Mechanical Characterization and Modeling of CF-SMC Materials / Author Martin Tiefenthaler, BSc

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Author Martin Tiefenthaler, BSc Submission Institute of Polymer Product Engineering Thesis Supervisor Univ-Prof. Dipl-Ing. Dr. Zolt´an Major

Assistant Thesis Supervi-sor Dipl.-Ing. Philipp Siegfried Stelzer September, 2020 JOHANNES KEPLER UNIVERSIT¨AT LINZ Altenbergerstraße 69

Mechanical

Characterization and

Modeling of CF-SMC

Materials

Master’s thesis

to confer the academic degree of

Diplom-Ingenieur

in the Master’s Program

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EIDESSTATTLICHE ERKLÄRUNG

Ich erkläre an Eides statt, dass ich die vorliegende Masterarbeit selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Masterarbeit ist mit dem elektronisch übermittelten Textdokument identisch.

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Acknowledgement

First of all, I wish to thank Univ.-Prof. Dipl.-Ing. Dr. mont. Zoltán Major, head of the Institute of Polymer Product Engineering at Johannes Kepler University in Linz, Austria, for giving me the opportunity to write this thesis and to work at his institute. I also owe particular thanks to my colleague Dipl.-Ing. Philipp Stelzer, for all his support and encouragement, for always being available for any questions, concerns or suggesti-ons.

Furthermore, I want to give thanks to my colleague Michael Lackner, for all the support and always having best advise regarding experimental tests. Also, I want to thank the Transfercenter für Kunststofftechnik GmbH (Wels, AUT) for support in performance of experimental tests. The project was funded by the Austrian Research Promotion Agen-cy (FFG) and the Climate and Energy Fund of the Austrian Federal Government under the program “Energieforschungsprogramm 3. Ausschreibung” with the grant number KR16VE0F13251.

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Abstract

Carbon fiber sheet molding compound (CF-SMC) composites are an attractive solution for several lightweight applications in the automotive and aircraft industries. The pro-duction process of randomly oriented platelets being compression molded in a low flow setup leads to a complex meso-structure with high scattering of mechanical properties. The objective of this thesis was the characterization and modeling of the mechanical behavior of CF-SMC materials. Experimental tests include a tensile test study with varying specimens dimensions in width and thickness, 3-point bending (3PB) tests, and short beam bending tests were carried out adopting coupon level specimens. Re-sults from these tests exhibited high scattering, which was considered to derive on the one hand from the stochastic meso-structure and on the other hand from specimen width (W < 50 mm) being below the length of platelets (L = 50 mm), which reduces the probability of having full size platelets in a coupon and induces an unpredictable randomization of platelet dimensions. To overcome this problem, subcomponent level 3PB and 4-point bending (4PB) tests were realized. Findings from these experiments exhibited significantly less scattering than tests performed at coupon-level.

Four numerical modelling approaches (quasi-isotropic, stochastic with random fiber orientations, stochastic with random representative volume elements (RRVEs), and sto-chastic with RRVEs and a Cohesive Zone Model (CZM)) were adopted to describe the mechanical behavior of CF-SMC structures using the software suite Abaqus. Four mate-rial models were developed by using experimental data directly in Abaqus or indirectly as input parameters for the software package Digimat to calculate material model pa-rameters. Virtual 3PB and 4PB tests were carried out utilizing 3D-solid and 2D-shell models of the subcomponent. The solid model was used to perform quasi-isotropic simu-lations adopting a Tsai-Wu failure criterion and progressive damage behavior. All other modeling approaches have been carried out using a shell model. A python script was written capable of (i) partitioning of the hat into RRVEs with a composite ply layup and a predefined material model; (ii) generating composite layups with random or uniform thickness values; (iii) assigning of random, or quasi-isotropic ply orientation values.

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as the stochastic, and stochastic with RRVEs modeling approaches over-predicted the CF-SMC strength values due to neglection of interlaminar effects. Therefore, a cohesive zone model was implemented to the model for consideration of the matrix dominated failure behavior of CF-SMC materials in terms of interlaminar effects, as platelet de-bonding and pullout. Stiffness and strength prediction was accomplished by using the stochastic model with RRVEs and CZM having random ply orientation values and a uniform ply thickness utilizing fracture toughness values from a previous work.

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Kurzfassung

Die vorliegende Arbeit beschäftigt sich mit der experimentelle Charakterisierung von Carbon Fiber Sheet Molding Compounds (CF-SMC) und der Entwicklung eines nu-merischen Simulationsmodells. CF-SMCs gehören zur Klasse der Faserverbundwerk-stoffe und stellen eine attraktive Lösung für verschiedenste Leichtbauanwendungen der Automobil- und Flugzeug-Industrie dar. CF-SMC Werkstoffe weisen eine kom-plexe Meso-Struktur auf, welche erzeugt wird durch ihren Produktionsprozess. CF-SMC Strukturen werden erzeugt durch zuschneiden von UD-Prepreg-Matten zu kleinen Plättchen mit festgelegten Dimensionen von 50 mm x 8 mm und anschließender zufäl-ligen Verteilung dieser Plättchen in einem Werkzeug, wo der Prozess mit einer finalen Aushärtung mittels Druck und Temperatur durch Fließpressen beendet wird. Die so-mit erzeugte heterogene Materialstruktur führt zu hohen Streuungen der mechanischen Eigenschaften. Eine Zugprüfungs-Studie wurde durchgeführt mittels des Testens von Prüfkörpern mit abweichenden Dicken und Breiten. Außerdem wurden 3-Punkt Biege-versuche (3PB) und Short-Beam BiegeBiege-versuche mit einfachen Prüfkörpern angewendet. Resultate von diesen Experimenten wiesen hohe Streuungen auf welche einerseits durch die stochastische Meso-Struktur des Materials und andererseits durch die Tatsache er-klärt wurden dass die Prüfkörperbreite (W < 50 mm) geringer war als die Länge der Plättchen (L = 50 mm). Dieser Umstand reduziert die Wahrscheinlichkeit vollständi-ge Plättchen in einem Prüfkörper zu haben, was zu einer unvorhersagbaren zufällivollständi-gen Verteilung der Plättchendimensionen führt. Um diesem Problem zu begegnen und re-präsentativere Ergebnisse zu erzielen wurden 3PB und 4-Punkt-Biegeversuche (4PB) mit CF-SMC Subkomponenten (Hut-Profil) statt konventionellen Prüfkörpern durch-geführt. Diese Tests führten zu Ergebnissen mit deutlich geringerer Streuung gegenüber den Resultaten von konventionellen Prüfkörpern.

Vier numerische Modellierungsstrategien (quasi-isotrop, stochastisch mit zufälligen Faserorientierungen, stochastisch mit Random Representative Volume Elements (RR-VEs), und stochastisch mit RRVEs und einem Cohesive Zone Model (CZM) wurden verwendet um das mechanische Verhalten von CF-SMC Strukturen mittels der finiten

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mit a) direkter Implementation von experimentellen Daten in Abaqus, oder b) indirek-ter Verwendung von empirisch ermittelten Daten als Eingangsdaten zur Berechnung von virtuellen Materialmodellen mittels des Softwarepakets Digimat erstellt. Compu-tersimulationen von 3PB und 4PB Subkomponenten Tests wurden mit 3D-Solid und 2D-Shell Modellen des Hut-Profils durchgeführt. Ein quasi-isotroper Modellierungsan-satz bestand aus einem Solid-Modell in Kombination mit dem Tsai-Wu Kriterium und Schädigung. Für alle anderen Modellierungs-Ansätze wurde das Shell-Modell des Hut-Profils verwendet. Zur Generierung der verschiedenen Shell-Modell wurde ein Python-Programm mit folgenden Funktionen entwickelt: (i) Partitionierung des Hut-Profils in RRVEs welche eine Laminatstruktur mit einem vordefinierten Materialmodell besitzen; (ii) Zuweisung von zufälligen oder gleichen Schichtdicken; (iii) Generierung von zufäl-ligen oder quasi-isotropen Orientierungen für die einzelnen Schichten. Quasi-Isotrope Simulationen lieferten Werte mit guter Übereinstimmung zu den experimentell ermit-telten Daten. Rein stochastische und stochastische Modelle mit RRVEs überbewerteten die mechanische Festigkeit von CF-SMC Materialien aufgrund der Vernachlässigung von interlaminaren Effekten. Deshalb wurde das stochastische Modell mit RRVEs durch die Berücksichtigung von interlaminaren Effekten in Form eines CZM erweitert, implemen-tiert mithilfe von Bruchfestigkeitswerten, welche in einer zuvor verfassten Arbeit für dasselbe CF-SMC Material ermittelt wurden. Dieses Simulationsmodell lieferte gute Vorhersagen von CF-SMC Steifigkeiten und CF-SMC Festigkeiten.

