Mechanical characterization of fibres-reinforced elastomers

Mechanical characterization of fibres-reinforced elastomers: a study on continuous glass fibres-Sylgard 184 composite

TESI DI LAUREA MAGISTRALE IN

MATERIALS ENGINEERING AND NANOTECHNOLOGY

INGEGNERIA DEI MATERIALI E DELLE NANOTECNOLOGIE

Author: Dario Magni

Student ID: 944489 Advisor: Claudia Marano Co-advisor: Tiziana Bardelli Academic Year: 2020-21

i

Abstract

Fibres-reinforced elastomers (FRE) are progressively becoming materials of larger interest and application in soft robotics and biomedical field. This growing interest is justified by the peculiar mechanical behaviour showed by this type of composites, characterized by a high deformability, which in combination with their anisotropic mechanical response, can be exploited to mimic natural complex movements through the application of simple stimuli. This thesis work is aimed at studying the mechanical behaviour of a FRE constituted by continuous glass-fibres embedded in a polydimethylsiloxane (PDMS) matrix of Sylgard 184. The mechanical properties of a lamina have been investigated, focusing on the evaluation of the transversal modulus, E2, and in-plane shear modulus, G12, for different fibres volume fractions. For E2 measurement, quasi-static tensile tests have been conducted on unidirectional laminates while, for G12, symmetric ±45° angle-ply laminates were used. Moduli values higher than the ones predicted by the commonly employed micromechanical models were obtained, suggesting that the rubber molecules mobility reduction, typically occurring in filled elastomers, and rubber confinement effects, are responsible for a significant matrix stiffening. Since the matrix is an elastomer, in the studied FRE strains larger than those of conventional composites are reached, thus further investigations on different laminates have been performed. Monotonic and cyclic tensile tests have been carried out on symmetric ±30°, ±45°, ±60° and ±75° angle-ply laminates. The tensile stress-strain curves are highly non-linear, qualitatively recalling that of the neat elastomeric matrix, characterized by a linear behaviour at small strains, followed by an intermediate “softened” region extended for a wide strain range and by an almost constant curve slope, and a final “hardening” region at high strains, characterized by an increasing curve slope. Fibres re-orientation have been evaluated and resulted to be related also to changes observed in laminate’s thickness. A fibre angle threshold below which the fibres contribution to the composite stiffness is predominant have been also identified. Due to the highly non-linear response, to significant changes in fibres orientation, and to the lack of adequate predictive models, deeper studies on FREs are necessary for the proper design of products to be used in soft robotics applications. This thesis work is a first step in this direction.

Key-words: Fibres-Reinforced-Elastomers; polydimethylsiloxane-glass fibres laminates; shear modulus; fibres re-orientation; non-linear behaviour.

ii | Abstract

iii

Abstract in italiano

I compositi elastomerici fibro-rinforzati (FRE) stanno progressivamente diventando materiali di ampio interesse, con applicazioni nel campo biomedico e della soft robotics. Questo crescente interesse è giustificato dal peculiare comportamento meccanico mostrato da questo tipo di compositi, in quanto caratterizzati da un’elevata deformabilità, la quale, in combinazione con la loro risposta meccanica anisotropa, può essere sfruttata per imitare movimenti naturali complessi tramite l’applicazione di stimoli semplici. Questo progetto di tesi ha lo scopo di studiare il comportamento meccanico di un FRE costituito da fibre di vetro continue integrate in una matrice polidimetilsilossanica (PDMS) di Sylgard 184. Le proprietà meccaniche della lamina sono state analizzate, focalizzandosi sulla determinazione del modulo trasversale, E2, e del modulo a taglio, G12, per diverse frazioni volumetriche di fibra. Per la misura di E2, sono state eseguite prove di trazione uniassiale su laminati unidirezionali mentre, per G12, sono stati impiegati laminati simmetrici angle-ply a ±45°. Sono stati ottenuti valori dei moduli superiori a quelli predicibili dai modelli micromeccanici comunemente utilizzati; ciò suggerisce che la riduzione di mobilità delle molecole elastomeriche, tipica negli elastomeri caricati, e l’effetto del confinamento della gomma tra le fibre, siano responsabili del significativo irrigidimento della matrice. Considerando che la matrice è un elastomero, nel FRE in esame sono ottenibili deformazioni maggiori rispetto a quelle raggiunte dai compositi convenzionali, ragion per cui ulteriori studi su vari laminati sono stati condotti. Sono state eseguite prove uniassiali monotoniche e cicliche su laminati simmetrici angle-ply a ±30°, ±45°, ±60° and ±75°. Le curve sforzo-deformazione sono fortemente non-lineari, e richiamano qualitativamente quelle dell’elastomero puro. Le curve sono caratterizzate da un comportamento lineare a basse deformazioni, seguito da una zona intermedia per un ampio range di deformazioni, caratterizzata da una minor rigidezza e da una pendenza della curva pressoché costante, e da una terza zona di irrigidimento finale, ad alte deformazioni, in cui la pendenza della curva incrementa progressivamente. È stata valutata la re-orientazione delle fibre, la quale è risultata essere anche legata a variazioni nello spessore del laminato. Un valore limite di angolo delle fibre è stato identificato, oltre il quale il contributo delle fibre alla rigidezza del composito è predominante. Considerando la significativa risposta non lineare, la forte re-orientazione delle fibre, e la mancanza di modelli predittivi adeguati, studi più approfonditi sui FREs

iv | Abstract in italiano

risultano essere necessari, in particolare modo in vista della progettazione di prodotti da impiegare in applicazioni nel campo della soft robotics. Questo lavoro di tesi è da considerarsi come un primo passo in questa direzione.

Parole chiave: Elastomeri fibro-rinforzati; laminati polidimetilsilossano – fibre di vetro; modulo a taglio; re-orientazione delle fibre; comportamento non-lineare.

vii

Contents

Abstract ................................................................................................................................. i

Abstract in italiano .......................................................................................................... iii

Contents ............................................................................................................................ vii

Introduction ........................................................................................................................ 1

1. THEORETICAL BACKGROUND ........................................................................ 5

1.1 Composite materials ........................................................................................ 5

1.2 Fibres-reinforced elastomers, FREs ............................................................... 8

1.3 Modelling of the mechanical behaviour of fibres-reinforced polymers . 11

1.3.1 Micromechanics of a lamina .............................................................. 13

1.3.2 Macromechanics of a lamina ............................................................. 17

1.3.3 Classical Laminate Theory, CLT ....................................................... 20

1.4 Overview of laminates shear testing procedures ...................................... 24

1.4.1 Uniaxial tension of a symmetric ±45° laminate ............................... 26

1.4.2 Ten degree off-axis tensile test .......................................................... 32

1.4.3 Uniaxial tension of a 45° off-axis laminate ...................................... 36

1.4.4 Other shear testing procedures ......................................................... 38

1.5 Elastomers ....................................................................................................... 42

2. EXPERIMENTAL DETAILS ................................................................................ 46

2.1 Materials and specimens preparation ......................................................... 46

2.1.1 Constituent materials .......................................................................... 46

2.1.2 Laminate preparation ......................................................................... 48

2.1.3 Specimens preparation ....................................................................... 51

2.2 Testing setup ................................................................................................... 55

2.2.1 Evaluation of the transversal modulus, E2....................................... 56

2.2.2 Evaluation of the in-plane shear modulus, G12 ............................... 58

viii | Contents

2.2.3 Laminates mechanical behaviour characterization ........................ 64

3. EXPERIMENTAL RESULTS AND ANALYSIS ............................................... 66

3.1 Lamina characterization ................................................................................ 66

3.1.1 Transversal modulus, E2 ..................................................................... 66

3.1.2 In-plane shear modulus, G12 .............................................................. 72

3.2 Laminate analysis ........................................................................................... 89

3.2.1 Quasi-static tensile tests ..................................................................... 90

3.2.2 Quasi-static loading-unloading tests .............................................. 113

3.3 Preliminary Finite Element Analysis (FEA) ............................................. 119

3.3.1 Model development .......................................................................... 119

3.3.2 Model M_UD90 – Results................................................................. 122

4 CONCLUSIONS AND FUTURE DEVELOPMENTS ................................... 127

Bibliography ................................................................................................................... 131

A Appendix A ........................................................................................................... 139

List of Figures ................................................................................................................. 145

List of Tables .................................................................................................................. 153

1

Introduction

The present master thesis is part of a more comprehensive PhD project, carried out by T.Bardelli, aimed at the development and characterization of a glass fibres-reinforced silicone elastomer (FRE). The present work focuses on the evaluation of the FRE transversal modulus, E2, and of the in-plane shear modulus, G12, as well on a wider study of its mechanical behaviour.

From World War II to the recent years, the development of new composite materials has gained an increasingly research effort, particularly in those fields where high efficiency and performances have to be accompanied by lightweight designs. In those fields, conventional materials like metals, alloys or ceramic materials, are progressively coming to their application limits. On the other hand, composite materials show an outstanding combination of high bearable loads and reduced weight, which permits them to replace conventional materials in many applications. Several composite materials have been developed in the years, characterized by different constituent materials and types of reinforcement dependent on the application area. Among them, fibres-reinforced polymers, FRPs, are widely applied in structural applications, where they are mainly used as strengthening members of existing elements, in particular of concrete beams, or as an alternative to conventional materials in the design of different structures, like wind turbines or car frames. Considering these applications, fibres-reinforced composite characterized by an elastomeric matrix, FREs, may result quite unusual and inadequate. As a matter of fact, the elastomeric matrix, although been characterized by good flexibility, damping and absorption performances [1], when alone is limited by its low strength and stiffness, which prevent its application in those fields where load bearing is required. Nevertheless, the integration of stiff reinforcing elements in elastomeric matrices would allow to enhance their mechanical properties, retaining at the same time flexibility and the mentioned characterizing properties of elastomers.

Actually, FREs are not really a novelty in the composite world, as a matter of fact car tires and conveyor belts have been using these materials for years, but their applications, apart from these few fields, have always been overlooked. In recent years, the growing field of soft robotics has started to look at FREs, in particular for biomimetic applications. Biomimetics could in principle be applied in many fields, thanks to the diversity and complexity of biological systems, and the number of

2 | Introduction

features that might be imitated is essentially infinite, from locomotion to morphing, to the development of prothesis or exoskeletons. However the biomimetics basic idea of “learning from nature”, has clashed with the need to find suitable materials. Here come these unnoticed, but highly promising, elastomeric composite, able to respond to the demanded features of lightweight, load bearing and high flexibility.

Especially for soft robotics applications, the combination of elastomers with suitably oriented reinforcements, as the continuous fibres in the present case, can generate composites with strongly direction-dependent properties and high flexibility, which may be exploited to achieve complex deformational responses under simple stress states. In particular, the application of an external stimulus, provided by an embedded active component (generally shape memory alloys that respond to mechanical, thermal or electric inputs), would allow to achieve mechanical response not attainable by conventional composites. By the integration of actuators in FREs, the so-called Smart Soft Composites, SSCs, are obtained. The complex displacements obtained as response to the applied stimulus are the result of a properly designed stacking sequence employed in the composite fabrication, and of the high matrix deformability. The stacking sequence itself determines the types of response, in particular the type of displacements, that the FRE element will undergo at the application of simple stimulus, and it is the result of load-coupling effects common to all fibres-reinforced polymers. As an example, bend-twist or bend-extension displacements can be obtained at the application of a tensile load. However, it is the intrinsically high flexibility of FREs that actually allows to exploit these responses. The high deformability of the FREs components allows to cover the wide range of motion demanded by the different applications (like the movements of an artificial finger), which instead, cannot be achieved by conventional FRPs, characterized by much stiffer matrices that would fail before than concluding the desired movement. This high deformability can be regarded as the most important characteristic and the greatest advantage of FREs over conventional FRPs.

Since SSCs field is quite a novelty, the state of the art on the developments of these stimuli-responsive materials with load-coupling effects is still quite limited. Literature mainly reports on preliminary attempts of SSCs applications, like the rehabilitation glove showed by Chen et al. [2], the “muscle glove” of Yao et al. [3] or the bio-inspired autonomous underwater vehicle of Kim et al. [4], and on the studies performed on the analysis of the displacement fields reachable by these composites [1,5–9]. The attention is mainly focused on the possible uses of SSCs and on their actuation mechanisms, while the mechanical properties of the constituent FRE are considered to a limited extent. In particular, regarding unidirectional elastomeric composites reinforced with continuous fibres, very little investigations

| Composite materials 3

have been carried out [10,11]: some literature, limited in this case too, is available on cord-rubber composites and woven-reinforced elastomers [12–14], which can be considered similar to the FREs of interest in this work. In-depth of knowledge of the deformational response of fibres-reinforced elastomers and related deformation mechanisms as well as suitable characterization methods for determining their mechanical properties is lacking, reason why the present experimental work has been thought to be of interest and useful for further work.

The experimental activity carried out during this thesis work will be described in the following chapters, with the presentation of the obtained results and the related observations. Chapter 1 is dedicated to the theoretical background. An introduction to fibers reinforced plastics (FRPs), and in particular to fibres reinforced elastomers (FREs) will be first provided, followed by a condensed explanation of the theoretical basis and models required to understand and characterize the mechanical behaviour of fibres-reinforced composites. These more general sections will be followed by an extended section reporting a critical review of the testing methods generally adopted for the evaluation of a composite in-plane shear modulus, G12.

Chapter 2 will focus on the description of the experimental activity performed. Details on the FRE in exam will be provided, as well as the fabrication procedures employed during the work. A description of the testing approaches will follow, for both the mechanical characterization of the lamina, and therefore for the evaluation of the composite transversal modulus, E2, and in-plane shear modulus, G12, and for the study of the mechanical behaviour of laminates.

In Chapter 3 experimental results obtained will be provided. A first section will focus on the lamina characterization results, with an analysis of the trend of the mechanical properties of interest with respect to the reinforcement content. Commonly employed constitutive models will be used to compare the experimental results with the theoretical predictions. A second section will instead focus on the mechanical characterization of the laminate behaviour. Experimental results of further tests will be reported and analysed in detail. In particular, the role of the elastomeric matrix and of fibres orientation change on laminate mechanical response will be investigated, and a possible explanation of the recorded results accounting the unique features of FREs will follow. A final short section will be dedicated to some preliminary numerical investigations performed with a finite element software, aimed to provide a first basis for further possible activities.

A summary of the results and observations reached in this thesis work will be reported in the final Chapter 4, within some possible hints on future works.

4 | Introduction

5

1. THEORETICAL BACKGROUND

In this chapter, a short overview about the theoretical background relative to composite materials will be reported. A more extensive description of fibres-reinforced polymeric composite will follow, with a summary of the theoretical approaches commonly used in the evaluation of the lamina mechanical properties. A second part of the chapter will subsequently focus on the testing procedure for the experimental evaluation of the lamina mechanical properties. In particular, this will mainly focus on the mechanical properties considered in the performed work, namely the transversal modulus, E2, and the in-plane shear modulus, G12.

1.1 Composite materials With the name composite materials we refer to a wide family of materials characterized by different constituents and properties. In general, a composite is a structural material that consists of two or more constituents, not soluble in each other, characterized by quite different mechanical properties, and that are combined at a macroscopic level [15]. Examples of composite systems include concrete reinforced with steel, epoxy reinforced with graphite fibres, wood, etc. One of the constituents is called reinforcing phase and the one in which it is embedded is called matrix. The matrix phase is generally continuous and its functions are manifold, but three main ones can summarize its role: support of the reinforcements, protection of the reinforcements and stress-transfer between fibres. Matrices are generally characterized by lower densities with respect to the reinforcing phase, especially if polymeric matrices are considered, and in most of the cases they are also characterized by a lower elastic modulus. Regarding the reinforcing phase material, it may be present in the composite in different forms, like fibres, particles or flakes, which differently affect the composite characteristics and mechanical responses. The main role of the reinforcement is to improve the loading carrying capacity of the matrix material, increasing both its stiffness and strength. On the overall, the composite material will be characterized by a weighted combination of the properties of matrix and reinforcing materials. This can be also considered the major advantage of such materials over traditional construction materials. Indeed, they essentially show the same (or even better) performances of traditional materials at

6 1-THEORETICAL BACKGROUND

much lower weights [16]. An example of this consideration can be highlighted by the chart reported in Figure 1.1 where the Young’s modulus, E, is plotted versus the density, ρ, for different classes of materials. Composite materials may show a Young’s modulus similar to that of metals and some technical ceramics, but are characterized also by a lower density.

Composite materials can be classified accordingly to the type of matrix and to the geometry of the reinforcement. In the first case, we can mainly distinguish between composites based on metal matrices, ceramic matrices or polymer matrices, these last further distinguished between thermoplastic and thermosetting matrices.

In relation to the type of the reinforcement, we can distinguish between [15]:

- particulate composites, - fibres reinforced composites - nanocomposites

In the present research, the composites of interest are the fibres-reinforced ones, and specifically those ones having a polymeric matrix, therefore only them will be considered in the following.

The family of fibres-reinforced polymers (FRP) accounts a large number of possible composite materials, characterized either by different matrix materials and/or types of fibres. The polymeric matrices can be both thermosetting, like epoxy, polyester or phenolic, and thermoplastic, like polypropylene, ABS or polycarbonate. The

Figure 1.1 - Example of material property chart. Comparison of Young's modulus of different material with respect to their density [Ansys Granta Edupack]

| Composite materials 7

reinforcing fibres can be made of different materials. Among them, the most largely used reduce essentially to carbon-fibres, glass-fibres and aramid-fibres, characterized by different mechanical, chemical and thermal properties [15]. In addition, the fibres embedded in the matrix can be distinguished on the base of their length and orientation, resulting in different final properties of the composite material produced. Shortly summarizing and simplifying the FRPs commonly employed, it is possible to distinguish between the main types reported in Figure 1.2 [15] as:

- Unidirectional discontinuous fibres composites, characterized by fibres oriented in a common direction and by a reduced fibres length, as reported in Figure 1.2b

- Randomly oriented discontinuous fibres composites, characterized by fibres oriented in random directions, but again by a reduced fibres length, as reported in Figure 1.2c. These can be considered as quasi-isotropic composites, for which the material properties are almost equal in every material direction

- Unidirectional continuous fibres composite, characterized by fibres oriented in a common direction and by a fibres length comparable to that of the component produced. A representation is reported in Figure 1.2d.

- Cross-ply or fabric continuous fibres composite, reported in Figure 1.2e, characterized by fibres oriented in two common direction and by a fibres length comparable to that of the component produced

- Multidirectional continuous fibres composite characterized by continuous fibres arranged in a precise three-dimensional array, as schematically represented in Figure 1.2f.

In the present work, only FRP laminates characterized by a sequence of continuous and unidirectional reinforcing fibres properly stacked have been used, therefore the further explanation will focus on them. To be even more precise, the family the analyzed material belong to is the one of fibres-reinforced elastomers, FRE.


1.2 Fibres-reinforced elastomers, FREs With FRE we refer to a fibers-reinforced polymer in which the matrix is made of an elastomeric matrix. These peculiar FRPs are part of the wider family of the so-called flexible composites, namely the composites based upon elastomeric polymers. These are characterized by an usable range of deformation much larger than that of the conventional thermosetting or thermoplastic polymer-based composites [13]. The most common example of FRE, present in the everyday life, are the car tires, where a continuous and unidirectional metal cord is embedded in a rubber matrix.

In the past years, modelling, development and application of FREs were prevalently limited to cord-rubber composites, thus mainly to pneumatic tires. From a wider perspective, the larger family of flexible composite materials is actually employed in many other applications, like conveyor belt constructions, air-supported or cable-supported building structures, tents, parachutes, decelerators in high speed airplanes, bullet-proof vests, inflated structures such as escape slides [13]. In these cases, mainly fabrics coated by a soft polymeric material have been used for. Nowadays, novel application fields for flexible composites, and in particular FREs, begin to be studied. Indeed, non-tire fibres-reinforced elastomers show promise for

Matrix +

Continuous Fibers Discontinuous Fibers

or Whiskers Particulate Filler

(a) Particulate composite

(b) Unidirectional discontinuous

fibers composite

(d) Unidirectional

continuous fibers composite

Unidirectional (e)

Crossply or fabric continuous fibers

composite

(c) Randomly oriented

discontinuous fibers composite

(f)

Multidirectional continuous fibers

composite

Quasi-isotropic composites

Figure 1.2 - Schematic representation of common types of composites

| Fibres-reinforced elastomers, FREs 9

use in a broad range of applications, including flexible underwater vehicles [4], variable camber wings [17–19] or inflatable aerospace structures [20]. Another relevant and highly promising application field is relative to the development of bio-inspired devices, like “rubber muscle” actuators or flexible robotic “skeletons” [3,21,22]. Other bio-inspired devices and structures that exploit FRE aim to mimic the abilities of soft-bodied animals. These can indeed maintain posture, squeeze through narrow spaces, or more in general move just using forces transmitted by muscles to a hydrostatic skeleton, thus with forces transmitted not through rigid skeletal elements rather through the application of an internal pressure [23]. Properly designed FREs allows to imitate these motions, in particular when they are combined with shape memory alloys, SMA, with these lasts that act as mechanical actuators. The mimic of the locomotion and manipulation that can be achieved by means of these devices allows to develop soft-robots characterized by a light weight and an high adaptability [23]. Under this aspect, in literature are reported some examples of possible bio-inspired devices based upon a FRE as reported in the following figures. Figure 1.3 reports the motion of a crawling robot, similar to that of a worm [21]. Figure 1.4 reports the bending and rotations that can be generated in a soft continuous arm fabricated with a FRE [24]. Figure 1.5 reports the motion of a turtle-like robot, where the real movements of the turtle fins are imitated using suitably stacked laminae of a FRE [4].

Figure 1.3 - Motion of a crawling robot [21]


The increasing interest in flexible composites and FREs is mainly related to their advantages over “conventional” composites. Indeed, FREs behave very differently from “conventional” rigid FRPs. FREs differ from the “conventional” FRPs under two main aspects. First of all, a large difference in the constituents mechanical properties of a FRE is present. Indeed, if one considers the ratio between the Young’s moduli, E, of the constituent materials 𝐸

𝐸 of a conventional FRP, this usually ranges between 10 to 100. In the case of a FRE, due to the much lower elastic modulus of the matrix, the same ratio can reach values much higher, in the order of 104 to 106. A second aspect regards the matrix itself. As a matter of fact, an elastomeric matrix is intrinsically characterized by a higher deformability with respect to the polymeric matrices generally employed in FRPs. Therefore, the ability of flexible composites to sustain large deformation and fatigue loading and

Figure 1.5 - Swimming motion of a turtle-like robot [4]

Figure 1.4 - Bending and rotations induced in a soft continuous arm [24]

| Modelling of the mechanical behaviour of fibres-reinforced polymers 11

still provide high load-carrying capacity allows the application of these composites to new fields, where the conventional ones generally could not be employed [25]

Other advantages include increased damping and the ability to tailor mechanical characteristics such as elongation, nonlinearity and stiffness over a much broader range. This enhanced tailoring ability is able to provide increased capability to adaptive structures [25].

In addition, FREs show a low shear modulus and hence large shear distortion, which allows the reinforcing fibers to change their orientations under loading, as well reaching a much larger elastic deformations than those of conventional composites [13].

1.3 Modelling of the mechanical behaviour of fibres-reinforced polymers

Generally speaking, FRPs consist of two or more constituents, therefore their analysis and description will differ from that of homogeneous and isotropic materials. The most common approach to deal with the description of the mechanical behaviour of a fibre-reinforced composite is based on a two-level treatment of the overall material [16], as schematically represented in Figure 1.6. First, from the knowledge of the properties of the constituents of the composite material, the matrix and the fibres, the properties of the lamina, the fundamental unit of continuous fibres reinforced composites, also called ply or layer, are derived [15]. This approach is referred to as micromechanics of the lamina: the composite material behaviour is analysed at a microscopic scale, considering also the interactions between the constituent materials, and their effects on the overall composite material properties [16]. Then, material’s stiffness and overall stress-strain relationship, as well material’s thermal expansion coefficients, can be obtained for the lamina considered as a homogeneous orthotropic material. This approach is referred to as macromechanics of the lamina. This assumption allows


to consider how the constituent materials properties affects the average apparent macroscopic properties of the composite material. In general, the use of single a lamina is generally not sufficient in many engineering applications. To overcome this limitation, the stacking of several laminae is the most common solution, to optimize the design of the final element. The structural element obtained from the stacking of different laminae is called laminate. The proper design of laminated composites requires the knowledge of basic relations between the mechanical features of the plies and the final laminate, thus a “third-level” treatment has to be introduced when laminate structures are considered. This further mechanical behaviour description is in this case related to the final laminate, and it is most commonly described by the so-called classical lamination theory, CLT.

To characterize a lamina, four independent elastic constants are required: the elastic modulus E1, in the direction parallel to the fibres, the elastic modulus E2, in the direction perpendicular to the fibres, the in-plane shear modulus, G12 and the in-plane Poisson’s ratio, ν12 [16]. Indeed, the design of a laminate often requires an optimization of the stacking sequences, which have to be based on the knowledge of the mechanical properties of each lamina. This is necessary to account for the maximization of the load bearing capacity of the composite structure. These properties can be theoretically estimated through the lamina micromechanics approach, or experimentally evaluated. In particular, to avoid the compromise of material performances, and to obtain the desired mechanical response from the

Figure 1.6- Schematic of analysis of laminated composites (Adapted from [15])


material, is of foremost importance to accurately evaluate the mechanical properties of the lamina. From these considerations stem the need of reliable and accurate methods for the determination of the four independent elastic constants of the composite material.

