20] focussed their research on an accurate charge updating using the electronegativity equalisation method (EEM) after each bond formation. As multiple EEM updates are computationally intensive, a parameterised EEM-based charge assignment was developed, that leads to reasonable results with a decrease in computational cost. Demir et al. [6] proposed a related molecular modeling approach, with the difference of using the Charge Equilibration (QEq) method and an increasing reaction cut-off during the crosslinking. Crosslinking of novolac-type phenolic system was investigated by Monk et al. [25, 26] with respect to the chain motif (ortho-ortho or ortho-para) and the chain length. The results show, that the type of hydrogen bonding (interchain or intrachain) was strongly affected by the chain motif and lead to differing physical properties of the crosslinked system. Shudo et al. [38] followed up on a study of Izumi et al. [15] and constructed and characterised the network structure of phenolic resins by MD simulation. They presented a good agreement with experimental results, obtained by small- and wide-angle X-ray scattering, in terms of the branching structure of the phenolic units and the methylene linkages, the molecular weight distributions and densities for various conversion rates. The effect of water on the crosslinking of epoxy resins was investigated with MD simulations by Sharp et al. [37]. The obtained results were in agreement to experimental data and showed that water increases the molecular diffusion, which lead to an increase in the cure rate for low degrees of crosslinking. Tam et al. [41] presented a study on the molecular modeling of an epoxy resin with various water

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concentrations, with results indicating that the bonding behaviour and the resultant network structure is significantly influenced by the moisture concentration. The methodology of curing nanocomposites, composed of epoxy systems and nano-sized filler materials is also considered in literature [5, 12, 30, 39, 44]. Coco et al. [5] focussed on the molecular modeling of epoxy/carbon nanotube nanocomposites and Vo et al. [44] investigated the effect of the atomistic modeling on the morphology and thermomechanical properties of thermosetting epoxy/clay nanocomposites. Molecular modeling of an epoxy-POSS (polyhedral oligomeric silsesquioxane) nanocomposite was presented by Song et al. [39] and Neyertz et al. [30]. The discussed molecular modeling procedures follow the common method to use a dynamic step-wise crosslinking algorithm with equilibration steps between the bond formation. A comparison and validation of these molecular modeling methods with experimental results of the curing process is still pending in literature. Because of the difficulty of obtaining experimental insight into the curing process, the procedures presented are validated indirectly by comparing simulated material properties with experimental values. In this publication, a molecular modeling method for crosslinking epoxy resins is presented that is able to realistically describe the curing mechanism and to generate simulation models that are in good agreement with experimentally analysed cured epoxy resins. In order to reduce simulation time, and with this the computational cost, a parameter study was conducted to increase the efficiency while keeping the crosslinked model well equilibrated. By using the method of in-situ near-infrared spectroscopy (NIR), the entire time evolution of instantaneous concentrations of reactive groups, epoxy and either amine or anhydrite curing groups, can be observed, instead of only the concentrations of the final cured polymer. It has been shown that this method is well suited for analysing the curing process and for characterising the fully hardened epoxy system [8, 9, 34]. Thus, NIR measurements of the polymerisation reaction of an epoxy system give valuable insights into the curing process that can be used to significantly improve the model generation towards reality. Experimentally obtained measurement data of the curing process is used to compare and validate the presented numerical crosslinking method with respect to the crosslinking procedure and the resulting network structure. The molecular modeling method is incorporated into the Molecular Dynamic Finite Element Method (MDFEM) [27–29] framework that can be used for deriving physical material properties [11, 17].

14.2 Near-Infrared Spectroscopy 14.2.1 Material System A commercially available amine-cured epoxy system from Olin Epoxy was chosen for the study. The epoxy resin was of type AIRSTONE 880E and as a hardener AIRSTONE 886H was chosen, both made for the infusion process of large and

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Table 14.1 Overview of the chemical composition of the epoxy system (names, mixing ratios and CAS identification numbers). Corresponding chemical structures are shown in Fig. 14.1 Name Parts (wt%) CAS-No. Resin (AIRSTONE 880E)

Bisphenol A diglycidyl ether

1,4-Butanediol diglycidyl ether Alkyl (C12-C14) glycidyl ether Hardener (AIRSTONE α,ω-poly(oxypropylene)diamine 886H) 3-(aminomethyl)-3,5,5-trimethylcyclohexanamine 2,4,6-tris[(dimethylamino)methyl] phenol

75–100

25068-38-6

10–25 1–5 50–75

2425-79-8 68609-97-2 9046-10-0

25–50

2855-13-2

1–5

90-72-2

Fig. 14.1 Chemical structures of all molecules of the investigated epoxy system: a–c chemical components of the epoxy resin, d–f chemical components of the hardener

thick composite parts, such as wind turbine blades. An overview of the chemical composition of the epoxy system is given in Table 14.1 and the chemical structures of all molecules are shown in Fig. 14.1 (all information is taken from material data sheets [2, 3]).

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Fig. 14.2 Principle polymerisation reactions

The two main chemical components of the epoxy resin, Bisphenol A diglycidyl ether and 1,4-Butanediol diglycidyl ether, each consist of two epoxy groups as the reactive groups. The reactive groups of the two main hardener molecules, α,ω-poly (oxypropylene)diamine and 3-(aminomethyl)-3,5,5-trimethyl-scyclohexanamine, are the amine groups (NH2 ), from which each can form two covalent bonds to an epoxy group. The respective polymerisation reactions, from primary to secondary amine and from secondary to tertiary amine, are shown in Fig. 14.2a and b. First, the epoxy ring is opened, resulting in an additional hydroxyl group and a reactive methylene group, which then is the reaction partner for the amine group, forming a bond. Beside this dominant reaction of amines and initially epoxy groups, also an etherification reaction between a hydroxy group and an epoxy group is possible. The existing hydroxyl groups as well as the newly created hydroxyl groups, can be a reaction partner for the etherification reaction, shown in Fig. 14.2c. NIR measurements of the curing process were conducted on the standard stoichiometric mixture (100 parts epoxy resin: 31 parts hardener), as well as on mixtures with a shortage of hardener (100:15) and with an excess of hardener (100:60). The near-infrared spectroscopy was conducted during the 5 h isothermal curing process at 80 ◦ C with an additional heat-up process of 50 min prior to the curing. The overall curing and measurement time is 350 min.

14.2.2 Methodology Spectroscopy methods allow to identify the chemical composition of a material and by repeated measurements during a chemical reaction, the changes in the molecule

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structure can be observed. Common spectroscopy methods for polymer characterisation are the near-infrared (NIR), mid-infrared (MIR) and Raman spectroscopy. Specific vibrations and rotations are the consequence of the specific wavelength absorption by covalent bonds, which is measured. Poisson et al. [35] compared NIR and MIR spectroscopy with respect to measurements of the epoxy curing conversion and showed that NIR is very well suited to analyse the curing kinetics. This result was underlined by Lachenal et al. [18], whose results showed a very good agreement of NIR results with differential scanning calorimetry (DSC) and size-exclusion chromatography (SEC) results. They concluded that NIR spectroscopy allows for a valuable real-time analysis of epoxy systems. Pandita et al. [34], Duemichen et al. [8] and Erdmann et al. [9] successfully realised the idea of NIR spectroscopy of in-situ curing kinetics of epoxy resin. In the NIR, light with wavelengths between 800 and 2500 nm is used, which include mainly the less intensive overtones and combination vibrations compared to the fundamental signals in mid-infrared. As a consequence, the spectra often give unspecific structural information, but due to the lower intensities of these overtones compared to the fundamentals, thicker samples can be measured. Because glass does not absorb in NIR wavelength range, samples can be measured through glass, hence, we can use optical cuvettes, analysing the change of chemical structure during the complete curing process from the liquid up to the fully cured state. The experimental setup as well as the relevant experimental parameters are described in detail in the work of Duemichen et al. [8] and Erdmann et al. [9]. Exemplarily selected spectra of the studied epoxy system during the complete curing process are shown in Fig. 14.3. The decrease of epoxy and amine absorption band signals during the curing process is illustrated as well as the formation of hydroxyl groups. For the determination of signal evolution rates of each absorption band, integration limits are chosen and a corresponding basis line is defined as a lower boundary. Based on the integration limits and the basis lines, the respective peak area is integrated, quantifying the absorbance of each absorption band over the curing time. An overview about the absorption bands and the corresponding signal assignment as well as the integration limits are given in Table 14.2. The integrated

Fig. 14.3 Individual, exemplarily spectra of the studied epoxy system during the curing process

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Table 14.2 NIR measurement assignment of absorption bands and the corresponding evaluation parameters Absorption band Assignment Integration limits (cm−1 ) Epoxy a Epoxy b Amine a Amine b Hydroxy a Hydroxy b Aromatics

Combination of CH stretching and CH2 deformation Overtone from CH stretching Combination of primary NH2 stretching and bending vibration Overtone of primary NH2 plus secondary NH stretching Combination of OH Overtone of OH stretching Combination of aromatic C=C and aromatic CH stretching

4560–4497 6110–6036 4971–4887 6610–6410 4851–4750 7090–6901 4702–4658

Table 14.3 Relation between functional groups and absorption bands Functional group Assignment of absorptions bands Unlinked epoxy Primary amine Secondary amine Hydroxy

Epoxy a Amine a Amine b–Amine a Hydroxy a

peak area of the absorption bands can be processed to determine the normalised time-dependent change of reactive groups. For the present system, we focus on the time-dependent change of epoxy rings, the primary, secondary and tertiary amines, as well as the hydroxyl groups. The aromatic signal was chosen as an invariant reference signal, because no influence of curing chemistry should affect this signal. The relations between the functional groups and the absorption bands are shown in Table 14.3. The epoxy absorption band concentration is directly related to the concentration of reactive epoxy groups. In the following, only Epoxy a is considered as the respective epoxy absorption band, neglecting the overtone signals from CH stretc.hing (Epoxy b). The same holds for the respective hydroxy absorption bands, for which only Hydroxy a is considered as the functional hydroxy group. The absorption band of Amine a directly corresponds to the functional primary amine group. As the absorption band of Amine b consists of both primary and secondary amine signals, the difference of Amine b and Amine a is calculated as the quantity of secondary amines.

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14.2.3 Experimental NIR Results—Absolute Absorbance The development of the relevant absorbance band signals as a function of time is presented in Fig. 14.4b–e together with the corresponding temperature history (a). Additionally, the absorbance band of the aromatics is added as a reference (f). The quantity of the aromatics does not change throughout the curing process, since the aromatics do not react during the curing process. This allows the change of the absorbance band signal to be used as an indicator for the temperature dependency. Due to different integration limits and peak values, the absolute signal intensities are not quantitatively comparable. However, a qualitative analysis is useful as it gives insights into the influence of the mixing ratio on the curing reaction and to identify the importance of the etherification reaction between epoxy and hydroxy in general. The comparison of all curves with the reference signals at the beginning and the end of the curing procedure shows that especially the amines and the hydroxyl groups are influenced by the temperature. For the amines, this effect is small in comparison to the absolute signal. The hydroxyl groups are stronger influenced by the temperature, as a temperature change has a direct effect on hydrogen bonding,

Fig. 14.4 Development of curing temperature (a) and relevant absolute absorption band signals (b)–(f) as a function of curing time for different stoichiometric mixtures

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which in turn influences the absorption band signals. The evaluation of absolute signal intensities of the standard mixture shows that after consumption of around 50% of epoxy groups, 90% of primary amines are already reacted and the signal intensity of secondary amines is at its peak value. At this stage, the formation of secondary amines starts to decrease, which means that from this point, the reaction of secondary to tertiary amines is the dominant one and faster, compared to the reaction of primary to secondary amines. Towards the end of the curing time, the converged values of the standard mixture indicate that the system is fully cured. The absolute signal intensities of epoxy and primary amine converge against zero. The absolute signal intensity of secondary amines does not reach its starting value of zero, indicating that there is a small amount of secondary amines left in the cured system. With an excess of hardener molecules, all epoxy groups are fully reacted as indicated by the signal intensity of epoxy groups that is converging towards zero. The complete consumption of epoxy is detectable after ≈120 min, whereas remaining signals of primary and secondary amines are observable. Due to the shortage of epoxy, not all hardener molecules can react, thus a small amount of primary and a substantial amount of secondary amines are left in the system. The constant signals of amines and hydroxyl groups from ≈120 min on clearly show that the complete consumption of epoxy stops the reaction in the mixture and no relevant contribution from reactivity between the hydroxyl and amine groups is observable (i.e., condensation reactions by ether formation). The third measured mixing ratio with a shortage of hardener results in a cured system with a substantial amount of non-reacted epoxy groups left. The final signal intensities of primary and secondary amines, both converging against zero, indicate that all hardener molecules are fully reacted to tertiary amines. Compared to the standard mixture between 100 and 350 min, the decay of secondary amines signal and the formation of the hydroxyls signals shows a very similar pattern. This means the curves are only shifted in the vertical positions, due to different integration values, but they do not cross or converge to each other. This can be seen as an indicator, that even at a lack of amines, no hydroxyl groups react additionally by etherification with epoxy groups. The most likely reason is that the hydroxyl groups do not react at all with the epoxy, as it was also stated by Strehmel [40] for amine-cured systems without an accelerator. However, it should be noticed that the reaction of epoxy with hydroxyl groups does not result in additional hydroxyl groups, because ether is formed, hence, the concentration of hydroxyls keeps constant. In general, it seems that even at the end of the reaction still slight changes are observable, although the main changes of all signals occur during the first half of the curing time. The increasing viscosity as well as the gel formation significantly prevents the transport of reactive groups to reach others, therefore the reaction is significantly slowed down at a certain level and finally stopped.

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14.2.4 Experimental NIR Results—Normalised Change of Reactive Groups For an analysis of the curing process and the following validation of the molecular modeling, it is useful to derive the normalised change of reactive groups from the absorption band measurements. This is achieved by first normalising the timedependent change of the absorption bands (Fig. 14.4) and second, by calculating the normalised change of reactive groups from the absorption bands. The normalisation can be done for the absorption bands of Epoxy a, Amine a and Amine b, as these bands are related to functional groups that are only decreasing due to the chemical reaction (see Table 14.2). Therefore, the initial state refers to the reference value of the uncured system that is related to 100% of each absorption band. Based on the integration parameters, an integrated peak area of zero refers to 0% of the absorption band. The normalisation of the hydroxyl absorption band is rather difficult, as there is already an unknown quantity of hydroxyl groups at the initial state and the absorption band is increasing during the chemical reaction. Thus, it is not possible to determine a reference value either for the initial or the final state of hydroxyl groups. However, as the previous analysis of the time-dependent change of absorbance bands has shown, the hydroxyl groups are not significantly involved in the curing process and thus neglected for the following analysis. The normalised change of the absorption band is shown in Fig. 14.5a. A smoothing is applied to reduce oscillations and minor corrections of the NIR signal are applied to correct measurement inaccuracy. The concentration of the Epoxy a and both amine absorption bands is decreasing over time, as it is expected due to the curing reaction. The oscillations in the absorption band signals are likely caused by the chemical shrinkage of the epoxy system. The chemical shrinkage induces thermal stress in the glass cuvette, which leads to minimal motion of the cuvettes within the NIR device and finally to broken cuvettes at the end of the curing process. This phenomenon results in an oscillating base line offset that influences the integrated peak area.

Fig. 14.5 Normalised NIR measurement data of the standard mixing ratio (100:31)

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After the normalisation, the concentration of the functional groups can be calculated from the normalised absorption band concentration. The relations between the functional groups and the absorption bands have already been shown in Table 14.3 and hold for the normalised absorption bands. In addition, for the normalised values it is possible to calculate the concentration of tertiary amines as the differences of Amine b to the initial value of 100%. The normalised changes of the relevant reactive groups for the standard mixing ratio are shown in Fig. 14.5a. The concentration of unlinked epoxy groups is steadily decreasing, as the curing only happens irreversibly with the reaction of epoxy and amine groups to crosslinked monomer hardener molecules. Therefore also the concentration of the primary amines is decreasing continuously, with over 90% reacting to secondary amines within the first 100 minutes of the curing process. This can also be seen by the increase of secondary amines until a peak is reached at ≈75 min, after which the quantity of secondary amines decreases. The progress of the secondary amines indicates, that the reaction from primary to secondary amines is dominant within the first 75 min. The reaction from secondary to tertiary amines becomes dominant only after the majority of primary amines reacted to secondary amines. Finally, the curing reaction results in a large amount of ≈90% tertiary amines in the final state. After 350 min, the system is fully cured, indicated by the converged concentration of unlinked epoxy groups. Figure 14.6a shows the normalised change of reactive groups for the mixing ratio with a hardener shortage (100:15). The general evolution of reactive groups is qualitatively the same as for the standard mixing ratio. Due to the shortage of hardener, nearly all of the amine groups are fully reacted to tertiary amines, whereas only ≈60% of the epoxy groups are reacted. Although the normalised concentration of secondary amines is lower than the concentration of primary amines, both amines can be interpreted as fully reacted, due to the shortage of hardener. The normalised change of reactive groups for the mixture with an excess of hardener (100:60) is shown in Fig. 14.6b. Due to the excess of hardener, the epoxy groups are fully reacted after ≈150 min, with a substantial amount of non-reacted amine groups left.

Fig. 14.6 Normalised change of reactive groups of modified mixing ratios: a shortage of hardener (100:15) b excess of hardener (100:60)

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It is also worth mentioning that for all mixing ratios a strong decrease of primary amine groups without a corresponding decrease of unlinked epoxy groups can be seen within the first 30 min. This physically unlikely measurement effect needs to be interpreted with respect to the heating of the specimen and is also partly related to measurement inaccuracy. NIR absorption is temperature-dependent, with an increasing temperature leading to a decrease of signal intensity, as it can be seen in Fig. 14.4. For further information on the temperature dependency the reader is referred to Duemichen et al. [8] and Erdmann et al. [9]. Nevertheless, it is appropriate to use the measurement data obtained during heating, in order to observe the full curing process.