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1 Introduction and Objectives

CF-SMC materials are based on discontinuous and randomly oriented UD-prepreg platelets or fiber bundles dispersed in a resin. Stacked CF-SMC sheets are the com-pression molded to produce components. This results in a complex, stochastic micro-and meso-structure, causing several challenges (e.g. platelet waviness micro-and thickness, heterogeneous fiber distributions, ...) for virtual modeling of CF-SMC materials. How-ever, the main advantages of this material are its high mechanical properties and their insensitivity to holes and defects, its relatively low labor costs, and the enabling of UD-tape-scrap reutilization. Due to these advantages, CF-SMC composites are an at-tractive solution for several applications in the automotive and aircraft industries. The current thesis deals with the mechanical characterization and modeling of CF-SMC materials. A general methodology followed in this thesis is given in Fig.1.1.

Figure 1.1: Schematic of general methodology for mechanical characterization and mod-eling of CF-SMC materials

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material behavior. This was done by performing tests as 3-point bending (3PB), short beam bending and tensile tests utilizing coupon-level specimens. When trying to char-acterize the mechanical behavior of CF-SMC materials, many influences need to be considered, as they are dependent on platelet -orientation, -dimension, -distribution, and material defects. Therefore, 3PB and 4-point bending (4PB) tests were carried out using subcomponent-level specimens.

In the following step, four material models were developed with two models describing 2D-Shell and two models describing 3D-Solid material behavior, respectively. Tradi-tionally, experimental data is implemented directly as a material model in combination with literature values in a finite element software. This was done to create a 2D-Shell and a 3D-Solid material model. Furthermore, the software package Digimat was used for creation of virtual solid- and shell- material models by utilizing fiber and matrix properties as input parameters.

In the next step, four different modeling approaches: (i) quasi-isotropic (QI); (ii) stochastic with random fiber orientations; (iii) Stochastic with Random Representa-tive Volume Elements (RRVEs); and (iv) Stochastic with RRVEs and a Cohesive Zone Model (CZM), have been applied for prediction of the CF-SMC material behavior using the finite element analysis (FEM) software Abaqus. This modeling strategies were uti-lized in combination with the developed material models on virtual subcomponent-level tests. Therefore the subcomponent was discretized as 3D-Solid part and as 2D-Shell part, respectively. A python program was written, for generation of 2D-shell parts with the features needed for the modeling approaches. The validation of the simulation models was performed by comparison of the numerical results from virtual 3PB and 4PB subcomponent-level tests with the experimental data. The structure of the thesis is as follows. Chapter 2 gives an overview of CF-SMC material modeling approaches carried out by various authors. Experimental work performed in this thesis is presented in Chapter 3. CF-SMC material modeling and simulation approaches adopted for this thesis are presented in Chapter 4. Chapter 5 gives an overview of the obtained simula-tion results compared to experimental data for validasimula-tion. Finally, the overall findings of this work are given in Chapter 6.

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2 Literature review

2.1 Carbon Fiber Sheet Molding Compounds (CF-SMC)

Manufacturing method and the material system determine the performance of a com-posite part. Comcom-posites can be categorized into continuous and discontinuous fiber architectures. Industrially used conventional production processes for composite ap-plications are (i) injection molding for discontinuous short fiber reinforced polymers; (ii) draping of unidirectional (UD) or woven carbon fiber prepregs with subsequent curing for manufacturing of continuous composite laminates; (iii) wet layup process, where fibers are placed in a cavity, impregnated with a resin and subsequently cured. Compression molding and thermo-stamping are further used for curing of laminate layups. Injection molding enables time- and cost-efficient production of complex dis-continuous composite parts. However, it also limits the fiber length and weight content (∼ 40 wt.%). Furthermore, orientations are predetermined through the melt flow in the cavity. Continuous composite laminates are multilayer structures of bonded lay-ers, where individual plies consist of continuous fibers bonded together with a polymer matrix. Some drawbacks are the high raw material costs, the production of scraps through cutting of the preforms, and the high labor costs. Moreover, the draping pro-cess limits the complexity of produced parts to shell-like structures. A recent developed approach combines the processability of discontinuous composites with the mechanical properties of continuous composites. The materials are manufactured by compression molding of randomly oriented prepreg platelets, strands, tows, or bundles impregnated with a polymeric matrix. Parts produced this way exhibit fiber volume contents of 50-60 vol.%, which leads to high mechanical properties [1], [2]. Materials produced from platelets might be termed "high performance CF-SMC" [3]. Thus, mechanical properties of CF-SMC are determined by the bonding strength between the platelets and the amount of resin pockets [2], [7], [8]. However, the composite system exhibits a complex meso-structure with controlled fiber volume fractions and dimensions. The composite system exhibits several advantages: (i) the architecture of CF-SMC results in higher fiber volume contents and therefore in better mechanical properties [1]–[4];

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(ii) the CF-SMC production process enables automated fabrication of complex three-dimensional geometries, which leads to a significant reduction of labor costs since this could not be accomplished with conventional draping techniques using continuous fiber laminates [3], [4], [9], [10]; (iii) molding with prepreg platelets opens wide opportunities for the scrap reutilization, which plays a significant role in the reduction of costs and the environmental impact of composite manufacturing [11]–[13]; (iv) mechanical prop-erties of CF-SMC parts are relatively insensitive to holes and inserts as they tend to fail at the weakest section [4], [14]. Due to these advantages, CF-SMC composites are an attractive solution for several applications in the automotive industry. Tradition-ally, properties of continuous laminate parts were superior to CF-SMC parts, but in a recently published work, Alves et al. [15] presented a CF-SMC consisting of ultra thin, high-strength carbon fiber platelets with mechanical properties that are comparable to continuous composite laminates with quasi isotropic layup. CF-SMC failure is a matrix dominated event [2]–[6], as they predominately fail due to platelet debonding or pullout. The load transfer efficiency between a single fiber and the surrounding matrix is a function of fiber and matrix mechanical properties, fiber dimensions, fiber volume content, and the interfacial bonding strength. A critical fiber length is suggested by the Kelly-Tyson [16] model in Eq. 2.1,

Lc=

dσf

2τ (2.1)

where Lc represents the critical fiber length, σf is the ultimate fiber’s tensile strength,

d is the fiber’s diameter and τ is the shear strength of the matrix. The Kelly-Tyson model introduces Lc, where the fiber’s tensile strength is fully utilized. Fiber lengths

below Lcwould lead to weaker structures due to not using the full potential of the fiber.

Chang et al. [17] proposed that maximum fiber efficiency is achieved when fiber length is greater than 50 times the critical fiber length Lf 50 × Lc. For the definition of

CF-SMC failure, however, the critical aspect ratio is used for defining failure instead of Lc. The classical aspect ratio for single fiber reinforced composites, which is defined

as diameter l/d [18], is replaced by width l/w ([3]) or length-to-thickness l/t ratio ([3], [9]) for CF-SMC materials in order to consider the geometric differences between a single fiber and a platelet. If the measured aspect ratio is higher than the critical value, failure is determined through fiber fracture. Otherwise, the material will fail rather due to platelet pullout.