This section will focus over the general theoretical description of a lamina and of a laminate mechanical behaviour. In Section 1.4, an overview of the experimental testing procedures available aimed at the evaluation of the in-plane shear modulus, G12, the elastic constant of interest in the present work, will be provided. This work focuses mainly on the evaluation of this property since it will be part of a more comprehensive experimental research which has been run in parallel.

1.3.1 Micromechanics of a lamina

The objective of a micromechanical approach is to determine the stiffnesses or compliances of a composite material in terms of the elastic moduli of its constituent materials taking into account the interaction between the composite components [16]. To deal with the lamina micromechanics, some relevant assumptions must be introduced [16]:

- The lamina is macroscopically homogeneous, orthotropic and initially stress-free.

- The fibres are isotropic and homogeneous, with a linear elastic behaviour. Moreover, they are regularly spaced in the lamina, perfectly aligned and perfectly bonded to the matrix. This last assumption means also that the strains in the fibres direction are the same in the fibres as in the matrix.

- The matrix is isotropic and homogeneous, with a linear elastic behaviour. Moreover, it is assumed to be void-free.

In the microscopic description, the heterogeneity of the composite must be accounted for. This means that stresses and strains vary depending on the specific constituent considered. To do so, a representative volume element is generally defined, containing a single fibre surrounded by the matrix material. Differently, as briefly anticipated, in a macroscopic description, the composite material is considered homogeneous, and then stresses and strains distribution can be regarded as uniform from a macroscopic point of view.

The results of the micromechanics studies of composite materials with unidirectional fibres is in general represented in terms of each mechanical property dependency on the fibres volume fraction. The fibres volume fraction, 𝑣 can be defined simply as:


𝑣 =𝑉

𝑉 (1.1)

where 𝑉 and 𝑉 are respectively the volume of fibres and the volume of the composite.

The basic approaches to predict the apparent orthotropic moduli of unidirectional fibres-reinforced composite materials make use of simple mechanical models [16].

The longitudinal modulus of the composite material in the fibres direction, E1, can be determined by means of the so-called rule of mixture, which is the result of a springs-in-parallel model [16]. It assumes that the average stress, 𝜎 , results from an applied force, F1, that acts on the representative volume element cross-section, is shared between fibres and matrix, and that the strains in fibres, 𝜀 , matrix, 𝜀 , and composite material, 𝜀 , are equal (𝜀 = 𝜀 = 𝜀 ) (Figure 1.7). Then, under the assumption the constituents have an elastic behaviour , the following relation can be derived [26]:

𝐸 = 𝐸𝐴

𝐴+ 𝐸

𝐴

𝐴= 𝐸 𝑣 + 𝐸 𝑣 (1.2)

where A is the cross-section of the representative element, AFibres and AMatrix are the cross-section of the fibre and matrix respectively, vf is the fibres volume fraction, vm is the matrix volume fraction and EFibres and EMatrix are respectively the fibre and matrix Young’s modulus. The elastic modulus in the longitudinal direction varies linearly with the fibres volume fraction, and it is essentially a fibres-dominated property (since 𝐸 is generally much lower than 𝐸 ) [16].

Figure 1.7 - Schematic representation of the longitudinal modulus, E1, evaluation [26]


The transversal modulus of the composite material in the direction orthogonal to the fibres, 𝐸 , can be determined by means of a spring-in-series model [16], represented in Figure 1.8. In this case, the stresses acting on the matrix, σMatrix, and on the fibres, σFibres, due to the application of the force F2, is the same equal to the stress on the composite, 𝜎 (𝜎 = 𝜎 = 𝜎 ). The matrix and fibres deformations differ accordingly to the elastic modulus of the constituent materials. The relation that can be derived is the following:

𝐸 =𝐸 𝐸

𝐸 𝑣 + 𝐸 𝑣 (1.3)

where vf, vm, EFIbres and EMatrix are again the fibres and matrix volume fractions and the fibre and matrix Young’s modulus respectively. The elastic modulus in the transversal direction, E2, is a matrix-dominated property. For FREs the higher deformability of the matrix plays the major role in determining the composite 𝐸 modulus value [26].

The major Poisson’s ratio 𝜐 of the lamina is obtained by means of an approach similar to that of E1, so that the springs-in-parallel model gives again a rule of mixture of the type:

𝜐 = 𝜈 𝑣 + 𝜈 𝑣 (1.4)

where 𝜈 and 𝜈 are respectively the fibre and matrix Poisson’s ratio.

3

Figure 1.8 - Schematic representation of the transversal modulus, E2, evaluation [26]


The shear modulus G12 is obtained by means of an approach similar to that of E2, so that the springs-in-series model gives:

𝐺 =𝐺 𝐺

𝐺 𝑣 + 𝐺 𝑣 (1.5)

where 𝐺 and 𝐺 are respectively the shear modulus of fibre and matrix.

The relationships reported for the four elastic constants give in general a rough approximation of the real values, especially for what concerns the transversal and shear moduli. The springs-in-series and the springs-in-parallel models can be intended respectively as the lower and upper bounds to the real quantities, which are actually affected by the fibres shape and arrangement in space, and by their capacity of effectively reinforce the composite material [16]. To account for these aspects, and in particular for the strong dependence of G12 and E2 on the actual microstructure of the material, different approaches have been followed and several models have been developed, mostly based on energy considerations [27]. Generally speaking, these models result in complex equations difficult to use in the design phase. A particularly simplified approach was developed by Halpin and Tsai [28]. Strongly summarizing the concepts behind it, Halpin-Tsai (HT) equation is a mixed parallel and in series springs-model in which the influence of the microstructural parameters is embedded in a reinforcing factor ξ, that essentially is a scale parameter between the models in parallel and in series, which allows to better estimates the reinforcing action of the fibres in the composite. This esteem depends on the fibres geometry, on the packing geometry and loading conditions. The resulting Halpin-Tsai equation can be expressed as [16]:

𝑀 = 𝑀1 + 𝜉𝜂𝑣

1 − 𝜂𝑣 (1.6)

with

𝜂 =

𝑀𝑀 − 1

𝑀𝑀 + 𝜉

(1.7)

and where 𝑀, 𝑀 and 𝑀 are respectively the composite, fibres and matrix material modulus or Poisson’s ratio. There, η can be interpreted as a reduced fibres volume fraction dependent on the constituent material properties [16]. The values of ξ can be either determined theoretically from the knowledge of the fibres


geometry and arrangement, but also, with a more empirical approach, from a curve fitting of the determined composite constants versus the fibres volume fraction.

1.3.2 Macromechanics of a lamina

The macromechanics can be described as the study of composite material behaviour wherein the material is assumed homogeneous and averaged properties of the composite material, depending on the constituent materials, are considered.

The relationship between stresses and strains can be expressed by means of the generalized Hooke’s law [16]:

𝜎 = 𝐶 𝜀 (1.8)

Where 𝜎 and 𝜀 are respectively the stress and strain components and 𝐶 is the stiffness matrix. It can be demonstrated that [16], for an anisotropic body, the stiffness matrix is characterized by 21 independent terms, arranged in a symmetric matrix. These reduce to 9 for an orthotropic material, to 5 for a transversely isotropic one (like a lamina) and to only 2 if an isotropic material is considered. The same considerations hold if the strain-stress relationship is considered, in which the compliance matrix 𝑆 is introduced instead of 𝐶 .

The components of the stiffness and compliance matrices can be related to more useful engineering constants performing experimental material characterizations, applying either a known load or displacement in a suitable direction.

Unidirectional lamina falls under the transversely isotropic material category. If the lamina is thin and does not carry any out-of-plane loads, one can assume plane stress conditions for the lamina (𝜎 = 𝜏 = 𝜏 = 0). Moreover, if it is considered that the direction of the applied load or displacement coincides with the principal directions 1-2, the stiffness and compliance matrices simplify, and they will be characterized by 4 independent constants only [16]:

𝜀𝜀𝛾

=𝑆 𝑆 0𝑆 𝑆 00 0 𝑆

𝜎𝜎𝜏

(1.9)

𝜎𝜎𝜏

=𝑄 𝑄 0𝑄 𝑄 0

0 0 𝑄

𝜀𝜀

𝛾 (1.10)


where 𝑄 is the so-called reduced stiffness matrix. Equations (1.9) and (1.10) show the relationship between stresses and strains through the stiffness and compliance matrices, which can be expressed in terms of engineering elastic constants as:

𝜀𝜀

𝛾=

⎣⎢⎢⎢⎢⎢⎡

1

𝐸−

𝜐

𝐸0

−𝜐

𝐸

1

𝐸0

0 01

𝐺 ⎦⎥⎥⎥⎥⎥⎤

𝜎𝜎𝜏

(1.11)

𝜎𝜎𝜏

=

⎣⎢⎢⎢⎡

𝐸

1 − 𝜐 𝜐

𝜐 𝐸

1 − 𝜐 𝜐0

𝜐 𝐸

1 − 𝜐 𝜐

𝐸

1 − 𝜐 𝜐0

0 0 𝐺 ⎦⎥⎥⎥⎤

𝜀𝜀

𝛾 (1.12)

Whenever the directions x-y of the applied stresses and strains do not coincide with principal directions 1-2, as schematically reported in Figure 1.9, a transformation of the stress-strain relations from a coordinate system to another is possible. The notation here in use identifies with the reference axis 1-2 the principal material coordinates, with direction-1 aligned with the direction of the fibres, and with the reference axis x-y the global reference system.

x

y

2

1

Figure 1.9 - Representation of the principal material directions, 1-2, and reference system directions, x-y, for a unidirectional lamina

θ


The transformation between the two reference systems passes through the definition of a transformation tensor 𝑇 dependent on the angle 𝜃 between the two reference systems, defines as follow:

𝑇 =𝑚 𝑛 2𝑚𝑛𝑛 𝑚 −2𝑚𝑛

−𝑚𝑛 𝑚𝑛 𝑚 − 𝑛

(1.13)

with 𝑚 = cos 𝜃 and 𝑛 = sin 𝜃. Hence, the stress transformation and the strain transformation can be written as:

𝜎 , = 𝑇 𝜎 , (1.14)

𝜀 , = 𝑇 𝜀 , (1.15)

These transformations must be accounted when a load (or a displacement) is applied in a global reference system not coincident with the material reference system. It can be easily demonstrated that, through the application of these transformation rules, the resulting transformed reduced stiffness matrix, 𝑄 , , and

compliance matrix, 𝑆 , , are [16]:

𝑄 , = 𝑇 𝑄 , 𝑇 (1.16)

𝑆 , = 𝑇 𝑆 , 𝑇 (1.17)

Again, if we consider the conversion of the component of the two transformed reduced matrices to the corresponding engineering elastic constants, the transformed reduced compliance matrix can be written as [16]:

𝜀𝜀𝛾

=

⎣⎢⎢⎢⎢⎢⎡

1

𝐸−

𝜐

𝐸

𝜂 ,

𝐺

−𝜐

𝐸

1

𝐸

𝜂 ,

𝐺

𝜂 ,

𝐸

𝜂 ,

𝐸

1

𝐺 ⎦⎥⎥⎥⎥⎥⎤

𝜎𝜎𝜏

(1.18)

where the terms 𝜂 , and 𝜂 , are the so-called coupling coefficients or coefficients of mutual influence of Lekhnitskii, which relate respectively a shearing in the ij plane caused by a normal stress applied in the i-direction and a stretching in the i-direction caused by a shear stress in the ij-plane [16].


1.3.3 Classical Laminate Theory, CLT

The properties of a laminate constituent laminae can be employed in the determination of the elastic properties of the laminate itself. Laminates are the result of the stacking of a number of laminae (or plies), characterized either by different fibres orientation and/or different constituent materials. To identify the resulting laminate properties, and its consequent behaviour upon the application of a load (or of a displacement), relations between the mechanical properties of the laminate and of its constituent laminae can be obtained from the classical lamination theory CLT [15].

The CLT is based on some general hypotheses [15]:

- The laminate consists of perfectly bonded laminae, so that there is no mutual shear under applied loads: this means that there is continuity of displacements and strains at the interfaces of two contiguous plies

- Every ply of the laminate is homogeneous and transversally isotropic - The plies are thin enough to assume a plane-stress condition - The Kirchhoff’s hypothesis holds: this states that, under the effect of generic

external actions, the laminate undergoes deformation. The segment P0P, normal to the laminate midplane π, as reported in the representation of Figure 1.10, throughout the deformation process remains straight and normal to the laminate midplane and its length remains constant, as schematically represented in Figure 1.11.

Figure 1.10 - Schematic representation of the laminate


Under these hypotheses, a generalized strain vector can be written as follow:

𝜀 = 𝜀 + 𝑍𝜅 (1.19)

where the two vectors that compose it are the middle surface strain vector 𝜀 and the middle surface curvature vector 𝜅, and Z is the distance from the midplane π. From the knowledge of the strains at the midplane, the stresses for the specific ply, 𝜎 , can be obtained too as:

𝜎 = 𝑄 ′ 𝜀 = 𝑄 ′𝜀 + 𝑍𝑄 ′𝜅 (1.20)

where 𝑄 ′ is the transformed reduced stiffness matrix for the specific ply. It is

important to note that the strains vary linearly along the laminate thickness without any dependency on the orientation of the plies, since the two components of the vector 𝜀 do not depend on the specific ply considered. Differently, the stresses depend on the ply in exam, since the transformed reduced stiffness matrix depends on the specific orientation of the ply considered and may differ for each ply.

A relationship can now be established between the strains at the midplane and the forces and moments applied to the laminate cross-section. The resultant forces and moments acting on a laminate are obtained by integration of the stresses in each layer through the laminate thickness:

𝑁 = 𝜎 𝑑𝑍

𝑀 = 𝑧 𝜎 𝑑𝑍

(1.21)

Deformation process P0

P

P0’

P’

Mid-plane trace

Deformed mid-plane

trace

x x

z z

Figure 1.11 - Representation of the mid-plane deformation process accordingly to Kirchhoff's hypothesis


where 𝑁 is the vector representing the resultant forces and 𝑀 is the vector representing the resultant moments. Inserting the previous Equation (1.20), we can obtain the following relationships:

𝑁 = 𝐴 𝜀 + 𝐵 𝜅

𝑀 = 𝐵 𝜀 + 𝐷 𝜅 (1.22)

where the matrices 𝐴, 𝐵 and 𝐷 are respectively the:

- membrane stiffness matrix, defined as

𝐴 = 𝑄 ′ (ℎ − ℎ ) (1.23)

- membrane-flexural coupling stiffness matrix, defined as:

𝐵 =1

2𝑄 ′ (ℎ − ℎ ) (1.24)

- flexural stiffness matrix, defined as:

𝐷 =1

3𝑄 ′ (ℎ − ℎ ) (1.25)

where hk is the distance between the k-lamina top-surface and the laminate mid-plane, and hk-1 is the distance between the k-lamina bottom-surface and the laminate mid-plane, as represented in Figure 1.12.

Figure 1.12 - Representation of the laminate [15]


These three sub-matrices constitute the laminate stiffness matrix 𝐴𝐵𝐷, a symmetric matrix that relates the forces 𝑁 and moments 𝑀 acting on the laminate cross-section to the strains 𝜀 and curvatures 𝜅 of the laminate midplane.

𝑁

𝑀=

𝐴 𝐵𝐵 𝐷

𝜀

𝜅 (1.26)

The different components of the laminate stiffness matrix result in different behaviour of the laminate when the loads are applied. To better understand this concept, the extended version of the 𝐴𝐵𝐷 matrix is reported in Equation (1.27):

⎣⎢⎢⎢⎢⎢⎡

𝑁𝑁

𝑁

𝑀𝑀

𝑀 ⎦⎥⎥⎥⎥⎥⎤

=

⎣⎢⎢⎢⎢⎡𝐴𝐴𝐴𝐵𝐵𝐵

𝐴𝐴𝐴𝐵𝐵𝐵

𝐴𝐴𝐴𝐵𝐵𝐵

𝐵𝐵𝐵𝐷𝐷𝐷

𝐵𝐵𝐵𝐷𝐷𝐷

𝐵𝐵𝐵𝐷𝐷𝐷 ⎦

⎥⎥⎥⎥⎤

⎣⎢⎢⎢⎢⎡

𝜀𝜀𝜀𝜅𝜅𝜅 ⎦

⎥⎥⎥⎥⎤

(1.27)

Different responses at the application of the different loads are obtained in relation to the 𝐴 , 𝐵 and 𝐷 terms values. The different responses, schematically represented in Figure 1.13, can be summarized as follow:

- when the terms 𝐴 and 𝐴 are not null, a shear-extension coupling is expected

- when the terms 𝐷 and 𝐷 are not null, a bend-twist coupling is expected - when the terms 𝐵 are not null, a bending-extension coupling is expected

The presence of the different terms is directly related to the specific stacking sequence of the laminate, so that, by means of a proper design, a desired response

Figure 1.13 - Examples of load-coupling effects


of the laminate under specific applied loads can be obtained. These load-displacement couplings are in most of the cases unwanted features in laminates to be used in structural application, although they started to be employed in some cases [29], but can be of greatest interest in the case of FRE. Upon the application of a simple load, in general by means of either a mechanical or piezoelectric actuator, complex displacement fields can be generated in the laminate. These displacements can be fully exploited when an elastomeric matrix is used for the composite, since the laminate will not fail immediately thank to the high matrix deformability, which allows to reach large displacements. Despite this field have not largely explored yet, some simple but clarifying examples of applications of these concepts can be found in literature [7–9].

1.4 Overview of laminates shear testing procedures As previously said, in order to characterize a lamina, four independent elastic constants are required: the elastic modulus, E1, relative to the direction parallel to the fibres, the elastic modulus, E2, in the direction perpendicular to the fibres, the in-plane shear modulus, G12 and the in-plane Poisson’s ratio, ν12. This section will focus on the experimental procedures aimed at the evaluation of the in-plane shear modulus, G12. A deeper analysis of the experimental procedures that would be of interest in our case for the determination of G12 will be provided, followed by an overview about some other testing procedures available.

Considerable experimental and analytical efforts have been expended in the development of in-plane and through-thickness (out-of-plane) shear test methods for the determination of shear modulus and strength of fibres-reinforced polymer composites. Nevertheless, a limited number of methods still exist nowadays [30]. Indeed, one of the principal difficulties in the development of a test method for the measurement of shear properties is the provision of a pure shear stress state in the specimen [30]. Ideally, for quantitative shear measurements, the shear test method should provide a region of pure and uniform shear stress in the test section of the specimen throughout the linear and non-linear response regimes. In addition, a unique relationship should exist between the applied load and the magnitude of the relevant shear stress acting in the test section [30]. The difficulty of inducing a pure shear load increases with increasing anisotropy and inhomogeneity of the material: as these characteristics increase, the complex stress states arising at, or near, the loading zones become more and more dominant, particularly for continuous unidirectional laminates containing high modulus and high strength fibres. In these materials, it is difficult to obtain adequate regions of uniform shear stress free of extraneous stress components within the specimen, even if the production of the

| Overview of laminates shear testing procedures 25

specimen and test alignment are perfect [30]. The difficulties encountered in producing a state of pure shear in composite specimens have resulted in a limited number of test methods being incorporated into national and international standards. Actually, there is no universal method suitable for the accurate evaluation of the shear properties for the extensive range of material architectures encountered in composite technology [30]. All the shear methods, whether standardized or otherwise, have physical and geometrical limitations. These limitations become even more relevant whenever a novel composite material has to be characterized, since other problems may arise due to the characteristics of the material itself.

Following a general approach, most of the shear tests have been developed with the objective of maximising shear stress and minimising extraneous induced stresses, which would compromise the determination of the shear properties. Still, it is possible to measure the shear stress-strain response of a composite in the presence of stresses that are not of shear, provided the magnitude of the shear stress is considerably larger than these other stress contributions [16]. Moreover, although the present work will focus only on the methods for the determination of the in-plane properties, a full characterization requires generally also the determination of the properties in the 1-3 and 2-3 planes, which may lead to different measured values. For these measurements, the development of reliable test methods becomes even more complex, since both an in-plane and a through-thickness uniform shear loading must be obtained, with a consequent further reduction of the testing procedure feasible. Since the present work is limited to the determination of the in-plane shear modulus, G12, only, the concerns regarding the properties in the other material directions are of no direct interest, as the ones related to the shear strength measurements.

As explained, there is not a large availability of shear testing methods that can be performed and considered fully reliable. The commonly used methods for the evaluation of the shear properties are:

I) Uniaxial tension of a symmetric ±45° laminate [31]

II) Uniaxial tension of a 10° off-axis laminate [32]

III) Uniaxial tension of a 45° off-axis laminate [33]

IV) Two rail or three rail shear test [34]

V) Iosipescu shear test [35]

VI) Twisting of a flat laminate [36]


VII) Torsion of a thin-walled tube [37]

In the following sections, a brief description of these tests will be provided, within the possible difficulties that may be encountered when applied to the material under investigation in our case, that is a fibre reinforced elastomer, FRE. A deeper description will be provided for the uniaxial tension of symmetric ±45° laminates and for the uniaxial tension of 10° and 45° off-axis laminates, which have been chosen as possible testing methods for the material in exam.

1.4.1 Uniaxial tension of a symmetric ±45° laminate

This is probably the most used method for what concerns the assessment of the shear properties of a composite, and specifically of the in-plane shear modulus G12. It is a standardized procedure which is available in the international standards as ASTM 3518-3518D [31].

This test method allows to determine the in-plane shear response of polymer matrix composite materials reinforced by high modulus fibres. It employs laminates characterized by a symmetric stacking sequence of plies oriented at ±45° with respect to the load direction, which have to be tested through a quasi-static uniaxial tensile test. The use of a simple tensile test configuration, in which no particular fixtures or apparatus are necessary, is the main reason for the widespread use of such methodology. The test procedure described in this standard employs rectangular specimens with a preferred width of 25 mm, length of 200 to 300 mm and thickness of 2 mm, with a recommended number of layers between 16 and 24 [31]. The use of tabs is generally not required but can be considered in accordance with the ASTM 3039-3039D [38].

Whenever such a laminate is loaded in uniaxial tension, the different laminae are subjected to a biaxial stress state, where the shear stress in the lamina coordinate system is related only to the load applied in x-direction. As proposed by Rosen [39], the magnitude of the ply shear stress is calculated from the average applied tensile stress σx, from the simple relation:

𝜏 =1

2𝜎 (1.28)

The magnitude of the lamina shear strain γ12 is derived from the values of the longitudinal and transversal strains, εx and εy respectively, as:


𝛾 = −𝜀 + 𝜀 (1.29)

εx and εy have to be measured either with extensometers or strain gauges. The in-plane shear modulus, G12, can be evaluated as the ratio:

𝐺 =𝛥𝜏

𝛥𝛾 (1.30)

considering therefore the slope of the shear stress-shear strain curve, over a recommended shear strain range of 0.1% to 0.5% [31].

Equations (1.28) and (1.29) are based on classical lamination theory, CLT (see section 1.3.3), for the particular case of a symmetric and balanced ±45° laminate. With the laminate loaded in uniaxial tension in the x-direction, the stresses in each lamina (in terms of lamina strains) are given by:

𝜎𝜎𝜏

=

𝑄 𝑄 𝑄

𝑄 𝑄 𝑄

𝑄 𝑄 𝑄

𝜀𝜀𝛾

(1.31)

where Qij (with i,j = x, y, s) represent the terms of the transformed reduced stiffness matrix, derived from the transformation rule:

𝑄 , = 𝑇 𝑄 , 𝑇 =

= 𝑚 𝑛 −2𝑚𝑛𝑛 𝑚 2𝑚𝑛𝑚𝑛 −𝑚𝑛 𝑚 − 𝑛

𝑄 𝑄 0𝑄 𝑄 0

0 0 𝑄

𝑚 𝑛 𝑚𝑛𝑛 𝑚 −𝑚𝑛

−2𝑚𝑛 2𝑚𝑛 𝑚 − 𝑛

(1.32)

with m and n equal respectively to the cosine and sine of the fibres angle, θ, and the term Qkm (with k,m = 1, 2, 6) representing the stiffness constants. Note that a slight change in the notation of the terms have been introduced to avoid misunderstandings. The reported τs and γs corresponds to τxy and γxy showed in Section 1.3.3.

In this particular case, since no load is applied in the y-direction, σy is null, and, since the laminate is balanced, also the term γs is zero. Considering this, using stress transformation equation, the shear stress in the principal coordinate system becomes:


𝜎𝜎𝜏

=𝑚 𝑛 2𝑚𝑛𝑛 𝑚 −2𝑚𝑛

−𝑚𝑛 𝑚𝑛 𝑚 − 𝑛

𝜎𝜎𝜏

𝜏 = −𝑚𝑛𝜎 + (𝑚 − 𝑛 )𝜏

(1.33)

Assuming a fibres angle of 45°, the equation simplifies to the one previously reported (1.28). Substituting these results in Equation (1.12), the relation for the shear strain can be derived too.

This test method yields reliable results for what concerns the initial stress-strain measurement and for the evaluation of the shear modulus but has to be adopted with caution when the shear strength has to be determined, since it does not generate a state of pure shear stress [40].

This method was firstly proposed by Petit [41], and it is a modification of a previous testing method suggested by Tsai [33]. The procedure proposed by Petit, instead of using unidirectional test specimens with fibres oriented at 45° test specimen with respect to the load direction, as proposed by Tsai, employs as said balanced and symmetric ±45° laminates. This improvement allows to eliminate the shear/extension coupling that is a common issue of off-axis specimens, resulting in a shear stress-strain curve more indicative and not influenced by undesired stresses.