14.3 Molecular Modeling Method for Crosslinking 14.3.1 Generation of the Unlinked Model The unlinked model, as a representation of the uncured epoxy system, is randomly generated based on the molecule models of the monomer and curing agent. The chemical composition of the epoxy system (AIRSTONE 880E & AIRSTONE 886H from Olin Epoxy) is presented in Table 14.1, the corresponding chemical structures are shown in Fig. 14.1. Both the epoxy resin and the hardener are a mixture of three molecules, from which in both cases one molecule with a weight fraction 90%) can be achieved by cut-off radii larger or equal then 4 Å with a respective increase of computational time. Furthermore it shows, that larger values (>10 Å) would not lead to an further increase in efficiency, as there is a negligible difference between the results obtained by values of rc = 8 Å and rc = 10 Å. For insights into the effect on the curing mechanism, the evolution of reactive amine groups for the rc values investigated is shown in Fig. 14.9. The relative change of reactive amine groups as a function of the degree of crosslinking is presented by different symbols for each of the investigated cut-off radii. The results show a perfect agreement without an identifiable deviation in the curing mechanism for all cut-off values, with the only difference being the maximum degree of crosslinking that is reached within the defined time (3 ns). The maximum degree of crosslinking is shown with a dashed vertical lines for the respective values of rc . Figure 14.9 shows that the choice of rc has no influence on the curing mechanism, but only on the efficiency i.e., the computational time needed to reach a desired degree of crosslinking. From the simulation results, shown in Figs. 14.8 and 14.9, it can be concluded that the curing process predicted by the present molecular modeling is not sensitive to the chosen cut-off value. However, a larger cut-off value provides us with higher efficiency without any difference in the curing mechanism and without any loss in accuracy. However, for subsequently analysis of the physical material properties, it is crucial to ensure the polymer system to be in an equilibrated state. Figure 14.10a shows the distribution of bond lengths of all crosslinks formed at the end of the crosslinking simulation. The equilibrium distance of newly formed bonds between hardener nitrogen atoms and epoxy carbon atoms has a value of rN−C = 1.462 Å, indicated by a vertical dashed line. The distribution of the bond length is in very good agreement

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Fig. 14.9 Simulation results of the curing process in terms of the relative change of reactive amine groups as a function of the degree of crosslinking for all investigated values for rc for the default mixing ratio of 100:31. Vertical dashed lines indicate the maximum degree of crosslinking for the respective values of rc within the defined time of 3 ns

Fig. 14.10 Characteristics of the crosslinked system: a Distribution of bond length of newly formed crosslink bonds only. b Radial distribution function of all nitrogen atoms to all other carbon atoms in the system

and shows a reasonable distribution with its peak value very close to the equilibrium bond length. Figure 14.10b shows the radial distribution function (RDF) of all nitrogen atoms to all other carbon atoms in the system. The RDF also depicts the main peak at the distance of the equilibrium bond length. Both results prove that the system is in a well equilibrated state and that all newly formed bonds, with initial bond lengths up to rc = 10 Å, have reasonable bond lengths. The investigation of the influence of the cut-off distance clearly shows that with the largest investigated cut-off value of rc = 10 Å, a well equilibrated and fully crosslinked system can be achieved with a significant reduction of the simulation time, compared to lower values of the cut-off distance. These results show that a cut-off distance of rc = 10 Å leads to the most efficient molecular modeling of a well equilibrated and fully crosslinked epoxy system. Therefore, this value is used for the following analysis and comparison of crosslinking simulation with NIR data.

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14.4.2 Experimental Validation Using NIR Measurements As the time scales of the NIR-measured curing (≈5 h) and the molecular modeling method (≈10−9 s) differ by magnitudes, a direct comparison of the time-dependent change of reactive groups is difficult. However, an analysis of the change of reactive groups relative to the degree of curing, specified by the decrease of unlinked epoxy groups, allows a time-independent comparison of the numerical method and the experimentally gained NIR data. Figure 14.11 shows the curing results obtained by the NIR experiment (solid line) and the molecular modeling (dashed line). The final degree of crosslinking is comparable with values of ≈94% for the molecular modeling and a degree of crosslinking up to 100% for the NIR measurement, as it can be seen in Fig. 14.5b. The general behaviour of the numerical results is in qualitatively good agreement with the experimental data. The strong reaction of primary to secondary amines without a reduction of unlinked epoxy groups observed in the NIR results is physically unlikely as it was explained in the NIR section. Considering this effect of the NIR measurement, caused by measurement inaccuracy and temperature dependency, the curves for the primary amines are shifted vertically and show a very good agreement. Furthermore, the peak value of the secondary amines is very well represented in the numerical results. Beside the curing process of the epoxy system with the default mixing ratio, curing with shortage and excess of hardener were also investigated. The respective results are shown in Fig. 14.12. Figure 14.12a shows the results for the mixing ratio with a shortage of hardener (100:15). The general qualitative curing behaviour is in fair agreement, although the strong reaction of primary to secondary amines at the beginning of the curing process without any reaction of epoxy groups leads to differences. These deviations decrease within the crosslinking process and the finally crosslinked system is in good agreement. An interpretation and explanation for this physically unlikely experimental data is given in the last paragraph of Sect. 14.2.4. The molecular modeling method yields a final degree of curing of 56%, which is close to the experimental degree of curing of 60% indicated by the NIR measurements. In

Fig. 14.11 Simulation and experimental results of the curing process in terms of the relative change of reactive amine groups as a function of the degree of crosslinking for the default mixing ratio of 100:31

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Fig. 14.12 Simulation and experimental results of the curing process in terms of the relative change of reactive amine groups as a function of the degree of crosslinking for the modified mixing ratios: a shortage of hardener (100:15) b excess of hardener (100:60)

both cases, all hardener has fully reacted to 100% tertiary amines, resulting in 0% of primary and secondary amines after the curing process. However, the molecular modeling is not able to cover the curing mechanism precisely compared to the results obtained with the default mixing ratio. Especially the asymptotic behaviour of tertiary amines, which lead to a plateau, is only poorly predicted. Figure 14.12b shows the results for the mixing ratio with an excess of hardener (100:60). In both crosslinking procedures, the experimental and the molecular modeling method, a full reaction of all epoxy groups is achieved, since an excess of hardener is available. The results, which are shown in terms of normalised concentration of amine groups, are in good agreement with the NIR measurements.

14.4.3 Discussion The present parameter study reveals the influence of the cut-off distance on the crosslinking mechanism and the final degree of curing with respect to the simulation time. It was shown that different cut-off distances do not change the qualitative evolution of reactive groups, but the overall time needed to reach a certain degree of crosslinking. The final choice of rc = 10 Å results in an increase of efficiency without any loss of accuracy, as shown by the distribution of bond lengths. However,

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a further increase of the cut-off distance may lead to a long relaxation time in order to reach the equilibrium state. Estridge [10] used MD for her research on competitive primary and secondary amine reactivity focussing on the structural evolution and derived material properties of an amine-cured epoxy system. The study points out that it is important to include an accurate kinetic crosslinking mechanism, as it can have a significant effect on the material properties predicted. Nevertheless, no comparison or validation of the crosslinking kinetics were presented by Estridge [10], which makes it difficult to assess or define a specific reaction probability. The present comparison of the curing details of the molecular modeling method with experimental results of the in-situ curing confirm the assumption that no specific reaction probability for the reaction from primary to secondary and from secondary to tertiary is needed to cover the realistic curing mechanism. The results underline that due to the steric effects and physical interactions, a reaction mechanism in close agreement with experimental results can be achieved. The focus of the present study is the experimental validation of the proposed molecular modeling of epoxy resin crosslinking. Research on the size of epoxy clusters or the behaviour of bond percolation was out of scope for the present study and interested readers are referred to [10, 21, 43, 48]. The proposed molecular modeling method leads to results that are in good agreement with the experimental results for the default mixing ratio (100:31) and an excess of hardener (100:60) and to results that are still in fair agreement for the mixing ratio with a shortage of hardener (100:15). The main cause for the deviation between simulation and experimental results is the strong reaction of primary to secondary amines measured by NIR at the beginning of the curing process. However, this strong reaction does not correspond to the minor reaction of epoxy in the same time period. As already discussed in the NIR section, this effect must be interpreted as measurement inaccuracy as well as an unfiltered temperature influence on the signal intensity. The epoxy system considered with a shortage of hardener (100:15) consists of an increased quantity of high-molecular epoxy molecules. Compared to the other monomers, the main epoxy resin component Bisphenol A diglycidyl ether is a stiff and inflexible molecule. Thus the increased quantity of this molecule leads to a higher viscosity and less molecular movement in the system. This phenomenon influences the curing mechanism, as the low-molecular curing agents are disproportionately prevented to reach the reactive epoxy groups, which is related to a decrease in reaction evolution. The molecular modeling method cannot precisely cover this effect, since the artificial bond formation is not influenced by a changed viscosity. This explains the fair agreement of the results with a shortage of hardener (100:15) compared to the good agreement obtained for both other considered mixing ratios (100:31 and 100:60). Differences in the results may also arise from slightly different mixing ratios of epoxy and hardener components, for which the average value was chosen based on the specified range in the corresponding data sheets. With the proposed method, a molecular modeling is presented that is validated against experimental results and transferable to other material systems or curing conditions. It allows us to investigate the influence of temperature or pressure on the

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curing process and thus helps to understand and predict curing mechanism of epoxy resins.

14.5 Conclusions The curing procedure of thermosetting epoxy resins has an influence on the final network structure and thus on the material properties. Therefore, an accurate representation of the crosslinking process is crucial for understanding the underlying mechanism and deriving material properties with MD simulations. We contributed to the challenge of generating realistic representations of the atomistic network structure of epoxy resins by presenting the first experimentally validated molecular modeling method for the crosslinking of epoxy resins. The method presented, which implements a step-wise bond formation based on the distance between reactive groups, is studied by applying it to a commercially available amine-cured epoxy system. Numerical studies on the curing process lead to a reasonable choice of rc = 10 Å for the cut-off distance of the crosslinking algorithm in terms of efficiency and accuracy. The results of the parametric study show that this value decreases the simulation time significantly needed for a realistic crosslinking while still maintaining a well equilibrated system. It was shown that NIR is a valuable analysis method for the characterisation of the curing process of epoxy systems, especially due to the possibility of identifying the time-dependent evolution of reactive groups. Experiments with three different mixing ratios were carried out for a validation of the molecular modeling method. Crosslinking simulations with the corresponding mixing ratios were conducted and a comparison to experimental NIR results showed that the fundamental curing mechanism is well represented. The present molecular modeling method provides a precise prediction of the chemical reactions and the corresponding evolution of reactive groups. Additionally, the final degree of curing is in good agreement with NIR results for the mixing ratios considered. This is achieved by considering only the inherent physical interactions and steric effects without specifying an explicit reaction probability for the different types of amine. The proposed method enables material properties derived by MD simulations to be a reliable prediction, because the atomistic network structure is experimentally validated as a realistic representation of the cured polymer material. Moreover, the validated method allows us to gain deeper insights into the curing mechanism of epoxy resins. By deriving the influence of the curing process on the atomistic network structure, MD simulations effectively contribute to the virtual material development of thermosetting polymers by improving existing and designing new epoxy systems. Acknowledgments The authors would like to express special thanks to Hannah Quantrell who conducted the NIR results during her internship at the Federal Institute for Materials Research and Testing. The authors acknowledge the support by the LUIS scientific computing cluster, which is funded by Leibniz Universität Hannover, the Lower Saxony Ministry of Science and Culture (MWK)

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and the DFG. This work originates from the research project ‘Hybrid laminates and nanoparticle reinforced materials for improved rotor blade structures’ (‘LENAH—Lebensdauererhöhung und Leichtbauoptimierung durch nanomodifizierte und hybride Werkstoffsysteme im Rotorblatt’), funded by the Federal Ministry of Education and Research of Germany. The authors wish to express their gratitude for the financial support. Copyright Notice: This chapter is a reprint from the publication by Unger et al. [42], with permission from Elsevier.

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Chapter 15

Permeability Characterization and Impregnation Strategies with Nanoparticle-Modified Resin Systems Dilmurat Abliz and Gerhard Ziegmann

Abstract Impregnation processes are dominated by the preform permeability and the resin viscosity. During the impregnation with NP-modified matrices, the NPs may influence not only the resin viscosity, but also the preform permeability significantly due to the increased particle-particle, particle-resin and particle-fiber interactions— in some cases leading even to filtration of the particles and influencing significantly the flow speed and length. The chapter investigates the impregnation characteristic of NP-modified resin systems and provides concepts for impregnation strategies based on experimental and simulation study of the influence of different NP concentration and surface modifications on the impregnation process.

15.1 Introduction The filtration of NPs may greatly influence the impregnation quality and speed, possibly compromising the impregnation process and the final properties of FRPs. Two main filtration mechanisms, cake filtration and deep bed filtration, could be distinguished during LCM processing of particle-filled suspensions [13, 22]. Cake filtration is the volumetric clogging or sieving due to particles that are larger than the inter-fiber channels [15]. Deep bed filtration, on the other hand, describes the gradual capture of particles smaller than the pore channels due to the surface interactions between fibers and particles [11, 25]. The ratio between the size of pore channels— i.e. filament distance—and the particle size is pointed out to be the most critical factor which governs the flow behavior and filtration mechanism [16, 17, 23]. According to Sakthivadivel et al. [24] cake filtration occurs when the ratio of filter grain to particle diameter (D/d) is smaller than 10. For 10 < D/d < 20 retention takes place and if D/d > 20, no or limited retention is expected. There are also similar classificaD. Abliz (B) · G. Ziegmann Technische Universität Clausthal, Clausthal-Zellerfeld, Germany e-mail: [email protected] G. Ziegmann e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_15

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tions describing the influence of particle size on the retention mechanism [12, 22], stating that for smaller particles (d ≤ 1 µm) surface interactions are predominant, whereas for bigger particles (d ≥ 30 µm) volume phenomena prevail over surface phenomena. In-between these ranges, both, volume and surface phenomena, may occur. Many studies concentrating on the processing of fibrous (carbon nanotubes, carbon nanofibers) or sheet-structured (graphite, graphene) NPs, with at least one dimension being dozens of microns, reported severe filtration effect during the impregnation [7, 29, 30]. These presumably occurred because the particle size of up to dozens of microns led to a critical D/d ratio. In contrast to the fibrous or sheet-structured particles, quasi-spherical NPs, with all dimensions similar in nano scale, are expected to be more suitable for processing in LCM processes and to show no retention. Nevertheless, Louis et al. [14] reported a remarkable filtration of quasi-spherical nano-alumina (Al2 O3 ) NPs during Resin Transfer Moulding (RTM) process even at a fiber volume fraction (V f ) of only 41%. In a more recent study, Louis et al. [15] investigated different material and process parameters and reported that the factors with the biggest impact on the filtration behavior of NPs are particle size (esp. agglomerates) and V f . It is interesting to note that even quasi-spherical NPs can show a remarkable filtration behavior even at low V f during LCM processes, indicating that the filtration is a critical aspect to be scrutinized. The particle-size distribution is strongly dependent on the preparation method, surface characteristics of NPs and the suspension stability [18, 19]. The different synthesis and fabrication strategies of nanomaterials are categorized as “top-down” and “bottom-up” approaches [6]. Attrition, milling, or the dispersion of NPs are “topdown” methods, whereas atom-by-atom synthesis to get a colloidal dispersion is a good example of a “bottom-up” approach. Accordingly, it is to be expected that the NPs, depending on the approach that is applied, show different morphology, surface characteristics and size distribution. This chapter focuses on investigating the flow behavior two quasi-spherical NPs, boehmite (AlOOH) and silica (SiO2 ), during LCM processes. They are prepared by different industrial masterbatch technologies. The foci lie on the investigation of flow characteristics regarding different dispersion processes and NP surface characteristics. Furthermore, different impregnation strategies for the application of NPmodified resin systems are put forward, with focus on the influence on NPs on the permeability and maximum impregnation length.

15.2 Materials and Methodologies For the first NP system, boehmite NPs (primary particle size of 14 nm and an average agglomerate size of around 25 µm according to the manufacturer, DISPERAL HP14, SASOL Germany) are firstly dispersed in the epoxy matrix with a high-performance laboratory kneader (HKD-T0.6, IKA, Germany) to produce masterbatches with 40 wt% boehmite NPs (kneader batch), thereafter it is further dispersed in three-roll-

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miller (kneader-miller batch). For further details of the dispersion process, please refer to the Chap. 11. As a second NP system, a commercial Diglycidyl Ether of Bisphenol A (DGEBA) epoxy-based silica nanosuspension (NANOPOX F400, 40 wt%, epoxy equivalent weight: 295 g/eq) from Evonik Industries AG is used. In contrast to the boehmite nanosuspension, the silica nanosuspension is manufactured in-situ in the epoxy resin by a modified sol-gel process. It has an average particle size of 20 nm and a very narrow particle size distribution [26]. In order to study the particle size after cure, cured epoxy samples are broken after cooling down in liquid nitrogen to obtain a cryo-fracture surface. The surface is coated with carbon and observed with scanning electron microscopy (Helios Nanolab 600, FEI).

15.2.1 Flow Experiments For the flow experiments, the masterbatch is diluted to the desired particle concentration with a planetary mixing machine (Thinky Mixer ARV-310) at room temperature in two run series with the following parameters: (1) 1.5 min at 0 rpm under 5 kPa absolute pressure; (2) 1.5 min at 1000 rpm under 10 kPa absolute pressure; (3) 1.5 min at 2000 rpm under 1 kPa absolute pressure. Before injection, the NP-epoxy matrix system is filtered with a fine sieve (mesh size approx. 190 µm) to remove contaminations. Considering the possible effects of curing on the suspension stability and so the NP reagglomeration behavior, the experiments are carried out using both unreactive (without hardener) and reactive (with hardener) diluted NP-epoxy suspensions, respectively. The flow experiments are carried out with a custom-developed RTMtool, as illustrated in Fig. 15.1. For the experiments with nonreactive suspensions, the fluid samples are extracted during the impregnation. For the experiments with reactive systems, the textiles are punched in each 5 cm distance before impregnation, so matrix samples could be taken after fully curing for the particle concentration determination. The samples are extracted at five locations between the inlet and the outlet in every 5 cm distance. The NP concentration in the extracted samples is measured by thermo-gravimetric analysis on a TGA Q5000 from TA Instrument (Ramp: 30–800 ◦ C @ 10 K/min, air condition). Because V f is very critical for the flow and retention of the particles, the cavity height is checked after each experiment with the help of wax pellets, which are placed along the sides of the textile during the experiment. Furthermore, a sealant prevents the occurrence of any race tracking and distortion at the textile edges.