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The CF-SMC production process leads to discontinuous composite systems with: (i) randomly distributed fiber platelets resulting in fiber-rich regions and resin pockets ([2], [4]); (ii) out of plane platelet waviness due to shear and bending deformations during the low flow molding process [2], [8], [14], [19].

This results in a complex, stochastic micro- and meso-structure, causing several chal-lenges (e.g. platelet waviness and thickness, heterogeneous fiber distributions, ...) for virtual modeling of CF-SMC materials. In order to characterize their behavior, a wide range of experimental testing has been done by various authors, whose research findings will be discussed in the following paragraphs.

Feraboli et al. [3] carried out tensile, compressive and flexural tests using CF-SMC carbon-fiber/epoxy composite specimens. They found high variations of mechanical properties deriving from heterogeneous material structure. Influence of platelet aspect ratios l/w and l/t on strength values was higher than on elastic modulus. Furthermore, failure was shown to be a matrix dominated event with two prevalent failure modes of interlaminar platelet cracking and intralaminar platelet delamination. The authors of [3] concluded that module values are comparable to conventional CF composites, whereas strength values were found to be significantly lower (between 30-60%) with tensile strength being the lowest, followed by compressive and flexure strengths. Se-lezneva et al.[20] carried out tensile, compressive and shear tests with thermoplastic composites consisting of carbon fiber platelets embedded into a PEEK matrix. They found high scattering of mechanical properties caused by material heterogeneity and a dependence of mechanical performance on platelet length. Damage was located at the interface between platelet ends, caused by stress concentrations due to the discontinu-ous nature of the material. Failure was thought to derive from this damage initiation followed by platelet debonding and pullout. Nicoletto et al. [21] analyzed CF-SMCs and identified the occurrence of final failure rather due to platelet debonding perpen-dicular to load direction, than due to fiber fracture. Li et al. [5] examined the influence of platelet thickness on mechanical properties. Two different platelet thicknesses were used for specimen production and thicker platelets were found to have lower strength and modulus values. The strength reduction was attributed to the earlier onset of platelet pullout due to thick platelet configurations. Wan et al. [22] found an increase in modulus and strength with higher fiber volume content. Fluctuations of mechanical properties were considered to derive from variations of fiber volume content. The au-thors of [22] also observed a dependence of CF-SMC composite mechanical parameters on platelet thickness, as strength increases with the decline of platelet thickness.

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Compression molded CF-SMC specimens have shown that stiffness values are compa-rable to those of quasi-isotropic CF laminates [3], [20], hence making them an effective alternative to continuous fiber composites for stiffness-driven designs. Another notable feature of CF-SMCs is that their mechanical properties are fairly insensitive to notches, holes, inserts, or other initial material defects. Feraboli et al. [4] explored this behavior by comparing tensile and compression tests of notched and un-notched specimens. The results revealed that the macroscopic response of a CF-SMC composite specimen is notch-insensitive. According to the authors, a possible explanation for this behavior could be internal stress concentrations resulting from the meso-structure’s heteroge-neous nature. Boursier et al. [14] determined notch-insensitivity by testing specimens containing molded-in defects, visible damage from impacts, and incidental damages. The heterogeneous material structure of randomly dispersed platelets greatly delayed crack propagation in comparison to classical CF laminates. Furthermore, initial cracks created while testing did not correlate with final failure location. In previous research [6], Mode I and Mode II tests of CF-SMC composites were carried out. Unlike con-ventional CF composites, discontinuous crack propagation effects were observed. The failure behavior was assumed to be matrix dominated. A similar behavior was also reported by Guo et al. [23].

Another recently published article by Martulli et al. [24] depicts the effects of a high-flow induced morphology on mechanical response of CF-SMC composites. High-high-flow compression molding induces high distortion of platelets at outer surface regions and manufacturing defects as voids or cracks. The authors of [24] also found that the mate-rial was insensitive to manufacturing defects, whereas platelet orientation was identified as a key factor for stiffness and strength values. An anisotropic material behavior was detected with varying failure mechanisms for different specimen directions. The main tensile failure mechanism for 0° specimens was recognized as platelet pullout, failure, and splitting. In comparison, 45° and 90° specimens exhibited a matrix-dominated fail-ure. By high-flow compression molding, mechanical properties were slightly increased in longitudinal flow direction and significantly decreased for transverse directions.

As reported by [3], [20], CF-SMC composite materials exhibit modulus values com-parable to conventional CF composites, whereas strength values are 30-60% below. In a recently published work by [15], researchers successfully improved strength val-ues to high levels of conventional CF composites by utilizing ultra-thin tapes of high modulus carbon-fibers. Tensile tests were carried out using the modified material com-position, and results revealed a rise of stiffness and strength (> 100%) up to values

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these mechanical properties are enhanced stress transfer mechanisms enabled by thin platelets, which lead to fiber fracture failure mechanisms rather than fiber pullout. Alves et al. [15] suggested the production of thin wall structures with the proposed material as mechanical properties decline with increasing part thickness. As a con-sequence, they developed a high performance material offering more design flexibility than conventional CF-SMC materials.

2.2 CF-SMC Modeling Approaches

In general, three approaches were identified in the literature to model the complex be-havior of CF-SMC composites: (i) analytical; (ii) numerical; and (iii) explicit modeling of platelets. Several modeling strategies by various authors from these categories are summarized in the following sections of this chapter.

2.2.1 Analytical modeling

The main advantage of analytical models is the reduction of computation times com-pared to other approaches. Therefore, existing models of random oriented discontinuous fiber (ROF) composite materials have been further developed in order to fit the char-acteristics of CF-SMC materials. However, analytical models can be subdivided into two categories: (i) mean field homogenization models and (ii) laminate analogy models. Examples of these strategies are presented below.

2.2.1.1 Mean field homogenization models

Kirupanantham [25] described analytical methods in order to predict modulus and tensile strength of CF-SMC materials by utilizing mechanical properties of fiber, matrix, and interface as input parameters. He identified the major problem of the applicability of existing analytical ROF models to CF-SMC materials to be the assumption of a homogeneous fiber distribution. Therefore, he modified an existing ROF model by implementing material heterogeneities through the adoption of the Voigt and Reuss rule of mixture. For the modified model at the mesoscale, CF-SMCs were considered as randomly distributed and oriented fiber platelets. The platelets were approximated by a single large individual fibers embedded in a cylindrical matrix (see Fig. 2.1).

A shear lag model was used to predict platelet stiffness within a laminate whilst considering the effect of fiber length by implementing the aspect ratio l/d. Fiber volume fractions and length variations were incorporated by modifying the rule of mixture

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Figure 2.1: Simplified platelet model (reprinted from [25])

using a homogeneity efficiency factor derived from [26], [27]. For consideration of the random platelet orientations, a theoretical orientation factor of η0 = 0.375 (proposed

by Krenchel [28]) was used. Finally, a model developed by Kelly and Tyson [16] was adopted for strength prediction.

In a different work, Jain et al. [29] developed a model for stiffness calculation of CF-SMC using a mean field homogenization scheme proposed by Lielens [30]. Finite element analysis (FEA) models were built to generate dilute strain concentration con-ditions, which were used as input for the mean field homogenization scheme. The dilute strain concentration relates the strain inside the inclusion to the applied strain in the matrix. This was calculated by FEA using a model with a large cuboid containing a single inclusion. This cubic model was subjected to six uniaxial unit strains, and the average strain in the inclusion is calculated. Each FE calculation was used to populate a horizontal row of the dilute strain concentration matrix, leading to the 6×6 matrix. The homogenization model was interpreted as an interpolation of the Mori-Tanaka (MT) mean field theory and inverse MT formulations. The inverse MT formulation is a good approximation when the volume fraction of the inclusions is high, while the MT formulation is a good approximation for low inclusion contents.