In its work [41], Petit highlights also the two major approximations of the procedure. The first approximation is related to the lack of a pure shear stress or strain state in the ±45° test specimen. This is due to the slight difference existing in

εx εy

γxy/2 + 45° laminate

- 45° laminate

Figure 1.14 - Mohr's circle of strains


the strains in the longitudinal and transversal direction, εx and εy respectively, which results in a laminate Poisson’s ratio νxy not precisely equal to 1. This means that the state of stress created in the specimens is not of pure shear, rather a small contribution coming from tensile stress is present. A useful representation of this can be given making use of the Mohr’s circle, as reported in Figure 1.14. The presence of a tensile stress can be considered negligible if its magnitude is much lower than that of the shear one, but it must be considered when the difference between εx and εy increases. For this reason, this type of test is not considered fully reliable to determine the material shear strength. The second approximation is strictly related to the classical lamination theory CLT, on which this testing method is essentially based. CLT assumes that the strains through the specimen thickness are a linear function of the distance from the laminate mid-plane. This is essentially a consequence of the Kirchoff hypothesis (see section 1.3.3), which guarantees the strain compatibility between the different laminae of the laminate. In practice, an in-plane shear stress is present in each constituent lamina whenever a tensile stress is applied to the laminate. The sign of this shear stress alternates in each lamina oriented at ±θ, as it can be highlighted in Figure 1.15, resulting in a null shear stress for the laminate.

This assumption holds in the majority of the lamina surface, exception done at the specimen free-surfaces, where the in-plane shear stress must go to zero. In this boundary zone the previous considerations do not hold anymore, since no applied shear stress is present there, as reported in Figure 1.16. In addition, Murthy and Chamis [42] showed that also the magnitude of σx changes across the specimen width, with a substantial reduction near the free-edges. Therefore, a complex

Figure 1.15 - Stress state in the laminae


distribution of the stresses is present at the free-edges, making this region inadequate for the analysis.

Note however that this non-uniformity of the in-plane shear stress can be neglected if specimens with a sufficiently large width are used [42,43], and if the strains are measured as close as possible to the laminate half-width.

It is also important to highlight that all the procedures and calculations are based on an average value of the axial stress applied to the laminate, which must be as homogenously distributed as possible in the laminate. Therefore, the greater the degree of material inhomogeneity, such as materials with significant resin-rich or fibres-rich regions, the greater the potential for inaccuracies in the measured response.

In addition, also the region where the strains are measured must be carefully considered. As pointed out by Beter [1], three main zones can be identified in the clamped specimen, as reported in Figure 1.17:

- The zone A, definable as clamping stress free zone, where fibres displacement and deformation are unaffected by specimen gripping and are connected only through the surrounding elastomeric matrix such that their deformation is constituted only by the present shearing.

- The zone B presents fibres rigidly clamped in the fixtures, hindered in displacement, and the strains have not to be measured in this zone.

- The zone C contains both constrained and unfixed fibres, thus a sort of mixed shear and tensile loading takes place. As for zone B, also this specimen region cannot be exploited for the shear modulus evaluation.

Figure 1.16 - Lamina shear log effect [41]


A further aspect to consider is the effect of fibres rotation. This point was partially overlooked by Petit, but actually it becomes relevant in the shear modulus determination. The present technique lays its foundation in the use of a symmetric laminate with fibres oriented at ±45°, angle which guarantees the application of the previously reported equation. However, the fibres will progressively re-orientate themselves during the test, modifying, as consequence, the response of the material, which will show a stiffer behaviour as the level of strain increases. A major point to be accounted for, is the invalidity of the basic assumption and founding equation of the method. Indeed, as the fibres angle changes, the stress state developed in the laminae is modified too, and a state of nearly pure shear stress cannot be considered anymore. This will be reflected on Equations (1.28) - (1.33), which won’t hold anymore.

Fibres re-orientation becomes a relevant concern at high strain levels. In general, an angle change below 1.5° is considered acceptable in the shear modulus determination [31]. The effect of the fibres re-orientation for the material analysed in this work will be delved deeply in the further sections.

Finally, some consideration should be added also regarding the number of plies constituting the laminate. The ASTM 3518-3518D [31] reports a preferable number of plies of 16 to 24, however, a previous study [43] has showed that the effect of the number of plies becomes relevant only when shear strength measurements have to

Figure 1.17 - Schematic representation of the 3 main zones present in a ±45° clamped laminate


be performed: in the shear modulus determination, a number of plies lower than the standard suggested one should not produce remarkable differences in the results, in sight of the limited strain range in which the measure is performed.

Despite this method is far from being free from defects, it has been considered adequate for the characterization of the composite in exam, and it has been chosen as preferred method. Indeed, although it does not allow shear strength measurements, when compared to other techniques it demonstrated to bring reliable results in terms of shear modulus measurements, which are comparable to the much more complex technique of torsion of a thin-walled tube (ASTM D5448 [37]) [43]. In addition, no complex fixtures or instrumentations are required to perform the test, and the specimens can be produced with ease, without the need of complex machining.

1.4.2 Ten degree off-axis tensile test

The 10°-off axis tensile test represents an alternative and convenient technique in the determination of the in-plane shear properties of composite materials [32]. Indeed, alike the tension of a [±45°]s laminate, also this type of test is one of the most largely employed in the evaluation of the shear properties of composite materials.

This test method makes use of rectangular specimens constituted by laminates characterized by unidirectional fibres oriented at 10° with respect to the loading direction, as represented in Figure 1.18. When the specimen is subjected to a uniaxial tensile load, a biaxial stress state is induced in the material’s principal coordinate system. This biaxial stress state consists of a combination of longitudinal σ1, transversal σ2 and in-plane shear τ12 stress state. The stress state in the specimen can be easily derived from the transformation equation, resulting in:

𝜎 = 𝜎 cos 𝜃

𝜎 = 𝜎 sin 𝜃

𝜏 = 𝜎 cos 𝜃 sin 𝜃

(1.34)

where the stresses with numerical subscript refer as usual to the principal material coordinate system, and σx is the applied tensile stress. The strains are measured in the x-y reference system, employing either a strain-gauge rosette or a digital image correlation software, and are transformed in the strains in the 1-2 reference system making use of the equations showed in section 1.3.2. The shear curve is obtained plotting the evaluated τ12 versus γ12, and the shear modulus is evaluated from the slope of the tangent to the initial portion of the shear curve [32].


The test has the advantages of adaptability to conventional tensile testing (reason why it is largely used), uniformity of through-thickness shear stress, no residual stresses and ease of specimens manufacturing. However, it requires to measure three strains at a point and to transform stresses and strains to another coordinate system. Moreover, the test is very sensitive to deviation in fibres orientation from the 10° set value (as small as 1°), which may occur also in the machining of the specimen resulting in large errors in shear measurements [32].

This test method is not registered either as an ISO or ASTM standard, thus there are no commonly used or agreed specifications for specimen dimensions and specimen preparation. Still, some literature [32] reports as preferable dimensions 25 mm for the width, 250 mm for the length and 2 mm for the thickness.

Although this test can be regarded as a simple and quite reliable test procedure, several concerns are still related to it. In particular, high attention must be posed to end-constrains effect, which are also related to the specific off-axis angle, to the specimen geometry and to the degree of material anisotropy [44]. When a transversely isotropic tensile specimen is tested under off-axis loading conditions, the off-axis orientation effectively transforms it into an anisotropic specimen in the coordinate system coincident with the load axis. If the ends of the coupon are rigidly gripped and prevented from rotation, a highly inhomogeneous state of deformation

σx

σx

σ1

σ1

σ2

σ2

τ12

τ12

Figure 1.18 - Schematic depicting loaded 10° off-axis tensile test specimen and stresses at element at 10° plane (adapted from [32])


will be induced along the specimen. This has been referred to as the end-constraint effect [45].

That’s due to the presence of the extension-shear coupling, which generates rotations and transverse strains at the extremities of the specimen. Under an applied uniform normal stress σx, if the material would be able to deform freely at its extremities, a rectangular specimen will deform in a parallelogram as shown in Figure 1.19a. However, the presence of a fixed gripping system would result in an in-plane bending as illustrated in Figure 1.19b. Such non-uniform deformation of the coupon is in turn related to an inhomogeneous stress field, which may strongly affect the analysis.

Several authors have analyzed the influence of the constrains imposed by the clamping as in [44] on the stress state generated in the specimens. Pagano and Halpin [46] attributed the main cause of the non ideal deformed shape to the restriction of the lateral strain. On the other hand, Rizzo [45] and Wu and Thomas [47] experimentally found that the dominant factor is related to the restriction of the rotations at the extremities of the specimen.

To overcome the occurring of such “s-shape” bending, several strategies have been developed. Generally speaking, two types of approach can be followed:

- the use of correction factors to account the inhomogeneous stress field generated in the specimen;

- the use of alternative gripping systems and modifications in the specimen characteristics.

Figure 1.19 - Effect of shear-extension coupling. (a) Ideal specimen deformation, (b) specimen deformation when clamped ends are considered [44]

(a) (b)


Among the authors that followed the first path, Pagano and Halpin [46] proposed a first close-form analytical solution to account the effect of end constrains. Pindera and Herakovich [44] presented a milestone work, in which they derived an analytical solution to correct the apparent shear stress and modulus measured. Mujika [48] proposed two further analytical approaches to correct the experimental results.

Despite the validity of these approaches, it was generally found easier to introduce changes in the testing equipment and in the design of the specimens, to allow a uniform strain field in the gauge. Among the solutions proposed following this strategy, it can be found the use of rotating grips [47], the use of pinned-end fixtures [49], or the use of oblique-shaped end-tabs able to ensure a uniform state of strain and stress in the gauge length of the off-axis specimen [50].

Despite the test is called “10° off-axis”, 10° is not always the optimal off-axis angle. It is generally chosen around 10°-15° for most of the conventional composites, since it allows to minimise the effects of longitudinal and transverse stress components σ1 and σ2 on the shear response since at an angle of 10° the shear strain γ12 approaches a maximum value. This can be understood if the magnitude of the material strains ε1, ε2 and γ12, normalized with respect to the specimen strain εx, are considered as shown in Figure 1.20.

Still, it is fundamental to note that the optimal angle at which the relative magnitude of γ12 with respect to the global strain εx is at its maximum, is strongly influenced by the ratios E1/E2 and E1/G12: as showed by Longana et al. [51], high values of these ratios have the effect of reducing the optimal angle maximizing γ12.


1.4.3 Uniaxial tension of a 45° off-axis laminate

An alternative to the uniaxial tension of 10° off-axis laminates reported in Section 1.4.2 is the tensile test of unidirectional laminates with fibres oriented at 45° with respect to the load direction, as schematically represented in Figure 1.21.

This approach was proposed by Tsai [33], and essentially exploits the mathematical relations existing between the elastic properties of unidirectional composites and their fibres orientation to identify the shear modulus and the shear strength. In particular, the relation between the lamina elastic properties and the laminate modulus can be employed to evaluate the lamina shear modulus. Indeed, recalling the transformation equation for the uniaxial stiffness of an orthotropic material:

1

𝐸=

𝑐𝑜𝑠 𝜃

𝐸+

1

𝐺−

2𝜐

𝐸𝑐𝑜𝑠 𝜃𝑠𝑒𝑛 𝜃 +

𝑠𝑒𝑛 𝜃

𝐸 (1.35)

the shear modulus, G12 can be evaluated knowing the transformed uniaxial stiffness, Ex, and the other lamina elastic properties, namely the axial stiffness, E1, the transverse stiffness, E2 and the major Poisson’s ratio, ν12.

γ12

/εx

ε1/ε

x

ε2/ε

x

Figure 1.20 - Variation of the strain in the material axes against the load direction (Adapted from [32])


Inverting the above equation, for a 45° unidirectional laminate, the relations that is obtained reduces to:

1

𝐺=

4

𝐸−

1 − 𝜐

𝐸−

1 − 𝜐

𝐸 (1.36)

where the minor Poisson’s ratio, 𝜐 can be obtained from the relation 𝜐𝐸 =

𝜐𝐸 . Therefore, simply measuring the uniaxial stiffness, Ex, for θ = 45°, the shear

modulus can be obtained. Despite the simplicity of this method, the problem related to the coupling between shear strains γxy and normal stresses σx is still present [16]. In addition, the ends of the lamina should be free to deform otherwise the lamina would be restrained from shearing deformation and it would undergo to an in-plane bending as previously showed in Figure 1.19. However, provided that the specimen is long and thin enough, the deformation in its middle is similar to the shearing and extension of an unrestrained lamina. The requirement of a high length-to-width ratio is related also to the evaluation of the uniaxial stiffness, Ex. A sufficiently high ratio grants that a pure uniaxial strain is obtained, since the boundary conditions at the specimen end grips are of no consequence, so that it holds:

𝜎 = 𝐸 𝜀 (1.37)

If not, what is effectively measured would be the transformed reduced stiffness 𝑄 , which may lead to an underestimation of G12.

Considering all these aspects, and in particular the dependence of the optimal off-axis angle on the material properties, which will be analysed in details in Section

Figure 1.21 - Uniaxial loading of a unidirectional specimen at 45° to the 1-direction [16]


2.2.2.2, the off-axis tensile test on specimens with unidirectional fibres oriented at 45° was chosen as possible alternative to the uniaxial tension of symmetric ±45° laminates, always having in mind all the issues related to the shear coupling.

1.4.4 Other shear testing procedures

In addition to the above reported procedures, other shear testing methods commonly employed are the Iosipescu shear test, the two rails or three rails shear test, the twisting of flat laminates and the torsion of a thin-walled tube. In this sub-section, a brief description of these methods will be provided.

1.4.4.1 Iosipescu shear test

The Iosipescu shear test, initially developed for metals, has been standardized for testing composites as ASTM D5379 [35]. This test makes use of a specimen in the form of a rectangular flat strip with two symmetrical centrally located v-notches, loaded in a mechanical testing machine by a special fixture, as reported in Figure 1.22. The specimen is inserted into a particular fixture with the notches plane aligned to the loading direction and the two halves of the fixture are moved into opposite directions while monitoring force. In practice, the Iosipescu shear test ideally produces a state of pure shear loading at specimen mid-length by application of two counteracting moments. Then, the loading can be idealized as an asymmetric bending, where, through the relative displacement between the two fixture halves, a constant shear load in the middle section of the specimen is obtained. The shear response must be measured by means of two strain gauge

Figure 1.22 - Iosipescu test fixture [35]


elements, oriented at ±45° to the loading axis placed in the middle of the specimen. The presence of the notches is required, since they allow the shear strain to occur along the loading direction, making the strain distribution more uniform than that would be seen without the notches. As explained by ASTM D5379 [35], in addition to the in-plane shear modulus, also the shear modulus in other material directions can be obtained, provided to use the correct stacking sequence. Fixture design and specimen characteristics are reported in the standard too.

Although the Iosipescu shear test can provide appreciable results, there are still present several issues and concerns which have been reported in literature [52–54]. These are mainly referred to the inhomogeneous stress and strain state in the test region, twisting of the specimen, crushing at loading points, and unwanted failure modes. Many of these are interconnected, which make them difficult to investigate, thus an in-situ optimization of the different aspects should be performed. Another drawback of the method is relative to the specimen preparation itself, which requires manufacturing techniques with a high degree of precision [55]. Considering all these aspects, the adoption of such method to determine the shear modulus of the soft composite in analysis has been disregarded. Severe machining operation would introduce an excessive amount of defects in the specimens, and the soft nature of the matrix may lead to an incorrect loading condition of the specimen. Also, the fixture required for the testing should be optimized for the specific material in use.

1.4.4.2 Two (or three) rails shear test

Another possible shear testing procedure is the two (or three) rails shear test. This test method allows only to evaluate the in-plane shear properties of a composite laminates. Two test configurations can be adopted to impose an edgewise shear load, making use of a two rail or three rail fixture. The two test configurations, two-rail shear and three-rail shear, and the associated test specimen geometries are specified in ASTM D4255 [34]. In both the configurations, a flat laminate is clamped through bolts between two or three parallel steel loading rails as represented schematically in Figure 1.23a and Figure 1.23b respectively. When loaded in tension, these fixtures introduce shear forces in the specimen, and a strain gauge is used to measure shear strain. Generally, the three rail configuration can reproduce a state of stress closer to the one of pure shear stress.


This test method is affected by some drawbacks too. First of all, the specimen preparation and testing are time consuming and expensive, and it is difficult to machine the holes necessary in the specimens with the provided tolerances. In addition, damages are commonly introduced during the drilling of holes, and longitudinal fibres splitting can occur if a soft matrix is considered. Numerical analysis [56] has shown that the length to width ratio can have a major effect on stress distributions and that it depends on the laminate properties, however, respecting the specimen dimensions specified in ASTM D4255 [34], it can be ensured a uniform shear stress distribution over a large region in the central part of the specimen, producing valid shear moduli values for all laminates.

The most relevant problem in our case is the complexity of specimen’s preparation. Indeed, in view of the particular nature of the material in analysis, characterized simultaneously by a soft matrix and a much more rigid reinforcement, the machining operation would probably introduce several damages and defects in the prepared specimens. In addition, also fibres misalignments would be introduced, resulting in a non-reliable testing.

1.4.4.3 Twisting of flat laminates

The twisting of flat laminates [36] is another possible testing procedure standardised for wood products by the ASTM D3044-7 [57], but adopted also for some composite materials. However, it has been applied only to materials characterized by shear modulus values ranging from 0.29 GPa to 88.2 GPa, and it can provide information only about the shear moduli [36]. The test is conducted by deflecting the diagonal corners of a square (or rectangular) plate specimen, and supporting two diagonally remaining opposite corners, as reported in Figure 1.24. The total load is recorded as a function of the applied displacement. It is assumed

(a) (b)

Figure 1.23 - Schematic representation of the: (a) two rails shear test, (b) three rails shear test


that the loading points are located exactly at the specimen corners, but this is in general experimentally difficult to be achieved.

However, as reported by Sims [36], if the load is applied in a more comfortable position close to the corners, the in-plane shear modulus can be evaluated applying some correction factors. The main advantages of this test method are both a simple loading condition (bending mode) and the use of a simple loading jig, commonly used for bending tests. Further the specimen preparation is simple, the large displacements at low loads can be achieved (allowing use of crosshead movement to measure displacement), and the shear stress distribution is uniform. The test is suitable for unidirectional composite material and avoids the necessity of preparing cross-ply test plates. The main drawback of the method is the restricted range of shear moduli that can be measured, and the absence of a properly defined standard for composite materials. In addition, the requirement of relatively big and thick specimens, and the particular type of loading conditions to be applied make the present testing method not adeguate to test the material in exam.

1.4.4.4 Torsion of a thin-walled circular tube

A last shear testing procedure of relevance is the torsion of a thin-walled circular tube [37]. It is a method that directly applies shear load to fibre-reinforced plastic composites, and from an applied mechanics viewpoint it is the most desirable one for shear characterisation. In this test, an approximate state of pure shear stress is induced in a thin-walled cylindrical tube subjected to pure torque about the longitudinal axis of the specimen, as represented in Figure 1.25, such that the shear stress is uniformly distributed around the circumference and along the specimen length. For this test, the specimen is made of a thin-walled hoop wound cylinder with fibres oriented at approximately 90° to the longitudinal axis, adhesively bonded into two concentrically fitting circular end fixtures, which are inserted at each end of the specimen. The specimen/fixture assembly is mounted in the testing

Figure 1.24 - Schematic representation of the plate-twist test geometry [36]


machine and monotonically loaded in in-plane shear while recording force. Shear strain is measured by means of two bonded triaxial strain gauges.

The main disadvantage of this method is the cost and difficulty associated with tubular specimen fabrication and testing. Prohibitive material fabrication costs and the need for specialised testing and gripping equipment have restricted the use of this test method. Actually, a similar test method based on the application of a torque to the specimen has been developed by Tsai [58]. In this case, the test does not make use of a cylindrical specimen, rather of a unidirectional prismatic coupon. For it, the shear moduli G12, G13 and G23 can be determined by conducting selected tests on this simpler specimen, measuring the torque and angle of twist. However, a direct measurement of the overall angle of twist does not yield accurate results because of and effects from the specimen tabs and grips, and thus, a more complex post-processing analysis and correction of the recorded data will be required.

1.5 Elastomers Since the composite material in exam is constituted by an elastomeric matrix, a brief overview about elastomers will be provided too.

Elastomers are typically amorphous polymers, characterized by a continuous Brownian motion of the long and flexible macromolecules at normal temperatures due to thermal agitation. These macromolecules arrange in random coil conformations characterized by a high conformational disorder (when the polymer is undeformed),and are bonded together at relatively large distances. This bonding, named crosslinking, may be both chemical and physical, resulting in a flexible molecular network that produces a soft elastic solid [59]. The most important physical characteristic of elastomeric materials is the high degree of deformability exhibited under the action of comparatively small stresses [60], and the capability of recovering their original shape after being stretched to great extents. The roots of these characteristics have to be searched in the molecular structure and arrangement of the elastomers. To better understand this aspect, the concept of

Figure 1.25 - Torsion of a thin-walled circular tube [37]

| Elastomers 43

enthalpic and entropic elasticity should be introduced. Most solids show an enthalpic-elastic behaviour, in which the elastic recovery after an imposed deformation is driven for the majority by an enthalpy reduction associated to displacement of atoms, that wants to return to their equilibrium positions. Instead, elastomers are characterized by an entropic-elastic behaviour. When an elastomer is stretched, its molecular conformation changes, with the macromolecules moving progressively from a disordered random coil configuration to a more ordered arrangement with chains aligned in the direction of the imposed stretch. Such progressive increase in the order of the system corresponds to a reduction of the entropy of the system, which, at the removal of the imposed deformation, has to be recovered, with the return of the macromolecules to a random coil configuration. A representation of this concept is reported in Figure 1.26 [61]. For a more detailed description of the thermodynamic origin of these concepts, a lot of literature is available, in particular with [59] giving an at the same time an exhaustive clear explanation.

As mentioned, one of the main characteristics of elastomers is the high deformability at low applied stresses. As a matter of fact, typical Young’s modulus for elastomers are in the order of 106 Pa, while the strains at brake can reach and overcome 103 %. This results in a typical stress-strain curve as the one reported in Figure 1.27 [59]. As visible, the curve is highly non-linear, so that the mentioned Young’s modulus can be actually defined in the small strain regime.

Initial state Final state after load removal

Figure 1.26 - Representation of the entropic-elasticity of a cross-linked elastomer. In the initial state a random coil conformation is showed. At the application of the load, chains

stretch with a consequent entropy reduction. At the load removal, the random coil configuration is recovered.


However, some features of the elastomers’ behaviour can be understood only when the response at large deformations is considered, like the strain-hardening that develops at high strains, when elastomer chains are fully stretched. A theory for large elastic deformations is therefore required and it was first developed by Rivlin [62]. Since the treatment of such topic would be too extensive and out of the scopes of this work, a complete description won’t be provided. Strongly summarizing the concepts, elastomers are assumed isotropic in elastic behaviour and very nearly incompressible. The elastic properties of an elastomer can then be explained in terms of a strain energy function, U, based on the strain invariants J1, J2 and J3 defined as [59]:

𝐽 = 𝜆 + 𝜆 + 𝜆

𝐽 = 𝜆 𝜆 + 𝜆 𝜆 + 𝜆 𝜆

𝐽 = 𝜆 𝜆 𝜆

(1.38)

where λi are the principal stretches. This theory essentially offers a mathematical description, based on continuum mechanics, to describe rubbery behaviour. In this approach, stress and strain analysis problems may be solved independently of the microscopic system or molecular concepts and the elasticity theory can be the starting point of any kind of modelling effort as follow [63]:

Figure 1.27 - Typical stress-strain curve of an elastomeric material

| Elastomers 45

𝑈 = 𝑓(𝐽 , 𝐽 , 𝐽 ) (1.39)

which, for incompressible materials like elastomers, becomes:

𝑈 = 𝐶 (𝐽 − 3) (𝐽 − 3) (1.40)

Elastomers can then be defined by as hyperelastic materials, namely materials for which the stress – strain relationship derives from a strain energy function, U. The coefficients, Cij, in these functions should be determined by uniaxial, biaxial and shear test data. The main issue is to determine the most suitable strain energy function able to fit adequately the experimental data. Various forms of the strain energy function for modelling an incompressible and isotropic elastomer exist, with different coefficients, required experimental data and different ranges of validity and/or applicability. Among the most used and known, of great relevance are the Ogden model [64], the Mooney-Rivlin model [62], the Yeoh model [65], the Neo-Hookean model and the Arruda and Boyce model [63,66]. All these models present advantages and disadvantages, as they may be based on fitting procedures or on physically-based considerations. An extensive review about the efficiency of different hyperelastic models has been provided by Marckmann and Verron [67].

46 2-EXPERIMENTAL DETAILS

2. EXPERIMENTAL DETAILS

In the present chapter, the materials and the methods employed in the production and testing of the fibres-reinforced elastomeric composite in exam will be reported. A first section will focus on the material and specimens preparation, followed by the sections regarding the experimental testing for the evaluation of the material properties of interest and for the characterization of the laminates.

2.1 Materials and specimens preparation

2.1.1 Constituent materials

The composite analyzed in the present work is a fibres-reinforced elastomer, constituted by continuous and unidirectional fibres embedded in an elastomeric matrix. Specifically, glass-fibres are used as the reinforcing component, embedded in a silicone-based matrix.

The fibres adopted are obtained from commercial glass-fibres fabrics. Information on the producer are not available. The fabrics are constituted by two sheets of E-glass fibres oriented at ±45°, and maintained in position by cotton stitches. The fabrics adopted in the production of the material are characterized by two areal weights, of 300 g/m2 and 400 g/m2. The required unidirectional layers of fibres are obtained from the separation of the sheets constituting the fabrics. The fibres are not pre-impregnated with any resin. No information are available about the presence of some sizing, and no test has been performed to verify it.