15.2.2 Permeability Test The permeability tests are carried out on a permeability test rig for the permeability test and flow front tracking, which is equipped with sensors to monitor pressure

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Fig. 15.1 Sample extraction for the NP flow and retention investigation

(a)

(b)

Fig. 15.2 Example image of a 1D linear flow (a) and b 2D radial-flow experiment and evaluation steps of the digital image processing algorithm (left to right)

and temperature and allows to optically track the flow front. Besides, the test rig is also equipped with heating elements so that the measurement can be carried out also with special high-viscous fluids (NP-filled epoxy etc.) up to 120 ◦ C, with an accuracy of ±2 ◦ C. A proprietary pressure and temperature recording software and an in-house Matlab script are used to conduct the experiment. The Matlab script provides a graphical user interface to conduct the experiment. It sets the injection pressure (0.5–6 bar) and tracks the flow front by taking photos with a resolution of 2560 × 1920 pixels and up to 3/fps. The Matlab script also features an algorithm to detect the flow front on-line by using difference images, as shown in the Fig. 15.2. Special data evaluation and calculation algorithms are incorporated in the Matlab script considering the flow and filtration behavior of the NP-filled fluids.

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15.3 Experimental Results 15.3.1 Nanoparticle Systems Flow and Retention According to the results for NP flow and retention, as shown in Figs. 15.3 and 15.4, it can be found that at 40 and 50% V f , the concentrations of both NP types keep almost constant or decreases very slightly along the flow length. This indicates no retention or only a slight deep-bed retention behavior. The same trend applies also to silica NP even at high 60% V f . In contrast to this, at 60% V f , boehmite NPs that are dispersed in kneader show a remarkable retention. The severe filtration of the boehmite NPs at the 60% V f indicates a different filtration mechanism: cake filtration, with a significant influence on the flow speed and maximum flow length so that the flow speed decreases towards zero at a flow length of 100–150 mm. As seen in the microscopic investigations of the fracture surface in cured epoxy samples (Fig. 15.5), there are plenty of fine dispersed NPs in both NP systems to be observed at high magnification (left). Interestingly, the boehmite-epoxy samples show some amounts of coarse-particles in microns (right). The amounts of the coarse

Fig. 15.3 Boehmite NP concentration along flow length by different V f . Left: Kneader batch, nonreactive suspension; Right: Kneader-Miller batch, reactive suspension Fig. 15.4 Silica NP concentration along flow length by different V f

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Fig. 15.5 Particle size distribution in the cure sample (10 wt%) Magnification from left: 1µm; right: 20 µm

particles decrease significantly after the three-roll-milling process (kneader-miller batch). Similar to the kneader-miller batch, silica-epoxy sample did not show any detectable coarse particle or agglomerates. The possible reason for the occurrence of coarse particles in the kneader batch are the residual agglomerates that were not fully dispersed during the dispersion process. The dispersion of the particles in a kneader is achieved by the shear stress acting on the particles, which is dependent on the local shear rate within the kneader chamber [20].

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The local shear rate differs considerably within the kneader chamber due to varying gap widths and relative velocities of neighboring blade surfaces. Hence, kneaders have a broad distribution of stress intensity. The final size of a particle is determined by the maximum stress intensity and residence time. A uniform dispersion state is, therefore, a matter of the statistical residence time distribution of each fluid element within the zone of the kneader where it is exposed to the maximum shear rate [18]. As a result, a complete dispersion can only be achieved at an impractically long dispersion time. Therefore, it is reasonable to assume that the coarse particles will most likely be residual agglomerates that are not fully dispersed, due to the inherent processing limits by kneader process. It is interesting to note that, after the subsequently dispersion with three-rollmilling, the number of coarse particles is remarkably reduced, by which the shortcoming of the kneading process can be effectively compensated. The major advantage of the three-roller mill is that all material that leaves the processing zone has been stressed at least once. This reliably removes coarse particles from the suspension. Nevertheless, solely using the three roller mill for production could come with the certain disadvantages that the minimum reachable particle size is limited, as the maximum applicable viscosity and hence the stress intensity is smaller compared to the kneading process due to wall slip phenomena on the three roller mill [8]. Therefore, combining the kneader with a further dis-integration of coarse particles via threeroller milling of a diluted master batch seems be a good practice for preparing the NP-masterbatches.

15.3.2 Permeability Figure 15.6a and b show the viscosity development of the NP-epoxy suspensions (without hardener) as functions of temperature, shear rate, surface functionalization and particle concentration for the permeability tests. The viscosity of the boehmite (K: neat boehmite; KS: surface modified with stearic acid) NP-epoxy suspensions with concentration of up to 5 wt% shows an exponential decrease with increasing the temperature. However, for 10 wt% there is a sudden increase of viscosity at a temperature of about 80 ◦ C. This sudden change is caused by an irreversible chemical reaction between the particles and the neat epoxy resin [4]. Correspondingly, the viscosity of the suspensions of up to 5 wt% shows a Newtonian behavior (Fig. 15.6b), whereas the viscosity of the suspension with 10 wt% boehmite NPs shows a strong shear rate dependency. Therefore, in order to exclude the influence of the reaction between the NP and epoxy and the shear rate on the viscosity, the permeability tests with boehmite NPs are only conducted with KS particle concentrations up to 5 wt%. The viscosity results of silica NP-epoxy suspension did not show such a reaction and showed a Newtonian behavior. During the permeability tests, for one-dimensional flow experiments with standard incompressible fluids (e.g. oil, neat resin) or suspensions without NP filtration effect (kneader-miller batch), the development of the squared flow front (x 2f ) versus time

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(a) Viscosity vs. temperature

(b) Viscosity vs. shear rate

Fig. 15.6 Characteristics of boehmite-epoxy test fluids

(a) Incompressible flow without NP filtration

(b) Compressible flow with NP filtration

Fig. 15.7 Flow front development in a permeability measurement

(t) shows a linear increase with a constant slope m (Fig. 15.7a). In contrast to this, if filtration occurs, the squared flow front versus time curve deviates from linearity for particle-filled systems (Fig. 15.7b). This deviation is caused by the retention of NPs, which influences the suspension viscosity and the local permeability. These changes render the common analytical solution for permeability [5, 28] nonapplicable unless the local changes of matrix viscosity, preform porosity and pressure gradient are known. Accordingly, the permeability is measured applying only the kneader-miller batches—suspensions without filtration effect. Figure 15.8 shows the permeability change according to Vf applying different test fluids. The tests are repeated at the lowest and highest V f 45 and 65% so that the test error could also be taken into account. It can been seen that, if the statistical test error is considered, the change of the test fluid on the permeability lies within the range of test error, indicating that, given no filtration effect of NPs, there is not a significant influence of the applied NP-epoxy suspensions on the permeability of the textiles.

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Fig. 15.8 Permeability versus different applied test fluid

15.4 Simulation of the Impregnation Length Figure. 15.9 shows the flow simulation and validation results with the modeling and simulation methods that are described in Chap. 3 for a standard incompressible flow—without NPs. As shown, the simulation can quite reliably predict the flow front development, indicating that the applied numerical implementations are feasible. As it is shown that for the NPs that are post-dispersed with three-roll-mill, there is not any significant change in the preform permeability. Therefore, impregnation simulation is carried out to determine the maximum flow length considering the influence of different NP concentration and surface modification on the cure kinetics and rheology. According to the Fig. 15.10, the maximum flow length and time is gradually

Fig. 15.9 Flow simulation and validation (without NP)

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Fig. 15.10 Maximum flow length and time depending on boehmite particle concentration

Fig. 15.11 Maximum flow length and time depending on particle surface modification

decreased with increase of NP concentration. Nevertheless, this is quite dependent on the surface modification of the NP, as shown in Fig. 15.11. The suspension with the stearic acid modified NPs showed only a small decrease in the maximum flow length with 10 wt% compared to the neat resin, whereas the suspension that is modified with pure boemite shows the highest decrease in the maximum flow length.

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15.5 Impregnation Strategies for Gradual Functionalization with NPs In a standard impregnation strategy design, the target function mostly would be impregnation time or impregnation quality (impregnation velocity, converging flow fronts etc.) with the aim of getting low porosity. Corresponding to that goal the boundary conditions are the impregnation time/length of the matrix depending on the permeability. However, to fabricate FRP structures with NP-filled matrix, the flow velocity and maximum flow length is strongly dependent on the concentration and modification of applied NPs. Therefore, it is important to design proper process step sequences and impregnation strategies to optimally apply the NP-modified matrix systems to produce and functionalize FRP structures. First of all, two different impregnation methods, in-plane and out-of-plane, can be differentiated. By the in-plane impregnation process, as illustrated in Fig. 15.12a, the textiles are placed in a closed mold, and the matrix is injected from one side as linear (1D) or from the middle as radial flow (2D) form. The fluid flows inplane homogenously within the textile to the outlet direction. In comparison, by the out-of-plane impregnation process (see Fig. 15.12b), a gap or a high permeable flow medium is applied above/under the textile, by which the fluid will quickly flow along the flow media and impregnate the laminate in the thickness direction due to the high permeability difference between the compacted textiles and the flow media. There are different variations of out-of-plane impregnation methods, including Compaction Resin Transfer Moulding (CRTM) or Gap Impregnation etc. For impregnation of complex geometries, in-plane impregnation is preferable, where the maximum flow length is important for designing the inlet/outlet positions. Based on the maximum flow length, optimal injection strategies regarding structure complexity and application requirements can be designed. The different in-plane injection strategies could be illustrated with the following two most representative injection strategies, as shown in Fig. 15.13 a and b: sequential and parallel injection. By sequential injection, multiple injection gates may be designed with optimal distance, according to the NP gradient degree and flow length. If the flow front reaches the nearby injection gate, the first gate closes, and the second gate opens. In comparison, by the parallel injection, all the injection gates can be opened simultaneously.

(a) in-plane

Fig. 15.12 Illustration of impregnation processes

(b) out-of-plane

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(a) sequential injection

(b) parallel injection

Fig. 15.13 In-plane linear impregnation strategy

Furthermore, impregnation with selective NP-modified epoxy matrix systems (different NP types or concentrations) enables an optimal gradient-functionalized FRP structure corresponding to the stress gradient during application. Some representative examples of the functional-gradient materials from nature can be found in bone, teeth and bamboo [9, 10, 21, 27]. They, after thousands of years of natural evolution, are optimized for maximum structural properties with optimized distribution of each component. For example, the cross section of a bamboo culm shows a functional graded distribution of the cellulose fibers—more fibers near to the outer region, which provides the optimum resistance to bending moment from its own weight and wind load. The application of different NP-modified systems for gradual functionalization of FRPs may provide an optimal tradeoff between the performance and fabrication cost of FRPs.

15.6 Summary In this chapter, flow and retention studies are carried out with different NPs: boehmite and silica that are produced differently. It is found out that the NP retention behavior is strongly dependent on the dispersion quality of the NPs in the epoxy, which is closely dependent on the production method of the masterbatch. It was shown that the nanosuspensions that are dispersed with combination of kneader and three-rollmiller or directly prepared “bottom-up” in the matrix show almost no retention and no significant influence on the permeability of the fiber preform. Flow simulations are carried out to anticipate the maximum flow length and time with respect to the different NP-modified matrix systems. It was shown that the maximum flow length and time is strongly dependent on the NP concentration and surface modification. With respect to the influence of NP concentration and surface modification on the impregnation process and final properties, different impregnation strategies with selective NP-modified matrix systems may be applied to achieve an optimal tradeoff between the processing and properties.

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Copyright Notice This chapter is based on Abliz’s Ph.D. thesis [1] and the articles [2, 3].

References 1. Abliz D (2017) Functionalization of fiber composites with nanoparticle-modified resin systems. Dissertation, Technische Universität Clausthal 2. Abliz D, Berg DC, Ziegmann G (2019) Flow of quasi-spherical nanoparticles in liquid composite molding processes. Part II: Modeling and simulation. Comp Part A: Appl Sci Manuf 125:105,562 (2019). https://doi.org/10.1016/j.compositesa.2019.105562 3. Abliz D, Finke B, Berg DC, Schilde C, Ziegmann G (2019) Flow of quasi-spherical nanoparticles in liquid composite molding processes. part i: Influence of particle size and fiber distance distribution. Comp Part A: Appl Sci Manuf 125:105,563 (2019). https://doi.org/10.1016/j. compositesa.2019.105563 4. Abliz D, Jürgens T, Artys T, Ziegmann G (2018) Cure kinetics and rheology modelling of boehmite (alooh) nanoparticle modified epoxy resin systems. Thermochimica Acta 669:30– 39. https://doi.org/10.1016/j.tca.2018.06.017 5. Arbter R, Beraud JM. Binetruy C, Bizet L, Breard J, Comas-Cardona S, Demaria C, Endruweit A, Ermanni P, Gommer F, Hasanovic S, Henrat P, Klunker F, Laine B, Lavanchy S, Lomov SV, Long A, Michaud V, Morren G, Ruiz E, Sol H, Trochu F, Verleye B, Wietgrefe M, Wu W, Ziegmann G (2011) Experimental determination of the permeability of textiles: a benchmark exercise. Comp Part A—Appl Sci Manuf 42(9):1157–1168 6. Cao G (2004) Nanostructures and nanomaterials: synthesis, properties, and applications. Imperial College Press 7. Da Costa E, Skordos AA, Partridge IK, Rezai A (2012) RTM processing and electrical performance of carbon nanotube modified epoxy/fibre composites. Comp Part A—Appl Sci Manuf 43(4):593–602 8. Finke B, Nolte H, Schilde C, Kwade A (2019) Stress mechanisms acting during the dispersing in highly viscous media and their impact on the production of nanoparticle composites. Chem Eng Res Des 141:56–65. https://doi.org/10.1016/j.cherd.2018.10.002 9. Ghavami K (2005) Bamboo as reinforcement in structural concrete elements. Cement Concrete Comp 27(6):637–649. https://doi.org/10.1016/j.cemconcomp.2004.06.002 10. Ghavami K, de Souza Rodrigues C, Paciornik S (2003) Bamboo: functionally graded composite material. Asian J Civil Eng (Build Housing) 4(1):1–10 11. Ghidaglia C, De Arcangelis L, Hinch J, Guazzelli E (1996) Hydrodynamic interactions in deep bed filtration. Phys Fluids 8(1):6. https://doi.org/10.1063/1.868810 12. Herzig JP, Leclerc DM, Goff PL (1970) Flow of suspensions through porous media— application to deep bed filtration. Indus Eng Chem 5(62):8–35 13. Lefevre D, Comas-Cardona S, Binétruy C, Krawczak P (2007) Modelling the flow of particlefilled resin through a fibrous preform in liquid composite molding technologies. Comp Part A: Appl Sci Manuf 38(10):2154–2163. https://doi.org/10.1016/j.compositesa.2007.06.008 14. Louis BM, Maldonado J, Klunker F, Ermanni P (2014) Measurement of nanoparticle distribution in composite laminates produced by resin transfer molding. In: 16th European conference on composite materials (ECCM), 16th European conference on composite materials (ECCM), Seville, Spain 15. Louis BM, Maldonado J, Klunker F, Ermanni P (2018) Particle distribution from in-plane resin flow in a resin transfer molding process. Polym Eng Sci. https://doi.org/10.1002/pen.24860 16. Maroudas A, Eisenkla P (1965) Clarification of suspensions—a study of particle deposition in granular media: Part I—some observations on particle deposition. Chem Eng Sci 20(10):867– 873

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17. Maroudas A, Eisenkla P (1965) Clarification of suspensions—a study of particle deposition in granular media: Part II—a theory of clarification. Chem Eng Sci 20(10):875–888 18. Müller F, Peukert W, Polke R, Stenger F (2004) Dispersing nanoparticles in liquids. Int J Mineral Proces 74:S31–S41. https://doi.org/10.1016/j.minpro.2004.07.023 19. Nolte H, Schilde C, Kwade A (2010) Production of highly loaded nanocomposites by dispersing nanoparticles in epoxy resin. Chem Eng Technol 33(9SI):1447–1455. https://doi.org/10.1002/ ceat.201000096 20. Raasch J (1961) Beanspruchung und Verhalten suspendierter Feststoffteilchen in Scherströmungen hoher Zähigkeit. PhD thesis, T.H. Karlsruhe 21. Ray AK, Mondal S, Das SK, Ramachandrarao P (2005) Bamboo–a functionally graded composite-correlation between microstructure and mechanical strength. J Mater Sci 40(19):5249–5253. https://doi.org/10.1007/s10853-005-4419-9 22. Reia Da Costa EF, Skordos AA (2012) Modelling flow and filtration in liquid composite moulding of nanoparticle loaded thermosets. Comp Sci Technol 72(7):799–805. https://doi. org/10.1016/j.compscitech.2012.02.007 23. Sakthivadivel R (1969) Clogging of a granular porous medium by sediment. Hydraulic Engineering Laboratory, College of Engineering, University of California, Berkeley 24. Sakthivadivel R, University of California B, Hydraulic EL (1969) Clogging of a granular porous medium by sediment. Hydraulic Engineering Laboratory, College of Engineering, University of California, Berkeley 25. Santos A, Araújo JA (2015) Modeling deep bed filtration considering limited particle retention. Transp Porous Media 108(3):697–712 (2015). https://doi.org/10.1007/s11242-015-0496-7 26. Sprenger S (2014) Improving mechanical properties of fiber-reinforced composites based on epoxy resins containing industrial surface-modified silica nanoparticles: review and outlook. J Comp Mater 49(1):53–63. https://doi.org/10.1177/0021998313514260 27. Tan T, Rahbar N, Allameh SM, Kwofie S, Dissmore D, Ghavami K, Soboyejo WO (2011) Mechanical properties of functionally graded hierarchical bamboo structures. Acta Biomaterialia 7(10):3796–3803. https://doi.org/10.1016/j.actbio.2011.06.008 28. Verne N, Ruiz E, Advani S, Alms JB, Aubert M, Barburski M, Barari B, Beraud JM, Berg DC, Correia N, Danzi M, Delaviere T, Dickert, M, Di Fratta C, Endruweit A, Ermanni P, Francucci G, Garcia JA, George A, Hahn C, Klunker F, Lomov SV, Long A, Louis B, Maldonado J, Meier R, Michaud V, Perrin H, Pillai K, Rodriguez E, Trochu F, Verheyden S, Wietgrefe M, Xiong W, Zaremba S, Ziegmann G (2014) Experimental determination of the permeability of engineering textiles: Benchmark II. Comp Part A—App Sci Manuf 61:172–184 29. Yum SH, Lee WI, Kim SM (2016) Particle filtration and distribution during the liquid composite molding process for manufacturing particles containing composite materials. Comp Part A: Appl Sci Manuf 90:330–339. https://doi.org/10.1016/j.compositesa.2016.07.016 30. Zhang H, Liu Y, Huo S, Briscoe J, Tu W, Picot OT, Rezai A, Bilotti E, Peijs T (2017) Filtration effects of graphene nanoplatelets in resin infusion processes: problems and possible solutions. Comp Sci Technol 139:138–145. https://doi.org/10.1016/j.compscitech.2016.12.020

Part V

Structural Mechanics of Fiber Reinforced Nanocomposites

Chapter 16

Nanoscaled Boehmites’ Modes of Action in a Polymer and Its Carbon Fiber Reinforced Plastic Christine Arlt, Wibke Exner, Ulrich Riedel, Heinz Sturm, and Johannes Michael Sinapius Abstract Laminates of carbon fiber reinforced plastic (CFRP), which are manufactured by injection technology, are reinforced with boehmite particles. This doping strengthens the laminates, whose original properties are weaker than prepregs. Besides the shear strength, compression strength and the damage tolerance, the mode of action of the nanoparticles in resin and in CFRP is also analyzed. It thereby reveals that the hydroxyl groups and even more a taurine modification of the boehmites’ surface alter the elementary polymer morphology. Consequently a new flow and reaction comportment, lower glass transition temperatures and shrinkageindexresin shrinkage, as well as a changed mechanical behavior occur. Due to a structural upgrading of the matrix (higher shear stiffness, reduced residual stress), a better fiber-matrix adhesion, and differing crack paths, the boehmite nanoparticles move the degradation barrier of the material to higher loadings, thus resulting in considerably upgraded new CFRP. This chapter published already in 2013 marks a starting point of the collaborative research of the Research Unit FOR2021.