2.2.1.2 Analytical laminate analogy models

Rasheed [31] developed a systematic approach to handle the non-deterministic nature of the CF-SMC, by combining the phenomenology of failure and concepts of statis-tics. A tortuosity-based approach was applied to handle the random overlaps and local meso-structure simultaneously, which was considered as the main influence on the failure. Furthermore, a Metropolis Monte-Carlo (MMC) model was adopted for the generation of a randomly overlapping meso-structure using 2D platelet dimensions (length×thickness) as input parameters. Since platelet thickness variations were

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ne-(a).

Figure 2.2: (a) Realizations of RVE with overlapping platelets using Metropolis Monte-Carlo algorithm; (b)computed values of shortest path in pixels, colors indi-cate distance from starting point (reprinted from [31]).

Matlab®’s geodesic distance transformation function was applied to calculate the shortest path of crack propagation through a virtual specimen, leading to outputs as presented in Fig. 2.2 (b). The difference between the geodesic length and the specimen’s thickness was set equal to the value of L as the geodesic path inherently includes the specimen thickness. Load transfer was assumed to be dependent on interface shear stress profiles. Consequentially, strength is controlled by platelet overlapping lengths (l0) [32], [33]. A critical overlapping length (Lc) was defined and used for identification

of the failure mode: (i) for overlap lengths of l0 < Lc, the platelets are not fully loaded

and a pullout mode of failure was expected; (ii) for l0 > Lca higher load is transferred to

the platelets resulting in interface failure. Rasheed [31] considered failure to be matrix dominated. Shear strengths were included in form of links between parallel platelets. When a local shear stress exceeded the interface strength, a crack was initiated. A tortuosity measure was defined to characterize the crack path based on the assumption of cracks avoiding reinforcements. A non-dimensional tortuosity value T is defined in Eq. 2.2.

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T = L

ts (2.2)

It includes information on random overlaps and local through thickness structure, where ts is the thickness of the specimen, and L is the total length of the crack, which

is the sum of the overlap lengths. Rasheed [31] assumed a constant overlap between two adherents and neglected matrix hardening effects. Subsequently, the effective stress σef f that a unit joint can withstand was calculated according to Eq. 2.3.

σef f = T τmy (2.3)

τ_my is the yield strength of the interface matrix, and σef f is the effective strength of a

cross section of the specimen with a tortuosity T . Using Eq. 2.3, a simple phenomeno-logical failure criterion for platelet reinforced materials is given by Eq. 2.4.

σi

σef f

≥1 (2.4)

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Nakashima et al. [34] proposed a model assuming random in-plane orientation of platelets. Overlapping regions (see (Fig. 2.3)) resulting in defects or resin-rich areas were ignored in the developed model.

Figure 2.3: Resin rich areas and defects due to overlaps (reprinted from [34]) Platelets were considered as square plies containing information about the size and orientation. An example transformation of a platelet into an equivalent square ply is shown in Fig. 2.4. The square plies were arranged over the defined specimen area without any overlap. The number of tapes in a specimen was kept constant as the area of an equivalent square ply is the same as a platelet. The orientation angle of a tape with respect to the global longitudinal axis, (θ), was defined as the square ply fiber orientation, while the square itself remained orthogonal to the global coordinate system (see Fig. 2.4).

Figure 2.4: Transformation of tape into equivalent square ply (reprinted from [34]) The equivalent square plies were combined to form the model specimen. As shown in Fig. 2.5, the combination was conducted along the z-axis (the obtained cube was termed as "cell"), y-axis, and x-axis. Stiffness matrices of each cell were calculated. Subsequently, the resulting stiffness of aligned cells in the y-axis direction was obtained, and finally, flexural stiffness and strength values for the virtual specimen were evaluated using the Classical Laminate Theory (CLT).

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Figure 2.5: Schematic diagram of the compounding process in the model (reprinted from [34])

Sato et al. [46] carried out a model for tensile test simulations of CF-SMC composites using the Monte Carlo Method with assuming the specimen thickness to be 10× of a single UD ply (see Fig. 2.5 (a). Furthermore, the plane direction was defined for the UD-tapes without gaps in between for approximation of UD-tapes to the squares of the same area (see Fig. 2.5 (b).

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Figure 2.6: Modeling methods of tensile test CF-SMC specimens for (a) width and (b) thickness, carried out by [46].

A random orientation value (θi) was assigned to each tape. Then the elastic

mod-ulus (Em) was calculated using the laminate theory for each square. These moduli

were combined by superposition principle to the modulus (E0

m) in width direction (see

Eq. 2.5). E_m0 = n X 1 aiEi (2.5)

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Subsequently, the modulus in longitudinal direction was determined using the super-position principle to get the elastic modulus of the measurement range (E).

Visewarth et al. [35] proposed a semi-empirical model that utilizes a stochastic platelet placement procedure and analytical expressions for the interlaminar shear and bending stress distributions for an arbitrary CF-SMC-hybrid laminate. The authors of [35] considered the consolidation pressure to cause reduction of platelet thickness. This reduction, dependent on the platelet flow and consolidation pressure, was set as semi-empirical parameter in the model. The platelet placement procedure started by dividing the plate thickness into n in-plane layers, which were subdivided into parti-tions. Subsequently, platelets with random orientations were generated and placed at random locations within these partitions on each in-plane layer (see Fig. 2.7 (a)).

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Figure 2.7: Top view of an in-plane layer (a) without platelet dilation; (b) with platelet dilation of 33 % (reprinted from [35]).

As the volume of the platelets was unchanged, the reduction in their thickness was compensated by the transverse-to-fiber dilation of platelets (see Fig. 2.7 (b)). The dilation was computed according to Eq. 2.6.

Wd= Wo

tn

tmean (2.6)

Wdand Woare the platelet width values of a dilated and original (undilated) platelets,

while tn and tmean stand for the nominal and mean platelet thickness values,

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regions. Outputs were generated in form of plates, visible in Fig. 2.7 (b). Specimens were selected from these plates to replicate real test coupon trimming from an actual plate. Subsequently, mechanical properties were calculated using CLT. Fig. 2.8 illus-trates the longitudinal stiffness scattering within a virtual specimen. The resin-rich areas in generated specimens account for 2 − 2.5 vol%.

Figure 2.8: Longitudinal stiffness of an CF-SMC specimen obtained using classical lam-inate theory on partition lamlam-inate (reprinted from [35]).

Selezneva et al. [36] proposed a purely analytical model for strength prediction of CF-SMC. The model is structured into three stages (see Fig. 2.9): (i) creation of a micro-structure consisting of randomly oriented platelets; (ii) creation of possible failure paths by considering platelet delamination and fracture; (iii) calculation of strength by finding the weakest path and summation of its strength components.

The proposed model generated a 2D micro-structure to represent the length-wise section of a tensile specimen, which can be seen in Fig. 2.10. The modeled cross-section included specimen dimensions, the platelet size and their off-axis orientation. Several micro- and macro-structural features were ignored: (i) specimen width; (ii) variability in through thickness number of platelets; (iii) platelet waviness; (iv) defects as resin rich areas or voids. As Selezneva et al. [36] only considered 2D cross sections neglecting the width, effective platelet lengths and thicknesses, which are dependent on orientation angles (θ), the cut locations (y) were defined for an accurate representation of a real specimen. A comparison between an actual and a simulated cross-section of an CF-SMC modeled micro-structure is given in Fig. 2.10 b).