Table 2.1 reports the main properties of glass fibres. The glass fibres density, ρFibres, and elastic modulus, EFibres, have been measured in a previous work [68], the Poisson’s coefficient, νFibres, has been taken from literature [69], the shear modulus, GFibres, has been calculated using the equation for isotropic materials:

𝐺 =𝐸

2(1 + 𝜐 ) (2.1)

| Materials and specimens preparation 47

Property Value Unit of measure

Density, ρFibres 2.663 ± 0.004 g/m3

Elastic Modulus, EFibres 72000 MPa

Shear Modulus, GFibres 30000 MPa

Poisson’s Coefficient, νFibres [69] 0.23 - Table 2.1 - Glass-fibres properties of interest

The matrix of the composite in exam is an elastomeric one, specifically a polydimethylsiloxane, PDMS. PDMS is a chemically crosslinked elastomer with alternating silicon and oxygen atoms as backbone and side methyl groups. Its solidification process consists in an exothermic and irreversible polymerization and crosslinking. The PDMS employed is the Sylgard 184, a commercial PDMS produced by Dow Corning®. It is a two component system, that are the elastomer base (component A) and the curing agent (component B). The exact composition of Sylgard 184 is proprietary, anyway, the materials safety data sheet states that the component A contains mainly dimethyl siloxane, dimethylvinylsiloxy-terminated and dimethylvinylated and trimethylated silica and the component B contains mainly siloxanes and silicones, dimethyl, methylhydrogen, dimethyl siloxane, dimethylvinylsiloxy-terminated, dimethylvinylated and trimethylated silica and methyl-vinylcyclosiloxane [70].

According to the datasheet, prepolymer and curing agent have to be mixed with a mass ratio 10 to 1. The cure reaction begins after mixing the two parts with a pot life, the time in which the material remains processable, of about 1.5 h at 25°C. The progression of the polymerization reaction is evidenced by a gradual increase of the viscosity of the solution. According to the datasheet, the curing can be performed in different ways, either at room temperature for a time of 48 h, or in an accelerated way, by means of heating [71].

Table 2.2 reports the main material properties of the Sylgard 184. The density, ρMatrix, and the elastic modulus, EMatrix, have been measured by T.Bardelli [68], the shear modulus, GMatrix, has been evaluated using Equation (2.1), and the Poisson’s coefficient, νMatrix, has been experimentally measured in a previous thesis work [72]. Note that the mechanical properties of Sylgard 184 change depending on the thermal history followed during the curing [73], but only one thermal history has been employed in the present work. The properties reported in Table 2.2 refer to the thermal history followed, for a 10:1 mix ratio.


Property Value Unit of measure

Density, ρMatrix (cured) 1.004 ± 0.003 g/m3

Elastic Modulus, EMatrix (cured) 3.3 MPa

Shear Modulus, GMatrix (cured) 1.1 MPa

Poisson’s Coefficient, νMatrix (cured) [72] 0.49995 - Table 2.2 - Sylgard184® mechanical properties of interest

It is important to highlight the large difference existing between the values of the constituent materials elastic moduli, that, dealing with FREs, is much higher than the one present in conventional composites.

2.1.2 Laminate preparation

The composite material has been produced employing a casting procedure, in which a hand-layup of the fibres layers is performed.

This method is essentially a hand-layup process, in which the fibres layers are arranged in an aluminium mould with the desired orientation and the mixture of the two matrix components, A and B, is poured into the mould and let to crosslink. The procedure is actually composed by five main steps, as schematically sketched in Figure 2.1:

I. Glass-fibres layers preparation. As previously reported, the layers of unidirectional glass fibres are obtained cutting the fibres tape which constitute the fabric. In the used fabrics, as previously reported, two unidirectional fibres tapes are superimposed and maintained in position by cotton stitches. The unidirectional fibres layers are obtained from the removal of the stitches and the separation of the two sheets. The fibres orientation after the stitch’s removal is maintained using an adhesive tape. This allows the subsequent fibres layers handling. This procedure is necessary to avoid influences and interferences in the mechanical response of the composite, deriving from the presence of the stitches. This is not required for conventional composites, characterized by matrices stiffer than the cotton stitches, but is a requirement for the material in exam characterized by a much softer matrix.

II. Matrix preparation. A ratio elastomer base – curing agent of 10:1 by weight has to be adopted according to Sylgard 184 datasheet [71]. The amounts of elastomer base and curing agent are tuned on the volume of the mould adopted.


The mixture is immediately subjected to a mechanical mixing, performed at Tamb for 10 min. Depending on the amounts of matrix required, the mixing can be performed at different speeds to guarantee a uniform mixture, ranging from 200 to 400 rpm. After the mechanical mixing, the matrix is de-gassed in a vacuum chamber for approximately 20 min at 400 ÷ 500 mmHg, to allow the evacuation of the air entrapped during the mixing.

III. Composite layup. The prepared fibres layers and uncured matrix are manually placed in the mould cavity. To favour a homogenous impregnation of the fibres layers, an alternate deposition of layers and uncured PDMS is performed. The maximum number of fibres layers that can be deposited in the mould is strictly related to the depth of the cavity in use.

IV. Composite de-gassing. The mould filled with fibres sheets and matrix is subjected to a further de-gassing in the vacuum chamber, to remove the majority of the air bubbles entrapped during the layup phase. The degassing is performed with the mould open, for approximately 1 h with a vacuum of 500 ÷ 600 mmHg. At the end of the degassing the mould is closed.

V. Curing. The curing of the matrix is performed in two steps. An initial curing at constant temperature of T = 25°C for 20 h is followed by a curing step in the oven at a temperature of T = 150°C for 1.5 h. It has been investigated that a reduction of the curing time of 4 h, hence reaching an overall time of 16 h at ambient temperature, does not modify the final composite properties.


Different aluminium moulds, identified as moulds 1, 2, 3 and 4 have been used. A representation is reported in Appendix A, within the specific dimensions. Moulds 1, 2 and 3 have a 80x80 mm cavity and differ in the cavity depth, that is of 1, 1.5, and 2 mm respectively. Mould 4 has 350x260 mm cavity and a tunable cavity depth in the range 0.5 ÷ 3.5 mm.

As just described, the composite production method used is based on a manual cutting and placing of the fibres layers and on the matrix mixture casting. As it can be expected, this may introduce some misalignments in the fibres orientation, which could affect the experimental results.

I II

III IV

T=25°C for 20h T=150°C for 1.5h

V

Figure 2.1 - Representation of the steps required for the composite production. I: Glass-fibres layers preparation, II: Matrix preparation, III: Composite layup, IV: Composite de-gassing, V: Curing


Another composite production procedure has been adopted during the experimental activity. This can be intended as a sort of modified vacuum-bag assisted layup, in which a vacuum bag is used to flow the polymer mixture through the piled up fibre layers and favour as well the air-removal and the sample compaction. This method was developed to produce larger laminates (about 290x210 mm) useful to prepare specimens with as high dimensions as the one required for some standard testing procedures.

In the vacuum bag-based method, steps I, II and III remained unaltered with the only difference that the fibre layer layup is performed over a flat plate of aluminium rather than in a mould; that plate surface was preliminary sprayed with PTFE to favour an easier detachment of the sample. A second plate is kept on the top of the composite sample during “vacuum” application and the whole curing process to guarantee the flatness of the sample and to favour the evacuation of the excess of matrix. The base plate is enclosed in a vacuum-bag and both a peel ply and some layers of bleeder are used too, to favour the detachment of the vacuum bag and to soak up the excess matrix. The vacuum was applied with a vacuum pump at different values of vacuum pressure that was maintained for different times, from 30 min up to 20 h. Then the vacuum system was removed and the polymer curing was completed in a oven at 150°C for 1.5 h.

Regarding the time under vacuum, different attempts have been performed. This was required to find the optimal combination in terms of vacuum pression and time under vacuum to control the final sample thickness since no data are available in literature. The methodology followed was then a sort of “trial and error” approach, in which different combinations of vacuum and time were tested to obtain different sample thicknesses. Despite different attempts were tried, a satisfactory combination of vacuum and time was not found. All the samples prepared with this procedure were too thin (with a too small fraction of matrix) and even not homogeneous in the thickness. In some cases, zones rich in entrapped air bubbles were present, and the delamination of some fibres layers was observed in some samples. Therefore, this method was not further considered for the sample production, and the data regarding the prepared samples won’t be reported.

2.1.3 Specimens preparation

Following the procedure reported in the previous section, square or rectangular shaped composite samples were produced. From samples obtained with mould 1, 2 and 3, four specimens 80x12.5 mm can be obtained by punching while, from samples obtained from mould 4, seven specimens 351x25 mm can be cut (Figure 2.2).


The specimen in Figure 2.2a was adopted for the testing of unidirectional laminates with fibres at 90° with respect to the specimen’s length, and for symmetric laminates with fibres at ±30°, ±60° and ±75°. The specimen in Figure 2.2b was adopted for the testing of symmetric laminates with fibres at ±45° and unidirectional laminates with fibres at 45° with respect to the specimen’s length.

The mean fibres volume fraction of the specimens was evaluated from the fibres tapes areal weight, provided by the supplier and experimentally verified and equal to 𝑚 , = 200 g/m2, from the fibres density 𝜌 = 2.663 g/m3 that was experimentally evaluated, and on the measured volume of the sample or specimen:

80mm

12.5mm

351mm

25mm

(a)

(b)

Figure 2.2 - Specimens dimension. (a) Example of UD90, obtained by punching. (b) Example of SAP45, obtained by cutting.


𝑣 =𝑉

𝑉=

𝑚 ,𝜌

𝐿𝑊𝑠=

𝑚 , 𝑛 𝐿𝑊𝜌

𝐿𝑊𝑠

=𝑚 , 𝑛

𝑠𝜌

(2.2)

where L, W and s are respectively the specimen’s length, width and thickness, and 𝑛 is the number of fibres layers.

The summary of the produced laminate samples with the corresponding fibres volume fractions, is reported in Table 2.3 and Table 2.4. They will be identified by the following notation:

- UDθ_i/j = laminate samples with unidirectional fibres oriented at an angle θ with respect to direction of the load that will be applied in a uniaxial tensile test, which will correspond to the specimen longitudinal axis.

- SAPθ_i/j = laminate samples with a symmetric stacking sequence of unidirectional fibres oriented at ±θ with respect to the direction of the load that will be applied in a uniaxial tensile test, which will correspond to the specimen longitudinal axis. These are the conventionally called symmetric angle ply laminates [±θ]ns.

where i identifies the sample code number, and j the sample’s fibres volume fraction. For specimens UDθ, θ is equal to 90° (UD90) and 45° (UD45) and the fibres volume fraction was varied between 0.146 and 0.464. For the specimens SAPθ, θ is equal to ±30°, ±45°, ±60° and ±75° with respect to a direction perpendicular to the loading one. They are respectively identified by the acronyms SAP30, SAP45, SAP60 and SAP75.

Laminate Sample Average Fibres Volume Fraction, 𝒗𝒇 STD Dev. 𝒗𝒇

UD90_A/0.28 0.281 ±0.002

UD90_B/0.29 0.290 ±0.001

UD90_C/0.29 0.289 ±0.001

UD90_D/0.29 0.288 ±0.001

UD90_E/0.29 0.288 ±0.004

UD90_F/0.22 0.216 ±0.001

UD90_G/0.23 0.225 ±0.001


UD90_H/0.24 0.241 ±0.004

UD90_I/0.15 0.151 ±0.001

UD90_J/0.46 0.464 ±0.006

UD90_K/0.44 0.442 ±0.009

UD90_L/0.35 0.352 ±0.005

UD45_A/0.27 0.265 ±0.029

UD45_B/0.20 0.201 ±0.003

UD45_C/0.13 0.134 ±0.002

UD45_D/0.15 0.146 ±0.006

UD45_E/0.20 0.200 ±0.010

Table 2.3 - Summary of the unidirectional laminate samples (UDθ) prepared

Laminate Sample Average Fibres Volume Fraction, 𝒗𝒇 STD Dev. 𝒗𝒇

SAP45_A/0.27 0.269 ±0.008

SAP45_B/0.31 0.308 ±0.003

SAP45_C/0.28 0.276 ±0.009

SAP45_D/0.38 0.382 ±0.017

SAP45_E/0.23 0.234 ±0.014

SAP30_A/0.30 0.296 ±0.005

SAP60_A/0.30 0.301 ±0.001

SAP75_A/0.30 0.304 ±0.001

Table 2.4 - Summary of the symmetric angle ply laminate samples (SAPθ) prepared

Different specimen thickness, in the range between 0.48 mm and 2.31 mm were obtained changing number of fibres layer employed in the sample preparation, which, finally, affected the fibres volume fraction obtained. No end-tabs were required. A gage length of 50 mm was considered for UD90, with a clamped specimen length of 15 mm from each side, while for SAP30, SAP60 and SAP75, a gage length of 60 mm and a clamped specimen length of 10 mm were used. This was required to have

| Testing setup 55

a sufficient analysable area of the specimens, avoiding interferences from specimens clamping, which would hinder fibres movement (see Section 1.4.1 and Figure 1.17). For UD45 and SAP45 a gage length of 210 mm and a clamped specimen length of approximately 50 mm were used.

To allow a subsequent Digital Image Correlation (DIC) analysis for local strains measurement, speckle pattern was created on each specimen surface. First, a thin layer of a commercially available crack detector (VMD 139) was sprayed on a specimen surface. A matte black acrylic paint (VMD 100) has been over sprayed to create a fine random pattern. The use of the crack detector base layer was necessary to reduce the material transparency and to minimise undesired light reflections during the testing coming from the glass fibres. An example of the specimen surface appearance is reported in Figure 2.3, showing a frame taken from the recorded video of the test at the beginning of a tensile test.

2.2 Testing setup Uniaxial tensile tests were performed on an Instron 5967 dynamometer with a load cell of 2kN at a constant nominal strain rate of 𝜀̇ = 0.12 min-1. Nominal forces, P, and crosshead displacements, ΔL, were automatically recorded by the dynamometer.

All the tests were video recorded with a uEye camera UI-5490SE equipped with a photographic lens Nikon 28-105 to perform a DIC analysis for local strains measurement. A video acquisition speed of 5 fps was considered for all the tests.

Post-processing operations of the recorded videos were performed using the VIC-2D digital image correlation software (version 2009.1.0) provided by Correlated Solution Inc. The subset and step sizes were chosen accordingly to the image resolution and speckle dimension/dispersion. The region of interest (ROI) analysed depends on the specific specimen in exam, and it is directly related to the specific fibres orientation. For UD90 specimens, the ROI corresponds almost to the whole specimen surface: only the boundary regions and the zones close to the gripping zone were not included in the ROI. For UD45, SAP30, SAP45, SAP60 and SAP75, a smaller ROI was considered to avoid any edge effect as well any effect of the

Figure 2.3 - Frame of a video reorded test, showing the spakle patter created on a specimen surface


specimen clamping, which hinders fibres movement (see Section 1.4.1). An example of the ROI analysed for the different specimens, characterized by different fibres orientations, is reported in Figure 2.4.

2.2.1 Evaluation of the transversal modulus, E2

As standardized testing method for the evaluation of the transversal modulus, E2, is common use to refer to ASTM D3039 [38] for the experimental determination of the tensile properties of reinforced polymers. It has been considered as a basis for the development of the testing procedure followed. What is fundamental is to guarantee a uniform stress state in the lamina, required to characterize unequivocally the mechanical response of the material. Indeed, differently from isotropic materials, the orthotropy of composite materials may introduce coupling

(a) UD90

(b) SAP30

(c) SAP60

(d) SAP75

(e) SAP45

Fibres orientation Analyzed ROI

Figure 2.4 - Area of interest (ROI) analyzed for each type of specimen. (a) UD90 specimen, (b) SAP30 specimen, (c) SAP60 specimen, (d) SAP75 specimen, (e) SAP45 specimen

x

y

| Testing setup 57

between stresses and strains, which may affect the measured response [16]. The specimen is never composed by just one lamina, because it would be too fragile and difficult to handle, rather it is a stacking of more unidirectional laminae.

In the present study, for the determination of the transversal modulus, E2, UD90 specimens were used (see section 2.1.3), with fibres oriented at 90° to the load direction, as schematically reported in Figure 2.5. Quasi-static tensile tests are performed, where the tensile stress, σx, is evaluated as the force, P, measured by the dynamometer divided by the cross-sectional area in the gage section of the undeformed specimen, A:

𝜎 =𝑃

𝐴 (2.3)

The test is conducted under displacement control. The engineering strains, εx and εy are obtained from the post-processing DIC analysis of the recorded video images, and averaged over the whole ROI. Since the principal material directions 1-2 are aligned with the local reference system x-y, the measured engineering strains εx and εy correspond respectively to ε2 and ε1. In the analysis of UD90 specimens, only the strain in the loading direction, εx, is considered.

Figure 2.5 - Schematic representations of UD90 specimen employed in the transversal modulus, E2, evaluation

12.5mm

x, 2

y, 1

80mm

50mm

x

y z

P

P

2

1

A


The transversal elastic modulus, E2, is evaluated as the slope of the engineering stress-strain curve in its initial linear regime, as schematically reported in Figure 2.6, employing Equation (2.4):

𝐸 =∆𝜎

∆𝜀 (2.4)

For most of the specimens, the measure is performed up to εx = 0.01. For specimens with high fibres volume fractions (vf > 0.35), a reduced εx range is considered, due to a shorter linear portion of the σx -εx curve. Unless otherwise specified, the values of E2 that will be reported are obtained from a linear fitting of the stress-strain curve up to εx = 0.01.

Note that care must be posed in the specimens preparation and machining. It is fundamental to avoid as much as possible fibres misalignments, to guarantee a uniform and homogeneous fibres distribution in the specimens, and to prevent the presence of matrix-rich zones.

2.2.2 Evaluation of the in-plane shear modulus, G12

As anticipated in Section 1.4, three procedures were preliminary considered to experimentally evaluate the in-plane shear modulus, G12: the uniaxial tension of ±45° laminates [74], the uniaxial tension of unidirectional 10° off-axis laminates [32] and the uniaxial tension of unidirectional 45° off-axis laminates [33]. Nevertheless, the experimental approaches actually followed were only the uniaxial tension of ±45° laminates and of unidirectional 45° off-axis laminates. The reason for discarding the uniaxial tension of unidirectional 10° off-axis laminates method will be explained in Section 2.2.2.2. In addition, in the following sections, some other considerations about the uniaxial tension of ±45° laminates will be reported.

Regarding the experimental setup employed in the evaluation of G12, the same equipment and testing parameters presented in Section 2.2 have been adopted.

εx

σx

E2

Figure 2.6 - Schematic representation for the evaluation of the transversal modulus, E2

∆𝜎

∆𝜀

| Testing setup 59

SAP45 specimens and UD45 specimens have been used, characterized by stacking sequences of [±45]ns (with n = 1, 2) and [45]3 respectively.

2.2.2.1 Considerations on the uniaxial tension of ±45° laminates method [74]

For the determination of the in-plane shear modulus, G12, following the standardized procedure ASTM D3518 [74], SAP45 specimens were used. As reported in the standard, symmetric angle ply laminates, with fibres oriented at ±45° to the load direction, have to be tested with a quasi-static tensile test. Recalling the equations of interest presented in Section 1.4.1, the shear stress, τ12, the shear strain, γ12, and the in-plane shear modulus, G12, are determined as:

𝜏 =𝑃

2𝐴=

1

2𝜎 (2.5)

𝛾 = −𝜀 + 𝜀 (2.6)

𝐺 =∆𝜏

∆𝛾 (2.7)

where P is the recorded load, A is the cross-sectional area in the gage section of the undeformed specimen, σx is the engineering tensile stress in the x-direction of the specimen reference system, as represented in Figure 2.7, and εx and εy are the nominal deformation locally measured along the x and y directions. As for the transversal modulus, the local deformations εx and εy are obtained from a DIC analysis of the video recorded images. The in-plane shear modulus, G12, has been evaluated as the slope at low strains of the shear curve obtained from the plotting τ12 versus γ12, up to γ12 = 0.01, as schematically represented in Figure 2.8.

Figure 2.7 - Definition of specimen and material axes [74]

0.01 γ12

τ12

G12

Figure 2.8 - Schematic representation for the evaluation of the in-plane

shear modulus, G12


As reported in ASTM D3518 [74], fibres re-orientation during specimen elongation must be accounted for in the analysis of the data, since, if an excessive fibres angle modifications would occur, the basic assumptions of the test method would be no more valid (see Section 1.4.1). Data to be used for the G12 should therefore be truncated at a maximum shear strain γ12 = 0.05 , corresponding approximately to a fibres re-orientation of 1.5°. It is therefore of interest to evaluate the magnitude of fibres re-orientation during the specimen extension. This can be done by means of a geometric approach. Consider an undeformed lamina at a time t = t0, characterized by an initial width, w0, and with the fibres oriented at an angle θ0, with respect to the loading direction, as reported in Figure 2.9. All the fibres have length d and they can be considered inextensible (EFibres >> EMatrix). As the lamina is stretched, at a generic loading time ti, its width is reduced (w < w0) and the fibres, which do not deform, will be re-oriented accordingly. The actual fibres angle reached can be related to the deformation measured perpendicularly to the loading direction, εy, as:

𝜀 =𝑤 − 𝑤

𝑤=

𝑑 sin 𝜃 − 𝑑 sin 𝜃

𝑑 sin 𝜃 (2.8)

𝜃 = sin 𝜀 + 1 sin 𝜃 (2.9)

For the specific case of fibres oriented at ±45°, the equation reduces to:

𝜃 = sin 𝜀 + 1√2

2 (2.10)

w0 d θ0

θ d w

t = t0

t = ti

Figure 2.9 - Schematic representation of fibers re-orientation in a lamina of the specimen as consequence of the elongation.

P P

| Testing setup 61

2.2.2.2 Considerations on the uniaxial tension of 10° off-axis laminates method [32]

Recalling the concepts introduced in Section 1.4.2, this test method is based on the uniaxial tension of laminate specimens characterized by a stacking of unidirectional plies with fibres oriented at 10° off-axis with respect to the loading direction. As stated by Chamis and Sinclair [32], the 10° off-axis angle is the optimal angle to conduct the test for most of the conventional composites. Still, as furtherly explained by Longana et al. [51], the optimal off-axis angle should be evaluated for the specific composite material in exam, since its value strongly depends on the ratios E1/E2 and E1/G12. Accounting this consideration, a first esteem of the optimal testing angle for the material in exam has been performed. To do so, E1, E2, G12 and ν12 were initially evaluated using Equations (1.2) to (1.5) reported in Section 1.3.2. The esteem was performed considering a fibres volume fraction of 0.3. The estimated values are reported in Table 2.5. Using the reported values, the ratios ε1/εx and ε2/εx between the strains in the principal material directions 1-2 (see Figure 1.18), and the strain, εx, in the loading direction, as well the ratio γ12/εx, were evaluated for several fibres angle, θ, using the following equations.

𝜀

𝜀=

𝑐𝑜𝑠 𝜃1

𝐸− 𝑠𝑖𝑛 𝜃

𝜐𝐸

𝑐𝑜𝑠 𝜃1

𝐸+ 𝑠𝑖𝑛 𝜃

1𝐸

+ 𝑐𝑜𝑠 𝜃𝑠𝑖𝑛 𝜃1

𝐺− 2

𝜈𝐸

(2.11)

𝜀

𝜀=

𝑠𝑖𝑛 𝜃1

𝐸− 𝑐𝑜𝑠 𝜃

𝜐𝐸

𝑐𝑜𝑠 𝜃1


1𝐸


𝐺− 2

𝜈𝐸

(2.12)

𝛾

𝜀=

𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃1

𝐺

𝑐𝑜𝑠 𝜃1


1𝐸


𝐺− 2

𝜈𝐸

(2.13)

The variation of the normalized strains, ε1/εx, ε2/εx and γ12/εx with the off-axis angle, θ, is reported in Figure 2.10. As explained in Section 1.4.2, the optimal fibres angle at which the test should be performed, is the one for which the shear strain in the principal material axes γ12 is maximum.


Property Estimated Value

E1 2.16 GPa

E2 4.71 MPa

G12 1.58 MPa

ν12 0.407

Table 2.5 - Estimated mechanical properties for the glass fibres-PDMS lamina in exam, considering a fibres volume fraction of 0.3

Considering the values reported in Table 2.5, an optimal fibres angle of 0.5° has been found. This off-axis angle results to be excessively small to be experimentally obtained during the specimen preparation and/or machining.

A further esteem of the optimal angle has been performed at a later time once the material properties have been evaluated experimentally within both this thesis project (as far as E2 and G12) and Bardelli PhD thesis project (as far as E1 and ν12) [68]. The properties values are reported in Table 2.6 with their standard deviation for a fibres volume fraction equal to 0.3. The values of E1 is the same reported in Table 2.5: as shown in [68], the measured value of E1 is pretty close to Equation (1.2) prediction. Figure 2.11 reports the normalized strains versus the off-axis angle considering Table 2.6 properties values.

Optimal Angle = 0.5°-10

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

ε ij/ε

x[-

]

Off-Axis Angle, θ [°]

γ12/εx ε1/εx ε2/εx Optimal Angle

Figure 2.10 - Variation of the normalized strains, εij/εx, versus the off-axis angle, θ. Calculated assuming the estimated material properties, reported in Table 2.5

| Testing setup 63

Property Measured Value

E1 2.16 GPa

E2 20 MPa

G12 17 MPa

ν12 2.5

Table 2.6 – Measured mechanical properties of the material in exam, considering a fibres volume fraction of 0.3

Despite the experimental values of E2 and G12 are one order of magnitude higher than the estimated ones, the optimal off-axis angle remains too small to allow the sample production. Accounting these results, the samples required in this G12 evaluation method are impractical to be fabricated. Consequently, this approach has been considered not adequate to characterize the material in exam, and it was discarded in favour of the one proposed by Tsai [33], based on uniaxial tension of unidirectional 45° off-axis laminates.