This chapter was previously published in: M. Wiedemann and J. M. Sinapius (Eds.), Adaptive, tolerant and efficient composite structures, Springer-Verlag Berlin, ISBN 978-3-642-29189-0 (2013). C. Arlt · W. Exner · U. Riedel German Aerospace Center, Braunschweig, Germany e-mail: [email protected] H. Sturm Bundesanstalt für Materialforschung und -prüfung (BAM), Berlin, Germany e-mail: [email protected] J. M. Sinapius (B) Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_16

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16.1 Challenges of Future Carbon Fiber Reinforced Plastics Increasing ecological awareness as well as quality and safety demands, which are present, for instance, in the aerospace and automotive sectors, lead to the need to use more sophisticated and more effective materials. Thereby cost efficiency is indispensable. CFRPs have successfully been established in lightweight construction due to their excellent mechanical performance, while having a low specific weight. Currently the injection—and the prepreg (pre-impregnated fibres)—technology are the widest spread manufacturing techniques for processing high performance CFRPs. The injection of liquid resin into dry fibres has the advantages of lower manufacturing costs and higher potential of volume production. However, prepreg manufactured CFRPs still have the higher property level. Typically, CFRP-structures are dimensioned via their fibers under external loading conditions. But it is well known, that laminate failure is often determined by the matrix behavior under miscellaneous loading conditions. Especially for laminates manufactured by injection technology the remaining residual stresses in the laminate and limited matrix properties hinder the development of these high performance materials to their full potential. Hence the idea of strengthening the injection matrix for CFRP is obvious. One approach of strengthening the matrix properties is the incorporation of fillers. In numerous projects several materials as metal oxides, glasses or carbonates have been investigated [3]. Initially micro scaled particles were used for reinforcement. Problems occurred due to brittleness of the polymer, increased viscosity and filtration of the particles by the fiber. In order to overcome these problems nanoscaled fillers became the focus of research [9]. Improved chemical, physical and mechanical material properties have been realized by many researchers. The accomplished changes mostly depend on the material characteristics of the filler, the particle size and shape, the dispersion quality, the filler content and the particle-matrix interaction [12, 13]. By altering the nanoparticles surface molecules, the particle-matrix interaction can be tailored and therewith the polymer network be formed [1]. Various research results show an increase of mechanical performance with improved resin-particle bonding [7]. Especially covalent bonding is addressed to enhance the load transfer and make the nanocomposites more resistant [4, 7, 10, 14, 17]. A limiting factor of the influence of a strong interphase is the network mobility. With a progress of cross linking density the network mobility decreases and the influence the particle-resin interphase becomes less influential [11]. For that reason general statements are difficult. So, the challenge of this research is to figure out relationships between the particles’ surface molecules, nano-polymer properties and the properties of corresponding CFRP.

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16.2 Resin-Particle Interactions For the reinforcement of carbon fiber reinforced plastics (CFRPs) nanoscaled boehmite is used. The particles have a cubical shape and a primary size of 14nm. Boehmite is a aluminum oxide hydroxide (γ -AlO(OH)). Therefore it posses a high number of hydroxyl groups at its surface, which allow a surface modification. In the presented work pure boehmite (HP14) and taurine modified particles (HP14T) are analyzed. According to literature [8] the taurine molecule probably reacts in an acid base reaction with the boehmite and forms an outer-sphere complex. Figure 16.1 presents the formed complex. By the means of BET measurements and elemental analysis the specific surface area and the atomic composition of the particles are tested. With the help of these results a surface coverage of 16% with taurine is calculated for the particles HP14T. The processed particles were provided by the company Sasol Germany GmbH as a dry powder. In the dry state the particles are clustered into agglomerates and aggregates. To obtain single nanoscaled particles, the powder was dispersed into the resin. This was done with a three-roll mill (Exakt, 80E) for HP14 and with a bead mill for HP14T. In both machines the particles get separated by mechanical loads. To analyze particle size and distribution scanning election microscope (SEM) picture were made of the nanocomposites. In Fig. 16.2 two representative pictures are presented. The results show a fine and homogeneous particle distribution. Only a small number of minor agglomerates are found. While processing the liquid nanocomposites a major difference in viscosity of both systems becomes obvious. Figure 16.3 shows the viscosity during cure at 80 ◦ C. Comparing the nanocomposite of pure boehmite with the pure resin, a significant increase in viscosity is observed. The taurine modified particles in contrast affect the flowability hardly. Further experiments show for the masterbatch of HP14 a thixotropic behavior, while the masterbatch of HP14 only have a slight shear thinning. The different viscosity of both liquid systems indicates a different particle-resin interaction caused by the different surface properties. While the interaction of the pure boehmite and the

Fig. 16.1 Outer-sphere complex between Sulfonat und Böhmit (detail view of crystal)

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Fig. 16.2 SEM-pictures of cured nanocomposites with a filler content of 15 wt% of boehmite; left: HP14; right: HP14T Fig. 16.3 Viscosity of nanocomposites with a filler content of 7.5 wt% during cure at 80 ◦ C

resin seems strong, the modification with taurine weakens this interphase. To verify this assumption both particles were processed with 4-tert-butylphenyl glycidyl ether. The 4-tert-butylphenyl glycidyl ether is used as a model substance for the resin. After the reaction with the model substance the particles got thor-oughly washed and analysed by means of attenuated total reflection (ATR) and nuclear magnetic resonance spectroscopy (NMR). Results showed an opening of the epoxy group. A clear product could not be identified, but strong interactions between the particles and the resin can be assumed. Taurine molecules can not re-act with the resin. For that reason the act as a spacer, which results in significant lower viscosities. The astonishing difference in flowing behavior caused by the modification is essential for the manufacturing of CFRPs by injection technology. Only with the particles HP14T this manufacturing method is realizable.

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16.3 Particle–Polymer Interphases The first approach of detecting changed particle-resin interactions and inter-phase strength by the altering the boehmites’ surface molecules showed that tau-rinemodified particles seem to interact less strong with the epoxy than the un-modified analogon (viscosity profiles, Sect. 16.2, Fig. 16.3). Now the resulting changes in the polymer network and therewith its Young’s modulus especially in the close-up range of the particles HP14 and HP14T should be considered. For that reason, samples of cured nanocomposites are prepared using abrasive grinding paper followed by polishing using diamante. First inspection using Atomic Force Microscopy (AFM) operated in the Force Modulation Mode (FMM) showed partially contamination with a soft, viscous surface layer, which was gradually removed by a magnetically coupled plasma (air, 10-1 mbar). Force Modulation Microscopy (FMM) is a variation of the AFM contact mode. The sample is mounted on a piezo (resonance frequency 6 MHz, d/U 2 nm/V, both for the unloaded case) and a sufficient drift range must be realized to sup-press acoustic near field effects, which in our case was realized by the sample thickness itself (≈1.5 cm). The actuator is excited with an AC voltage leading to a z of the surface of 90%). The relative change of reactive amine groups as a function of the degree of curing is also presented in Fig. 17.7. The increase of tertiary amine groups by increasing the degree of cross-

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linking confirms the artificial carbon-nitrogen bond formation associated with the curing simulations. For further analysis, the radial distribution function (RDF) of all nitrogen atoms to all other carbon atoms in the polymer network is shown in Fig. 17.8. The RDF depicts the main peak at the equilibrium distance of the carbon-nitrogen bond, proving that the system is cross-linked and is at a well equilibrium state. The simulations are repeated to generate three cross-linked epoxy samples. Next, an NPT simulation at 300 K and 1 bar is conducted for 20 ns for each sample, and two configurations are extracted for the subsequent analysis every 5 ns during the last 10 ns of the simulation. Finally, six cross-linked epoxy samples are prepared. The average number of rigid links between two cross-links represented by N in Eq. (17.28) is obtained to be 2.44 for the cured epoxy resins as presented in Table 17.2.

17 Viscoelastic Damage Behavior of Fiber Reinforced Nanoparticle . . . Table 17.2 Material parameters predicted by MD simulations Parameter Equations Viscoelastic dashpot Hyperelastic spring

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Value 1.43 × 10−7 0.79 205.50 2.44

Modeling of Cured Epoxy Under Tensile Loading

Once the cured samples are prepared, they are independently subjected to tensile deformations along the x−, y−, and z− directions at various strain rates and room temperature T = 298 K. A 40% axial tensile strain is applied to the periodic polymer system with zero-valued pressures in the lateral directions to allow for the natural Poisson contraction. The simulations of the cured epoxy subjected to tensile loading enables us to study the rate-dependent viscoelastic behavior of the material. The characterization of the viscoelastic flow rate of the epoxy resin will be possible through the analysis of the simulation results along with the Eyring model presented in Eq. (17.33). Based on the transition state theory, Eyring [20] showed that the deformation of a polymer is a thermally activated process involving the motion of segments of chain molecules over energy barriers. An external applied stress or strain allows a molecule rearrangement to an more energy-efficient state by providing the necessary energy to overcome the energy barrier. The pre-exponential factor and the activation volume required in model are then obtained from the resulting Eyring plot. A representative stress-strain curve of the cured epoxy system at strain rate ε˙ = 1 × 108 s-1 is shown in Fig. 17.9. The simulation data points are fitted by a piecewise cubic spline interpolation with an optimized knot. The knot point (1/Tk ) of the bicubic fit is determined by minimizing the least square error between the fit and the simulation data. The maximum observed on the stress-strain curve, which is the yield point, has been marked by an arrow in the figure. The yield point is associated with a sudden increase in the amount of strain which relaxes the stress. From Fig. 17.9, the yield stress at the strain rate equal to 1 × 108 s-1 is calcuated to be 282.27 MPa. An estimation of the viscoelastic flow rate can be obtained from the Eyring plot according to Eq. (17.33). For this, the deviatoric part of the yield stress (dev [σY ]) at multiple strain rates varying from 5 × 106 to 5 × 108 s-1 is calculated. The resulting Eyring plot shown in Fig. 17.10 suggest a linear behavior from which the preexponential factor and the activation volume are obtained to be 1.43 × 10−7 s−1 and 0.79 nm3 , respectively. The values are listed in Table 17.2. It is noteworthy that polymer network structures are ideally homogeneous in the simulations, while voids and cavities are created during the synthesis process of polymer materials. The presence of voids in polymer composites results in reductions in the matrix-dominated mechanical properties. Therefore, the evolution of damage state measured from experimental tests is different with that obtained using

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True stress (MPa)

400

300

200

100 ˙ = 1 × 108 s−1 Cubic spline interpolation

0

0

5 · 10−2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

True strain

Fig. 17.9 Tensile stress-strain behavior of an cured epoxy system at strain rate ε˙ = 1 × 108 s−1 21 20

ln ˙ ( ˙ : s−1 )

19 18 17 16 15 14 160

Simulation data Eyring equation (Eq.((33)))

165

170

175

180

185

190

dev [σY ] (MPa)

Fig. 17.10 Strain rate dependence of the yield stress

MD simulations. Therefore, in order to take into account the damage behavior in the composite polymers, a phenomenological damage model presented in Eq. (17.35) is considered. The damage model is calibrated using uniaxial tensile tests as discussed later. All the simulations are performed with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [42].

17 Viscoelastic Damage Behavior of Fiber Reinforced Nanoparticle . . .

395

Stress (MPa)

Fig. 17.11 Schematic illustration of the cyclic perturbations of tensile/compressive stresses at a constant shear strain (left), and Loading/unloading cycles versus time (right)

200 0 −200 0

200

400

600

800 1,000

Time (ps)

17.2.4.3

Modeling of Cured Epoxy Under Shear Loading

As discussed in Sect. 17.2, the connectivity and stretching of the polymer network impart the hyperelastic behavior and a generalized neo-Hookean model presented in Eq. (17.28) captures this behavior. Here, we present a MD simulation-based method to analyze the long-term configuration change of cured epoxy polymer subjected to a shear deformation. It allows us to study the stress relaxation behavior of the epoxy under a constant strain and to capture the time-independent response of the epoxy by which the corresponding time-independent shear modulus of the material that is required in Eq. (17.28) is obtained. Shear strains of 1% are initially applied to the cured samples in the xy-, yz-, and xz-planes. The simulation box is deformed at a strain rate of 107 s−1 and room temperature under NPT ensemble to control normal stresses along the x-, y- and z-directions at zero. To study the long-term response of the material, we utilized a method similar to that recently introduced to simulate an artificial relaxation [57]. In this method, starting from the deformed configuration, the system is subjected to small cyclic perturbations of tensile/compressive stresses along the x-, y- and z-directions ±σ p around zero pressure, while the shear strain is held constant. The cyclic stresses with period 100 ps are applied to the polymer system using using a series of NPT simulations at room temperature. Fig. 17.11 shows a cyclic tensile/compressive stresses at σ p = ±200 MPa. These small perturbations of stresses allow the system to overcome energy barriers to atomic motion in a computationally feasible time scale. The shear stress is averaged over a number of the cycles. It is worth noting that the relaxation process does not depend on the choice of σ p [57] since the sum of the cyclic stresses is zero over the cycles. Figures 17.12 present the variation of the shear modulus with respect to the number of stress perturbation cycles applied to three different cured epoxy configura-

B. Arash et al.

Shear modulus (MPa)

396 Piecewise linear fit 2,000

1,000

0

0

500

1,000

1,500

2,000

2,500

3,000

Shear modulus (MPa)

Number of loading/unloading cycles

Piecewise linear fit 2,000

1,000

0

0

500

1,000

1,500

2,000

2,500

3,000

Shear modulus (MPa)

Number of loading/unloading cycles

Piecewise linear fit 2,000

1,000

0

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

Number of loading/unloading cycles

Fig. 17.12 Shear modulus with respect to the number of loading/unloading cycles at σ p = 200 MPa and room temperature. The initial shear strain is 0.01

tions. The simulation results show that although the trend of the artificial relaxation varies for different configurations, the shear modulus reaches a plateau at µ = 205 ± 30 MPa after up to 2000 of the cycles (i.e., 200 ns). The plateau represents the time-independent response of epoxy resin under shear loading. A piecewise linear curve with two optimal knots is fitted to the simulation data points to estimate the shear stress at the relaxed configuration.

17 Viscoelastic Damage Behavior of Fiber Reinforced Nanoparticle . . .

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17.2.5 Finite Element Analysis Next, we use the constitutive model to develop a continuum mechanics incremental formulation of nonlinear problems with respect to finite element solution variables. It would make it possible to evaluate the applicability of the proposed atomistically informed constitutive model. In the following section, the continuum mechanics incremental and finite element equations are derived.

17.2.5.1

Total Lagrangian Formulation

The motion of a general body is considered in a stationary Cartesian coordinate system and the aim is to evaluate the equilibrium positions of the body at the discrete time points 0, Δt, 2Δt, . . . , t where Δt is an increment in time. The equilibrium of the body at time t + Δt using the principle of virtual displacements is expressed as ! t+Δt t+Δt V

Ti j δ t+Δt ei j d t+Δt V = t+Δt R,

(17.37)

where t+Δt Ti j is the Cauchy stress tensor, t+Δt ei j is strain tensor corresponding to virtual displacement, t+Δt V is volume at time t + Δt, and t+Δt R is the external virtual work. Equation (17.37) cannot be solved directly since the configuration at time t + Δt is unknown. A solution to the equation is obtained by referring all variables to the initial configuration at time 0 of the body that is called the total Lagrangian (TL) formulation. In the formulation, we consider the following equilibrium equation for the body in the configuration at time t + Δt ! t+Δt t+Δt 0 t+Δt R, (17.38) 0 Si j δ 0 εi j d V = 0V

where t+Δt Si j is the 2nd Piola-Kirchhoff stress tensor, t+Δt εi j is the Green-Lagrange 0 0 strain tensor. and the deformation-independent loading t+Δt R is given by ! t+Δt

R=

0A

! t+Δt s 0 0 f i δu i d A

+

0V

t+Δt b 0 0 f i δu i d V

(17.39)

The linearized equilibrium equation in the TL formulation is then obtained !

! 0V

0 0 C i jr s 0 er s δ 0 ei j d V +

0V

! 0 t 0 Si j δ 0 ηi j d V

= t+Δt R −

0V

0 t 0 Si j δ 0 ei j d V ,

(17.40) where 0 ei j and 0 ηi j are the linear and nonlinear incremental strains which are referred to the configurations at times 0.