Possible failure paths of the generated micro-structure were generated with the as-sumption that failure occurs as a result of combined platelet fracture and pullout ef-fects. The strength of each platelet was calculated based on its orientation and the surrounding plies by using the classical laminate theory and Hashin’s failure criteria [37]. The platelet layup – and therefore strength values – exhibited variations along specimen length. However, a single strength value was used for the determination of

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Figure 2.9: Overview of the modeling scheme (reprinted from [36]). platelet color illus-trates various fiber orientations.

Figure 2.10: Comparison between a) actual and b) modeled micro-structure (reprinted from [36]).

at several locations within a platelet, and a final estimation value was computed as the average mean of these values. This step is illustrated in Fig. 2.11, where the black dashed areas correspond to localized layups, and the red section shows a single platelet

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affected by these layups.

Figure 2.11: Effect of layup variability on strength of a particular platelet (reprinted from [36]).

Strength approximation between the adjacent platelets (interface) was added to the model according to Pimenta et al. [38]. Authors of [38] proposed a comprehensive modeling approach for composites with discontinuous stacked platelets, as shown in Fig. 2.12, by considering a strength-based and a toughness-based criterion for matrix failure.

Figure 2.12: (a) Unit cell for representation discontinuous staggered inclusions; (b) De-tail of (a); (c) Effect of overlap length on strength (reprinted from [36]). The strength based failure description is a modified version of the Kelly-Tyson [16] ex-pression, where Lo is the overlap length, and tb is the platelet thickness. The toughness

failure criterion is based on the assumption that the matrix fails due to the propaga-tion of a Mode-II crack initiated at the overlap end due to high strain concentrapropaga-tions. Eq. 2.8 shows the mathematical formulation of the toughness criterion, where Eb is the

platelet modulus approximated using CLT on a quasi isotropic carbon fiber laminate. tb represents the platelet thickness, and GIIc,IL is the matrix Mode-II interlaminar

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σc= Loτi tb (2.7) σc= s 2E bGIIc,IL tb (2.8) If platelets do not fracture, the combined effect of these two criteria on the strength of a composite is shown in Fig. 2.12 (c). The existence of a strength plateau caused by the toughness criterion helps to explain why the strength of CF-SMC composites is significantly lower than that of quasi isotropic carbon fiber laminates, even though platelets are sufficiently long to achieve high fiber length efficiency. To implement this concept into the model, the notion of limit length (Llim) was introduced. In the

proposed model, for short overlaps (i.e. Lo < Llim) the strength-based criterion was

used, and for long overlaps (i.e. Lo> Llim) the toughness-based criterion was used. In

the final stage of the model, failure strength of the structure was determined by finding the path of least-resistance for the crack to grow by using shortest path algorithms. The actual sequence of crack initiation and propagation was not modeled, and only the overall load required to break the cross-section was estimated. For finding the path of least-resistance, the micro-structure is represented by a set of nodes (i.e. platelet ends) and the links between them (i.e. overlap and platelet strengths). Selezneva et al. defined ultimate failure to emerge when a crack passes at least one platelet end while propagating through the whole specimen thickness (from side A to side B in Fig. 2.13). A micro-structure is shown in Fig. 2.13 with nodes (1-3) and the corresponding links. In this scenario, two failure paths are possible to occur, as the platelet can either be pulled out or break and debond. The developed model then calculates the shortest path of these two possibilities and converts it into laminate strength.

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Figure 2.13: Effect of layup variability on strength of a particular platelet (reprinted from [36]).

Li et al. [39] proposed multi-scale modeling approaches to predict the tensile strength and failure envelopes of CF-SMC composites by representing the actual composite as an equivalent ply-by-ply laminate. Several modeling approaches were considered for the different scales, including (i) a stochastic bi-linear shear-lag formulation account-ing for the random location of platelet-ends and matrix crackaccount-ing, (ii) a novel failure criterion for a discontinuous unidirectional ply accounting for the interaction between platelet pullout and transverse failure, and (iii) a ply-discount method or a maximum strain energy criterion for the final failure of the composite [39]. An equivalent ply-by-ply laminate was used to decouple the platelet effects of the discontinuous nature from their random orientations. Researchers adopted models from Pimenta and Robinson[40] and Henry and Pimenta[41]. These models were modified by replacing the individual bricks or fibers with platelets, and the matrix with a thin platelet-to-platelet interface. Within this model, plastic yielding and brittle fracture of the interfaces were incor-porated considering randomly located platelet ends. The shear-lag stresses, which are transferred along a platelets cross-sectional perimeter from its surrounding elements, build up tensile stresses (see Fig. 2.14). The equivalent characteristic thickness for the shear-lag model was defined according to Eq. 2.9.

tchar =

wt∗ tt

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This definition of the characteristic thickness effectively reduces the 3D problem shown in Fig. 2.14 to a 1D problem, regardless of the relative position and platelet surfaces (where shear stresses are transmitted).

Figure 2.14: Mesoscale C-SMC model of randomly oriented platelet ends. The image in the middle right position pictures a 3D representation, the upper and lower image a transformed 1D representation of the interactions between the middle platelet (orange) with its neighboring elements. The left image shows a cross-sectional laminate structure. The plotted length dimensions correspond to the overlapping lengths that define the interfacial strength (reprinted from Li et. al [39])

With the simplifications made, the stress-strain response of an RVE was computed using CLT assuming uniform remote strain fields. Unlike classical composite lami-nate structures, CF-SMC may fail under longitudinal tension by platelet pullout. Li et. al [39] considered this effect in an interactive tension-shear criterion including the longitudinal-tension versus the in-plane shear failure envelope. For any global load unit vector, failure of the equivalent laminate is defined by the global failure factor. For its calculation, [39] proposed two different methods: a ply-discount method and a strain-energy approach. In the Ply-Discount Method (PDM), the global proportional loading factor applied to the laminate was progressively increased until a ply failed. Subsequently, the local stiffness matrix of the damaged ply was reduced by the damage

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variable d. Although the PDM being physically-based, it required calculations of stress state and local loading factor for each ply of the equivalent laminate with updating of ply failures until global failure occurred. The second method assumed the strain energy required for the equivalent laminate to fail being equal to the sum of strain energies required for each ply to fail. This phenomenological method requires one time analyzing of strain energies and subsequent calculation of local failure, and therefore, computation time was significantly reduced.

Alves et al. [8] proposed a novel modeling approach consisting of a computationally-efficient micro-structure generator that is able to recreate the 3D features of platelet-based discontinuous composites (intrinsic waviness, fiber volume fraction, and thickness variations), coupled with an analytical stiffness model capable to quantify the effect of platelet waviness on the modulus of these materials. The micro-structure of CF-SMC specimens was generated including (i) the waviness due to nesting of the platelets; (ii) local fiber volume fraction variations; (iii) random and/or preferential in-plane orientations of the platelets. Therefore, a 3D specimen was defined with a pattern of grid points (see Fig. 2.15 a)), where each point contained information about local platelet count and orientations. Subsequently, platelets were randomly placed on the specimen according to a sequential adsorption algorithm. Orientations were sampled using a second order orientation tensor aij. Then, thickness values of platelets were

normalized (see Fig. 2.15 (b), and Fig. 2.16 (a)) at the grid points, and platelets crossing specimen borders were trimmed. The result was a specimen as shown in Fig. 2.15 (b). Next, the central tortuous path of each platelet was determined by discretization into segments of different levels (see Fig. 2.16 (b)) leading to raw waviness shown in Fig. 2.17 (a). Subsequently, the waviness was smoothed to obtain a more realistic wavy path, as shown in Fig. 2.17 (b).

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(a) (b)

Figure 2.15: Specimen generation: (a) platelet placement in a grid; (b) normalized platelet thickness variation and platelet waviness along the cross-section of a virtually generated specimen (reprinted from [8]).

(a) (b)

Figure 2.16: (a) Distribution of the platelets normalized thickness, with a mean value equal to tt_{; (b) A platelet i with its local coordinates (x}

1, y1),

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(a) (b)

Figure 2.17: (a) Raw waviness of a platelet; (b) Filtered waviness of a platelet (reprinted from [8]).