Optimal Angle = 1.6°

-10

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

ε ij/ε

x[-

]

Off-Axis Angle, θ [°]

γ12/εx ε1/εx ε2/εx Optimal Angle

Figure 2.11 - Variation of the normalized strains, εij/εx, against the off-axis angle, θ. Calculated assuming the measured material properties, reported in Table 2.6


2.2.2.3 Considerations on the uniaxial tension of unidirectional 45° off-axis laminates [33]

The theoretical background relative to the approach proposed by Tsai [33] has been reported in Section 1.4.3. Unidirectional laminate specimen with fibres oriented at 45° to the load direction have been employed in the G12 evaluation, following the Tsai’s approach [33]. The employed in this case have been identified as UD45 specimens, with a length of 351 mm and a width of 25 mm, as reported in Figure 2.2 and in Section 2.1.3, were used.

As for the other cases, quasi-static tensile tests have been performed, with the same testing parameters previously reported. Differently from the method reported in Section 2.2.2.1, there is no necessity to evaluate the shear stress, τ12, and shear strain, γ12, since the Tsai’s approach [33] exploits the transformation equations reported in Section 1.3.2. In the present case, only the applied load, σx, and the local engineering strain in the load direction, εx, are required. The stress is determined as:

𝜎 =𝑃

𝐴 (2.14)

where P is the applied load and A is the cross-section surface of the undeformed specimen. εx is determined by means of a DIC analysis of the video recorded images. The laminate modulus in the loading direction, Ex, can thus be evaluated employing Equation (1.37). As reported by Tsai [33], the in-plane shear modulus can thus be obtained as:

1

𝐺=

4

𝐸−

1 − 𝜐

𝐸−

1 − 𝜐

𝐸 (2.15)

Note that the longitudinal and transversal moduli, E1 and E2, and the Poisson’s ratio, ν12, have to be known a priori.

2.2.3 Laminates mechanical behaviour characterization

In addition to the above detailed evaluation of different laminates E2 and G12 moduli, a characterization of the mechanical behaviour of several laminates having a symmetric stacking sequence of angle plies and differing in fibres orientation was performed too. The laminates specimens characterized by fibres oriented at ±30°, ±45°, ±60° and ±75° to the loading direction, corresponding to SAP30, SAP45, SAP60 and SAP75 respectively, as reported in Table 2.4, were tested.

A quasi-static tensile test under displacement control has been performed, using the same equipment and testing parameters previously described. The force, P, has

| Testing setup 65

been automatically recorded by the dynamometer, while the strains εx and εy have been determined with the DIC analysis of the test video recorded images.

For some tests, a side-view video recording was also performed, with the camera of a smartphone, while the uEye camera was recording the frontal side of the specimen. The side-view videos were used mainly for an almost qualitative evaluation of thickness variation during the test.

In addition to the quasi-static tensile tests, SAP45_E/0.23 specimens have been tested in uniaxial loading-unloading cycles. The approach employed in these cases is summarized in the following:

- Case_A and B = 10 loading-unloading cycles were first performed up to a crosshead displacement of 20 mm, corresponding approximately an overall strain in the loading direction εx = 0.1. The specimen was removed from the clamping system and let to recovery at room temperature for 5 h. Then further 10 loading-unloading cycles were carried out up to a crosshead displacement of 65 mm, corresponding approximately to εx = 0.31 for the Case A, and up to 45 mm of constant displacement, corresponding approximately to εx = 0.21, for the Case B.

- Case_C = loading-unloading cycles with an incremental crosshead displacement of 5 mm, until reaching a final displacement of 70 mm, corresponding approximately to εx = 0.35.

In all three cases, the test was performed at 𝜀̇ = 0.12 min-1.

A single specimen was used for each test case. At the end of the cyclic tests, the specimens were removed from the testing machine and left to recover some days at the end, and then tested up to failure in a quasi-static uniaxial test. In all the tests, at the same nominal strain rate 𝜀̇ = 0.12 min-1.

For the laminate characterization, the quantities of interest were the measured engineering strains, εx and εy, and the applied tensile stress, σx, which has been evaluated as reported in Equation (2.14).

66 3-EXPERIMENTAL RESULTS AND ANALYSIS

3. EXPERIMENTAL RESULTS AND ANALYSIS

In the following chapter, the experimental results obtained will be reported. The chapter will be subdivided into two major sections. The first section will focus on lamina characterization, with the results relevant of the tests for the transversal modulus, E2, and in-plane shear modulus, G12 evaluation The second section will instead focus on the study of laminates behaviour, with the experimental results obtained from the symmetric angle ply laminates specimen, SAPθ. Finally, a small section dedicated to some preliminary Finite Element (FEM) simulations performed will be reported.

3.1 Lamina characterization

3.1.1 Transversal modulus, E2

As reported in Section 2.2.1, quasi-static tensile tests have been carried out on UD90 specimens. Figure 3.1 reports the σx - εx curve for the tested specimens. The dashed lines reported in Figure 3.1 identify the upper limit of the εx range over which E2 has been evaluated. Despite for almost all the UD90 specimens tested, the curve is linear up to εx = 0.02 approximately, to be conservative, E2 is evaluated as the curve slope up to εx = 0.01. Exception to this point applies to specimens UD90_J/0.46 and UD90_K/0.44, for which the range of linearity is limited to a smaller εx range due to their higher fibres volume fractions, which results in a stiffer material. To account this aspect, the transversal modulus, E2, is evaluated as the slope of the linear portion of the curve up to εx = 0.003 for UD90_J/0.46 and up to εx = 0.06 for UD90_K/0.44), as reported in Figure 3.1c. Table 3.1 reports the mean value of the transversal modulus, E2, measured for the prepared laminate samples.

Some variability is present between the curves of specimens from the same sample. This can be reconducted to possible inhomogeneities in the fibres distribution or to local fibres misalignments. No differences have been recorded relative to the number of plies used in the sample fabrication. The main variable to consider in the analysis is the fibres volume fraction, vf, of the samples. A larger variability in the measured modulus E2 for the specimens characterized by an higher vf can be recognized, as visible in Table 3.1, due to the combination of an high fibres content and a reduced specimen thickness.

| Lamina characterization 67

0.00 0.05 0.10 0.15 0.20 0.25 0.300.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

s x [M

Pa]

ex [-]

UD90_A/0.28 UD90_B/0.29 UD90_C/0.29 UD90_D/0.29 UD90_E/0.29

0.00 0.05 0.10 0.15 0.20 0.25 0.300.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

s x [M

Pa]

ex [-]

UD90_F/0.22 UD90_G/0.23 UD90_H/0.24 UD90_I/0.15

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.0500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

s x [M

Pa]

ex [-]

UD90_J/0.46 UD90_K/0.44 UD90_L/0.35

(a)

(b)

(c)

Figure 3.1 - Experimental tensile stress-strain curves σx - εx of the tested specimens. (a) UD90_A/0.28 to UD90_E/0.29, (b) UD90_F/0.22 to UD90_I/0.15, (c) UD90_J/0.46 to UD90_L/0.35. Dotted lines

identify the upper limit of the εx range considered for the E2 evaluation. In (a) and (b), a common upper limit has been considered. In (c), depending on the samples, different upper limits have been

considered; the colours are associated to the corresponding curves


Laminate sample Fibres Volume Fraction, vf [-] Transversal Modulus, E2 [MPa]

UD90_I/0.15 0.151 ± 0.001 6.43 ± 0.66

UD90_F/0.22 0.216 ± 0.001 11.18 ± 0.80

UD90_G/0.23 0.225 ± 0.001 10.87 ± 0.26

UD90_H/0.24 0.241 ± 0.004 11.91 ± 0.55

UD90_A/0.28 0.281 ± 0.002 21.02 ± 0.52

UD90_C/0.29 0.288 ± 0.001 17.52 ± 0.47

UD90_D/0.29 0.288 ± 0.004 18.46 ± 0.60

UD90_E/0.29 0.289 ± 0.001 18.28 ± 1.84

UD90_B/0.29 0.290 ± 0.001 18.39 ± 0.93

UD90_L/0.35 0.352 ± 0.005 29.48 ± 3.69

UD90_K/0.44 0.442 ± 0.009 49.22 ± 4.08

UD90_J/0.46 0.464 ± 0.006 52.88 ± 2.34

Table 3.1 - Summary of the mean value of the transversal modulus, E2

Figure 3.2 reports the value of the transversal modulus, E2, obtained from each tested specimen, versus the fibres volume fraction, vf. The red and black dots refers respectively to tests performed in this thesis and to tests preliminary performed by T. Bardelli for her PhD thesis [68]. The value of elastic modulus of the pure matrix, EMatrix, identified by the blue dot at vf = 0, is reported too. As it can be expected from the usual behaviour of fibres-reinforced composite materials [16], an increasing trend of the transversal modulus with the fibres volume fraction is present: for a fibre volume fraction of about 0.45 the composite transversal modulus exceeds the matrix elastic modulus, EMatrix, of more than one order of magnitude.


Although such increasing trend of E2 with vf is a common feature of fibres-reinforced composites, what is interesting to highlight is that the magnitude of this increase for the material in exam cannot be described by the models usually employed for conventional composites. The simpler model that can be adopted is the springs-in-series (SiS) model reported in Section 1.3.1, for which the E2 can be determined as:

𝐸 =𝐸 𝐸

𝐸 𝑣 + 𝐸 (1 − 𝑣 ) (3.1)

where EMatrix and EFibres are the elastic moduli of matrix and fibres respectively. As shown in Figure 3.3 (blue line), this simple model cannot predict at all the experimental trend obtained. Actually, such model generally fails also in the E2 prediction when conventional composites are considered [16], reason why it is prevalently considered as a lower limit for the esteem of E2 values. As stated in Section 1.3.1, a more reliable model commonly employed to evaluate the trend of E2 vs vf, is the one developed by Halpin and Tsai [16]. As showed in Equations (1.6) and (1.7), this semi-empirical model fits the experimental E2 vs vf data using an empirical reinforcing factor, ξ, as reported in (3.2):

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30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tran

sver

sal M

odul

us, E

2[M

Pa]

Fibres Volume Fraction, vf [-]

Figure 3.2 - Experimentally determined transversal modulus, E2, versus fibres volume fraction, vf

This thesis Previous work EMatrix


𝐸 = 𝐸1 + 𝜉𝜂𝑣

1 − 𝜂𝑣 (3.2)

𝜂 =

𝐸𝐸 − 1

𝐸𝐸 + 𝜉

(3.3)

This model is found to fit reasonably well the E2 values for the majority of the conventional fibres-reinforced composites when a reinforcing factor ξ = 2 is considered, in particular when medium-to-low vf are considered [16]. However, in the case of the composite in exam, the experimental data are not fitted by the Halpin Tsai model with ξ = 2 , as it can be noted in Figure 3.3. A different value of ξ has been searched, through the best interpolation of the available experimental data with the Halpin-Tsai equation: the reinforcing factor ξexp = 12 was obtained, with a mean square error R2 = 0.755. As it can be noted in Figure 3.3, the Halpin-Tsai prediction with this value of ξ (orange line), still does not properly fit the experimental data at high vf, which result to be quite higher than the ones predicted. In addition, such value of the reinforcing factors extremely high for a fibre-reinforced composite (although it can theoretically span from 0 to ∞ [16]).

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160

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Tran

sver

sal M

odul

us, E

2[M

Pa]


This thesis

Previous work

SiS Model

HT _ ξ = 12

HT _ ξ = 2

Figure 3.3 - Experimental data and prediction of E2 dependence on vf according to the springs-in-series model, SiS, (blue line), and Halpin-Tsai model, HT with different reinforcing factor, ξ,

(green and orange line)


These results may be tried to be interpreted considering both the theoretical meaning of the fibres reinforcing factor, and the peculiar nature of the material in exam. The reinforcing factor can be described as a measure of the effectiveness of the reinforcing action produced by the fibres on the matrix [16]. High ξ values are therefore related to a strong reinforcement of the matrix material, low ξ values to a less important reinforcing action. A possible explanation of this result may be found if the elastomeric nature of the matrix is considered. Rubbers are commonly reinforced with the addition of fillers, generally in the form of particles [75,76]. The stiffening effect is not only related to the presence of an almost nondeformable material, but also to filler-elastomer interactions, which reduces the rubber mobility around the filler particles [76]. Further, geometry induced constrains are also taken into consideration especially for high filler contents [59]. In principle, such stiffening effects should be accounted also in the case of the material in exam, even if fibres are used as reinforcement instead of particles. If good fibres-matrix interactions are present and for high fibre content, a stiffer response of the elastomeric matrix, with respect to the unfilled matrix, can be expected, hence EMatrix would increase with the fibres content. Therefore, an esteem of E2 vs vf trend should also account the simultaneous increase of EMatrix with the fibres volume fraction. Practically, these “low mobility rubber” and “confined rubber” effects may be intuitively regarded as a possible contribute to the stiffer response of the composite, which however is not accounted in the usual predicting models, based instead on purely mechanical considerations. Since in literature there is not yet a complete agreement about a possible physical explanation of these effects, particularly for the “low mobility rubber” one, the esteem of the dependency of EMatrix on the reinforcement content is unpractical. An attempt to define an empirical model able to predict the dependence of E2 on the reinforcement content for the material in exam has been performed. The E2 results have been interpolated with different functions, to find the most suitable one. A power function, as the one reported in Equation (3.4), has been chosen to estimate the experimental values.

where A and B are empirical parameters to be obtained from the fitting of the experimental results. The only parameter which has a physical meaning is the 𝐸 lower bound value, at vf = 0, which must obviously be equal to the elastic modulus of the pure matrix EMatrix = 3.3 MPa. The interpolation of the experimental values using Equation (3.4) resulted in:

𝐸 𝑣 = 𝐸 + 𝐴𝑣 (3.4)

𝐸 𝑣 = 𝐸 + 353𝑣 . (3.5)


Figure 3.4 reports the comparison of the E2 experimental data with the best fitting power function found (R2 = 0.818). Differently from the fitting with HT, a good agreement can be observed in this case, with the fitting function able to predict adequately E2 values both at high and low vf. Also from the comparison of the R2 values of the two predictive models, corresponding respectively to 0.755 for HT, and 0.818 for the power function employed, it results that the power function found provide a better prediction of the experimental values.

3.1.2 In-plane shear modulus, G12

As reported in Section 2.2.2, the in-plane shear modulus, G12, of the material in exam has been evaluated by means of two approaches, the uniaxial tension of ±45° symmetric laminates and the uniaxial tension of unidirectional 45° laminates. The following sections will report the results obtained from the two types of experimental tests performed, within some considerations about them.

3.1.2.1 Uniaxial tension of symmetric ±45° laminates

Figure 3.5 reports the full τ12-γ12 curves for the SAP45 specimens tested. The shear modulus is evaluated as the slope in the initial linear portion of the shear curve. Figure 3.6 reports the magnification of the initial portion of the curve where G12 has been evaluated. As visible in these figures, and in particular in the enlarged initial portion reported in Figure 3.6, the linearity extends up to approximately γ12 = 0.02,

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

70

80

90

100

E 2 [M

Pa]

vf [-]

Figure 3.4 - E2 values versus vf. Black dots identify the experimental values, red line the fitting function developed


with some variations depending on the fibres volume fraction of the sample considered. G12 has been evaluated as the slope of the shear curve up to γ12 = 0.01 for all the specimens. Table 3.2 reports the mean measured G12 for the tested specimens, with the corresponding average fibres volume fractions, vf.

Laminate Sample Fibres Volume Fraction,

vf [-] In-Plane Shear Modulus,

G12 [MPa]

SAP45_E/0.23 0.227 ± 0.009 13.37 ± 0.51

SAP45_A/0.27 0.269 ± 0.008 13.96 ± 0.93

SAP45_C/0.28 0.276 ± 0.009 13.89 ± 0.7

SAP45_B/0.31 0.308 ± 0.003 17.25 ± 2.43

SAP45_D/0.38 0.385 ± 0.017 23.53 ± 2.24

Table 3.2 - Summary of the measured in-plane shear modulus, G12


(e)

(d) (c)

(a) (b)

Figure 3.5 - Experimental shear stress-strain curves τ12-γ12 of the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28, (d) SAP45_D/0.38, (e) SAP45_E/0.23

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.101

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3

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67

8

910

1112

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t 12 [

MPa

]

g12 [-]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.101

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1112

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t 12 [

MPa

]

g12 [-]

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t 12 [

MPa

]

g12 [-]

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1112

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t 12 [

MPa

]

g12 [-]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.101

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45

67

8

910

1112

13

14

t 12 [

MPa

]

g12 [-]


(e)

(d) (c)

(a) (b)

Figure 3.6 - Magnification of the initial linear portion of the experimental shear stress-strain curves τ12-γ12 of the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28,

(d) SAP45_D/0.38, (e) SAP45_E/0.23

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g12 [-]

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g12 [-]

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]g12 [-]


Figure 3.7 reports the G12 values obtained plotted versus the fibres volume fraction, vf. As expected, an increase of the shear modulus is observed as the fibres volume fraction increases. In Figure 3.7 also the shear modulus of the pure matrix, GMatrix, identified by the blue dot at vf = 0 is reported.

On the base of micromechanical considerations (see Section 1.3.1), similarly to E2, also for G12 the springs-in-series (SiS) model can be used to give a first esteem of its trend with respect to vf, still with the same considerations previously reported regarding its reliability [16]. The inadequacy of the model has been confirmed also in the case of the material in exam, as it is clearly visible in Figure 3.8 where the springs-in-series (SiS) model prediction, corresponding to the blue line, is reported together with the measured values of G12 .

To predict the G12 vs vf trend, the Halpin-Tsai (HT) model can be employed also in this case. For G12, from Equation (1.6), the general equation of the HT model becomes:

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35

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

In-P

lane

She

ar M

odul

us, G

12[M

Pa]


Figure 3.7 - Experimentally determined in-plane shear modulus, G12, versus fibres volume fraction, vf, of SAP45 specimens. Blue dot represents the shear modulus of the pure matrix


𝐺 = 𝐺1 + 𝜉𝜂𝑣

1 − 𝜂𝑣 (3.6)

𝜂 =

𝐺𝐺 − 1

𝐺𝐺 + 𝜉

(3.7)

where η and ξ have the same meaning previously described, and GFibres and GMatrix are the shear modulus of the fibres and of the pure matrix respectively.

Literature [16] reports that, for vf < 0.5, a generally suitable value of ξ for the G12 prediction is equal to 1, for most of the common fibres arrangement in space. However, as for E2, a first attempt to fit the experimental results considering this reinforcing factor value does not yield an adequate fitting. Indeed, as it can be seen in Figure 3.8, the green curve representing the Halpin-Tsai prediction of G12 values with ξ = 1, falls much below the experimental results. Thus, a direct interpolation of the experimental G12 has been performed, obtaining a ξExp = 31.4 (with a mean square error R2 = 0.739). As for the case of E2, such ξExp is extremely high. Nevertheless, in this case the experimental data are reasonably well fitted by the HT model with ξExp = 31.4.

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50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

In-P

lane

She

ar M

odul

us, G

12[M

Pa]


ExperimentalSiS ModelHT _ ξ = 1HT _ ξ = 31.4

Figure 3.8 - Experimental data and prediction of G12 dependence on vf according to the springs-in-series model, SiS, (blue line), and Halpin -Tsai model, HT with different

reinforcing factor, ξ, (green and orange line)


As already said, the results reported above have been evaluated considering the initial portion of the shear stress-strain curve. As it can be noted in Figure 3.5, for all the tested materials, the strain range considered for G12 evaluation is much smaller than the strain range of the whole shear stress-strain curve. This point is directly related to the elastomeric nature of the matrix, which allows to reach deformations much higher than the ones typically showed by conventional composites, based on a rigid polymer matrix, as shown in Figure 3.9, where curves taken from ASTM D3518/D3518M [74] are reported.

The higher matrix deformability of the composite material in exam allows then to explore regions of the stress-strain curves that generally cannot be investigated in conventional composites, which fail earlier. The reported experimental results can be further investigated considering the concepts and hypothesis introduced in Section 1.4.1. The first observation is related to the number of plies employed in sample fabrication. No relevant differences are present between the measured shear moduli G12 of sample SAP45_C/0.28 (4 plies) and SAP45_A/0.27 (8 plies), characterized by similar vf, confirming the observation of Kellas et al. [77]. The only difference present is the slightly lower shear stress at failure with respect in the case of SAP45_C/0.28 (4 plies), which however is of no concern since the present work focuses only on the shear modulus evaluation.

A second aspect investigated is related to the analysable area of the specimens. As reported in Section 1.4.1, the effect of specimen clamping must be considered when longitudinal and transversal strains, εx and εy, are evaluated with DIC analysis. This clamping effects in the zones adjacent to the gripping fixture, which hinder fibres movement, have been experimentally verified. The values of εx and εy have been evaluated far and close to the fixtures, observing significant differences between the values of the strain measured in the central and in the side zones of the specimen.

Figure 3.9 - Typical shear stress-strain curves of conventional FRP composites with: (a) low-ductility matrix, (b) ductile matrix [74]

(a) (b)


Thus a reduced Region Of Interest (ROI) for the DIC analysis has been adopted, focused in the central zone of the specimens.

Some important observations must be introduced regarding the validity of the present testing approach. This point can be investigated from two points of view that actually are interconnected: from the Mohr’s circle of strains [41] and from the fibres re-orientation.

Starting from the Mohr’s circle of strains, Petit [41] reports that the state of pure shear stress is not actually present in the laminae. This point can be analysed if the Mohr’s circle of strains is built with the data obtained from the specimens. The Mohr’s circle of strain can be built using the measured values of longitudinal and transversal strains, εx and εy, and considering negligible the in-plane shear strains, γxy. This last point has been experimentally verified. The centre ε coordinate, εc, and radius, R, of the Mohr’s circle can be evaluated using Equations (3.8) and (3.9) .

𝜀 =𝜀 + 𝜀

2 (3.8)

𝑅 = (𝜀 − 𝜀 ) + 𝛾 (3.9)

In principle, if a state of pure shear is achieved, the position of the center of the Mohr’s circle of strain should coincide to the point (0; 0) in the γxy/2 – ε plane, that is εx and εy must be equal and opposite in sign (εx = -εy). From an experimental point of view this is never achieved, rather, as the specimen is progressively stretched, the difference between the two measured strains increases. This means that the position of the center progressively shifts on the ε axis, and therefore compressive stresses develop in the laminae. A representation of this concept is reported in Figure 3.10, where the Mohr’s circle of strains obtained from the measured εx and εy, evaluated at five test times, t, are reported. Table 3.3 reports the corresponding center coordinates C(εc; γxy,c/2) of the Mohr’s circles reported in Figure 3.10 and the corresponding radius, R. As it can be observed, the circle center progressively shifts towards left, meaning that some compressive stresses actually are present in the laminae, and that therefore, a pure state of shear is not maintained during the whole test. This may result even more clear if the abscissa coordinate of the center, εc, is plotted versus the test time, t, as reported in Figure 3.11.


t = 10 s t = 30 s t = 50 s t = 100 s t = 150 s

C(εc; γxy,c/2) -0.0003; 0 -0.0025; 0 -0.0060; 0 -0.0234; 0 -0.0565; 0

R 0.0171 0.0522 0.0889 0.1879 0.3026

Table 3.3 - Mohr's circle of strains center coordinates, C(εc; γxy,c/2) and radius, R, evaluated at increasing times for a SAP45_B/0.31 specimen

γxy/2

ε

Figure 3.10 - Mohr's circles of strains evaluated at increasing test time for a SAP45_B/0.31 specimen. The dots on the ε-axis identify the circle’s centers

t = 10 s t = 30 s t = 50 s t = 100 s t = 150 s


At the beginning of the test, εc can be considered approximately equal to 0, as it can be observed in Figure 3.11, meaning that the compressive stresses present in the laminae are negligible. However, as the time increases, it can be observed that εc progressively reduces, namely the Mohr’s circle center shifts and compressive stresses develop. Since for G12 evaluation is fundamental to maintain a state of stress as close as possible to the shear one, only the initial portion of the test can be considered reliable, where compressive stresses are still negligible.

More accurate indications regarding the shear strain range in which this approach is valid can be obtained if the fibres re-orientation is evaluated. This must be evaluated to ensure that the range in which G12 has been determined falls in the validity region of the shear curve. As reported by ASTM D3518/D3518M [74], the assumptions over which the approach is based can be considered valid up to a load induced fibres orientation of maximum 1.5°. Fibres re-orientation is evaluated by means of Equation (2.10). Equation (2.10) has been experimentally verified. From the evolution of the fibres angle, θ, with respect to the specimen elongation, represented by the measured longitudinal strain εx, the shear strain γ12 (θ = 1.5°) is traced back: it corresponds to the maximum shear strain γ12Max, until which the shear curve can be considered valid. Figure 3.12 reports the evolution of the fibres angle, θ, with respect to εx for the tested specimens. Significant fibres re-orientation occurs during specimens elongation, with fibres angles that decreases from 45° to

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Abs

ciss

a co

ordi

nate

of t

he c

ente

r, ε c

[-]

Test Time, t [s]

Figure 3.11 - Evolution of the position of the Mohr's circle of strain, εc, with respect to the test time, t, for SAP45_B/0.31 specimen. The colored dots refer to the same times

reported in Figure 3.10


approximately 15 ÷ 20° when a longitudinal strain of about 0.35 is applied. Such high fibres re-orientation is not generally observed in conventional FRPs, due to the

(a) (b)

(c) (d)

(e)

Figure 3.12 - Fibres angle, θ, change with respect to the longitudinal strain, εx, for the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28, (d) SAP45_D/0.38, (e)

SAP45_E/0.23

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fact that these composites fail at smaller strains. This point will be analyzed further in Section 3.2.1.