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0 C i jr s is the incremental stress-strain tensors at time t referred to the configurations at time 0, which is obtained from the constitutive model presented in Sect. 17.2. However, a closed-form calculation of the tensor for the constitutive model is not a straightforward task. Here, we follow a numerical perturbation method to derive an approximate tangent stiffness matrix [52]. Since the constitutive model uses a TL formulation, the derivative of the stress increment function Δσ with respect to the strain increment tensor Δε takes the following form

∂Δσ ∼ δΔσ = = ∂Δε δΔε

t+Δt

σ

t e t 0 F ,0



˜ F i ,t +Δt F F − t+Δt σ t0 F e ,t0 F i ,t +Δt 0 0 δ

, (17.41)

where t +Δt F˜ is the perturbated deformation gradient and δ is the size of the pertur0 F must be bation. For a three-dimensional analysis, the deformation gradient t +Δt 0 perturbated six times to derive the Jacobian matrix. The reader is referred to [52] for a detailed discussion on the numerical approximation of the tangent stiffness matrix.

17.2.6 Finite Element Matrices Displacements are then discretized at the element level using uh = Nu, where the shape function matrix N interpolates the nodal values u. Substituting the element coordinates and displacement interpolations into Eq. (17.40), the following system of equations for a single element is obtained: t

0 KL

+ t0 K N L u = t+Δt R − t0 F,

(17.42)

where t0 K L and t0 K N L are the linear and nonlinear strain (geometric) incremental stiffness matrices, respectively. t+Δt R is the vector of externally applied nodal point loads at time t + Δt; t0 F is the vector of nodal point forces equivalent to the element stresses at time t; and u is the vector of increments in the nodal point displacements. The stiffness matrices and force vectors are obtained from the finite element evaluation as ! t T t t 0 (17.43) 0 BL 0 C 0 BL d V , 0 KL = 0V

! t 0KN L

=

! t+Δt

R=

NA 0A

T

0V

t T t t 0BN L 0S 0BN L

d 0V ,

(17.44)

! t+Δt s 0 f

d0A +

NT 0V

t+Δt b 0 f

d 0V ,

(17.45)

17 Viscoelastic Damage Behavior of Fiber Reinforced Nanoparticle . . .

! t

F=

0V

t Tt ˆ 0 BL 0 S

d 0V ,

399

(17.46)

where t0 B L and t0 B N L are the linear and nonlinear strain-displacement transformation matrices, 0 C is the incremental material property matrix, t0 S is a matrix of 2nd PiolaKirchhoff stresses, and t0 Sˆ is a vector of these stresses. All the matrices are defined at time t with respect to the configuration at time 0. The linearization presented in Eq. (17.40) introduce errors leading to solution instability. It is therefore necessary to iterate in each load step until the relative L 2 -norm of the residual in Eq. (17.38) is less than a given tolerance set as 10−5 . The equilibrium iterations corresponding to a modified Newton iteration in the TL formulation is t

0 KL

(i−1) + t0 K N L Δu(i) = t+Δt R − t+Δt . (i = 1, 2, 3 ldots) 0 F

(17.47)

Table 17.3 summarizes a step-by-step algorithm used to the system of non-linear equations presented in Eq. (17.47).

17.2.7 Results and Discussion As discussed earlier in Sects. 17.2 and 17.2.4, the material parameters associated with ˙ 0 and V ∗ ) the hyperelastic spring (i.e., μhe M ) and viscoelastic dashpot elements (i.e., ε are determined using MD simulations. Further to the atomistically measured parameters, those of the nonlinear elastic spring (i.e., μeM and λeM ), the damage variable (i.e., A) and the fiber parameters (i.e., a1 , a2 and a3 ) are obtained through a calibration process with experimental data. In this study, commercially available spray dried BNPs with a primary particle size of 14 nm (DISPERAL HP14, SASOL [48]) and GFs with an average diameter and length of 11.5 and 350 µm (FG 300 [49]) are used, respectively. The mass density of the neat epoxy, BNP and GF are respectively 1.2, 3.0 and 2.55 g/cc from which the BNP and GF volume fractions required in the constitutive model can be obtained. Nanoparticles have a tendency to undergo agglomeration. The aggregation of nanoparticles results in insufficient dispersal in the polymer matrix, degrading the material properties of the nanocomposites. To reduce the particle agglomeration by shear mixing process, BNPs are dispersed in the epoxy using a high-energy vacuum dissolver at an extremely high rotation speed of 5800 rpm. The dissolver provides high shear forces to break up the agglomerates in the liquid epoxy in vacuum. The GFs are dispersed in the epoxy with a moderate rotation speed in order to avoid breakage of the fibers. The nanocomposites are prepared by firstly dispersing the required amount of BNPs in the liquid epoxy and by subsequent introduction of the fibers in the BNP-filled epoxy compound. Afterwards, the compound is blended with the curing agents. Finally, the material is cast

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Table 17.3 Summary of step-by-step algorithm used for FE analysis 1. The value of tensors {t0 F,t0 Fe ,t0 Fv ,t+ Δt 0 F} is available at the beginning of each time increment 2. Calculate the trial elastic deformation gradient using Eq. (17.1) t+Δt Ftrial = t+Δt F t F−1 e 0 0 0 v 3. Perform the polar decomposition t+Δt Ftrial = t+Δt Vtrial t+Δt R trial e e e 0 0 0 4. Compute the trial elastic strain 

 (i) (i) (i) t+Δt Etrial = [i = 1]3 ln λtrial etrial ⊗ etrial e 0 (i) t+Δt Vtrial where λ(i) e 0 trial and etrial are the eigenvalues and eigenvectors of 5. Calculate the trail stress t+Δt Ttrial using Eq. (17.29) neq 6. Calculate the deviatoric part of t+Δt Ttrial neq v 7. Calculate the trial viscoelastic flow rate ε˙ trial using Eq. (17.33). trial from Eq. (17.32). 8. Calculate the trial viscoelastic stretching Dv 9. Update t+Δt 0 Fv using the backward Euler method and Eq. (17.34) t+Δt F = t F + Δt Dtrial t+Δt Ftrial i 0 0 i 0 i

−1 i t+Δt F where Δt is the time increment and t+Δt Fitrial = t+Δt Fetrial 0 0 0 " "t+Δt trial t+Δt " " F − F 10. If < tolerance then GOTO step 11 else GOTO step 2 v 0 0 i T using Eq. (17.28) 11. Calculate t+Δt eq 0 12. Calculate the damage variable at t + Δt using Eq. (17.35). t+Δt T 13. With t+Δt nqe and d obtained in previous steps calculate the total stress using 0 Tqe , 0 Eq. (17.27) t+Δt F and report the total stress t+Δt T 14. Store t+Δt i 0 Fe and 0 0 15. Compute of the tangent stiffness matrix using Eq. (17.41) 16. Solve the system of non-linear equations presented in Eq. (17.47) using the Newton-Raphson procedure The state of damage and the tangent stiffness matrix are updated at each Newton-Raphson iteration

in a preheated (80 ◦ C) casting tool (depending on the test method, a size of nearly DIN A4 with thicknesses between 2 and 5 mm) and cured for 4 h at 80 ◦ C for gelation and 4 h at 120 ◦ C for post-curing. The casting tool is made of stainless steel and to provide an easy release a water-based mold release system is used. GFs are in-plane randomly-oriented, which are by two families of  represented fibers with the initial fiber directions of a01 = 1 0 0 and a02 = 0 1 0 . In the following section, the constitutive model is first calibrated based on a comparison with tensile test results of neat epoxy, BNP/epoxy nanocomposites and GF/epoxy composites. The predictive capabilities of the calibrated model to predict the stressstrain response of GF reinforced BNP/epoxy nanocomposites is then validated by experimental data. Next, the model is implemented in the FEA and is validated by comparing the numerical and experimental results of a four-point bending test.

17 Viscoelastic Damage Behavior of Fiber Reinforced Nanoparticle . . .

401

Fig. 17.13 Experimental setup for tensile test of the nanocomposite specimens with extensometer to measure the elongation of the samples

17.2.7.1

Experimental Parameter Identification

The objective of this section is to identify the aforementioned material parameters using experimental data obtained from uniaxial tensile tests. Concerning the tensile tests, specimens with a thickness of 2.2 mm are produced and tested according to testing standard DIN EN ISO 527 2, using an extensometer to measure the elongation of samples (Fig. 17.13). The specifications of the specimens are highlighted in Fig. 17.14. The stress-strain curves are recorded with a test speed of 2 mm/min. To calibrate the material parameters, the stress-strain relationships of three types of specimens made of neat epoxy, BNP (20 wt%)/epoxy nanocomposites and GF (60 wt%)/epoxy composites obtained from the constitutive model is compared with those of experimental data. An object function of differences between model results and experimental data is numerically minimized using the single-objective genetic algorithm method available in the DAKOTA software package [1]. The experimentally identified parameters of the constitutive model are listed in Table 17.4. These parameters are obtained with ten experimental tests at engineering strain rate ε˙ = 2.90 × 10−4 s−1 and room temperature.

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Fig. 17.14 Planar dimensions of the specimens with a thickness of 2 mm used for tensile tests. All dimensions are in millimeters Table 17.4 Material parameters identified by experimental tests Parameter Equations Value μeM (MPa) λeM (MPa) A a1 a2 a3

(17.29) (17.29) (17.35) (17.35) (17.35) (17.35)

810 1100 200 9 1 1

Based on the material parameters presented in Tables 17.2 and 17.4, the stressstrain relationship can be obtained from equations presented in Sect. 17.2. Figure 17.15 presents the model predictions for neat epoxy, BNP (20 wt%)/epoxy nanocomposites and GF (60 wt%)/epoxy composites compared with experimental data. The mean values and standard deviations of the data calculated from ten uniaxial tensile tests are shown in the figure. The simulation results show that the proposed constitutive model is able to fairly predict the stress-strain behavior of nanoparticle and fiber reinforced epoxy. It should be noted that the parameters identification procedure adopted here aims to provide a unique set of material parameters for GF reinforced BNP/epoxy nanocomposites. The predictive capability of the constitutive model with the unique set of parameters is further evaluated in the next section.

17.2.7.2

Model Validation

The second investigation here considered deals with a series of uniaxial tensile tests of GF reinforced BNP/epoxy nanocomposites. It allows us to evaluate the model performance in predicting the stress-strain relationship of the nanocomposites in the simultaneous presence of GF and BNP reinforcements with the calibrated material parameters listed in Tables 17.2 and 17.4. Figure 17.16 shows the effect of GF and BNP weight fractions on the stress-strain behavior at ε˙ = 1.54 × 10−4 s−1 and room temperature under uniaxial tensile load-

17 Viscoelastic Damage Behavior of Fiber Reinforced Nanoparticle . . .

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80 70

Stress (MPa)

60 50 40 30

Epoxy (model) Epoxy (experiment) BNP(20 wt%)/epoxy (model) BNP(20 wt%)/epoxy (experiment) GF(60 wt%)/epoxy (model) GF(60 wt%)/epoxy (experiment)

20 10 0

0

0.01

0.02

0.03

0.04

0.05

0.06

Strain

Fig. 17.15 Stress-strain curves of reinforced epoxy resin at strain rate ε˙ = 1.54 × 10−4 s−1 and room temperature under uniaxial tensile loading. The mean values and standard deviations of the experimental data are obtained from ten uniaxial tests 80 70

Stress (MPa)

60 50 Epoxy (model)

40

Epoxy (experiment) BNP(20 wt%)/epoxy (model)

30

BNP(20 wt%)/epoxy (experiment) BNP(15 wt%)/GF(15 wt%)/epoxy (model) BNP(15 wt%)/GF(15 wt%)/epoxy (experiment)

20

BNP(10 wt%)/GF(30 wt%)/epoxy (model) BNP(10 wt%)/GF(30 wt%)/epoxy (experiment)

10

GF(60 wt%)/epoxy (model) GF(60 wt%)/epoxy (experiment)

0

0

0.01

0.02

0.03

0.04

0.05

0.06

Strain

Fig. 17.16 Effect of GF and BNP weight fraction on the stress-strain relationship of the nanocomposites at strain rate ε˙ = 1.54 × 10−4 s−1 and room temperature under uniaxial tensile loading. The mean values and standard deviations of the experimental data are obtained from ten uniaxial tests

ing. Both experimental and numerical results are presented in the figure. From the experimental data, the Young’s modulus increases from 2.95 to 4.10 GPa for neat epoxy and BNP (20 %wt)/epoxy nanocomposites, respectively. The Young’s modulus respectively increases to 4.75, 5.25 and 7.07 GPa by increasing the GF weight fraction from 15 to 30 and 60 wt%, indicating percentage increases of 61, 78 and

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B. Arash et al. 0.35

Softening variable d

0.3 0.25 0.2 0.15 0.1

5 · 10−2 0

0

0.01

0.02

0.03

0.04

0.05

0.06

Strain

Fig. 17.17 Evolution of the softening variable d with strain for the neat epoxy

140%, respectively. The results show that the ductile behavior of the nanocomposites significantly affected by the fibers such that the critical strain decreases from 0.06 for neat polymer to 0.01 for GF (60 %wt)/epoxy composite. From Fig. 17.16, an overall satisfactory level of accuracy between the proposed model and the experimental data can be observed, evidencing the practicability and representativeness of the developed constitutive model at different GF and BNP weight fractions. There are some discrepancies between the model prediction and experimental data, which might be caused by the unique set of material parameters and the free energy terms defined in Eqs. (17.24) and (17.25). Figure 17.17 shows the evolution of the damage variable d during this deformation course for the neat epoxy material, showing a monotonically increasing trend with strain.

17.2.7.3

Application of the Proposed Model

Finally, a four-point bending test of a beam made of GF reinforced BNP/epoxy nanocomposites is analyzed to examine the predictive capability of the developed constitutive model in a real application. The bending tests are performed according to DIN EN ISO 14125 at a constant speed of 2 mm/min and an initial load of 5 N, while load and displacement are recorded (see Fig. 17.18). The specifications of the specimens tested in the investigation are illustrated in Fig. 17.19. The width of the specimens is 10.1 mm. Due to the specimen symmetry, a FEA of half of the beam using symmetric boundary conditions would provide as complete a solution as that of the full model with less computational cost. The reduced model takes into account the symmetry at the mid-length of the specimen as illustrated in Fig. 17.19. Figure 17.20 shows the cor-

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405

Fig. 17.18 Experimental setup for four-point bending test P=2

P=2 12.0

(a) 2.2 21.0

38.0

21.0

P=2

6.0

(b) 2.2 21.0

19.0

Fig. 17.19 a Four-point bending test: geometry of the specimen and boundary conditions (top), b loading and boundary conditions imposed on half of the beam because of symmetry (bottom). The specimen thickness is 10.1 mm. All dimensions are in millimeters

responding finite element mesh with 1187 nodes and 1064 four-noded quadrilateral (Q4) elements. The mesh is refined toward the right side of the model because of high stress and strain concentrations in the area. The following simulations are performed under plane strain conditions and the load is applied via an imposed displacement at a constant rate of 2 mm/min.

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Fig. 17.20 Two-dimensional finite element model composed of 1064 Q4 elements with 1187 nodes 200

Epoxy (FEA) Epoxy (experiment) BNP(20 wt%)/epoxy (FEA) BNP(20 wt%)/epoxy (experiment) BNP(15 wt%)/GF(15 wt%)/epoxy (FEA)

150

BNP(15 wt%)/GF(15 wt%)/epoxy (experiment)

Load (N)

BNP(10 wt%)/GF(30 wt%)/epoxy (FEA) BNP(10 wt%)/GF(30 wt%)/epoxy (experiment) GF(60 wt%)/epoxy (FEA) GF(60 wt%)/epoxy (experiment)

100

50

0

0

1

2

3

4

5

Displacement (mm)

Fig. 17.21 Effect of the GF and BNP weight fraction on the force-displacement response in the four-point bending test of the nanocomposites

Figure 17.21 shows the numerical predictions of the four-point bending tests of samples made of neat epoxy and different combinations of the GF and BNP weight fraction. The numerical results are compared with experimental data. The mean values and standard deviations of the data calculated from ten four-point bending tests are presented in the figure. From the resulting force-displacement curves, it can be seen that the flexural resistance increases by incorporating the GF and BNP content in the epoxy matrix. The flexural modulus can be calculated from the linear portion of the curves by 23 F L 3 , determining the load F and its corresponding displacement d as E f = 108 wt 3 d where L, w and t are the loading span, the width and thickness of the specimen, respectively. The numerical results indicate that the flexural modulus increases from 3.61 GPa for neat epoxy to 5.32 GPa for BNP (20 wt%)/epoxy, showing a percentage increase of 47%. The flexural modulus further increases to 7.28 and 9.67 GPa by increasing the GF weight fraction to 15 and 60 wt%, respectively. The numerical results are consistent with experimental data. The resulting force-displacement curves clearly indicate a satisfactory level of agreement along the whole evolution, which again evidences the ability of the proposed constitutive model in predicting the rate-dependent behavior of GF reinforced BNP/epoxy nanocomposites. Finally, Fig. 17.22 shows the evolution of the damaged zone. Figure 17.22 correspond to imposed displacements of 3.0 and 4.0 mm, respectively. It is worth noting that the material softening is modeled using damage variables. Beyond the onset

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407

Fig. 17.22 Contour plots of damage (d) for 1064 Q4 elements and imposed displacement of 3 mm (top) and 4 mm (bottom)

of softening, strain localization happens in the classical continuum damage model, which leads to the loss of solution uniqueness [9, 19]. Consequently, the numerical solution obtained from the FEA will be mesh-dependent. To avoid the well-known problem, we only continue the finite element solutions before the onset of stress softening at Gauss points.

17.2.8 Summary and Conclusions A physically based constitutive model accounting for the nonlinear hyperelastic, time-dependent and softening behavior of GF reinforced BNP/epoxy nanocomposites at finite strain was proposed. The model adopts a composite-based hyperelastic model and a modulus enhancement model to take into account the effect of the fiber and nanoparticle weight fraction on the stress-strain relationship. The softening behavior was modeled by an internal variable, which is assumed to obey a saturation type evolution rule. A methodological framework based on MD simulations and experimental tests was developed to calibrate the proposed model. For this, MD simulations were performed to investigate the rate-independent equilibrium and rate-dependent viscoelastic material behavior of epoxy resins. Molecular simulations of epoxy resins under tensile loading at various strain rates allowed the identification of viscoelastic material parameters (i.e., ε˙ 0 and V ∗ ) required for the Eyring model. A MD simulationbased method was also presented to study the equilibrium response of epoxy resin subjected to shear loading. It enabled predicting the hyperelastic material parameters (i.e., μhe M ). Further to the atomistically predicted parameters, uniaxial tensile tests of GF and BNP reinforced epoxy were conducted to identify the elastic spring and softening and fiber parameters (i.e., μeM , λeM , A, a1 , a2 and a3 ).