The developed stiffness model predicts the stiffness of CF-SMC in five steps: (i) generation of a virtual specimen; (ii) usage of a shear lag model proposed by [38] for determination of the equivalent longitudinal Young’s modulus neglecting platelet wavi-ness and derivation of the equivalent transverse elastic properties of a platelet from the Halpin-Tsai general expression [47]; (iii) prediction of the local equivalent stiffness ten-sor of each individual platelet including in-plane rotations; (iv) rotation of the platelet stiffness tensors to the global coordinate system; (v) calculation of the overall stiffness tensor by averaging the contribution of all platelets in the specimen.

2.2.2 Numerical Modeling

Although numerical modeling normally leads to higher computation times compared to analytical models, detailed results can be obtained over the whole specimen.

2.2.2.1 Numerical laminate analogy models

Harper et al. [42] developed a 2D FE model of randomly oriented 1D fibers embedded in a 2D matrix. A novel meshing method was applied for the definition of the boundary conditions of the interface between fibers and matrix. Planar 2D fiber architectures have been generated by a modified random sequential adsorption scheme, where fiber bundles were deposited over a region of interest (ROI), as shown in Fig. 2.18. Fiber orientations were created using a random number generator.

Fiber intersections were ignored and therefore, no limitation was imposed on the fiber volume fraction. A line clipping algorithm was used to trim the fiber bundles to the RVE

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Figure 2.18: Schematic of random sequential adsorption model in the as-deposited state (reprinted from [42]).

the model according to the specified fiber length. Platelets were sequentially deposited until the volume of fiber met the target volume fraction for the inner RVE (shown as a black box in Fig. 2.18). Abaqus [43] was used for simulating the 2D architecture using 1D beam elements with a circular cross-section consisting of fibers and matrix as representation of platelets. The tensile stiffness of each beam was calculated using a rule of mixtures. The matrix material was modeled using a regular array of 2D plane stress continuum elements. Beam elements are fixed to the solid ones using the *EMBEDDED ELEMENT technique, a type of multi-point constraint within Abaqus [43]. Embedding eliminated the need for a complex meshing algorithm to pair the coincident nodes of fibers to the resin. The novel meshing technique significantly reduced computation time of mesh generation without affecting Von Mises stress results as illustrated in Fig. 2.19.

Figure 2.19: Von Mises tensile stress comparison between a model using an unstruc-tured mesh with tie constraints (left) compared with strucunstruc-tured embedded elements (right) (reprinted from [42]).

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In order to further reduce computation times, a Saint-Venant’s principle [44] was adopted, whereby the considered RVE was embedded into a tertiary material and the effective material properties were extracted from the inner region (see Fig. 2.20). Two

Figure 2.20: Saint-Venant’s principle: application of the boundary conditions to a RVE using at homogeneous material by Harper et al. [42] (reprinted from [19]). different approaches have been investigated: (i) a homogeneous material to model the host material around the boundary and (ii) a heterogeneous material, which was a continuation of the material within the inner RVE. The critical decay length for the heterogeneous approach was found to be two times the fiber length in all cases. This resulted in much larger models, but the effective properties extracted from the inner region after convergence were considered to be more reliable than the homogeneous approach.

Feraboli et al. [1] analyzed CF-SMC in terms of tensile tests combined with a Digital Image Correlation (DIC) system. Full field strain measurements of specimens varying in width revealed highly non-uniform strain distributions. A classical discretization of the specimen to a RVE was not possible due to randomness of strain distributions caused by the heterogeneous structure CF-SMC composites. A novel discretization method was developed by filtering full field strain measurements for critical strain regions, which lead to repetitive strain patterns of high peaks and low valleys. The authors found that the critical strain regions exhibit a repetitive pattern in their randomness. The area of these isolated regions was measured and utilized for discretization of the specimen into Random Representative Volume Elements (RRVEs).

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(a) DIC captured image (b) Filtered critical strain regions

Figure 2.21: Results from DIC used for discretization by Feraboli et. al [1] The properties of the platelets constitute the upper and lower bounds respectively for the discontinuous material. Since measured moduli of CF-SMC were between these bounds, the probability of all platelets in one RRVE having the same direction was con-sidered minimal. This laminate analogy allowed the calculation of mechanical proper-ties using CLT. An algorithm was designed using a stochastic process to model the ran-dom distribution of sizes and orientations for the platelets of the physical plate, where needed input parameters were platelet and specimen dimensions. The randomization algorithm created a random generated platelet fraction with a random orientation angle θassigned. All platelet fractions having the same θ are summed up to the total platelet area A(θ), dependent on orientation state. This process was continued in a loop until the total volume of fractions reached the volume of the specimen. The thicknesses and orientations of the virtual laminate were used with mechanical properties of platelets for calculation of elastic laminate parameters. Feraboli et al. [1] analyzed tensile tests specimens using FEA in combination with the described algorithm. Therefore, the specimen was partitioned into 48 regions of 0.25 × 0.25 in according to the measured RRVE size.

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Figure 2.22: An example of a tensile specimen divided into several regions, with inde-pendently generated material properties (reprinted from [1]).

Subsequently, the randomization algorithm was applied to each RRVE assigning random thickness values to predefined orientation angles. Once the specimen is fully defined, a FEA was performed using ANSYS to get the macroscopic properties of the virtual tensile test specimens.

2.2.3 3D explicit modeling of platelets

Kravchenko et. al [11] considered the local stress transfer interactions between platelets to define the macroscopic response of the composite. The strength was shown to be dependent on platelet dimensions, overlapping, and orientations. Therefore, researchers characterized CF-SMC material with focus on the mesoscale by conducting tensile tests with specimens varying in thickness in combination with a DIC system to evaluate mechanical performance. Results of Young’s-Modulus and tensile strength exhibited high variations, therefore statistical analysis was performed on the data sets, which indicates an existence of a relationship between mechanical properties and the meso-structure. Digimat FE was used to create virtual CF-SMC specimens by combining geometry and mesh generation in a voxel-based process, where a defined specimen volume is discretized with a pattern of rectangular three-dimensional units (=voxels) of selected size. This pattern (=voxel mesh) is filled sequentially with platelets of defined size and orientation. An example output of this process is shown in Fig. 2.23.

The progressive failure analysis at the local platelet level was based on the stiff-ness reduction scheme and the smeared crack approach. The developed computational mesoscale approach considered individual platelets having a transversal isotropic mate-rial behavior with the properties of unidirectional prepreg tape. On the platelet level, progressive damage was mathematically described using continuum damage mechanics (CDM). Both initiation and propagation of damage were predicted with CDM without changing the original FE mesh of the system. Kravchenko et. al [11] assumed primary local mesoscale failures of specimens including platelet in-plane and out-of-plane

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fail-Figure 2.23: Digimat FE generated CF-SMC specimen using a voxel-based process (reprinted from Kravchenko et. al [11])

model was implemented using Abaqus in combination with a material model.

Pan et al. [45] assumed uniformly distributed fibers having an elliptical cross-section. Furthermore, these fibers were set to be either curved or straight, as shown in Fig. 2.24 b). A RVE was generated by stacking three fiber matrix systems as shown in Fig. 2.24, where a single fiber matrix system consisted of 3-matrix and 2-fiber sub-layers. Fiber-rich sub-layers were filled with randomly generated fibers. Curved fibers were generated when an added fiber intersected with other fibers; afterwards, the intersecting part of the new fiber was moved to the next layer (see Fig. 2.24 b)). If no intersection occurred when adding a new fiber, it was set straight.

(a) (b)

Figure 2.24: (a) Layer arrangement of the RVE; (b) Schematic illustration of a side view of intersecting fibers (reprinted from [45]).