Table 3.4 reports the γ12Max = γ12 (θ = 1.5°), with the corresponding test time, tMax = t (γ12 = γ12Max). As it can be observed from the reported data, a fibres angle re-orientation of 1.5° corresponds to a shear strain of approximately γ12 = 0.05. Therefore the above proposed method for G12 evaluation can be considered valid and the evaluated G12 values reliable, since the shear strain range over which the shear modulus has been evaluated, i.e. γ12 = 0 ÷ 0.01, falls in the strain zone where the shear test is valid. In addition, these results can be related also to the Mohr’s circle of strains reported in Figure 3.10, and to the shift of its center, εc, with test time, represented in Figure 3.11. Indeed a significant εc shift develops at times longer than about 15 s, a time which is coherent with the evaluated times tMax reported in Table 3.4.

Laminate Sample Fibres Volume Fraction, vf [-] γ12Max [-] tMax [s]

SAP45_A/0.27 0.269 ± 0.008 0.053 ± 0.001 15.9 ± 0.7

SAP45_B/0.31 0.308 ± 0.003 0.053 ± 0.002 15.5 ± 1.0

SAP45_C/0.28 0.276 ± 0.009 0.051 ± 0.001 12.3 ± 0.8

SAP45_D/0.38 0.385 ± 0.017 0.053 ± 0.001 13.9 ± 2.0

SAP45_E/0.23 0.227 ± 0.009 0.053 ± 0.001 13.3 ± 1.3

Table 3.4 - Maximum shear strain, γ12Max, and test time, tMax, below which the shear curve can be considered valid

3.1.2.2 Uniaxial tension of 45° off-axis laminates

Figure 3.13 reports the σx - εx curve for the UD45 specimens tested, with the dashed lines identifying the upper limit of the εx range over which Ex, namely the axial laminate stiffness in the load direction, has been evaluated, equal to εx = 0.01. Table 3.5 reports the average measured values of the axial laminate stiffness, Ex, for all the tested specimens and the corresponding value of in-plane shear modulus, G12, obtained from Equation (1.36).


1

𝐺=

4

𝐸−

1 − 𝜐

𝐸−

1 − 𝜐

𝐸 (1.36)

The required values of E1 have been evaluated using the rule of mixture (see Section 1.3.1),

𝐸 = 𝐸 𝑣 + 𝐸 𝑣 (1.2)

while the values of E2 have been evaluated using the power function model developed in Section 3.1.1. The Poisson’s ratio, ν12, has been maintained constant (ν12 = 2.5) for the sake of simplicity, since it was observed that changes in its value have negligible effects on the final result. Figure 3.14 reports the evaluated G12 with respect to the fibres volume fraction of the tested specimens.

Laminate sample Transversal modulus, E2

[MPa]

Laminate Modulus, Ex

[MPa]

In-Plane Shear Modulus, G12

[MPa]

UD45_A/0.27 14.83 11.98 ± 4.47 6.89 ± 2.82

UD45_B/0.20 8.56 16.33 ± 1.72 6.66 ± 1.12

UD45_C/0.13 5.01 16.92 ± 3.10 6.15 ± 1.54

UD45_D/0.15 5.78 15.41 ± 3.40 5.39 ± 1.65

UD45_E/0.20 8.56 27.43 ± 4.90 10.76 ± 2.79

Table 3.5 - Summary of the calculated (power function model, see Section 3.1.1) transversal modulus E2, measured laminate stiffness, Ex, and evaluated in-plane shear

modulus, G12


(e)

(d) (c)

(a) (b)

Figure 3.13 - Tensile stress-strain curve σx-εx of the unidirectional 45° off-axis specimens tested. (a) UD45_A/0.27, (b) UD45_B/0.20, (c) UD45_C/0.13, (d) UD45_D/0.15, (e)

UD45_E/0.20

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1.1s x

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]

ex [-]

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s x [M

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ex [-]

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0.1

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1.1s x

[MPa

]

ex [-]


As it can be observed in Figure 3.13, and in Figure 3.14, a large variability is present between specimens from the same laminate sample. This can be considered the result of a combination of different factors. First of all, specimens belonging to the same laminate sample have a different vf, in particular for laminate with high fibre volume fraction. This could be related to some experimental difficulties during laminate fabrication encountered in the first uses of the type 4 mould, which resulted in inhomogeneous sample’s thickness. In addition, UD45 samples have been prepared using only three fibres layers. This aspect, combined with the complex handling of the fibres layers, resulted in samples characterized by some defects: in particular, matrix-rich zones and fibres misalignments were present with various extents in each sample. The reduced number of fibres layers employed, combined with the presence of these defectivities, led to variability in the experimental results.

A second aspect to account when considering these data, is related to the testing approach itself. As reported in Section 1.4.3, similarly to the uniaxial tension of 10° off-axis laminates, this approach has the intrinsic problem of the influence of end constrains, which lead to a non-uniform state of stress in the specimen and the consequent in-plane bending. This in-plane bending was experimentally observed

0

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8

10

12

14

16

18

20

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

In-P

lane

She

ar M

odul

us, G

12[M

Pa]


UD45_A/0.27

UD45_B/0.20

UD45_C/0.13

UD45_D/0.15

UD45_E/0.20

Figure 3.14 - Experimentally determined in-plane shear modulus, G12, versus fibres volume fraction, vf, of UD45 specimens


also in the testing of the UD45 samples: Figure 3.15 reports the specimen bending observed in some tests. As it can be noted, the in-plane bending that develops is relevant, hence data from UD45 specimens should be critically evaluated.

In principle, as reported in literature [16,44,46], if the specimen is long and slender, with a sufficiently high ratio length/width, l/w, in the center of the specimen the deformation would be similar to the shearing and extension of an un-restrained

t = t0 t = t0 t = t0 t = t1 t = t1 t = t1

(a) (b) (c)

x y

Figure 3.15 - Observed in-plane bending for some UD45 specimens. Images on the left show the initial shape of the specimens at time t0, images on the right the deformed shape at a time t1. Specimens from laminates (a) UD45_A/0.27C, (b) UD45_B/0.20, (c) UD45_C/0.13


lamina (see Figure 1.19) [16]. In the present case, the specimens employed were characterized by a l/w ratio equal to 8. This value was expected to be high enough for the testing of unidirectional laminates with fibres at 45° to the load direction. As a matter of fact, Pindera [44] reports that a l/w value around 10 should be sufficient to avoid significant errors.

However, when the present results are compared to the experimentally measured values of G12, obtained by means of the standardized method ASTM D3518/D3518M [74], some discrepancies can be observed. Figure 3.16 reports all the experimentally determined values of G12 versus vf. Black dots identify the data obtained from the unidirectional laminates (UD45) employed in the present approach, red dots the data obtained from the symmetric laminates (SAP45) employed in the previously approach (see Section 3.1.2.1). Data coming from UD45 specimens result lower than it could be expected. For example, most of the data obtained in the fibres volume fraction range vf = 0.2 ÷ 0.3 are lower with respect to the ones obtained in the same range from SAP45 specimens. Similarly, at lower vf, higher G12 values would be expected if the trend of data coming from SAP45 specimens is considered. The reasons for these discrepancies may be reconducted to the combination of a reduced number of fibres layers and to the presence of resin-rich zones in the specimens, which would result in lower values of the measured modulus. It should be considered too, that literature [16,44] reports a dependency of the optimal l/w ratio

0

5

10

15

20

25

30

35

40

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

In-P

lane

She

ar M

odul

us, G

12[M

Pa]


SAP45 Data

UD45 Data

Figure 3.16 - Experimental in-plane shear modulus, G12, versus fibres volume fraction, vf. Summary of all available data.

| Laminate analysis 89

on the degree of anisotropy, of the composite material, measured by the ratio E1/E2. Since the composite material in exam is characterized by a significant E1/E2 value, there is also the possibility that the l/w ratio employed for the UD45 specimens was not sufficiently high. However, investigations about this aspect were not performed. Indeed, the present approach was initially thought as a second option to evaluate the in-plane shear modulus, G12, of the composite material in exam, in addition to the standardized method ASTM D3518/D3518M [74]. Even though not being a standardized methodology, it was thought it would be able to bring reliable results without the need of complex fixture/testing apparatus, and with the further benefit of an easier and less expensive, in terms of material, sample fabrication. However, in view of the issues observed in specimen deformations, and of the obtained experimental results, the use of this approach should be accurately considered. A large number of aspects must be considered and/or designed at priori, like the choice of the l/w ratio or the adoption of rotating fixtures to avoid in-plane bending. Accounting these considerations, the following approach, and the derived results, has not been considered fully reliable, at least for the particular composite material in exam, and the use of the standardized method ASTM D3518/D3518M, and of its results, would be preferable in the evaluation of G12.

3.2 Laminate analysis The investigations about the mechanical behaviour of laminates will be reported in the present section. Here will be studied the behaviour at strains much higher than that investigated in Section 3.1.2, where the material shows a highly non-linear behaviour, qualitatively resembling that of an elastomer. From the atypical results obtained from these additional studies, the response of symmetric laminates with different off-axis angle was thought to be of interest. New SAPθ samples have been produced, specifically SAP30_A/0.30, SAP60_A/0.30 and SAP75_A/0.30, as reported in Section 2.1.3. These new samples have been tested with the same approach employed for SAP45 specimens. Furthermore, as reported in Section 2.2.3, some SAP45_E/0.23 specimens have been tested with load-unload cycles, to possibly obtain a more comprehensive analysis of the material response.

The present sub-chapter will focus on the experimental observations and results obtained from these additional investigations. A first section will be dedicated to the quasi-static tensile tests performed, while a second section will focus on the results obtained from the loading-unloading cyclic tests.


3.2.1 Quasi-static tensile tests

To investigate the peculiar response of SAP45 laminate sample, tensile tests were performed and the curves σx – εx have obtained. No additional data were required

(e)

(d) (c)

(a) (b)

Figure 3.17 - Experimental tensile curves σx – εx of the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28, (d) SAP45_D/0.38, (e) SAP45_E/0.23

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450

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s x [M

Pa]

ex [-]0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

0

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s x [M

Pa]

ex [-]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450

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s x [M

Pa]

ex [-]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450

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s x [M

Pa]

ex [-]

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s x [M

Pa]

ex [-]


with respect to the ones already recorded for the G12 evaluation. Figure 3.17 reports the σx – εx curves obtained from the each SAP45 specimen.

As it can be noted from the comparison with Figure 3.5, the particular shape of the τ12 – γ12 curves is retained also when σx – εx curves are considered. Accounting this appearance, these curves can be qualitatively subdivided into three main zones, as reported in Figure 3.18:

- Zone 1 = an initial region characterized by a stiff laminate response. It extends for a reduced strain range with respect to the whole curve, and remains linear up to approximately εx = 0.025, with some variations depending on the fibres volume fraction considered. This zone includes the region in which G12 has been previously evaluated.

- Zone 2 = an intermediate “softened” region, characterized by a less stiff response with respect to zone 1. It extends for wide strain range, with a curve slope that remains almost constant in a relatively wide range of strain.

- Zone 3 = a final stiffening region, characterized by a stiff laminate response. It develops at high εx levels, with a progressive increase of the curve slope.

Figure 3.18 - Tensile curves σx – εx of a SAP45_B/0.31 specimen. Three different zones can be identified: initial region at low εx (Zone 1), intermediate "softened" region (Zone 2) and final

stiffening region at high εx (Zone 3)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400

2

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18

20

s x [M

Pa]

ex [-]

Zone 1

Zone 2

Zone 3


To describe the curves shape, some preliminary observations can be introduced. First of all, a similarity with the tensile curves of the pure matrix can be observed. Figure 3.20 reports the nominal σx – εx curve of the pure Sylgard 184, which can be compared with that of SAP45 (see Figure 3.18). If the shape of Sylgard 184 curve is compared to that of SAP45, resemblances between the intermediate softened region (zone 2) of the composite and the initial softer response of Sylgard 184, and between the final stiffening zone (zone 3) and the strain-hardening behaviour of the elastomeric matrix are evident. On the other hand, zone 1 is not present in the Sylgard 184 curve. Other affinities can be observed from the comparison of σx – εx curves of filled elastomers, reported in Figure 3.19 [78]. There, also the initial stiff region can be recognized, with a curve trend resembling the one of the composite in exam. Note however that these tensile curves refer to particle-reinforced elastomer, in which different phenomena should be accounted for. Still, this may lead to think to the possible presence of reinforcement-matrix interactions, like the

Figure 3.19 - Example of tensile stress-strain curves σx – εx of particle-filled PDMS [78]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

1

2

3

4

5

6

7

8

9

10

s x [M

Pa]

ex [-]

Figure 3.20 - Tensile stress-strain curve σx – εx of pure Sylgard 184


previously mentioned “low mobility rubber”, which may result in the stress-softening effect observed in zone 2, or rubber confinement effects. Therefore, not only the typical behaviour of conventional FRP should be accounted, with the main dependency on the fibres orientation, but also the strain induced phenomena occurring in the elastomeric matrix which interacts with the reinforcing phase: these phenomena can be investigated by means of loading-unloading tensile tests.

To try to identify the contributes of fibres orientation on the overall response, SAP30_A/0.30, SAP60_A/0.30 and SAP75_A/0.30 specimens have been tested, with the resulting σx – εx curves reported in Figure 3.21 All these specimens were characterized by a fibres volume fraction of approximately vf = 0.3, therefore the results obtained from these specimens will be compared to the tensile curves of SAP45_B/0.31, characterized by a similar vf.

(c)

(b) (a)

Figure 3.21 - Experimental tensile curves σx – εx of the tested specimens: (a) SAP30_A/0.30, (b) SAP60_A/0.30, (c) SAP75_A/0.30

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.130

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s x [M

Pa]

ex [-]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

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8

s x [M

Pa]

ex [-]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

s x [M

Pa]

ex [-]


Figure 3.22 reports the comparison of the σx – εx curves of the SAPθ specimens tested (a mean curve is considered for each laminate sample). The three zones previously described for SAP45 can be recognized also in the stress-strain curve for the SAPθ specimens, even though the extension and magnitude of these zones is modified. For SAP30_A/0.30, the softened zone is not so evident. On the other hand, for SAP75_A/0.30, the stiffening zone is hardly recognizable, while the softened zone 2 extends on a much wider strain range. Further, a difference in the slope of the curves, at low strains can be observed: it can be directly reconducted to the different fibres orientation of the different specimens. Recalling the concepts introduced in Section 1.3.2, fibres angles, θ, determine the laminate response. A lower value of θ, corresponding to fibres more oriented in the loading direction, would produce an overall stiffer response of the laminate since fibres contribution to the load carrying capacity is higher, as in the case of SAP30_A/0.30. On the opposite, a higher θ value results in a softer material, since fibres contribute to a lesser extent, as in the case of SAP75_A/0.30. To have a better perception of the presence of the 3 zones, the tangent modulus, ET, of the tensile curves has been evaluated in different εx ranges in which the experimental curve showed a linear trend. Figure 3.23 to Figure 3.26 report the values of ET measured in different strain ranges, versus the mean value of each strain range considered for the SAPθ specimens. For SAP30_A/0.30, represented in Figure 3.23, a tangent modulus reduction can be observed, followed by a strong increase. The same trend can be observed in Figure 3.25, relative to

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

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20

22

24

s x [M

Pa]

ex [-]

SAP30_1 SAP45_2 SAP60_1 SAP75_1

Figure 3.22 - Summary of the tensile curves σx – εx for SAPθ specimen


SAP60_A/0.30. For SAP75_A/0.30 reported in Figure 3.26, the third zone, although hardly recognizable has been actually measured, since ET begins to increase at high εx, but probably due to the premature failure of the specimens, a clear increment is not completely visible. On the other hand, the differences in the extension and magnitude of the three zones for the SAPθ specimens can be better caught in Figure 3.27, where the ET vs εx curves of the different SAPθ are compared.

Figure 3.23 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP30_A/0.30 specimens (identified by different colors)

0

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100

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200

250

300

350

400

450

0 0.02 0.04 0.06 0.08 0.1 0.12

E T[M

Pa]

εx [-]


0

2040

6080

100

120140

160180

200

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

E T[M

Pa]

εx [-]

Figure 3.24 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP45_B/0.31 specimens (identified by different colors)

0

10

20

30

40

50

60

70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

E T[M

Pa]

εx [-]

Figure 3.25 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP60_A/0.30 specimens (identified by different colors)


0

50

100

150

200

250

300

350

400

450

0 0.1 0.2 0.3 0.4 0.5 0.6

E T[M

Pa]

εx [-]

SAP30_A/0.30

SAP45_B/0.31

SAP60_A/0.30

SAP75_A/0.30

Figure 3.27 - Comparison of the tangent modulus, ET, measured at different longitudinal strains, εx, for some SAPθ specimens

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

E T[M

Pa]

εx [-]

Figure 3.26 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP75_A/0.30_A/0.30 specimens (identified by different colors)


To examine more in details the possible roots of these regions, some additional experimental observations should be introduced. From the analysis of the recorded test videos, what was initially observed for SAP45 specimens, was that the specimens elongation was always accompanied by a relevant specimen’s width reduction, as it can be observed in Figure 3.28a, where the initial and almost final frames of the recorded test video of a SAP45_B/0.31 specimen are reported. It is evident that the specimen width strongly reduces, reaching in most of the cases a final width value, wf, halved with respect to the initial one, w0 as shown in Figure 3.28b, where the evolution in time, t, of the specimen’s width, w, is reported. The comparison of the two images in Figure 3.28a, clearly highlights the significant fibres re-orientation described in Section 2.2.2.1 that occurs during the specimen elongation. Plotting the absolute value of the measured longitudinal and transversal strains, εx and εy, versus the test time, t, (see Figure 3.28c) it can be observed that up to about 50 s, |𝜀 | and 𝜀 , are almost equal, but at higher times, the two values begin to diverge, with 𝜀 values progressively higher than |𝜀 |. This point was already underlined in Section 3.1.2.1, observing the time left-shifting of the strains Mohr’s circle center point on the 𝜀 axis.

In the same way, the response of SAP30_A/0.30, SAP60_A/0.30 and SAP75_A/0.30 specimens has been investigated. Figure 3.29 reports the initial and almost final frames of the recorded test videos of these SAPθ specimens while in Figure 3.30 the corresponding width evolution in time are plotted. Figure 3.31 reports the evolution in time of the absolute value of the measured strains, |𝜀 | and 𝜀 for the different specimens. It is interesting to observe the different behaviours of these laminates. Similarly to SAP45 (see Figure 3.28c), 𝜀 measured for SAP30_A/0.30 is higher than |𝜀 |, as it can be noted in Figure 3.31a. Conversely, for SAP60_A/0.30 and SAP75_A/0.30, as shown in Figure 3.31b and Figure 3.31c, |𝜀 | is always higher with respect to 𝜀 . This difference is particularly evident for SAP75_A/0.30 specimen, for which the measured 𝜀 are almost 1/10 of |𝜀 |, as the frame comparison reported in Figure 3.29c already suggests. The differences between |𝜀 | and 𝜀 are lower in the case of SAP60_A/0.30. However, it should be noted that for SAP60_A/0.30, the rate of increase in time of the two strains are quite different: 𝜀 shows an increasing rate progressively higher in time, while |𝜀 | shows an initially high increasing rate followed by a reduction with respect to the initial trend. This point should not be overlooked, since it will turn out to be a contribution to other laminate features.


x

y

w0

wf

t = t0 t = tf

(a)

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0,60

ex

e i

[-]

t [s]

ey

(b)

(c)

Figure 3.28 - (a) First and almost final frame of the video recording of a SAP45_B/0.31 specimen under tensile test. (b) Measured width reduction in time of the specimen

reported on the left. (c) Evolution in time of the absolute value of the measured εx and εy for the specimen on the left.


x

y

w0 wf

t = t0 t = tf

x

y

w0

wf

t = t0 t = tf

x

y

w0

wf

t = t0 t = tf

(a) (b) (c)

Figure 3.29 - First and almost final frames of the video recording of the tensile test of: (a) SAP30_A/0.30 specimen, (b) SAP60_A/0.30 specimen, (c) SAP75_A/0.30

specimen.


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w [m

m]

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w0wf(c)

(b) (a)

Figure 3.30 – Time evolution of the width, w vs t, of a: (a) SAP30_A/0.30 specimen, (b) SAP60_ A/0.30 specimen, (c) SAP75_ A/0.30 specimen


0 50 100 150 200 250 300 350 4000.0

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eye i [

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t [s]

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ey

e i [

-]

t [s]

ex

(c)

(b) (a)

Figure 3.31 - Time evolution of the absolute value of the measured longitudinal and transversal strains, |εx| and |εy| of a: (a) SAP30_A/0.30 specimen, (b) SAP60_A/0.30

specimen, (c) SAP75_A/0.30 specimen

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.80

ei [

-]

t [s]

ey

ex


The behaviour of the laminates has been further investigated, performing an analysis of the specimens deformation in the thickness direction (z-direction, see Figure 1.10). Draw ratios, λi, are introduced and evaluated from the measured engineering strains. The draw ratios, defined as the ratio between the deformed length Li of an elementary cube of material in the i-direction, and the corresponding undeformed length Li0 [61], can be evaluated in the specimen reference system x-y-z as:

𝜆 =𝐿

𝐿= 𝜀 + 1 (3.10)

The draw ratio in the longitudinal and transversal directions, λx and λy respectively, can be easily evaluated from (3.10), using the measured εx and εy. Due to the low resolution of the sideview camera, for the evaluation of the draw ratio λz, the assumption of constant specimen volume has been introduced and considered as the lower limit. Indeed, a specimen volume reduction is not expected, while a volume increase can be considered possible. Also, the constant volume hypothesis is corroborated by some literature related to cord-rubber composites [12,13,79,80], another type of FRE that can be considered similar to the material in exam. This hypothesis can be considered reasonable for the material in exam in sight of the peculiar nature of the two constituent materials. As a matter of fact, the matrix is constituted by an elastomeric material, which deforms without changes in volume [61]. Therefore:

𝐿 𝐿 𝐿 = 𝐿 𝐿 𝐿 = 𝐿 (3.11)

𝐿 𝐿 𝐿

𝐿= 𝜆 𝜆 𝜆 = 1 (3.12)

In addition, since a large difference is present between the elastic modulus of the fibres and that of the matrix (see Table 2.1 and Table 2.2), it is reasonable to assume that the fibres are practically inextensible and unaffected by the specimen elongation, meaning that negligible volume changes can be assumed for the fibres. If these two assumptions are accepted as reasonable, as for the resulting hypothesis of constant specimen volume, the evolution of λz can be obtained using (3.14):

𝜆 𝜆 𝜆 = (𝜀 + 1) 𝜀 + 1 (𝜀 + 1) = 1 (3.13)

𝜆 =1

𝜆 𝜆=

1

(𝜀 + 1) 𝜀 + 1 (3.14)


Figure 3.32a reports the evolution of λz and λy with respect to λx for a SAP45_B/0.31 specimen. It is notable the increase of λz, meaning that the specimen thickness progressively increases during the specimen elongation, while at the same time λy reduces, coherent with the observed width reduction. The same approach has been followed also in the analysis of the other SAPθ specimens, characterized by the previously reported different trends of the measured strains. Figure 3.32b, Figure 3.32c and Figure 3.32d report the evolution of λz and λy with respect to λx for the SAP30_A/0.30, SAP60_A/0.30 and SAP75_A/0.30 specimens. Similarly to SAP45,

1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.100.30.4

0.50.6

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0.80.91.0

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ly

l i [-

]

lx [-]

lz

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]

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lz

ly

l i [-

]

lx [-]

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1.21.31.4

1.51.6

1.7

ly

lz

l i [-

]

lx [-]

(a)

(c)

(b)

(d)

Figure 3.32 - Evaluated draw ratios in the transversal, λy, and thickness, λz, direction with respect to the draw ratio in the longitudinal direction, λx, for a: (a) SAP45_B/0.31 specimen, (b)

SAP30_A/0.30 specimen, (c) SAP60_A/0.30 specimen, (d) SAP75_A/0.30 specimen


SAP30_A/0.30 is characterized by a thickness increase. On the other hand, SAP75_A/0.30 shows a λz reduction, which is coherent with the measured εx and εy. Even more interesting are the results obtained for SAP60_A/0.30. As visible in Figure 3.32c, the trend of λz is not monotonic as for the other specimens. Initially λz decreases, subsequently it seems to stabilize on a plateau, followed by a final increase, meaning that specimen thickness initially reduces and subsequently increases during specimen elongation. This result is coherent with the previously described trend of εx and εy, characterized by the different increasing rate previously mentioned.