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The predictive capability of the constitutive model with the unique set of parameters for different fiber and nanoparticle contents was evaluated. The comparison of simulation results of uniaxial tensile tests at room temperature with experimental data confirms that the model is able to adequately capture the overall material behavior of GF reinforced BNP/epoxy nanocomposites. The applicability of the model was demonstrated through its implementation in the FEA of four-point bending tests. Experimental-numerical validation showed a satisfactory level of accuracy along the loading evolutions. It is worth noting that the proposed constitutive model predicts the stress-strain relationships at room temperature. In the future, it would be interesting to study the effect of temperature on the material behavior of the nanocomposites. The potential development would provide a more comprehensive model to better understand the thermoviscoelastic behavior of the nanocomposites. Furthermore, more experimental tests than those reported in this study are still required to see if the present model can predict the hysteresis behavior under cyclic loading. Finally, the present modeling framework can be equipped with coarse-grained force fields [5, 34] to allow simulations of polymer systems at larger length and time scales. Acknowledgements This work originates from the research project ‘Hybrid laminates and nanoparticle reinforced materials for improved rotor blade structures’ (‘LENAH—Lebensdauererhöhung und Leichtbauoptimierung durch nanomodifizierte und hybride Werkstoffsysteme im Rotorblatt’), funded by the Federal Ministry of Education and Research of Germany. The authors wish to express their gratitude for the financial support. Copyright Notice This chapter is a reprint from a previous publication by Arash et al. [4] with permission from Elsevier.

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Chapter 18

Effect of Particle-Surface-Modification on the Failure Behavior of Epoxy/Boehmite CFRPs Maximilian Jux and Johannes Michael Sinapius

Abstract This chapter deals with the effect of particle content (up to 15 wt%) and particle surface modification on the mechanical properties (tensile properties and fracture toughness) of epoxy/boehmite composites. Furthermore, the failure behavior of epoxy/boehmite carbon fiber reinforced polymers (CFRPs) is investigated with compression after impact (CAI) tests at a constant fiber volume fraction of approximately 60 vol%. The CFRPs are fabricated by the Resin Transfer Moulding (RTM) method. To investigate the effect of particle-matrix-interaction, boehmite nanoparticles with different surface modifications (carboxylic acids and (3-aminopropyl)triethoxysilane) are used. The epoxy/boehmite masterbatches used for the preparation of the specimens are characterized concerning particle sizes and surface loadings to ensure a comparability of the test results. The used masterbatches possess nearly the same size distributions and surface loadings. In addition viscosity measurements showed that the processability of the modified resins is strongly affected by modifying the surface of the boehmite particles. The mechanical tests show that the examined mechanical properties of the epoxy/boehmite composites as well as the failure behavior of the epoxy/boehmite CFRPs are mainly influenced by the filler content. The effect of particle surface modification is visible in particular for tensile modulus of the epoxy/boehmite composites. In addition, it is found that there is only a minor effect of surface modification on the compression strength after impact.

18.1 Introduction The failure behavior of fiber reinforced polymers (FRP) strongly depends on type and direction of the loading. One of the most important modes of failure is delamination due to impact loads. Impact loads are considered critical, since delaminations M. Jux (B) · J. M. Sinapius Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected]; [email protected] J. M. Sinapius e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_18

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are non-visible damages that lead to a significant degradation of structural properties. Especially stiffness and strength of a FRP decrease due to delaminations [1, 31]. Impact loads are transversal loads, oriented perpendicular to the fiber direction. Therefore, the failure due to impact loads is significantly affected by the matrix. The property of FRPs to resist impact loads depends on the property to absorb and convert impact energy. According to Dorey [14, 15] no damage will form, if the impact energy is converted into elastic deformation energy. Furthermore, Dorey states that a high stiffness of the matrix leads to a high resistance to fiber buckling and damage growth due to compression loads. Evans and Masters [17] extend the work of Dorey and describe that the plastic deformation of the matrix is highly relevant, too. The plastic deformation of the matrix is fundamentally important after the initiation of a damage, to stop the damage growth by energy dissipation. The work of Williams and Rhodes [58] also concentrates on the effect of the matrix on the impact resistance of CFRP. In accordance with the aforementioned work Williams and Rhodes [58] conclude that FRPs are especially resistant to impact loads, if the matrix possess high stiffness, ultimate strength and fracture strain. Experimental results conducted by Hirschbuehler [22] and Sohi et al. [49] are in accordance with the aforementioned work, too. Based on their results William and Rhodes [58] propose that the matrix should have at least the following properties: a stiffness above 3.1 GPa, an ultimate strength higher than 69 MPa and a fracture strain of at least 4%. The fracture toughness of the matrix should also have a significant effect on the impact resistance of FRPs, since damages due to impact loads lead to delaminations between fiber and matrix, that are a form of matrix cracks. The experimental work of Davies and Moore [10] confirm this assumption. Their results show that an increased fracture toughness of the matrix leads to an increase of the compression strength after impact. The experimental results of Cartié and Irving [8] also demonstrate that an improved fracture toughness leads to an increased CAI value. Overall, excellent mechanical properties of the matrix like high stiffness, high strength and high fracture toughness are essential to obtain excellent impact properties. Thus, the modification of the matrix is a reasonable approach to improve impact properties of FRPs. A possibility to improve the mechanical properties of the matrix is the modification with nanoparticles. Especially anorganic nanoparticles possess the potential to simultaneously increase stiffness and fracture toughness of the matrix. The effect of nanoparticles on the mechanical properties of the matrix depends on different influencing factors. These factors are the particle size, the particle content and the particle-matrix-interaction. The effect of particle size on the elastic modulus of 2-phase-composites is estimated to be low [19, 30, 48, 60]. However, on the nanoscales it is observed that the elastic modulus increases with decreasing particle size [23, 27, 34]. In contrast to the effect of particle size on the elastic modulus of 2-phase-composites, the particle size has by far a greater effect on strength and fracture strain of 2-phase composites. The strength as well as the fracture strain of 2-phase composites significantly improve with decreasing particle sizes [6, 27, 60]. The effect of particle size on the fracture toughness is not uniformly described in literature. On the microscale it is often observed that the fracture toughness increases with increasing particle size

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[28, 37]. Other investigations show that the fracture toughness increases with decreasing particle size [27, 48]. The effect of particle content on the elastic modulus of the matrix is considerably higher compared to the effect of particle size. Numerous investigations show that the elastic modulus approximately increases in linear relation with increasing particle content [34, 51, 54]. The strength of the matrix also increases with increasing particle content, if the particles possess a high specific surface area and a strong particlematrix-bonding [9, 42]. In contrast to the effect of particle size on the elastic modulus of 2-phase-composites, the fracture strain decreases with increasing particle content [47, 56, 61]. The degradation of the fracture strain with increasing particle content depends on the particle size. Particles with higher sizes cause a stronger degradation of the fracture strain. Analog to the effect of particle content on the elastic modulus, the fracture toughness of the matrix approximately increases in linear relation with increasing particle content [2, 13, 25, 57]. It can be assumed that the effect of particle content on the elastic modulus is closely related to the effect of particle content on the fracture toughness. Concerning the effect of particle-matrix-interaction it is mainly reported that there is only a minimal effect on the elastic modulus [11, 12, 45]. However, there are also investigations showing that the elastic modulus as well as the strength of the matrix increase with increasing particle-matrix-bonding [38, 43, 55, 59]. Furthermore, it is reported that for thermosetting polymers there is only a marginal effect of particlematrix-interaction on the fracture toughness [35, 50].

18.2 Materials and Specimen Preparation To realize the experimental investigations, 2-phase-composites (particle/matrix) and 3-phase-composites (particle/fiber/matrix) are prepared and tested. A possibility to fabricate 3-phase-composites is to disperse particles in resin and use the masterbatches in Resin Transfer Moulding processes (RTM). Dispersing processes are time-consuming. Thus, preferentially matrix systems with at least two components (resin/hardener) should be used, to prevent a curing during dispersing. Furthermore, matrix systems used in RTM processes should possess the lowest possible viscosity to ensure a homogeneous fiber impregnation. However, it is widely known that modifying resins with particles increases viscosity [3, 18, 32, 56]. Therefore, the characterization of rheological properties is fundamentally important concerning the evaluation of the resin processability. Widely used matrix systems in RTM processes are epoxy systems. Characteristics of epoxy systems are high thermal and chemical resistance, as well as a high stiffness and strength [16, 33]. A hot curing matrix system consisting of three components (Araldite LY556/Aradur 917/Accelerator DY070) supplied by Huntsman Corporation is used for the fabrication of the composite specimens. The resin component is diglycidyl ether of bisphenol A (DGEBA) cured by anhydride molecules. The opening of the epoxy groups is initiated by the accelerator (imidazole molecules) at a temperature of 80 °C. In addition to the curing effect

414 Table 18.1 Mixing ratio Components Araldite LY 556 Aradur 917 Accelerator DY 070

M. Jux and J. M. Sinapius

Parts by weight

Parts by volume

100 90 0.5−2

100 86 0.6−2.4

a catalytic effect is attributed to the imidazole molecules [21, 39, 46]. Table 18.1 shows the mixing ratio of the used epoxy resin system. As filler material boehmite particles with a primary particle size of 14 nm are used (DISPERAL HP 14 supplied by Sasol Germany GmbH). Boehmite particles are commercially available in form of powder. The numerous hydroxyl groups on the surface of boehmite can be used to apply various surface modifications. Sasol Germany GmbH supplies boehmite particles with small primary particle sizes that are modified with carboxylic acids, sulfates or silanes. The works of Arlt [3] and Shahid et al. [46] confirm that the mechanical performance of CFRPs is significantly improved by modifying the matrix with boehmite nanoparticles. The properties of boehmite are described in detail by Fankhänel et al. in Chap. 6. For the fabrication of the CFRPs an unidirectional carbon fiber material is used (ECC Style 796 with a grammage of 270 g/m2 provided by Cramer). Table 18.2 summarizes the examined surface modifications. The bonding between surface modification and boehmite particle is realized by carboxyl groups for the carboxylic acids and by oxysilan groups for the APTES molecule. The carboxylic acids differ concerning chain-length and reactivity. Lactic acid and 12-hydroxylstearic acid possess additional hydroxyl groups (OH) compared to acetic acid and stearic acid. The additional hydroxyl groups should lead to a bonding between surface modification and resin molecules. In contrast, the non-reactive acetic acid and stearic acid should prevent a bonding between surface modification and resin molecules leading to a reduced particle-matrix interaction. The APTES molecules possess an amino group (NH2 ) that should also lead to a bonding between surface modification and resin molecules. The fabrication of the test specimens can be divided into 4 process steps. The first process step includes dispersing the boehmite particles in epoxy resin (Araldite LY 556). The dispersing process is realized in a pre-dispersing step via kneader (Sela LTK 3 R) and a post dispersing step via three-roll-mill (EXAKT 80 E). To keep interactions between particle and resin during dispersing as low as possible the dispersing is realized at low temperatures (20–40 ◦ C). The produced masterbatches possess a particle content of 30 wt%. In the second process step the masterbatches are diluted and mixed with epoxy resin (Araldite LY 556), anhydride hardener (Aradur 917) and accelerator (DY 070) with a centrifugal mixer (Hausschild SpeedMixer DAC 700.2 VAC-P). Subsequently the mixtures are carefully degassed. The third process step contains the curing of the mixtures. For the 2-phase-composites the diluted mixtures are poured into a pre-heated mould (60 °C) and subsequently cured

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Table 18.2 Investigated surface modifications Name Structural formula O Acetic acid

H3 C

OH O

H3 C

OH

OH

Lactic acid

O

Stearic acid

H3 C

OH

OH

H3 C

OH O

12-hydroxyl-stearic acid

O

(3-aminopropyl)triethoxysilane (APTES)

O

Si

NH2 O

for 4 h at 80 °C and for 4 h at 120 °C. For the 3-phase-composites the diluted mixtures are infused (RTM) into a layer structure (carbon fibers) and cured in an autoclave at a pressure of 3 bar for 4 h at 80 °C and for 4 h at 120 °C. The resulting fiber volume content amounts approximately 60 vol%. In the last process step the cured 2-phaseand 3-phase-plates are milled and sawed into test specimen geometry.

18.3 Characterization of Particle Sizes and Surface Loadings To ensure comparability of the test results, it is important that the masterbatches possess a similar particle size distribution. Furthermore, the modified boehmite particles should have comparable surface loadings. The measurements of the particle size distributions are realized with a disc centrifuge (CPS Instruments DC24000). The method is based on light scattering and takes advantage of the particles sedimentation behavior [36, 40]. To enable size measurements, the masterbatches

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distribution sum Q3 [%]

100

unmodified acetic acid lactic acid stearic acid APTES

80 60 40 20 0 1 10

102 particle size [nm]

103

Fig. 18.1 Particle size distributions of the masterbatches

are solved in methyl ethyl ketone (MEK). For a statistical coverage 3 measurements per masterbatch are realized. Frequently, microscopic methods like scanning electron microscopy (SEM) are also used to determine particle sizes [29, 44, 53]. However, these methods only give local impressions of the examined composite surfaces and usually concentrate on investigating primary particles. Therefore, in the framework of characterizing the particle sizes SEM-investigations only are additionally used to provide an impression of the particle distribution in the cured 2-phase-composites. Figure 18.1 illustrates the average curve progressions (distribution sums) determined from the results measured via disc centrifuge. The curves are relatively close to one another indicating that the masterbatches possess approximately the same particle size distributions. Table 18.3 summarizes the average particle sizes determined from the size distributions. The x10 -values are ranged between 57 and 68 nm. Due to standard deviations up to 6 nm the results are overlapping for the most part. The x50 -values are ranged between 79 and 98 nm and possess standard deviations up to 8 nm leading to large overlaps of the results, too. For the x90 -values the standard deviations are also overlapping for most of the particle types. Analog to the curve progressions showed in Fig. 18.1 the average particle sizes are nearly identical for the examined masterbatches. Due to these results it can be assumed that the effect of particle size on the mechanical properties of the 2-phase-composites and the failure behavior (CAI) of the 3-phase-composites can be neglected for the examined particle types. Since the primary particle size of the boehmite particles is 14 nm and all measured sizes are above this value, it should be noted that according to the measured sizes the boehmite particles seem to be agglomerated for the most part. Figure 18.2 shows representative SEM-pictures of cured 2-phase-composite fracture surfaces. In accordance to the size measurements conducted with the disc centrifuge, Fig. 18.2a shows that the boehmite particles form small agglomerates

18 Effect of Particle-Surface-Modification on the Failure . . . Table 18.3 Average particle sizes xi of the masterbatches Surface modification x10 (nm) x50 (nm) Unmodified Acetic acid Lactic acid Stearic acid APTES

56.5 ± 4.4 68.3 ± 1.1 66.6 ± 2.8 57.3 ± 2.5 60.5 ± 6.0

82.2 ± 5.1 97.8 ± 4.0 93.9 ± 4.1 87.6 ± 8.1 79.0 ± 6.1

417

x90 (nm) 216.8 ± 72.4 239.8 ± 33.7 281.4 ± 15.6 218.4 ± 56.1 160.7 ± 25.0

Fig. 18.2 SEM pictures of cured 2-phase-composites with a particle content of 10 wt%. a stearic acid modified boehmite, b lactic acid modified boehmite, c 12-hydroxyl-stearic acid modified boehmite, d magnification of the section marked in picture (c)

(see highlighted areas). Next to these small agglomerates for all examined composites, larger agglomerates with sizes of approximately 1–2 µm (see Fig. 18.2b) are present. However, only a small amount of these large agglomerates are found. Thus, it can be assumed that these large agglomerates only have a marginal effect on the mechanical properties of the 2-phase-composites and the failure behavior (CAI) of the 3-phase-composites. One exception is 12-hydroxyl-stearic acid modified boehmite. For the composites with 12-hydroxyl-stearic acid modified boehmite large areas (>1000 µm2 ) are found which possess a different structure compared to the rest of the composite (see Fig. 18.2c). In these areas plastic deformation is considerably higher indicated by a higher roughness of the fracture surface. Looking

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Table 18.4 Surface loadings δ of the modified boehmite particles (m mass difference (TGA), M molar mass, n amount of substance, S A specific surface area) Surface m (wt%) M (g/mol) n (mol) S A (m2 /g) δ (molecules/nm2 ) modification Acetic acid Lactic acid Stearic acid 12-hydroxylstearic acid APTES

0.32 1.38 2.93 2.46

60.05 90.08 284.48 300.48

5.38 × 10−5 15.35 × 10−5 10.31 × 10−5 8.18 × 10−5

175 178 168 168

0.185 0.519 0.369 0.293

1.78

58.10

30.58 × 10−5 139

1.325

at these areas at a higher magnification, it is revealed that the particle content is significantly higher compared to the rest of the composite (see Fig. 18.2d). An explanation is that these areas consist of not completely mixed masterbatch. Thus, due to missing hardener molecules the crosslink density is reduced, leading to a reduced elastic modulus and a higher plastic deformation. Therefore, the overall elastic modulus and strength of the cured composites with 12-stearic acid modified boehmite should also be reduced. A method to determine the surface loadings of modified particles is to measure the mass losses of the particles via thermogravimetric analysis (TGA). By calculating the difference of the measured mass losses of the modified particles and the mass loss of the unmodified particles the masses of the surface modifications are obtained. Knowing the molar mass of the surface modifications and the specific surface area of the particles, the surface loading δ can be calculated according to Vrancken et al. [52] by multiplying the amount of substance n with the Avogadro constant N A (=6,02214076 × 1023 mol−1) and dividing the product by the specific surface area S A . Specific surface areas of particles are determined according to the BET method [5]. The molar mass of APTES is equivalent to the molar mass of the C3 -carbon chain and the amino group. The silica content of the APTES molecule is not considered since the temperatures used for TGA-measurements (up to 950 °C) are not high enough to thermally decompose silica. For the other examined surface modifications the molar mass correspond to the molecules listed in Table 18.2. For the carboxylic acids the calculated surface loadings are in the same order of magnitude (see Table 18.4). In contrast, the surface loading of the APTES modified boehmite particles is at least 2.6-times higher compared to the other modified boehmite particles. The calculated values are comparable to values determined in the framework of other work [7, 20, 52].