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Fig. 2.25 illustrates a RVE generated with the described method. A subsequent FEA of the RVE delivered its macroscopic linear elastic properties. Using this model, Pan et al. [45] were able to create RVEs exhibiting realistic fiber volume content levels.

(a) (b)

Figure 2.25: Example of an RVE created by the model proposed by Pan et al. [45]: (a) RVE with curved fibers; (b) RVE fiber orientation distribution.

Sommer et al. [48] described an integrated methodology for analysis of stochastic CF-SMC to develop process-structure-property relationships. Flow-induced fiber ori-entation distributions were predicted using an anisotropic viscous constitutive model implemented in a nonlinear, explicit FE solver. A FEA with an explicitly modeled platelet meso-structure was developed, wherein the platelets were treated as a homo-geneous orthotropic medium, using continuum damage mechanics to model the intra-platelet failure and a cohesive zone model for interlaminar disbonding. A schematic flow chart of the developed model is illustrated in Fig. 2.26.

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Figure 2.26: Simulation framework for process-informed progressive failure analysis of CF-SMC (reprinted from [48]).

The flow simulation was performed using Abaqus Explicit to predict the final platelet orientation distribution function (ODF) from a given partial charge coverage by record-ing of changes from local platelet directions. Digimat FE was used for the generation of the preform. Platelets in the preform were represented by a set of voxels exhibiting the local orientation of the platelet as depicted in Fig. 2.27 (a), where each color repre-sents one platelet. The platelets itself were modeled as an incompressible, anisotropic viscous medium. Subsequently, the voxel mesh was converted to SPH particles (see Fig. 2.27 (b)). A slow flow simulation was carried out neglecting platelet to platelet interface interactions and therefore inhibiting elongation of platelets in the fiber di-rection [49]. An anisotropic viscous constitutive model of an incompressible platelet was used to simulate flow behavior. The model was defined transversely isotropic with three parameters: the suspending fluid viscosity, the anisotropy ratio, and an elastic bulk modulus. Platelets oriented with the flow direction were observed to translate, rather than tending to stretch. Researchers from [48] accounted the large viscosity of the material for this behavior. Platelets were found to stretch rather than translate when oriented transverse to flow direction. These platelets exhibit reduced thickness (see Fig. 2.27).

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(a)

(b) (c)

Figure 2.27: (a) Top down view of the flow front progression at different time points, (b) discretization of the voxelized domain with uniformly spaced SPH parti-cles, and (c) evolution of the in-plane fiber orientation distribution at four time points (reprinted from [48]).

A computational damage mechanics approach was utilized in combination with a cohesive zone model to analyze a virtual tensile test in terms of progressive failure behavior. A Monte-Carlo approach was used in order to consider the stochastic distri-butions of effective tensile properties derived from discontinuous platelet orientations. The platelets were modeled as homogeneous orthotropic material with the properties of the parent UD prepreg tape. The virtual progressive failure analysis accounted for three local meso-structural damage mechanisms: (i) matrix fracture, (ii) fiber fracture, and (iii) interfacial disbonding. The smeared-crack continuum damage model approach, where internal damage variables were used to degrade the stiffness matrix components and for identification of locally failed regions ("smeared cracks"), was adopted from

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material.

Salmi [51] discussed a voxel method combined with a stacking algorithm for generat-ing RVEs of CF-SMC composites implemented in Digimat FE. Usgenerat-ing Digimat FE with the orientation tensor for each relevant location (obtained through X-ray computed tomographic scans), RVE was generated, and homogenized properties were computed by means of full field homogenization. Platelets and platelet interfaces were modeled using solid and cohesive elements, respectively. Platelet waviness and resin-rich zones at the platelet ends were accounted [19].

Kilic [52], [53] developed a global–local nonlinear modeling approach for CF-SMC composites at the part level. At the local level, the micro-mechanical unit-cell model generated an effective non-linear response from average responses of two UD layers with axial and transversal fiber orientations. In-plane isotropy assumptions and a weighted average of a matrix-mode layer and fiber-mode layer were used. At the global level, the developed micro-mechanical model was implemented into a 3D FE framework using Abaqus [43]. A 3D Tsai-Wu failure criterion was used for predicting the damage initiation, and the properties of the fiber unit-cells were set to zero while allowing the elastic and non-linear properties of the matrix unit cells to degrade gradually and to represent the matrix determining failure initiation and propagation.

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3 Experimental Work

For the investigation of the mechanical properties of CF-SMC, several experimental tests were carried out on the coupon and subcomponent level. An overview of the conducted mechanical tests is given in Tab. 3.1.

Table 3.1: Performed experimental testing

Coupon level tests Subcomponent level tests

Tensile 3PB

3-point bending (3PB) 4-point bending (4PB) Short beam

In the following sections, the specimen production, the test setups with measurement systems and the obtained results will be discussed in detail.

3.1 Methodology

All experiments were performed on a servo-hydraulic MTS 852 damper test system of MTS Systems Corporation (Eden Prairie, MN, USA). Selected samples were tested in combination with the Aramis 4M Camera System of GOM GmbH (Braunschweig, Germany), applying the Digital Image Correlation (DIC) method in order to get full field surface strain distributions.

3.1.1 Material

The HexMC®_{-i-2000 commercial prepreg platelet based CF-SMC from Hexcel}

Compos-ites GmbH & Co KG (Neumarkt, Austria) was used for the experimental investigations. The material system consists of randomly oriented, stacked prepreg platelets fabricated from a continuous unidirectional (UD) tape with nominal dimensions of 50 mm×8 mm. The UD tape is made of continuous high strength carbon fibers combined with Hexcel’s HexPly® _{M77 resin system resulting in a fiber volume content of 57 vol% and a density}

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3.1.2 Coupon level tests

The fabrication of a CF-SMC plate, schematically pictured in Fig. 3.1 (a)-(d), starts by slitting and cutting a UD tape into platelets, which are randomly distributed and pressed to form the CF-SMC mats. A CF-SMC plate is produced by (i) cutting the mats into blanks with the desired dimensions; (ii) stacking the blanks into a preform; (iii) placing the preform in a cavity with initially covering 80%; (v) curing with a low flow compression molding step at 150°C for 2 minutes.

(a) (b)

(c) (d)

Figure 3.1: CF-SMC: (a)UD tape platelet, (b) mats of randomly distributed platelets, (c) stacked blanks, (d) finished CF-SMC plate

The plates used for experimental tests were provided by Hexcel (Hexcel Composites GmbH & Co KG, Industriegelaende 2, Neumarkt A-4720, Austria). The thickness of cured parts is controlled by the number of stacked mats in the cavity (schematically shown in Fig. 3.1 (c)). Plates with different thicknesses were analyzed.

A tensile test study has been carried out according to ASTM D-3039 in order to inves-tigate the influence of coupon sizes on the mechanical properties of CF-SMC composites. Furthermore, 3PB and short beam bending tests were carried out for the characteriza-tion of the bending behavior and interlaminar shear strength. For the identificacharacteriza-tion of width and thickness influences, several coupons with varying dimensions according to Tab. 3.2 were produced by milling from CF-SMC plates with dimensions of 350 mm ×

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350 mm × 2.1 mm see Fig. 3.2 (a).

Table 3.2: Dimensions of tensile test coupons Width variations Thickness variations l (mm) × w (mm) × t (mm) l (mm) × w (mm) × t (mm) 200 × 15 × 2.1 200 × 30 × 2.1 200 × 20 × 2.1 200 × 30 × 4.3 200 × 25 × 2.1 200 × 30 × 7.8 200 × 30 × 2.1 200 × 40 × 2.1 200 × 50 × 2.1

The tensile specimens were modified using aluminum tabs (25 mm × Width × 2 mm) glued to the grip area in order to prevent failure at the clamping region. Furthermore, one face of each specimen was covered with a random speckle pattern using white and black spray paint (see Fig. 3.2 (c)) to measure surface strains with the DIC sys-tem. Monotonic tensile tests were carried out at room temperature (+ − 23◦_C_{) with 1}

mm/min actuator speed.