The reported results have been qualitatively validated by means of the video recording of the thickness evolution. As anticipated in Section 2.2.3, due to the absence of a further 3D recording system to be used contemporary to the one adopted for the front view recording, the specimen thickness video has been performed with the camera of a smartphone. Due to the small room for the positioning of the camera, the quality of recorded video is low and the video cannot be used to obtain accurate values of the thicknesses variation. In addition, also the time synchronization with the data measured by dynamometer and uEye camera is approximative. Still, the sideview videos can be exploited to validate at least qualitatively the theoretical results showed. Figure 3.33 reports the initial and almost final frames of the sideview video for the different SAPθ specimens. As it can be observed, SAP30_A/0.30 and SAP45_B/0.31 effectively undergo a thickness increase, while the thickness of SAP75_A/0.30 decreases. For SAP60_A/0.30 it is harder to catch the reported trend change of the thickness, but the final thickness increase results enough evident. This can be considered as a first confirm of the theoretical evaluation illustrated. Although inaccurate, the values of the real thickness have been tried to be measured from the video, making use of the ImageJ program. Figure 3.34 reports the normalized thickness values measured from the sideview video, SReal, and the normalized theoretical one, STheo, for different times, t. The normalized values have been evaluated as the measured (or calculated) i-th thickness value, zi, divided by the initial measured (or calculated) thickness value, z0:

𝑆 =𝑧

𝑧 (3.15)


These must be critically considered, since the video resolution and the time synchronization are far from being optimal. In addition, as particularly visible from SAP30_A/0.30 and SAP45_B/0.31, the specimen width contraction is not uniformly along the thickness, rather a sort of edge rounding develops in the z-direction, as schematically represented in Figure 3.35. Therefore, the theoretically evaluated thickness can be considered as a sort of average, which do not account the specimens rounding. Examining Figure 3.34, it can be observed that for SAP30_A/0.30, SAP45_B/0.31 and SAP60_A/0.30 the thickness trend expected considering a constant volume deformation is effectively confirmed. Also for SAP75_A/0.30 the thickness decreasing trend is recognizable, even though a greater scattering of the data is present, due to a poor video resolution.

t = t0 t = tf t = t0 t = tf t = t1 t = t2 t = t0 t = tf t = tf t = t0

(a) (b) (c) (d)

Figure 3.33 – Side view video frames of the undeformed (t = t0) and of the almost at break deformed specimen (t = tf) for: (a) SAP30_A/0.30 specimen, (b) SAP45_B/0.31 specimen, (c)

SAP60_A/0.30 specimen, (d) SAP75_A/0.30 specimen


0.600.801.001.201.401.601.80

0 10 20 30 40 50 60 70Nor

mal

ized

Thi

ckne

ss, S

[-]

Time, t [s]

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0.80

1.00

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[-]

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[-

]

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0.600.801.001.201.401.601.80

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mal

ized

Thi

ckne

ss, S

[-

]

Time, t [s]

(d)

(c)

(b)

(a) STheo

SReal

Figure 3.34 - Normalized thickness values, S, versus time, t, for: (a) SAP30_A/0.30 specimen, (b) SAP45_B/0.31 specimen, (c) SAP60_A/0.30 specimen, (d) SAP75_A/0.30 specimen. Black lines represent the normalized thickness evaluated from the constant

volume deformation assumption, red lines the normalized measured ones


Similarly to what have been showed in Section 3.1.2.1, also the evolution of the fibres angle, θ, with the specimens elongation has been investigated for the examined SAPθ specimens. Figure 3.36 reports θ values during SAPθ specimens stretching plotted versus the draw ratio in the longitudinal direction, λx. What can be observed is that a relevant fibres re-orientation occurs, especially for SAP45_B/0.31 and SAP60_A/0.30 specimens. These results can be combined with the results regarding the thickness evolution of the specimens. The first thing that can be observed is the trend of the draw ratio λz, (that is related to the thickness of the specimen) and that of the actual θ with respect to the draw ratio λx, (that is related to the specimen elongation) as reported in Figure 3.37. In this plot, the continuous lines identify the evolution of λz, the dashed lines the evolution of θ. The presence of a sort of threshold fibres angle, at θThreshold = 45°, at which the thickness trend changes can be noticed. As it can be observed, for θ < θThreshold, the value of λz increases, as it can be recognized if the data of SAP30_A/0.30 and SAP45_B/0.31 are considered. On the other hand, for θ > θThreshold, the value of λz decreases, as it can be recognized if SAP75_A/0.30 data are examined. This is even more evident if the λz trend for SAP60_A/0.30 is observed: for this specimens, fibres re-orient extensively, passing from angles greater than θThreshold to values below it. What occurs at a θ value around θThreshold = 45°, at which the trend of λz undergoes an inversion, from a decrease to an increase, is noteworthy. Therefore, fibres re-orientation must be accounted when the thickness change is investigated, since it would depend not only on the initial fibres angle, but also on the actual fibres angle that is reached during stretching.

y

z

(a) t = t0

(b) t = tf

Figure 3.35 - Schematic representation of cross section in a (a) undeformed and (b) stretched SAPθ specimen


1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1005

1015202530354045505560657075

q [°

]

lx [-]

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.705

1015202530354045505560657075

q [°

]lx [-]

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.705

1015202530354045505560657075

q [°

]

lx [-]

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.705

1015202530354045505560657075

q [°

]

lx [-]

(a)

(c)

(b)

(d)

Figure 3.36 - Fibres angle, θ, evolution with respect to the specimen elongation, λx. (a) SAP30_A/0.30 specimen, (b) SAP45_B/0.31 specimen, (c) SAP60_A/0.30 specimen, (d)

SAP75_A/0.30 specimen.


The second thing that can be investigated is the dependence of SAPθ tangent modulus, ET, on the actual fibre orientation angle θ during a tensile test, as reported in Figure 3.38. First, it can be observed that for all SAPθ an initial reduction of ET at increasing θ values is present, followed by a ET value plateau. However, this curve “softening” is in conflict to the fibres orientation, that should produce a stiffer laminate response. Exception done for SAP75_A/0.30, after the plateau, a ET value increase is recognizable for all the SAPθ, exception done for SAP75_A/0.30 which fail earlier. The plateau extension changes with the SAPθ considered, with a reduction of its extension moving from SAP75_A/0.30 to SAP30_A/0.30. For SAP45_B/0.31 and SAP30_A/0.30, the ET values after the plateau are higher than the initial ones, ET,0 = ET (θ = θ0). Regarding this point, it is recognizable a range of fibres-angle, ΔθTransition = 21° ÷ 26° at which ET > ET,0, and the values of ET start to strongly increase. Essentially, this ΔθTransition can be thought as the transition point between the softened zone 2 and the hardening zone 3 previously described (see Figure 3.18). This ΔθTransition can be expected also for SAP60_A/0.30, whose curve shows a

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.70.6

0.8

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1.8 lz

q

lx [-]

l z [

-]

15

20

25

30

35

40

45

50

55

60

65

70

75

q [°

]

θThreshold

Figure 3.37 – Cross-plot of the draw ratio in the thickness direction, λz, and of the fibres angle, θ, with respect to the specimens elongation, λx, for all the SAPθ. Continuous lines identify λz

evolution, dashed lines identify θ evolution. Short points line identifies the θThreshold

SAP30_A/0.30 SAP45_B/0.31 SAP60_A/0.30 SAP75_A/0.30


progressive increasing trend, which is however truncated by specimen failure. Practically, only when a sufficient fibres angle is reached, identifiable in the range ΔθTransition, the fibres begin to strongly contribute to the laminate stiffness (zone 3), while in zone 2 the matrix role seems to be preponderant. Matrix contributions can be present to some extent also in the hardening zone 3. Indeed, elastomers undergoes strain-hardening when they are subjected to relevant stretches [61], due to the extension of the polymeric chains in the load direction, resulting in a relevant increase of the stiffness (see Figure 3.20). This effect may be present also in the laminates tested, where the matrix is subjected to high strain level also due to the complex stress states developed inside the composite material. This can lead to a stretching of the matrix macromolecules, contributing in this way to the stiffening observed in the zone 3 of the curve (see Figure 3.18). However, a method to isolate the two contributions has not been found yet.

Also, it is worthwhile to note the differences between the evaluated ET, and the theoretical values of laminate stiffness that one would expect from the Classical Lamination Theory (CLT), at different θ. The latter ET values can be calculated as explained in Section 1.3.3. Figure 3.39 reports the value of ET experimentally obtained for the SAPθ specimens, represented by the continuous line, and of the theoretical ET,CLT, represented by the orange dashed line. Some important differences can be noted with respect to the experimental ones showed. First, the experimental ET values measured in the initial linear region of the tensile curves at low εx, are

Figure 3.38 - Measured tangent modulus, ET, of the different SAPθ specimens with respect to the fibres angle, θ. The orange area locates the ΔθTransition at which ET strongly increases

0

50

100

150

200

250

300

350

400

450

0102030405060708090

E T[M

Pa]

θ [°]

SAP30_A/0.30

SAP45_B/0.31

SAP60_A/0.30

SAP75_A/0.30

ΔθTransition


fairly predicted by means of the CLT. This can be better understood if the values theoretical and experimental values are compared as in Table 3.6.

The laminate stiffnesses of SAP30_A/0.30 is somehow lower than the one theoretically predicted, while the experimental value of ET for SAP60_A/0.30 is a bit higher. Less important differences can be observed for SAP75_A/0.30 and SAP45_B/0.31.

Laminate sample

Experimental Laminate Stiffness at low strains, ET (εx = 0 ÷ 0.01) [MPa]

Theoretical Laminate Stiffness, ET,CLT [MPa]

SAP30_A/0.30 218.59 ± 22.55 286.88

SAP45_B/0.31 57.72 ± 8.86 69.19

SAP60_A/0.30 54.00 ± 6.35 32.86

SAP75_A/0.30 21.29 ± 1.42 23.75

Table 3.6 - Experimental values, ET, at low strains, and theoretical prediction (from CLT) of the stiffness, ET,CLT.

Figure 3.39 – Experimental ET curves obtained for the SAPθ specimens, represented by the continuous line, and theoretical ET,CLT curve, predicted by Classical Lamination Theory (CLT)

050100150200250300350400450500550600

051015202530354045505560657075808590

E T[M

Pa]

θ [°]

CLT

SAP30_A/0.30

SAP45_B/0.31

SAP60_A/0.30

SAP75_A/0.30


3.2.2 Quasi-static loading-unloading tests

Quasi-static tensile loading-unloading tests have been performed on three SAP45_E/0.23 specimens, following the loading histories identified as Case A, B and C described in Section 2.2.3. After the cyclic loadings, the specimens were tested up to failure. This was aimed at further investigating the possible origins of the peculiar non-linear shape of SAPθ tensile curves. Indeed, as mentioned in Sections 3.1.1 and 3.2.1, the addition of a reinforcing phase to an elastomeric matrix may lead to responses that in principle are not expected in conventional composites, related to the presence of reinforcement-matrix interactions.

Case A stress-strain curves σx – εx, are reported in Figure 3.40a, for the SAP45_E/0.23 specimen cyclically tested. The σx – εx curves of a SAP45_E/0.23 specimen tested in a monotonic quasi-static tensile test is also reported for comparison, reproducing the behaviour of the “Virgin” material. Similarly, Figure 3.40b reports the case B stress-strain curves and Figure 3.40c the case C stress-strain curves. Due to some difficulties with DIC analysis, the strains considered are the ones calculated from crosshead displacement. Different levels of maximum strain, εx,crossehead, reached have been considered, as already mentioned in Section 2.2.3, and are reported in Table 3.7.

Specimen εx,crosshead 1° set of loading-unloading cycles

εx,crosshead 2° set of loading-unloading cycles

Case A ~ 0.1 ~ 0.31

Case B ~ 0.1 ~ 0.21

Case C ~ 0.35

Table 3.7 - Overall strain applied at the end of each cycle, εx,crossehead, , for the three loading cases. For case C only the strain reached in the final cycle is reported


What can be immediately observed from the reported curves, is (i) after the first cycle, lower stress is required to re-deform the material, (ii) the presence of a residual strain after each cycle. The stress reduction is much more evident in cases A and B where, already from the first cycle, higher displacements have been imposed, but it can be easily observed also in case C. This stress softening upon reloading may resemble the change in the material stiffness after a first elongational deformation commonly observed in particle filled rubber compound and referred to as “Mullins effect”. Similarly to what is observed in filled elastomers, also in this case a dependence of this effect on the maximum strain achieved seems to be present and, the material behaviour is the same of the virgin one when the material

Figure 3.40 – Stress – strain curves, σx – εx,crosshead, of the quasi-static uniaxial load-unload tests performed on SAP45_E/0.23 specimens. (a) Case A, (b) Case B, (c) Case C. In each plot

is reported the σx – εx,crosshead curve of the “Virgin” specimen, identified by the black line

(a)

(c)

(b)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400

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s x [M

Pa]

ex,crosshead [-]

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s x [M

Pa]

ex,crosshead [-]


is stretched up to strains larger then than the one applied in the previous loading-unloading step (see Figure 3.40c).

As mentioned in Section 2.2.3, between the two loading histories of cases A and B, the specimens have been left free to recover. Similarly for case C, after the last loading-unloading history, the specimen was left to recover for at least 3 days, before being deformed up to rupture. In all the cases, a complete recovery of the residual deformation never occurred. In all the cases, the specimens result to have a longer length and a slightly reduced width. The specimens have been finally tested to failure. Note that the presence of residual deformation is essentially accompanied by a change in the fibres angles, which should be re-evaluated before performing the final test to failure. These fibres angle, to be considered for the tests to failure, are reported in Table 3.8. For the sake of simplicity, the specimens tested to failure have been renamed as AC_A, AC_B and AC_C, where AC_ stands for “After Cycling case_”, to identify the specimens tested to failure and previously subjected to loading-unloading cases A, B and C respectively.

Specimen Initial Fibres Angle [°]

AC_A 42.31

AC_B 44.44

AC_C 43.01

Table 3.8 - Fibres angle values at the beginning of the tensile test up to failure, after the loading-unloading test

Figure 3.41 reports the superposition of these curves to the one of the “Virgin” specimen. What can be immediately observed from the curves of the specimens AC_A, AC_B and AC_C, is a change in the initial region, identified in Section 3.2.1 as zone 1. For the AC_A and AC_C specimens, the stiffness at low strains (zone 1) is much lower than the one characterizing the virgin specimen at the same strains, but is comparable to the zone 2 stiffness of the virgin specimen at higher strains. Instead, for specimen AC_B, zone 1 is characterized by a stiffer response, but still lower with respect to the one observed in the virgin specimen. This may result clearer if the magnification of the initial zone of Figure 3.41b is observed.


Then, it has to be observed that: (i) the cyclic loading histories are different for the three cases (A, B,C) considered to pre-straining the material, (ii) the maximum strains applied in the Cases A and C are comparable and larger than that of the Case B (see Table 3.7). The tensile curves of the of specimens AC_A and AC_C overlap and differ from that of AC_B. These results suggest that the pre-straining loading history does not affect the material behaviour, provided the applied pre-strain is the same. The strain induced material softening is strongly dependent on the maximum strain level imposed during the pre-straining of the material (loading-unloading cycles): this strain level determines the presence or not of a high rigidity region (zone 1).

This may lead to think that some interactions are effectively present between fibres and matrix. The stiffer response observed in zone 1 can be plausibly explained by the possible presence of something like a “low mobility rubber”. When the material is subjected deformed, this “low mobility rubber” is progressively softened, resulting in the softening observable in zone 2. In addition, it is possible that also some internal damages are generated during deformation, either between or inside the laminae. Indeed, a complete recovery of the imposed deformation have never been observed, and the initial fibres angle after the loading histories resulted always reduced.

Up to now the effect of the fibres orientation on the material mechanical response has not been considered yet. To do so, the same approach employed in Section 3.2.1

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,350

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sx

[MP

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Virgin AC_A AC_B AC_C

0,00 0,01 0,02 0,03 0,04 0,050,0

0,5

1,0

1,5

2,0

sx

[MP

a]

ex [-]

Virgin AC_A AC_B AC_C

(a) (b)

Figure 3.41 – Superposition of the tensile stress-strain curves of a virgin specimen (black line) and of the specimens previously subjected to load-unload cycles (red line for AC_A

specimen, green line for AC_B specimen, blue line for AC_C specimen). (a) Complete view, (b) Magnification of the first region, highlighted in (a) by the yellow box.


have been followed, making use of the measured tangent modulus of the σx - εx curves. The tangential modulus, ET, of the virgin and pre-strained materials (AC_A, AC_B and AC_C) of the have been plotted versus both the longitudinal strain, εx, and the actual fibres angle, θ, as reported in Figure 3.42 and Figure 3.43 respectively.

From Figure 3.42 it can be recognized again the trend of the tensile curves reported in Figure 3.41, with the strong reduction of the high rigidity zone 1 stiffness for the pre-strained materials AC_A and AC_C, characterized by ET value at low strains much lower than the corresponding one of the virgin material. The low strain ET value of pre-strained material AC_B, although still lower than the virgin one, is higher with respect to that of AC_A and AC_C. From Figure 3.42 it can be also observed that the tangent modulus measured in zone 2 for the virgin material, is comparable to the low-strain ones of that the pre-strained materials AC_A and AC_C, and also to that at slightly higher strains of AC_B.

Other interesting results are reported in Figure 3.43. What can be observed here is that a stiffening of the material is present at a relatively low value of the fibres angle, θ. Moreover, this stiffening does not depend on the strain εx applied: if a specific value of ET at the corresponding θ in Figure 3.43 are compared to the same values of ET in Figure 3.42. For example, a value of ET = 100 MPa, falls in a small range of fibres angle θ = 24° ÷ 26°, but to reach the same ET value, much different strains must be reached for the different specimens (the virgin one requires εx ≃ 0.29, AC_A and AC_C require εx ≃ 0.225, while AC_B requires εx ≃ 0.25).

The ET vs θ curves of the differently pre-stretched materials (AC_A, AC_B, AC_C) almost overlap with that of the virgin material, suggesting that it is the fibre angle that determine the material rigidity, irrespective of the applied strain.


.

0

50

100

150

200

250

300

350

400

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

E T[M

Pa]

εx [-]

Virgin

AC_A

AC_B

AC_C

Figure 3.42 - Tangent modulus, ET, measured at different longitudinal strains, εx, for the virgin specimen (black line), and AC_A (red line), AC_B (green line) and AC_C (blue line)

specimens

0

50

100

150

200

250

300

350

400

1520253035404550

E T[M

Pa]

θ [°]

Virgin

AC_A

AC_B

AC_C

Figure 3.43 - Measured tangent modulus, ET, of the different specimens versus fibres angle, θ. Virgin specimen (black line), and AC_A (red line), AC_B (green line) and AC_C (blue line)

specimens

| Preliminary Finite Element Analysis (FEA) 119

3.3 Preliminary Finite Element Analysis (FEA) One of the objectives of mechanical characterizations, and therefore of the present work, is to provide usable data for the design of complex laminates or components. This is particularly important for the present material, whose unique mechanical properties and overall deformational behaviour have to be exploited in peculiar application fields, like the one of soft robotics. As mentioned in Section 1.3, when dealing with laminae, four independent material constants have to be found by means of theoretical approaches and experimental testing, and the determined material constants are then employed in the design of laminates. To optimize laminates design, considering both complex loading and complex geometrical shapes, the use of Finite Element Methods (FEM) software is common, where Abaqus, Ansys and Comsol are among the most used FEM software. These softwares are able to simulate the mechanical, thermal and electromagnetic response of materials and/or components, provided that suitable constitutive laws (material models) are available. The modelling must account not only the applied loads and boundary conditions, but also the properties of the chosen materials (mechanical, thermal and/or electromagnetic). From these stems the requirement of having reliable material properties data and an adequate material model. That’s particularly true for FRE, for which not all the assumptions over which the CLT are based hold true.

The obtained experimental data (reported in Section 3.1) have been employed in some preliminary simulation attempts, to provide the base from which further and deeper investigations could be performed. The performed simulations focused on the transversal modulus, E2, prediction of simplified unidirectional laminates loaded in direction perpendicular to the fibres. The software chosen to perform the simulations is Abaqus FEA (2021HF8 (6.21-9)) released by Dassault Systèmes.

3.3.1 Model development

Due to the elastomeric nature of the matrix material, a preliminary model to predict the hyperelastic behaviour of Sylgard 184 has been developed. This was required to implement the elastomeric nature in the subsequent modelling of the composite material. Abaqus allows to perform a “material evaluation”, to identify the material constitutive model parameters, through the best fitting of provided experimental data. Depending on the constitutive model chosen for the material description, different sets of experimental data should be provided. Following a previous thesis work [72], the best constitutive model available to reproduce Sylgard 184 behaviour is the Ogden model with a strain energy potential of the third order [60,72,81]. The form of the mentioned constitutive energy potential model is:


𝑈 =2𝜇

𝛼𝜆 + 𝜆 + 𝜆 − 3 +

1

𝐷(𝐽 − 1) (3.16)

where U is the strain energy potential, 𝜆 are the deviatoric principal stretches, 𝐽 is the elastic volume ratio, and μi, αi and Di are the material parameters to be evaluated in order to properly define the elastomer behaviour. The experimental data provided to the software, specifically from uniaxial and biaxial tests, were obtained from a unpublished previous work of T. Bardelli [68]. Sylgard 184 Ogden model parameters evaluated by Abaqus are reported in Table 3.9.

Parameter i = 1 i = 2 i = 3

μi -0.07696090 0.14556000 1.03835668

αi 5.94333325 10.77066990 -0.13782582

Di 0.00181736 0 0

Table 3.9 - Ogden model parameters evaluated by Abaqus from the provided experimental data of Sylgard 184

Figure 3.44 reports the comparison between the experimental uniaxial stress-strain curve and the stress-strain curve simulated by the software employing the model parameters reported in Table 3.9. Good agreements between the two curves have been observed.


To investigate the composite material in exam, a model to simulate the response of UD90 specimens has been realized. The model has been named M_UD90. This model has been intended as a simplified representation of the specimens employed in the experimental activities, even though it is characterized by the same dimensions of the real specimens (see Section 2.1.3). In the model, the specimen is composed by four plies of fibres with fibres equally spaced in the laminae planes and oriented at 90° to the specimen longitudinal direction. The total number of fibres has been arbitrarily fixed equal to 100. The matrix material has been modelled as a 3D deformable solid body; fibres as 3D deformable trusses. A fictitious cross-sectional area of the fibres has been considered. This has been calculated in order to obtain a vf = 0.3, resulting in a value for each fibre of 0.3 mm2 much higher than real one, which is instead in the order of 0.0002 mm2. Fibres have been integrated in the matrix part by means of a “Tie” constrain, to simulate a realistic displacement of the fibres with the matrix. Since the matrix is an elastomer, simulations accounting geometric non-linearity (NLGEOM) have been performed. As boundary conditions, an “Encastre” has been considered, and a displacement in the longitudinal direction, of 20 mm, has been applied in order to simulate the test. An example of M_UD90 with the applied boundary conditions is reported in Figure 3.45. The matrix part has been meshed with C3D8RH elements (8-node linear brick, reduced integration with hourglass control, hybrid with constant pressure), for a total

Figure 3.44 - Comparison of uniaxial tensile stress-strain curves. White line represents the experimental curve, red line the curve obtained from the “material evaluation”

performed by Abaqus employing the Ogden model.


number of 20000 elements; fibres have been meshed using T3D2 elements (2-node linear displacement), for a total number of 4 elements for each fibre.

Regarding the material properties, different shots have been performed. In all the cases, the fibres have been considered as linear elastic bodies, characterized by the material properties reported in Table 2.1. For what concerns the matrix material, two preliminary trial simulations have been performed, first modelling the matrix as a linear elastic material, characterized by the material properties reported in Table 2.2, then modelling it as a hyper-elastic material. In the second case, the above mentioned Ogden model with the reported material parameters have been used. These two simulation cases will be identified with the notation M_UD90_LE and M_UD90_HE, to identify the simulation performed with a linear elastic and hyper-elastic matrix respectively. A constant fibres volume fraction, vf, has been maintained, equal to vf = 0.3.

3.3.2 Model M_UD90 – Results

The transversal modulus, E2, has been initially evaluated from the two trial simulations M_UD90_LE and M_UD90_HE. Figure 3.46 and Figure 3.47 report the deformed shapes for the two simulations, both at the maximum displacement value (20 mm). The E2 values obtained from these simulations are reported in Table 3.10, reporting also the E2 experimental value. Also for the simulation, the value of E2 has been evaluated as the slope of the stress-strain curve, considering as upper limit for the strains a value of εx = 0.1. the curve has been built plotting the εx value extracted from an element of the central zone of the model, where the strains are more uniform, versus the calculated σx. The stress has been evaluated as the ratio between the sum of the reaction forces values in the x-direction, RFx, acting on the specimen top surface (dimensions 12.5x2 mm), and the cross-section of the specimen.

Figure 3.45 – M_UD90 specimen modelled in Abaqus


Figure 3.46 - Deformed shape of the M_UD90_LE model considering the matrix as a linear-elastic material. (a) nominal strains in the x-direction (NE11), (b) reaction forces

in the x-direction (RF1), (c) magnification of the cross-section from which the RF1 have been extracted and used for stress evaluation

(a)

(b)

(c)


Figure 3.47 - Deformed shape of the M_UD90_HE model considering the matrix as a hyper-elastic material. (a) nominal strains in the x-direction (NE11), (b) reaction forces in the x-direction (RF1), (c) magnification of the cross-section from which the RF1 have

been extracted and used for stress evaluation

(a)

(b)

(c)


Case Transversal Modulus, E2 [MPa]

M_UD90_LE 3.34

M_UD90_HE 3.90

Experimental 18.93

Table 3.10 - Transversal modulus, E2, values obtained from simulation and from the experimental characterization

From the reported results two points are evident. First, no significant differences can be observed from the two simulation results. This can be seen even better if the stress-strain curve, σx – εx, obtained from the two simulations are compared, as reported in Figure 3.48. Due to the little differences between the two stress-strain curve prediction, to reduce the computational time, the matrix material can be considered as linear elastic for other possible simulations, also in view of the previously reported tensile curve of Sylgard 184 (see Figure 3.44), which showed a linear behaviour at the low strains of interest.