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18.4 Effect of Particle-Surface-Modification on the Processability The processability of resin systems is defined by their viscosity and pot life (time to reach a viscosity of 0.5 Pas). To determine these properties, the modified epoxy resins are investigated with a rotational viscometer (Gemini, Malvern Instruments). The measurements (plate-to-plate method) are realized at constant temperature (80 °C) and constant shear rate (40 Hz) with a gap size of 1 mm and aluminum plates possessing a diameter of 40 mm. To eliminate the effect of temperature, the viscosity measurements are started when measuring device and resin reached the measuring temperature. From here on, abbreviated designations (see Table 18.5) are used to denote the test results. Figure 18.3 shows the results of the viscosity measurements depending on surface modification of boehmite particles for a particle content of 15 wt%. First of all, it has to be mentioned that the examined boehmite particles lead to an increase of viscosity and to a reduction of pot life (catalytic effect). Furthermore, resin systems with reactive particles (12-hydroxyl-stearic acid, lactic acid and unmodified boehmite) lead to a drop of initial viscosity due to the onset of shear loading. This effect (thixotropy) is in weakened form also present for resin systems with non-reactive particles (stearic acid and acetic acid). An explanation for thixotropic behavior is that weak bonds between particles and between particles and resin break up due to shear loading leading to a reduced viscosity [41]. By comparing the drop of initial viscosity of boehmite particles modified with reactive carboxylic acids with unmodified boehmite it can be seen that the thixotropic behavior is stronger for modified particles. Furthermore, thixotropy increases with increasing chain length for the reactive carboxylic acids as it can be seen by comparing 12-hydroxyl-stearic acid with lactic acid. In contrast, thixotropy decreases by modifying boehmite particles with non-reactive carboxylic acids (stearic acid and acetic acid). The thixotropy decreasing effect also depends on chain length indicated by a weaker thixotropic effect caused by stearic acid compared to acetic acid. The results suggest that by modifying boehmite particles with reactive carboxylic acids the particle-particle-interactions

Table 18.5 Definition of abbreviated designations Abbreviated designation Meaning Unnmodified Acetic acid Lactic acid Stearic acid 12-hydroxyl-stearic acid APTES Reference

Resin/composite with unmodified boehmite Resin/composite with acetic acid modified boehmite Resin/composite with lactic acid modified boehmite Resin/composite with stearic acid modified boehmite Resin/composite with 12-hydroxyl-stearic acid modified boehmite Resin/composite with APTES modified boehmite Resin/composite without boehmite particles

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viscosity [Pas]

1.5 1 0.5 0

0

5

10

15

20

unmodified lactic acid 12-hydroxyl-stearic acid reference

25

30 35 40 time [min]

45

50

55

60

65

70

acetic acid stearic acid APTES

Fig. 18.3 Viscosity-time-curves of the examined resin systems depending on the particle surface modification for a particle content of 15 wt%

and the particle-resin-interactions are increased, whereas by modifying boehmite particles with non-reactive carboxylic acids the particle-particle-interactions and the particle-resin-interactions are decreased. The effect of surface modification on the pot life can be described as follows: compared to unmodified boehmite the pot life is reduced due to modification with reactive carboxylic acids and increased due to modification with non-reactive carboxylic acids. The reduction as well as the increase of pot life is higher for longer chained surface modifications. Thus, it can be assumed that the catalytic effect of the boehmite particles increases for the reactive carboxylic acids and decreases for the non-reactive carboxylic acids. The catalytic effect is presumably caused by hydroxyl groups of boehmite, lactic acid and 12hydroxyl-stearic acid [4, 24]. The boehmite particles modified with APTES clearly show a different behavior. On the one hand there is no thixotropic effect caused by the APTES modified boehmite. On the other hand the resin with APTES modified boehmite possesses considerably higher pot life. Therefore, it seems that APTES strongly reduces particle-particle-interactions, particle-resin-interactions as well as the catalytic effect of boehmite, what is a strong argument against the assumption that the amino groups of APTES enable particle-matrix-bonding.

18.5 Effect of Particle-Surface-Modification on the Tensile Properties of the 2-Phase-Composites The tensile tests are performed according to test standard DIN EN ISO 527 with a constant test velocity of 1 mm/min. The specimens possess a waisted shape with a cross section of 10 × 2 mm2 in the measuring range. The tensile measuring is

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realized with a mechanical displacement transducer (MTS 776) with a measuring range of 25 mm. Figure 18.4 shows the effect of particle content and particle surface modification on the tensile modulus of the 2-phase-composites. The tensile modulus increases nearly linear with an increasing particle content for all examined composites. This is implied by correlation coefficients (according to Bravais-Pearson) with values between 0.992 and 1.000. Compared to the unfilled reference the tensile modulus increases by 217 to 857 Mpa. This corresponds to an increase of 7–26%. The effect of particle surface modification on the tensile modulus is considerably lower. At a particle content of 5 wt% there is nearly no difference between the examined composites. However, there are apparent tendencies at particle contents of 10 and 15 wt%. Compared to unmodified boehmite the long-chain modifiers stearic acid and 12-hydroxyl-stearic acid lead to a reduced tensile modulus. This can be explained by a reduced particle-matrix-interaction for stearic acid modified boehmite. In contrast to stearic acid modified boehmite, 12-hydroxyl-stearic acid modified boehmite leads to the formation of soft areas. These areas presumably consist of unmixed masterbatch and possess a reduced crosslink density (see Sect. 18.3) leading to an overall reduced tensile modulus. Comparing the results of stearic acid modified boehmite and 12-hydroxyl-stearic acid modified boehmite it can be seen, that the tensile modulus is more reduced by the 12-hydroxyl-stearic acid. This leads to the conclusion that the effect of stearic acid on the particle-matrix-interaction is weaker than the effect of 12-hydroxyl-stearic acid on the formation of areas with reduced crosslink density. The short-chain modifiers acetic acid and lactic acid only lead to a marginal reduction of the tensile modulus. In contrast the short chain modifier APTES leads to a clearly reduced tensile modulus. The reduction of the tensile modulus due to APTES is even higher compared to 12-hydroxyl-stearic acid and can be explained by a strongly reduced particle-matrix-interaction (see Sect. 18.4). Furthermore, the

tensile modulus [GPa]

4.5

reference unmodified acetic acid lactic acid stearic acid 12-hydroxylstearic acid APTES

4 3.5 3 2.5 2

0

5 10 particle content [wt%]

15

Fig. 18.4 Tensile modulus depending on the particle content and the particle surface modification

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ultimate strength [MPa]

100

reference unmodified acetic acid lactic acid stearic acid 12-hydroxylstearic acid APTES

90 80 70 60 50

0

5 10 particle content [wt%]

15

Fig. 18.5 Tensile strength depending on the particle content and the particle surface modification

stronger effect of APTES can be attributed to the higher surface loading of the APTES modified boehmite compared to the other boehmite particles (see Sect. 18.3). Figure 18.5 illustrates the effect of particle content and particle surface modification on the ultimate strength of the 2-phase-composites. For the composites with unmodified boehmite, the ultimate strength slightly increases up to 4% compared to the unfilled reference. For the other examined composites, the ultimate strength is approximately at the same level as the ultimate strength of the unfilled reference. The composite with APTES modified boehmite forms an exception as for this composite the ultimate strength decreases by 1.3% compared to the unfilled reference. Furthermore, it can be seen that the particle content has no significant effect on the ultimate strength. For the examined composites the change in ultimate strength due to a change in particle content does not exceed 1 MPa. The effect of particle surface modification on the ultimate strength is very low. Apart from the results for the composites with unmodified boehmite there is no significant change of the ultimate strength due to particle surface modification. The existing differences are in the same order of the standard deviations. Figure 18.6 shows the effect of particle content and particle surface modification on the fracture strain of the 2-phase-composites. Generally, the standard deviations are very high with values up to 30%. Thus, there is a large overlap of the results. Considering only the average values, it can be seen that the fracture strain of the examined composites decreases by 3–25% compared to the unfilled reference. Furthermore, for most of the examined composites the average fracture strains decrease with increasing particle content. The composites with 12-hydroxyl-stearic acid modified boehmite are an exception. For these composites the average fracture strains increase with increasing particle content. An explanation is that the number of areas with a reduced crosslink density that are formed due to the 12-hydroxyl-stearic acid

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7

reference unmodified acetic acid lactic acid stearic acid 12-hydroxylstearic acid APTES

fracture strain [%]

6 5 4 3 2

0

5 10 particle content [wt%]

15

Fig. 18.6 Fracture strain depending on the particle content and the particle surface modification

increase with increasing particle content and thus, the overall fracture strain increases, too. Concerning the effect of particle surface modification on the fracture strain no clear tendencies can be identified. At a particle content of 5 wt% the composites with unmodified and acetic acid modified boehmite possess the highest average values, whereas these composites possess the lowest average values at a particle content of 10 wt%. Furthermore, since the standard deviations lead to a large overlap of the results it can be stated, that there is no significant effect of the particle surface modification on the fracture strain.

18.6 Effect of Particle-Surface-Modification on the Fracture Toughness of the 2-Phase-Composites The fracture toughness of the 2-phase-composites is measured according to test standard ISO 13586 with compact tension specimens (CT-specimens) and a constant test velocity of 10 mm/min. Initial cracks are introduced using a sharp blade. To ensure a comparability of the test results, the initial cracks must have a similar crack length. For the evaluation of the test results only CT-specimens with an initial crack length of 16 to 19 mm are used. The fracture toughness of a material is conventionally described by the critical stress intensity factor and the critical energy release rate. The critical stress intensity factor describes the value when the crack under load starts to enlarge. The critical energy release rate is obtained by relating the critical stress intensity factor to the elastic modulus of the material. Figure 18.7 illustrates the effect of particle content and particle surface modification on the critical energy rate of the 2-phase-composites. In general, the critical energy rate increases nearly linear

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energy release rate GIC [J/m2 ]

300

reference unmodified acetic acid lactic acid stearic acid 12-hydroxylstearic acid APTES

250 200 150 100 50

0

5 10 particle content [wt%]

15

Fig. 18.7 Fracture toughness depending on the particle content and the particle surface modification

with increasing particle content for all examined composites. This is implied by correlation coefficients (according to Bravais-Pearson) with values between 0.983 and 1.000. Compared to the unfilled reference the critical energy release rate increases by 47–142 J/m2 . This corresponds to an increase of 43–130%. Thus, the effect of the boehmite particles on the fracture toughness is more than 4-times higher than the effect of the boehmite particles on the elastic modulus. In contrast to the effect of particle content the effect of particle surface modification is considerably lower. At a particle content of 5 wt% the standard deviations are overlapping almost completely. At a particle content of 10 wt% only the composite with APTES modified boehimte shows an increased energy release rate compared to the other composites. Since boehmite particles modified with APTES are only an addditional product and only small amounts of these particles were available, the investigations with APTES modified boehmite particles are not realized to the full extent. More detailed work concerning APTES modified boehmite particles will be addressed in future work. According to Spanoudakis and Young [50] the increased energy release rate can be explained by a reduced elastic modulus due to a reduced particle-matrix-interaction that also causes a reduction of the strength. However, this behavior is not existent for the boehmite particles modified with acetic acid and stearic acid. These surface modifications should also lead to a reduced particle-matrix-interaction. Actually, the average energy release rates are smallest for the composites with stearic acid modified boehmite next to the composites with 12-hydroxyl-stearic acid modified boehmite. This is most evident at a particle content of 15 wt%. However, it has to be mentioned that for a particle content of 15 wt% the standard deviations are overlapping almost completely, too. At a particle content of 15 wt% the result for the composite with lactic acid modified boehmite is most striking. For this composite the average critical energy release rate is up to 35% higher compared to the other

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composites. However, the tensile tests did not show any abnormalities relating to the increased critical energy release rate for the composite with lactic acid modified boehmite. Therefore, it can be assumed that this is a measuring artefact. Overall, it can be noted that for the examined composites the particle surface modification has no significant effect on the fracture toughness.

18.7 Effect of Particle-Surface-Modification on the Mechanical Properties of the 2-Phase-Composites Table 18.6 gives an overview on the effect of particle surface modification on the measured mechanical properties of the 2-phase-composites. The values listed in Table 18.6 are the percentage changes of the mechanical properties of the examined composites with surface modified particles referred to the mechanical properties of the 2-phase-composites with unmodified particles. It can clearly be seen that nearly all mechanical properties decrease for all examined surface modifications. Additionally, it has to be mentioned that the changes due to surface modification of the particles are only minimal and possess for most of the examined composites the same order as the standard deviations. This especially applies for the tensile modulus, the tensile strength and the fracture toughness. The percentage changes of the fracture strain are for some composites considerably higher. However, the standard deviations are also considerably higher. Furthermore, there are equally positive and negative percentage changes of the fracture strain. Thus, the effect of particle surface modification on the fracture strain cannot explicitly be defined.

18.8 Effect of Particle-Surface-Modification on the Compression Strength After Impact of the 3-Phase-Composites The investigations of the compression strength after impact are performed based on the model of airbus test method AITM 1.0010 with a constant test velocity of 0.5 mm/min. The test specimens possess a geometry of 150 × 100 × 4 mm3 and a symmetric layer structure ([90◦ /−45◦ /0◦ /+45◦ /+45◦ /0◦ /−45◦ /90◦ ]S ) with 16 layers. The impacts are realized at low velocities (lower than 5 m/s) with a drop weight tower (Zwick Amsler HIT600F). The pre-damaged specimens are tested with a test kit provided by Zwick/Roell that guides all four sides of a specimen to prevent buckling of specimens during testing. It has to be mentioned that not for all examined boehmite particles the impact properties are characterized for filler contents of 10 and 15 wt% due to the strong effect of particle surface modification on the processabiliy of the used epoxy resin system.

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Table 18.6 Minimum and maximum percentage change of the average mechanical properties of the epoxy/boehmite composites with surface modified particles referred to the average mechanical properties of the epoxy/boehmite composites with unmodified particles (AMB acetic acid modified boehmite, LMB lactic acid modified boehmite, SMB stearic acid modified boehmite, HSMB 12hydroxyl-stearic acid modified boehmite, APMB APTES modified boehmite) Average AMB LMB SMB HSMB APMB mechanical property Tensile modulus min Tensile modulus max Tensile strength min Tensile strength max Fracture strain min Fracture strain max Fracture toughness min Fracture toughness max

−2.20

−2.29

−3.39

−5.70

−6.22

−0.61

−0.55

−1.69

−1.61

−6.22

−5.56

−4.39

−5.15

−5.04

−5.81

−3.73

−3.07

−4.71

−4.28

−5.81

−5.46

−15.82

−3.71

−10.55

+10.04

+0.78

+10.71

+5.46

+21.14

+10.04

−2.60

−0.60

−8.23

−6.03

+8.04

−0.60

+8.66

−3.02

−2.99

+8.04

Figure 18.8 illustrates the effect of particle content and particle surface modification on the compression strength of pre-damaged specimens impacted with 30 J. The compression strength increases with increasing particle content for all 3-phasecomposites examined at different particle contents (acetic acid, lactic acid and stearic acid). In addition, for 3-phase-composites with stearic acid modified boehmite the compression strength approximately increases in linear relation with increasing particle content indicated by a correlation coefficient (according to Bravais-Pearson) of 0.99. Compared to the reference the compression strength increases up to 80 MPa. This corresponds to an increase of 50%. Concerning the effect of particle surface modification it has to be noted that only unmodified boehmite leads to a statistical relevant change. At a particle content of 5 wt% the average compression strength for the 3-phase-composite with unmodified boehmite is up to 23 MPa higher compared to the other examined 3-phase-composites. At a particle content of 10 wt% the 3-phase-composite with acetic acid modified boehmite shows slightly increased compression strength. However, this can be neglected since at a particle content of 5 wt% the 3-phase-composite with acetic acid modified boehmite possesses the lowest average compression strength. For the other examined surface modifications the standard deviations are overlapping and the average compression strengths are nearly

18 Effect of Particle-Surface-Modification on the Failure . . .

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Compression strength [MPa]

250

reference unmodified acetic acid lactic acid stearic acid 12-hydroxylstearic acid

200

150

100

50

0

5 10 particle content [wt%]

15

Fig. 18.8 Compression strength after impact depending on particle content and particle surface modification for specimens impacted with 30 J

identical. Nevertheless, the results are in accordance especially with the tensile testsof the 2-phase-composites, that have demonstrated that the composites with unmodified boehmite possess the highest tensile modulus and ultimate strength, whereas there is nearly no difference for the composites with carboxylic acid modified boehmite particles.

18.9 Summary The investigations presented in this chapter focus on the effect of particle surface modification on the mechanical properties of 2-phase-composites (epoxy/boehmite) and on the failure behavior of 3-phase-composites (epoxy/boehmite/carbon fiber). The experimental tests are conducted depending on particle mass fraction with boehmite particles modified with different carboxylic acids and APTES. Furthermore, the processability of the modified resins is characterized by viscosity measurements. The viscosity measurements show that the processability of the modified resins impairs if boehmite is modified with reactive carboxylic acids (12-hydroxylstearic acid and lactic acid) and improves if boehmite is modified with non-reactive carboxylic acids (stearic acid and acetic acid) and APTES. The mechanical tests show that the examined properties are primarily affected by the filler content. In particular, tensile modulus, fracture toughness and CAI increase approximately linear with increasing particle content. Compared to the effect of filler content, the effect of particle surface modification is considerably lower for the examined composites. For the 2-phase-composites the effect of surface modification is most visible on the tensile modulus. However, it has to be mentioned that the tensile modulus as well as ultimate

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strength, fracture strain and fracture toughness decrease if the boehmite particles are modified with the examined surface modifications. The effect of surface modification on the compression strength after impact is for the most part negligible. Analog to the mechanical properties of the 2-phase-composites the compression strength after impact is highest for the 3-phase-composites with unmodified boehmite. Copyright Notice This chapter is based on Jux’s PhD thesis [26].

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Chapter 19

Surface Quality of Carbon Fibre Reinforced Nanocomposites: Investigation and Evaluation of Processing Parameters Controlling the Fibre Print-Through Effect Thorsten Mahrholz and Johannes Michael Sinapius Abstract The chapter investigates the effect of the essential manufacturing parameters on the surface quality of uncoated carbon fibre reinforced composites (CFRP) used as car body panels with visible surfaces (Class A properties). A series of CFRP laminates maunfactured by the RTM technique are investigated varying in the fibre volume content (30–60%), reinforcement material (woven fabrics vs. unidirectional fibre reinforcements), curing temperatures (40–120 ◦ C), additives (SiO2 nanoparticles as matrix fillers) and using a surface finish applied as an in-mould coating. The laminate surfaces are characterized by roughness analysis (white-light interferometry) and wave-scan measurement to quantify the influence of the different manufacturing parameters on the surface quality. Especially the used resins are intensively characterized concerning thermal properties (CTE) and total resin shrinkage. These results correlate very well with the performed analysis of surface roughness. It is found that the fibre print through effect is significantly reduced by realising low total resin shrinkage and an even distribution of resin and fibres at the surface. Thus, using of unidirectional fibre reinforcement (no weft or sewing threads; very fine filaments), low curing temperatures (slow curing processes) and an in-mould coating are most successful for reduction of fibre print through effect and getting surface similar to Class A properties. In addition, the surface quality is quite positively affected by the application of nanoparticles and also strongly controlled by roughness of tooling.