(a) (b) (c)

Figure 3.2: Tensile test specimens: (a) Pattern of milled tensile test specimens (200 mm ×30 mm × 2.1 mm) from a CF-SMC plate; (b) Milled, unprepared tensile test specimen; (c) Tensile test specimen with aluminum tabs and speckle

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The strain was calculated as the average strain over the full surface for each frame cap-tured by Aramis. The tensile strength was recorded at specimen failure. The Young’s Modulus was calculated as the slope of the stress strain curve within the strain range of 0.05 − 0.25 %.

Monotonic 3PB tests were carried out according to ASTM D790 with coupon dimen-sions of 128 mm ×30 mm × 7.8 mm. Coupons were milled from 0° (3 specimens) and 90° (2 specimens) positions of CF-SMC plates (see Fig. 3.3 (a)). The load was applied with small steel rolls (d = 2.3 mm), as pictured in Fig. 3.3 (b). Five experiments were performed at room temperature (23◦_C_{+ −1}◦_C_{) with an actuator speed of 1mm/min.}

Tests were stopped as soon as a displacement of 15 mm was reached.

(a) (b)

Figure 3.3: 3PB tests: (a) Milling pattern of 3PB coupons; (b) Setup of 3PB tests The flexural stress was calculated via Eq. 3.1.

σf lex = 3F L

2bd2 (3.1)

f lex = 6dD

L2 (3.2)

F is the applied load, L, b and d are specimen length, width and thickness, respec-tively. Flexural strain values were evaluated according to Eq. 3.2, where D is the axial displacement, L and d are the specimen length and thickness, respectively.

Interlaminar shear strength (ILSS) values of CF-SMC were obtained by adopting short beam bending tests according to ASTM-D790. Eight specimens with dimensions

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of 80 mm × 40 mm × 7.8 mm were milled from a CF-SMC plate using a diamond cutter with 4 specimens from the 0° and 90° positions, respectively. The load was applied monotonically with small steel rolls (diameter d = 2.3 mm), as depicted in Fig. 3.4. Experiments were performed displacement-controlled and stopped at 5 mm axial displacement with an actuator speed of 1 mm/min. ILSS was calculated according to Eq. 3.3.

Figure 3.4: Setup of short beam shear tests

τILSS = _4bd3F (3.3)

F is the applied load, b and d are the specimen width and thickness, respectively.

3.1.3 Subcomponent level tests

In addition to the coupon level tests, 3PB and 4PB tests were carried out using subcomponent-level specimens. The subcomponent geometry was an U-shaped pro-file termed "hat propro-file" in this work.

The hat profile production process starts by cutting out three rectangular CF-SMC blanks with dimensions of 360 × 145 mm, ensuring about 80 % initial cavity coverage. The next step is to stack the blanks by pulling off the protective foils (see Fig. 3.6 (a)) and pressing the uncovered sides together. The resulting stack is preformed into a U-shape (see Fig. 3.6 (b)-(d)), remaining protective foils are removed (see Fig. 3.6 (f)), and the preform is placed in the cavity.

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Figure 3.5: Geometry and dimensions (in mm) of the CF-SMC hat profiles

(a) (b) (c)

(d) (e) (f)

Figure 3.6: Preparation steps for a hat profile before compression molding: (a) CF-SMC blank tor preforming and manufactured hat profile for comparison; (b) Preforming; (c) Preforming; (d) Preformed specimen; (e) Preformed specimen; (f) Preformed specimen without protective foils, ready for the final molding and curing step

compression-molding step is performed at a temperature of 130◦_C _{and a clamping force}

of 700 kN, to fill the cavity and cure the component.

3PB and 4PB hat profile tests were carried out utilizing a special test setup, as shown in Fig. 3.7 (a)-(b). In each test procedure, 3PB (see Fig.3.7 (a)-(b)) and 4PB (see Fig.3.7 (c)), a set of eight hat profiles was tested monotonically at an actuator

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speed of 1 mm/min and at room temperature (23◦_C_{+ −1}◦_C_{). Tests were stopped as}

soon as the displacement limit of 20 mm was reached or when a load drop of >75% occurred.

(a) (b)

(c)

Figure 3.7: Hat profile test setups: (a) 3PB front view; (b) 3PB side view; (c) 4PB front view

A special test setup was carried out for the 3PB and 4PB subcomponent tests (see Fig. 3.7 (a)-(b)) in order to center the hat profile. This enables an exact load application and therefore leads to a minimization of unwanted shear forces due to uncontrolled twisting. The centering was done for the profile width direction using four screws (see Fig. 3.7 (b)) and for the length direction by controlling specimen-support overlaps. In 3PB tests, the striker was placed in the middle of the profile (see Fig. 4.7 (a), (c)) and the supports were positioned symmetrically with equal length from the middle. In 4PB tests, two strikers were positioned symmetrically with equal length from the middle. The support/striker distances represent the distance between the supports/strikers. An overview of 3PB and 4PB test setup parameters is given in Tab. 3.3.

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Table 3.3: 3PB/4PB test setup parameters 3PB 4PB Striker diameter (mm) 20 40 Striker length (mm) 100 100 Striker distance (mm) - 150 Support diameter (mm) 20 40 Support length (mm) 100 100 Support distance (mm) 360 360

The Aramis 3D measurement method, where two cameras are targeted at the surface from different angels, was employed to measure the surface strains on three selected specimens for both 3PB and 4PB. For DIC measurements, the subcomponents’ front surfaces were covered with white paint and a spray pattern of black dots.

3.2 Coupon level test results

Stress-strain curves from tensile tests of CF-SMC coupons showed an initial linear response followed by a non linear stiffness degradation for varying widths and thick-nesses, as pictured in Fig. 3.8 (a) and 3.9 (a). High-pitched noise audible during the experimental tests indicated the formation of cracks. High scatter of results (> 10%) was observed for obtained tensile strengths (tensile thickness tests (min (MPa); max (MPa); stdev (-)) = (177.3; 372.7; 50.8), tensile width tests = (176.2; 398.8; 55.1)), and moduli (tensile thickness tests (min (GPa); max (GPa); stdev (-)) = (31.26; 51.22; 5.19), tensile width tests = (36.13; 52.48; 4.19)). Although the results might indicate a rise of the tensile strength with increasing coupon thickness (see Fig. 3.9 (a) - (b)), it could not be verified as the number of performed tests was too low for a proper statistical analysis.

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0.0 0.5 1.0 1.5 0 50 100 150 200 250 300 350 W _2L_15 W _2L_20 W _2L_25 W _2L_30 W _2L_40 W _2L_50 T e n s i l e S t r e s s ( M P a ) Failure Strain (%) (a) 0.0 0.5 1.0 1.5 150 200 250 300 350 400 450 W _2L_15 W _2L_20 W _2L_25 W _2L_30 W _2L_40 W _2L_50 T e n s i l e S t r e n g h ( M P a ) Failure Strain (%) (b)

Figure 3.8: Tensile test results from coupons varying in width: (a) Stress-Strain curves; (b) Scatter-plot: tensile strength over corresponding strains

Mechanical Characterization and Modeling of CF-SMC Materials / Author Martin Tiefenthaler, BSc

Mechanical

Characterization and

Modeling of CF-SMC

Materials

Master’s thesis

Diplom-Ingenieur

EIDESSTATTLICHE ERKLÄRUNG

Acknowledgement

Abstract

Kurzfassung

Contents

1 Introduction and Objectives

2 Literature review

2.1 Carbon Fiber Sheet Molding Compounds (CF-SMC)

2.2 CF-SMC Modeling Approaches

3 Experimental Work

3.1 Methodology

3.2 Coupon level test results