The second aspect, more relevant, is that both Abaqus simulations strongly underestimate the experimental E2 value. A plausible reason for such different results can be related to an over-simplification of the model. As mentioned in Section 3.1.1 and Section 3.2, the presence of a reinforcement in an elastomeric

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225

σ x[M

Pa]

εx [-]

M_UD90_LE M_UD90_HE

Figure 3.48 - Stress-strain curves, σx-εx, obtained from the simulations with a linear-elastic (M_UD90_LE) and hyper-elastic (M_UD90_HE) definition of the matrix


material probably produces a “low mobility rubber” in the vicinity of the reinforcing fibres, modifying the mechanical behaviour of the elastomeric matrix, no more properly described by the constitutive laws valid for the neat rubber. A specific “submatrix” able to describe the reinforcing effect of the “low mobility rubber” should be somehow implemented in Abaqus for a better simulation prediction. Further, the possible effect deriving from the rubber confinement in between the fibres might be not accounted if a low number of fibres is integrated in the matrix. As it can be observed in Figure 3.46 and Figure 3.47, strains are indeed mainly localized in the matrix material present in between the fibres, which instead are just rigidly displaced in the x-direction. If more reliable results would be obtained, also these effects must be accounted.

These results essentially confirm that, the adopted composite model is not able to properly reproduce the FRE behaviour. Therefore, other simulation strategies should be attempted. Due to lack of time, other possible approaches have not been tried. Still, these results can be intended as a possible starting point for future works. This can easily become an interesting field of study, since a very limited number of previous works have been performed on FEM simulations of fibres-reinforced elastomers. Due to the increasingly interest and possible application areas in which these materials can be exploited, the capability to predict the response of FRE, even when complex shapes and/or forces are considered, would become of large importance.


4 CONCLUSIONS AND FUTURE DEVELOPMENTS

In the present work, a study on the mechanical behaviour of a fibres-reinforced elastomeric composite has been performed as a part of a wider research activity developed within a PhD thesis project. The composite in exam was constituted by continuous and unidirectional glass fibres embedded in a PDMS matrix. Among the four mechanical properties required to characterize a lamina, the transversal modulus, E2, and the in-plane shear modulus, G12, have been experimentally measured for a range of fibres volume fractions, vf, between 0.15 and 0.45. Unidirectional laminates (UDLθ) and symmetric angle ply laminates [±θ]ns (SAPθ), characterized by appropriate fibres orientation, have been prepared and tested in quasi-static tensile tests. Different experimental and theoretical approaches have been used, and the results obtained were analyzed and compared. It was found that, in the small strain range, the commonly employed springs-in-series model strongly underestimate the dependency of E2 and G12 on vf. Also the widely used Halpin-Tsai model does not predict accurately the experimental E2 values, while fairly esteems the G12 values. A possible explanation of these results has been attributed to the elastomeric nature of the matrix material constituting the composite. Due to the presence of rigid fibres, a reduction of “rubber mobility” may occur, as commonly observed in particle filled rubbers [59], as well as constrains effects occurring on fibres surfaces [59], resulting in an increase of the matrix elastic modulus, dependent on the reinforcement content.

The elastomeric nature of the matrix is responsible for the observed non-linear elastic behaviour and for the higher deformability of the overall composite material, allowing to reach deformations higher than those commonly reached by conventional composites. The observed strong non-linearity of the stress-strain curve is characterized by an initial linear behaviour extended for limited strains, followed by a strain induced softening that extends for a wide strain range, and then, at larger strains, a “hardening” occurs. To further investigate the peculiar material behaviour, different symmetric angle ply laminates (SAPθ) characterized by a fibres angle, θ, of 30°, 45°, 60° and 75° have been characterized. Different experimental observations followed. First, due to the high matrix deformability a significant fibres re-orientation during tensile tests was observed. In addition, it was observed a dependence of the laminate’s thickness on the actual θ value. For θ > 45° the lateral contraction is limited, and the thickness decreases. For θ < 45°, the lateral

128 4-CONCLUSIONS AND FUTURE DEVELOPMENTS

contraction is more significant, thus, due to the rubber incompressibility, the specimen’s thickness increases. The change in fibres orientation must be accounted to explain the non-linear behaviour, in addition to the elastomeric nature of the matrix. The laminate stiff response at low strains can be attributed to the presence of “low mobility rubber” and rubber confinement effects, which is then softened at higher strains, at which fibres orientation seems to play a minor role. When a sufficiently small fibres angle is reached at even higher strains, the final strain stiffening of the laminate seems to be mainly related to the actual orientation of the fibres. Nevertheless, the strain stiffening of the matrix cannot be overlooked also in this last strain region.

As possible foundation for a future work, some preliminary FEM simulations have been performed too. The results obtained from these preliminary simulations showed a significant underestimation of the experimental values. These results have been reconducted to the strongly simplified model employed and suggest that, in FREs, not just the mechanical behaviour of the components should be considered, but also the effect of fibres-matrix interactions, not generally accounted in conventional composites.

Considering the results and observations of this study, a deeper investigation is required. In this direction, a more systematic investigation on the mechanical behaviour of the laminates, for example through additional cyclic tests, could be carried out. Different testing parameters, maintained constant in the present work, can be modified: for example the deformation rate, the recovery time, as well as the fibres angle. Furthermore, the investigations can be moved to a smaller analysis scale performing, for example, nano-indentations, to try to experimentally quantify the actual reinforcing action produced by the fibres on the matrix. Another possibility, may be the testing of other FRE, changing the matrix and/or the fibres constituting the composite, to verify if the observations reported in this work hold true also for other material systems. This may help in the identification of the fibres and matrix contributes to the mechanical response. Then, the definition of a more complex constitutive model able to take into account the hyperelastic nature of the matrix and fibres re-orientation is a key point for the development of this research topic. Eventually, this may help to conduct adequate FEM simulations, to properly predict the FREs deformational response under complex loading conditions. This would further enlarge the areas of investigation, with the possibility to design and simulate the response of more complex laminates. This could be of great help in the development of bioinspired soft robots or prothesis, to reproduce complex movements of living beings, which cannot be replicated with more conventional materials.


131

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139

A Appendix A

Table A.1 reports the summary of the dimensions for the aluminium based-moulds in which the composite material has been produced.

Figure A.1 reports a drawing of the lateral and top view of already available moulds, identified as mould 1, 2 and 3, which differ in the cavity depth, i.

80mm

100mm

80mm 100mm

80mm 100mm

i

(i = 1, 1.5, 2mm)

(a)

(b)

Figure A.1 - Representation of moulds 1, 2 or 3. (a) Lateral view, (b) Top view

140 A

Figures A.2 and A.3 report the blueprints of mould 4 and its dimensions. This mould has been specifically designed within this thesis project for the fabrication of bigger laminates required for the shear modulus, G12, evaluation. The mould has been designed as composed by 3 parts:

- the base mould (Figure A.2): it has a cavity depth of 7 mm; 0.5 mm thick aluminium plates can be placed within the cavity to reduce its depth. In addition to these aluminium plates, the base mould accommodates the lamination plate.

- the lamination plate (Figure A.3): it is composed by two parts, an aluminium plate (Figure A.3) with an internal cavity depth of 1 mm, and a commercial mirror of 2 mm thickness, fixed in the cavity of the plate and required to favour an easier laminate detachment. On the overall, the lamination plate is characterized by an external frame of 25 mm width and 1 mm deeper with respect to the plane of the mirror, over which the laminate is produced. Lamination plate is inserted in the base mould, above the aluminium plates used to reduce the base mould depth. The purpose of this design is to accommodate the “taped” portion of the fibres layers, which are slightly thicker than the remaining portion of the fibres layer, in the deeper frame, to reach higher fibres volume fractions.

- the mould cover (Figure A.4): Similarly to the lamination plate, also for this part two cavities are present: they accommodate a mirror and the “taped” portion of the fibres layers.

Table A.1 - Dimensions of the employed moulds

Mould 1 Mould 2 Mould 3 Mould 4

Cavity Width [mm] 80 80 80 261.6

Cavity Length [mm] 80 80 80 351.6

Cavity Thickness [mm] 1 1.5 2 0.5 ÷ 3.5

A 141

Figure A.2 - Mould 4 blueprints, base mould: (a) top view, (b) lateral view

(a)

(b)

142 A

Figure A.3 - Mould 4 blueprints, lamination plate: (a) top view, (b) lateral view

(a)

(b)

A 143

Figure A.4 - Mould 4 blueprints, mould cover: (a) top view, (b) lateral view

(b)

(a)

144 A

145

List of Figures

Figure 1.1 - Example of material property chart. Comparison of Young's modulus of different material with respect to their density [Ansys Granta Edupack] ............ 6

Figure 1.2 - Schematic representation of common types of composites ..................... 8

Figure 1.3 - Motion of a crawling robot [21] ................................................................... 9

Figure 1.4 - Bending and rotations induced in a soft continuous arm [24] .............. 10

Figure 1.5 - Swimming motion of a turtle-like robot [4] ............................................. 10

Figure 1.6- Schematic of analysis of laminated composites (Adapted from [15]) ... 12

Figure 1.7 - Schematic representation of the longitudinal modulus, E1, evaluation [26] ...................................................................................................................................... 14

Figure 1.8 - Schematic representation of the transversal modulus, E2, evaluation [26] ............................................................................................................................................. 15

Figure 1.9 - Representation of the principal material directions, 1-2, and reference system directions, x-y, for a unidirectional lamina ...................................................... 18

Figure 1.10 - Schematic representation of the laminate .............................................. 20

Figure 1.11 - Representation of the mid-plane deformation process accordingly to Kirchhoff's hypothesis ..................................................................................................... 21

Figure 1.12 - Representation of the laminate [15] ........................................................ 22

Figure 1.13 - Examples of load-coupling effects........................................................... 23

Figure 1.14 - Mohr's circle of strains .............................................................................. 28

Figure 1.15 - Stress state in the laminae ......................................................................... 29

Figure 1.16 - Lamina shear log effect [41]...................................................................... 30

Figure 1.17 - Schematic representation of the 3 main zones present in a ±45° clamped laminate .............................................................................................................................. 31

Figure 1.18 - Schematic depicting loaded 10° off-axis tensile test specimen and stresses at element at 10° plane (adapted from [32]) ................................................... 33

Figure 1.19 - Effect of shear-extension coupling. (a) Ideal specimen deformation, (b) specimen deformation when clamped ends are considered [44] .............................. 34

146 | List of Figures

Figure 1.20 - Variation of the strain in the material axes against the load direction (Adapted from [32]) .......................................................................................................... 36

Figure 1.21 - Uniaxial loading of a unidirectional specimen at 45° to the 1-direction [16] ...................................................................................................................................... 37

Figure 1.22 - Iosipescu test fixture [35] .......................................................................... 38

Figure 1.23 - Schematic representation of the: (a) two rails shear test, (b) three rails shear test ............................................................................................................................ 40

Figure 1.24 - Schematic representation of the plate-twist test geometry [36] .......... 41

Figure 1.25 - Torsion of a thin-walled circular tube [37] ............................................. 42

Figure 1.26 - Representation of the entropic-elasticity of a cross-linked elastomer. In the initial state a random coil conformation is showed. At the application of the load, chains stretch with a consequent entropy reduction. At the load removal, the random coil configuration is recovered. ....................................................................... 43

Figure 1.27 - Typical stress-strain curve of an elastomeric material ......................... 44

Figure 2.1 - Representation of the steps required for the composite production. I: Glass-fibres layers preparation, II: Matrix preparation, III: Composite layup, IV: Composite de-gassing, V: Curing .................................................................................. 50

Figure 2.2 - Specimens dimension. (a) Example of UD90, obtained by punching. (b) Example of SAP45, obtained by cutting. ....................................................................... 52

Figure 2.3 - Frame of a video reorded test, showing the spakle patter created on a specimen surface ............................................................................................................... 55

Figure 2.4 - Area of interest (ROI) analyzed for each type of specimen. (a) UD90 specimen, (b) SAP30 specimen, (c) SAP60 specimen, (d) SAP75 specimen, (e) SAP45 specimen ............................................................................................................................ 56

Figure 2.5 - Schematic representations of UD90 specimen employed in the transversal modulus, E2, evaluation .............................................................................. 57

Figure 2.6 - Schematic representation for the evaluation of the transversal modulus, E2 .......................................................................................................................................... 58

Figure 2.7 - Definition of specimen and material axes [74] ........................................ 59

Figure 2.8 - Schematic representation for the evaluation of the in-plane shear modulus, G12 ...................................................................................................................... 59

Figure 2.9 - Schematic representation of fibers re-orientation in a lamina of the specimen as consequence of the elongation. ................................................................ 60

| List of Figures 147

Figure 2.10 - Variation of the normalized strains, εij/εx, versus the off-axis angle, θ. Calculated assuming the estimated material properties, reported in Table 2.5 ...... 62

Figure 2.11 - Variation of the normalized strains, εij/εx, against the off-axis angle, θ. Calculated assuming the measured material properties, reported in Table 2.6 ...... 63

Figure 3.1 - Experimental tensile stress-strain curves σx - εx of the tested specimens. (a) UD90_A/0.28 to UD90_E/0.29, (b) UD90_F/0.22 to UD90_I/0.15, (c) UD90_J/0.46 to UD90_L/0.35. Dotted lines identify the upper limit of the εx range considered for the E2 evaluation. In (a) and (b), a common upper limit has been considered. In (c), depending on the samples, different upper limits have been considered; the colours are associated to the corresponding curves .................................................................. 67

Figure 3.2 - Experimentally determined transversal modulus, E2, versus fibres volume fraction, vf ............................................................................................................ 69

Figure 3.3 - Experimental data and prediction of E2 dependence on vf according to the springs-in-series model, SiS, (blue line), and Halpin-Tsai model, HT with different reinforcing factor, ξ, (green and orange line) ............................................... 70

Figure 3.4 - E2 values versus vf. Black dots identify the experimental values, red line the fitting function developed ........................................................................................ 72

Figure 3.5 - Experimental shear stress-strain curves τ12-γ12 of the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28, (d) SAP45_D/0.38, (e) SAP45_E/0.23 ..................................................................................................................... 74

Figure 3.6 - Magnification of the initial linear portion of the experimental shear stress-strain curves τ12-γ12 of the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28, (d) SAP45_D/0.38, (e) SAP45_E/0.23 .................... 75

Figure 3.7 - Experimentally determined in-plane shear modulus, G12, versus fibres volume fraction, vf, of SAP45 specimens. Blue dot represents the shear modulus of the pure matrix .................................................................................................................. 76

Figure 3.8 - Experimental data and prediction of G12 dependence on vf according to the springs-in-series model, SiS, (blue line), and Halpin -Tsai model, HT with different reinforcing factor, ξ, (green and orange line) ............................................... 77

Figure 3.9 - Typical shear stress-strain curves of conventional FRP composites with: (a) low-ductility matrix, (b) ductile matrix [74]............................................................ 78

Figure 3.10 - Mohr's circles of strains evaluated at increasing test time for a SAP45_B/0.31 specimen. The dots on the ε-axis identify the circle’s centers .......... 80

Figure 3.11 - Evolution of the position of the Mohr's circle of strain, εc, with respect to the test time, t, for SAP45_B/0.31 specimen. The colored dots refer to the same times reported in Figure 3.10 .......................................................................................... 81


Figure 3.12 - Fibres angle, θ, change with respect to the longitudinal strain, εx, for the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28, (d) SAP45_D/0.38, (e) SAP45_E/0.23 .................................................................................... 82

Figure 3.13 - Tensile stress-strain curve σx-εx of the unidirectional 45° off-axis specimens tested. (a) UD45_A/0.27, (b) UD45_B/0.20, (c) UD45_C/0.13, (d) UD45_D/0.15, (e) UD45_E/0.20 ....................................................................................... 85

Figure 3.14 - Experimentally determined in-plane shear modulus, G12, versus fibres volume fraction, vf, of UD45 specimens ........................................................................ 86

Figure 3.15 - Observed in-plane bending for some UD45 specimens. Images on the left show the initial shape of the specimens at time t0, images on the right the deformed shape at a time t1. Specimens from laminates (a) UD45_A/0.27C, (b) UD45_B/0.20, (c) UD45_C/0.13 ....................................................................................... 87

Figure 3.16 - Experimental in-plane shear modulus, G12, versus fibres volume fraction, vf. Summary of all available data. ................................................................... 88

Figure 3.17 - Experimental tensile curves σx – εx of the tested specimens: (a) SAP45_A/0.27, (b) SAP45_B/0.31, (c) SAP45_C/0.28, (d) SAP45_D/0.38, (e) SAP45_E/0.23 ..................................................................................................................... 90

Figure 3.18 - Tensile curves σx – εx of a SAP45_B/0.31 specimen. Three different zones can be identified: initial region at low εx (Zone 1), intermediate "softened" region (Zone 2) and final stiffening region at high εx (Zone 3) .............................................. 91

Figure 3.19 - Example of tensile stress-strain curves σx – εx of particle-filled PDMS [78] ...................................................................................................................................... 92

Figure 3.20 - Tensile stress-strain curve σx – εx of pure Sylgard 184 .......................... 92

Figure 3.21 - Experimental tensile curves σx – εx of the tested specimens: (a) SAP30_A/0.30, (b) SAP60_A/0.30, (c) SAP75_A/0.30 ................................................... 93

Figure 3.22 - Summary of the tensile curves σx – εx for SAPθ specimen .................. 94

Figure 3.23 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP30_A/0.30 specimens (identified by different colors)........................................... 95

Figure 3.24 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP45_B/0.31 specimens (identified by different colors) ........................................... 96

Figure 3.25 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP60_A/0.30 specimens (identified by different colors)........................................... 96

Figure 3.26 - Tangent modulus, ET, measured at different longitudinal strains, εx, for SAP75_A/0.30_A/0.30 specimens (identified by different colors) ............................. 97


Figure 3.27 - Comparison of the tangent modulus, ET, measured at different longitudinal strains, εx, for some SAPθ specimens ..................................................... 97

Figure 3.28 - (a) First and almost final frame of the video recording of a SAP45_B/0.31 specimen under tensile test. (b) Measured width reduction in time of the specimen reported on the left. (c) Evolution in time of the absolute value of the measured εx and εy for the specimen on the left. ......................................................... 99

Figure 3.29 - First and almost final frames of the video recording of the tensile test of: (a) SAP30_A/0.30 specimen, (b) SAP60_A/0.30 specimen, (c) SAP75_A/0.30 specimen. ......................................................................................................................... 100

Figure 3.30 – Time evolution of the width, w vs t, of a: (a) SAP30_A/0.30 specimen, (b) SAP60_ A/0.30 specimen, (c) SAP75_ A/0.30 specimen ...................................... 101

Figure 3.31 - Time evolution of the absolute value of the measured longitudinal and transversal strains, |εx| and |εy| of a: (a) SAP30_A/0.30 specimen, (b) SAP60_A/0.30 specimen, (c) SAP75_A/0.30 specimen ........................................................................ 102

Figure 3.32 - Evaluated draw ratios in the transversal, λy, and thickness, λz, direction with respect to the draw ratio in the longitudinal direction, λx, for a: (a) SAP45_B/0.31 specimen, (b) SAP30_A/0.30 specimen, (c) SAP60_A/0.30 specimen, (d) SAP75_A/0.30 specimen .......................................................................................... 104

Figure 3.33 – Side view video frames of the undeformed (t = t0) and of the almost at break deformed specimen (t = tf) for: (a) SAP30_A/0.30 specimen, (b) SAP45_B/0.31 specimen, (c) SAP60_A/0.30 specimen, (d) SAP75_A/0.30 specimen ..................... 106

Figure 3.34 - Normalized thickness values, S, versus time, t, for: (a) SAP30_A/0.30 specimen, (b) SAP45_B/0.31 specimen, (c) SAP60_A/0.30 specimen, (d) SAP75_A/0.30 specimen. Black lines represent the normalized thickness evaluated from the constant volume deformation assumption, red lines the normalized measured ones ................................................................................................................. 107

Figure 3.35 - Schematic representation of cross section in a (a) undeformed and (b) stretched SAPθ specimen .............................................................................................. 108

Figure 3.36 - Fibres angle, θ, evolution with respect to the specimen elongation, λx. (a) SAP30_A/0.30 specimen, (b) SAP45_B/0.31 specimen, (c) SAP60_A/0.30 specimen, (d) SAP75_A/0.30 specimen. ....................................................................... 109

Figure 3.37 – Cross-plot of the draw ratio in the thickness direction, λz, and of the fibres angle, θ, with respect to the specimens elongation, λx, for all the SAPθ. Continuous lines identify λz evolution, dashed lines identify θ evolution. Short points line identifies the θThreshold .................................................................................... 110


Figure 3.38 - Measured tangent modulus, ET, of the different SAPθ specimens with respect to the fibres angle, θ. The orange area locates the ΔθTransition at which ET strongly increases ........................................................................................................... 111

Figure 3.39 – Experimental ET curves obtained for the SAPθ specimens, represented by the continuous line, and theoretical ET,CLT curve, predicted by Classical Lamination Theory (CLT) .............................................................................................. 112

Figure 3.40 – Stress – strain curves, σx – εx,crosshead, of the quasi-static uniaxial load-unload tests performed on SAP45_E/0.23 specimens. (a) Case A, (b) Case B, (c) Case C. In each plot is reported the σx – εx,crosshead curve of the “Virgin” specimen, identified by the black line .............................................................................................................. 114

Figure 3.41 – Superposition of the tensile stress-strain curves of a virgin specimen (black line) and of the specimens previously subjected to load-unload cycles (red line for AC_A specimen, green line for AC_B specimen, blue line for AC_C specimen). (a) Complete view, (b) Magnification of the first region, highlighted in (a) by the yellow box. ..................................................................................................... 116

Figure 3.42 - Tangent modulus, ET, measured at different longitudinal strains, εx, for the virgin specimen (black line), and AC_A (red line), AC_B (green line) and AC_C (blue line) specimens ...................................................................................................... 118

Figure 3.43 - Measured tangent modulus, ET, of the different specimens versus fibres angle, θ. Virgin specimen (black line), and AC_A (red line), AC_B (green line) and AC_C (blue line) specimens .......................................................................................... 118

Figure 3.44 - Comparison of uniaxial tensile stress-strain curves. White line represents the experimental curve, red line the curve obtained from the “material evaluation” performed by Abaqus employing the Ogden model. ......................... 121

Figure 3.45 – M_UD90 specimen modelled in Abaqus ............................................. 122

Figure 3.46 - Deformed shape of the M_UD90_LE model considering the matrix as a linear-elastic material. (a) nominal strains in the x-direction (NE11), (b) reaction forces in the x-direction (RF1), (c) magnification of the cross-section from which the RF1 have been extracted and used for stress evaluation .......................................... 123

Figure 3.47 - Deformed shape of the M_UD90_HE model considering the matrix as a hyper-elastic material. (a) nominal strains in the x-direction (NE11), (b) reaction forces in the x-direction (RF1), (c) magnification of the cross-section from which the RF1 have been extracted and used for stress evaluation .......................................... 124

Figure 3.48 - Stress-strain curves, σx-εx, obtained from the simulations with a linear-elastic (M_UD90_LE) and hyper-elastic (M_UD90_HE) definition of the matrix . 125

Figure A.1 - Representation of moulds 1, 2 or 3. (a) Lateral view, (b) Top view .. 139


Figure A.2 - Mould 4 blueprints, base mould: (a) top view, (b) lateral view ......... 141

Figure A.3 - Mould 4 blueprints, lamination plate: (a) top view, (b) lateral view 142

Figure A.4 - Mould 4 blueprints, mould cover: (a) top view, (b) lateral view ....... 143

153

List of Tables

Table 2.1 - Glass-fibres properties of interest ............................................................... 47

Table 2.2 - Sylgard184® mechanical properties of interest......................................... 48

Table 2.3 - Summary of the unidirectional laminate samples (UDθ) prepared ...... 54

Table 2.4 - Summary of the symmetric angle ply laminate samples (SAPθ) prepared ............................................................................................................................................. 54

Table 2.5 - Estimated mechanical properties for the glass fibres-PDMS lamina in exam, considering a fibres volume fraction of 0.3 ....................................................... 62

Table 2.6 – Measured mechanical properties of the material in exam, considering a fibres volume fraction of 0.3 ............................................................................................ 63

Table 3.1 - Summary of the mean value of the transversal modulus, E2 .................. 68

Table 3.2 - Summary of the measured in-plane shear modulus, G12 ......................... 73

Table 3.3 - Mohr's circle of strains center coordinates, C(εc; γxy,c/2) and radius, R, evaluated at increasing times for a SAP45_B/0.31 specimen ..................................... 80

Table 3.4 - Maximum shear strain, γ12Max, and test time, tMax, below which the shear curve can be considered valid ........................................................................................ 83

Table 3.5 - Summary of the calculated (power function model, see Section 3.1.1) transversal modulus E2, measured laminate stiffness, Ex, and evaluated in-plane shear modulus, G12 ............................................................................................................ 84

Table 3.6 - Experimental values, ET, at low strains, and theoretical prediction (from CLT) of the stiffness, ET,CLT. ............................................................................................ 112

Table 3.7 - Overall strain applied at the end of each cycle, εx,crossehead, , for the three loading cases. For case C only the strain reached in the final cycle is reported .... 113

Table 3.8 - Fibres angle values at the beginning of the tensile test up to failure, after the loading-unloading test ............................................................................................ 115

Table 3.9 - Ogden model parameters evaluated by Abaqus from the provided experimental data of Sylgard 184 ................................................................................. 120

Table 3.10 - Transversal modulus, E2, values obtained from simulation and from the experimental characterization ...................................................................................... 125

154 | List of Tables

Table A.1 - Dimensions of the employed moulds ...................................................... 140

155

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Mechanical characterization of fibres-reinforced elastomers

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