T. Mahrholz (B) German Aerospace Center, Braunschweig, Germany e-mail: [email protected] J. M. Sinapius Technische Universität Braunschweig, Braunschweig, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 M. Sinapius and G. Ziegmann (eds.), Acting Principles of Nano-Scaled Matrix Additives for Composite Structures, Research Topics in Aerospace, https://doi.org/10.1007/978-3-030-68523-2_19

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19.1 Introduction Fibre-reinforced plastics (FRP) are widely used in aerospace engineering. The primary advantage is the significant reduction of the mass of the components in comparison with conventional materials such as steel or aluminium. Moreover higher rigidity and strength as well as better corrosion performance are additional benefits of polymer composites [4]. These properties have increased the importance of composite materials also in the automotive industry [3, 21]. Light-weight construction using FRP plays a decisive role in the further development of electromobility. The development of various manufacturing technologies, in particular Resin Transfer Moulding (RTM), economically enabled the application of composite materials in small and medium scale production in the automotive industry [1, 4]. Therefore FRP are more and more used for the manufacturing of car bodies to reduce the mass of the vehicle. In addition to the interior, a large part of the body (exterior) is visible to the vehicle user. In the automotive industry these visible surfaces are called Class A surfaces [2]. Moreover, surfaces having a defined very high optical quality standard are also rated as Class A. For example, fibre-reinforced composites have a Class A surface if their optical properties are identical to or better than those of comparable painted series panels [15]. Typical surface defects in the production of fibrereinforced plastics are due to impregnation faults (dry spots), porosities (degassing

Fig. 19.1 Schematic illustration of fibre print through effect at the surface of fibre-reinforced plastics (FRPs) due to matrix shrinkage [16] and images of various CFRP panel surfaces manufactured in this study showing different degrees of fibre print through (here marked by arrows; sample code Table 19.1): a reference surface with low quality due to wavy patterns (CA-1); b and c high quality surface without pattern and high gloss indicating Class A properties (CA-7). CTE: coefficient of thermal expansion

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problems, air inclusions) and so-called fibre print through effects [8, 15]. The fibre print through effect considered in this investigation results from the heterogeneous distribution of fibres and matrix in the surface, i.e. there are high-resin and low-resin (high fibre content) zones. Resin shrinkage (reaction and cooling shrinkage) in highresin zones causes the fibre architecture to be visible on the surface. The carbon fibre itself is dimensionally stable, with a low thermal expansion [19]. Primarily the more pronounced resin shrinkage processes cause clear fibre print through effects in the production of fibre-composite components using the resin transfer moulding method. High-resin spots between the reinforcing fibres sink in deeper, which cause the reinforcing fibres and their corresponding architectures respectively to print through to the surface of the components. Figure 19.1 illustrates the principles of fibre print through effect by sketch and images of CFRP plates manufactured in this study with low and high surface quality. This surface profiling is sensed as visually disturbing and hence impairs the appearance of the surface. Conventional painting processes amplify this effect. Presently most fibre-reinforced car body parts (GRP/CFRP) are manufactured using high-pressure RTM [4, 14]. Due to the high degree of automation and the short process times, this technology enables cost-efficient production with a high quality of the components. Manufacturing technologies using lower pressures like the Quickstep technology yield laminate surfaces of lower quality [7]. Surface films [19], gelcoats [11] and in-mould coatings [17], which are also used to produce Class A surfaces, are not suitable for series production due to cost reasons. In the favoured high-pressure RTM process, a preform is impregnated with resin inside a two-part mould at an injection pressure of 100 bar. The advantages of this method are the automated and short product cycles of just a few minutes and the production of the components within close tolerances. A special painting process follows to ensure the component surfaces meet the Class A requirements. For the compensation of the fibre print through, a functional coat (sacrificial coat) is applied before painting the component. This production concept yields excellent results. However, in particular the painting process is very cost-intensive and, moreover, requires relocation of the FRP component out of the overall production process of the vehicle (off-line production). In view of this expenditure for post-processing, the question is whether the production of composites can be optimised so far that post-treatment of visible CFRP parts by means of additional painting steps can be avoided. This approach would have an enormous cost saving potential. The aim of the research of this chapter is to study the influence of fundamental manufacturing parameters on the surface quality of non-painted CFRP laminates (socalled visible CFRP laminates). The objects of the investigation are the surface effects based on the fibre reinforcement and the epoxy resin. Therefore, CFRP laminates with varying fibre volume fractions and fibre reinforcement architectures (fabric and unidirectional (UD) reinforcement) are prepared using RTM technique. Moreover, the total shrinkage of the epoxy resins as the decisive influencing factor is investigated in detail. The effect of curing cycles and nanoparticles as filler materials is studied to reduce the resin shrinkage. In particular the use of nanoscale silicon dioxide

High fibre volume fraction Low fibre volume fraction Fabric type

Slow curing

Fast curing

Filler (25 wt% SiO2 )

Gelcoat on woven fabric; pre-curing Gelcoat on woven fabric; pre- and post-curing Gelcoat on uni-directional fabric; pre-curing Gelcoat on uni-directional fabric; pre- and post-curing

CA-1 (ref.)

CA-4

CA-5

CA-6

CA-7

6

6 6

Woven fabrica

Woven fabrica

Woven fabrica

Unidirectional fabric (UD)b

6

6

6

Woven fabrica

Unidirectional fabric (UD)b

6

Unidirectional fabric (UD)b Woven fabrica

6

3

Woven fabrica

Woven

6

No. Plies

fabrica

Fabric type

62

58

59

59

58

62

59

62

32

61

FVFc (%)

b Style

a Style

475-5T: satin weave 41 ; Tenax-J HTA40; weight: 298 mg2 797-1: UD; Tenax HTA5131; weight: 280 mg2 c Fibre volume fraction quantified by TGA measurement d Epoxy resin without nanoparticles: Araldite LY564:Aradur 22962 = 100:25 wt% e Epoxy resin with SiO nanoparticles: Araldite LY556:Aradur 22962 = 100:23 wt% 2 f LARIT carbon Protect HT

CA-10

CA-9

CA-8

CA-3

CA-2

Investigated parameter

Sample code

LARIT f

LARIT f

LARIT f

LARIT f

–

–

–

–

–

–

Gelcoat

Table 19.1 Overview of materials and manufacturing parameters used for the preparation of the CFRP plates

Araldite LY 564 Aradur 22962d

Araldite LY 564 Aradur 22962d Araldite LY 564 Aradur 22962d Araldite LY 564 Aradur 22962d Araldite LY 564 Aradur 22962d Araldite LY 564 Aradur 22962d Araldite LY 556 Aradur 22962 25 wt% SiO2 e Araldite LY 564 Aradur 22962d Araldite LY 564 Aradur 22962d Araldite LY 564 Aradur 22962d

Resin

12 h 40◦ C; 2 h 70◦ C; 2 h 130 ◦ C

12 h 40◦ C; 2 h 70◦ C; 2 h 130 ◦ C 12 h 40 ◦ C

12 h 40 ◦ C

15 min 120◦ C; 2 h 130 ◦C 2 h 70◦ C; 2 h 130 ◦ C

12 h 40◦ C; 2 h 130 ◦ C

2 h 70◦ C; 2 h 130 ◦ C

2 h 70◦ C; 2 h 130 ◦ C

2 h 70◦ C; 2 h 130 ◦ C

Curing

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particles promises a significant reduction of the resin shrinkage [13]. Additionally the rheological properties as well as the thermal and mechanical characteristics of the epoxy resin systems are determined. Moreover, the influence of gelcoats to improve the surface quality of CFRP laminates is investigated. Roughness and wave-scan measurements are performed for a qualitative and quantitative characterisation of the produced surfaces. Based on the results of the surface analysis, the manufacturing parameters investigated here are discussed and measures for improving the surface quality in the production process are proposed.

19.2 Materials and Methods 19.2.1 Characterisation and Testing of Epoxy Resins The epoxy resin Araldite LY 564 (diglycidyl ether of bisphenol A) with the aminic curing agent Aradur 22962 (isophorondiamine) supplied by Huntsman served as the reference resin system for the studies. The mixing ratio of epoxy resin (Araldite LY 564) and curing agent (Aradur 22962) was 100:25 wt%. The commercially available epoxy resin Nanopox F400 from Hanse-Chemie (Germany) has been used to investigate the nanoparticle-filled epoxy resin. Nanopox is a master batch consisting of the epoxy resin Araldite LY 556 (diglycidyl ether of bisphenol A; by Huntsman) and 40 wt% SiO2 nanoparticles (diameter: 8–50 nm; manufacturer’s specification) with a reactive surface modification on a silane basis. According to the manufacturer, the epoxy resins Araldite LY 556 and Araldite LY 564 are almost identical which is proven by the similar epoxy equivalents (LY564: 176 g/eq; LY556: 183 g/eq). However, the LY564 is slightly less viscous due to a reactive diluent. To adjust different filler contents (0 to 35 wt% SiO2 ), the master batch Nanopox F400 was diluted with the pure epoxy resin Araldite LY556 and then mixed with the curing agent Aradur 22962. The mixing ratio of epoxy resin (Araldite LY556) and curing agent (Aradur 22962) was 100:23 wt% in the investigation. The determination of the mechanical properties of nanocomposites and analogue pure resins served to determine the influence of filler content and curing conditions on the material properties. The test plates manufacturing comprised of the liquid resin mixing at 40 ◦ C and briefly degassing, then pouring into the preheated casting moulds and finally curing in defined curing cycles. Table 19.1 summarizes the curing parameters. The test specimens for the thermal and mechanical measurements were milled out of the test plates. The determination of the flow properties (dynamic viscosity, pot life), the total resin shrinkage, the glass transition temperature, the coefficients of thermal expansion (CTE), as well as the mechanical properties (tensile strength, tensile modulus, elongation at break) provided the necessary characteristics of the epoxy resin systems.

0 0 0 0 15 25 35

12 h 40 ◦ C; 2 h 130 ◦ C 2 h 70 ◦ C; 2 h 130 ◦ C 15 min 120 ◦ C; 2 h 130 ◦ C 2 h 70 ◦ C; 2 h 130 ◦ C 2 h 70 ◦ C; 2 h 130 ◦ C 2 h 70 ◦ C; 2 h 130 ◦ C 2 h 70 ◦ C; 2 h 130 ◦ C 1.3 2.3 3.3 2.3 1.8 1.2 0.6

Shrinkagec [%] 134 135 135 136 135 134 134

Tg d [◦ C]

b Epoxy

a Epoxy

resin without nanoparticles: Araldite LY564:Aradur 22962 = 100:25 wt% resin with SiO2 nanoparticles: Araldite LY556:Aradur 22962 = 100:23 wt% c Total resin shrinkage detected by a rotational viscometer d Glass transition temperature (T ) determined by DSC g e Coefficient of thermal expansion (CTE) at various temperature ranges

LY564-1a LY564-2a LY564-3a LY556-0b LY556-15b LY556-25b LY556-35b

Sample code SiO2 content Curing cycle [wt%]

Table 19.2 Thermo-mechanical parameters for the prepared epoxy resins

70.9 72.2 67.3 63.6 54.1 48.9 45.0

−6 [ 10K ]

αT Tg e

2693 2722 2719 2837 3455 3909 4590

E t [MPa]

Tension σt [MPa] 77.6 77.3 77.9 80 84 87 89

t [%] 6.3 7.1 7.6 7.2 6.7 6.0 4.0

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A rotational viscometer (Gemini, Malvern Instruments) determined dynamic viscosity, pot life and total resin shrinkage. The viscosity measurement (plate/plate method) was taken isothermally in rotation mode in accordance with DIN EN ISO 3521 with a shear rate 1 Hz and a plate-to-plate distance of 1 mm. The diameter of the parallel aluminium plates was 40 mm. The time to reach the empirical limiting viscosity number of 0.5 Pas defined the pot life for the applied RTM technology. The change of the plate gap measured without force in oscillation mode for a chosen curing cycle yielded total resin shrinkage. According to this method, total resin shrinkage is quantified only from the time at which the resin starts gelling [6]. The adhesion of the resin to the plates is sufficient only at that point and the plate gap changes as the resin shrinks. A correction of the plate gap at the end of the measurement, taking into account the thermal expansion coefficient of the plates, resulted in the total resin shrinkage as the sum of reaction and cooling shrinkage. The Differential Scanning Calorimetry (DSC 822e, Mettler-Toledo) provided the glass transition temperature at a selected heating rate of 10 K/min. and the thermomechanical analysis (TMA/DSTA841; Mettler-Toledo) the coefficients of thermal expansion (CTE) according to DIN 51909. The heating rate in the measured temperature range from −20 to 200 ◦ C was 3 K/min. To determine the mechanical properties, tensile tests based on DIN EN ISO 5272/1A/1 were taken using an electromechanical tensile testing machine (type Zwick 1476; 5 kN). The crosshead velocity was kept constant at 1 mm/min.

19.2.2 Preparation of Fibre-Reinforced Composites The laminates produced consisted of the epoxy resin Araldite LY 564 as the reference resin system and Nanopox F400 as the filled epoxy resin. In both cases Aradur 22962 served as aminic curing agent. The carbon fibre reinforcements were the fabrics Style 475-5T (satin weave 1/4, Tenax-J HTA40; weight: 298 g/m2 ) and the unidirectional fibre reinforcement Style 797-1 (predominantly UD; Tenax HTA 5131; weight: 280 g/m2 ), both provided by C. Cramer (Germany). The fabric Style 475-5T was used as reference material. The manufacturing process for the CFRP laminates was vacuum assisted RTM (VA-RTM). The dry fibre reinforcement was inserted into the cavity of a two-part RTM tool. The resin injection occurred after evacuation of the mould in a heated press (pressure: 60 bar). The resin systems themselves have each been injected using a pressure cartridge with a pressure of 6 bar at 40 ◦ C. Prior to this, all resin systems were mixed at 40 ◦ C and briefly degassed. Table 19.1 lists the well defined curing cycles used for the laminates. In order to achieve a good surface quality of the laminates, highly polished metal sheet inserts were used (type Polystar, Ra 10.1 nm; Rq 11.8 nm (own measurement), Thyssen Krupp, Germany). Silicone resin-based release agents made by ChemTrend (Howell, USA) served to impregnate the mould surfaces.

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Six single layers in size of 250 × 250 mm of the carbon fabric Style 475-5T or the unidirectional carbon fibre reinforcement Style 797-1 were used in each case to prepare the CFRP laminates. The plate thickness was about 1.6 mm, the fibre volume fraction was about 60 vol.%. The reference cycle consisted of two hours pre-curing at 70 ◦ C and two hours post-curing at 130 ◦ C. Starting from these reference parameters, five process parameters were varied and their influence on the surface quality was examined. These parameters are described below and detailed in Table 19.1: • Fibre volume fraction (FVF) A laminate with a FVF of 30 vol.% was produced to investigate the influence of the fibre volume fraction in comparison to the reference laminate with 60 vol.% • Carbon Fibre material Fabrics (Style 475-5T) and unidirectional fibre reinforcements (Style 797-1) were used to examine the influence of the fibre reinforcement architecture on the surface quality. • Curing temperature/curing cycle In addition to the reference curing cycle (2 h at 70 ◦ C and 2 h at 130 ◦ C), a slow curing cycle (12 h at 40 ◦ C and 2 h at 130 ◦ C) and a rapid curing cycle (15 min at 120 ◦ C and 2 h at 130 ◦ C) were investigated. The heating rate was 1 ◦ C/min in each case. • Nanoscale fillers Use of 25 wt.% of SiO2 nanoparticles as matrix fillers was investigated to reduce total resin shrinkage. • Gelcoating LARIT Carbon Protect HT provided by Lange+Ritter (Germany) was used for in-mould coating of the mould surfaces. The gelcoat (resin to curing agent = 100:50) was applied to the highly polished mould surfaces (type Polystar) in six runs using a paint spraying gun with a 1.5 mm nozzle without thinner. Following four hours of curing of the gelcoat at room temperature, the metal sheet insert was put into the RTM mould. After placing the fibre stack on the gelcoat plate the injection and curing process were carried out at a temperature of 40 ◦ C. As the glass transition temperature of the gelcoat is just 51 ◦ C, the laminates were cured using the slower curing cycle. 12 h pre-curing took place in the mould at 40 ◦ C. After that, the laminate was removed from the mould to check the surface quality. Two hours post-curing took place in the oven at 130 ◦ C.

19.2.3 Surface Characterisation The basis for the characterization of the manufactured laminate surfaces was DIN 4760, which classifies surface form deviations into form deviations, waviness and roughness. Roughness and waviness of the laminates are reported in this chapter in order to evaluate the formation of fibre print through and the associated surface profile in dependence on the manufacturing parameters examined. The characteristic roughness values Ra and Rq were determined using the optical measuring method of white-light interferometry. In addition, Abbott curves of the CFRP laminates were derived for a further analysis and an estimate of the profile

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shape. The white-light interferometer type WYKO NT 9100 made by Veeco (USA) was used for the measurements. The arithmetic mean of the measured values was calculated from three individual measurements of the surface. Moreover, the surface profile was determined in VSI mode (vertical scanning interferometry). The waviness character of the surfaces produced was determined using the wavescan measuring method of BYK Gardner (Germany). This measuring method allows for a fast and meaningful assessment of high-gloss to medium-gloss surfaces as to their waviness character and is established in the automotive industry for assessing the surface quality (Class A) of painted series-production panels [9]. In the process, the backscatter behaviour of a light source is looked at and the waviness of the surfaces is quantified in terms of a short-term and a long-term waviness range [9, 18]. The limit values of waviness for Class A surfaces are not precisely defined. However, waviness values based on experience exist which are minimum requirements for painted Class A surfaces (acc. to Grünberg [5]: Short wave: