Finite Element Modelling of the Mechanical Behaviour ofGraphene Nanocomposites

Jorge Miguel Grossinho Araújo

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor: Prof. Nuno Miguel Rosa Pereira Silvestre

Examination Committee

Chairperson: Prof. Fernando José Parracho LauSupervisor: Prof. Nuno Miguel Rosa Pereira Silvestre

Member of the Committee: Prof. José Arnaldo Pereira Leite Miranda Guedes

December 2016

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Acknowledgments

First of all, the aspiration of doing my master degree dissertation on such remarkable theme began

some years ago, when I intended to perform a simple but encouraging presentation on nanomaterials

for my high-school colleagues. For giving me that opportunity and all the needed support to accomplish

my objective, I would like to express my enormous gratitude with Prof. Nuno Silvestre.

To my family, my mother Alice, my father Jorge and my sister Ana Filipa, there are not enough words

that may appear written here to reveal my greatest joy in giving them this so expected day in their lives,

as well on my own.

For accompanying me throughout the entire master course, I would like also to manifest all my

appreciation for having such faithful friends, Miguel Dias, Nuno Martins, David Palma and Joao Silva,

who started by my side the shortest but yet challenging journey of our academic lives. Certainly, I will

remember forever all the good conversations and all the late-night dinners, but most important is the

satisfaction I have by sharing those moments with such partners.

For all the days that I was less patient or fewer present, for all the love that I was given, for never

having refused to hear my long speeches about nanomaterials and finite elements, and for being always

there when I mostly needed, to my cherished girlfriend Rafaela Albino, my sincere apologies and my

biggest thank you I could fit into two words.

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Resumo

O grafeno e um material de ultima geracao, conhecido pelas suas propriedades mecanicas, termicas,

oticas e eletricas unicas. Por esta razao, as comunidades tecnica e cientıfica tem vindo a explorar a sua

imensidao de aplicacoes nas ultimas decadas. Presentemente, o excelente comportamento mecanico

do grafeno tem-no levado a ser empregue em materiais compositos com uma notavel capacidade de

otimizacao. Nesta dissertacao, e desenvolvida uma analise de elementos finitos composta em dois

nıveis para estudar o comportamento de materiais compositos reforcados com grafeno. Inicialmente,

as propriedades mecanicas da folha de grafeno sao extraıdas atraves de um modelo de elementos

finitos que simula a sua estrutura atomica, e onde as ligacoes covalentes sao consideradas como el-

ementos estruturais. Em seguida, a representacao atomica do grafeno e inserida num meio elastico

(matriz) representado num elemento de volume adequado para extrair os modulos elasticos do material

nanocomposito. A ultima parte do documento investiga as consequencias no comportamento elastico

do nanocomposito quando sao introduzidos defeitos atomicos no grafeno. A abordagem apresentada

e capaz de reproduzir as propriedades elasticas do grafeno em concordancia com outras metodolo-

gias, revelando que o comportamento ortotropico previsto anteriormente da lugar a isotropia em folhas

quadradas de maiores dimensoes. Adicionalmente, o modelo representativo do nanocomposito mostra

que o aumento da otimizacao mecanica e conseguido quando e considerada uma maior aderencia en-

tre o grafeno e a matriz envolvente. Por fim, foi identificado que o grafeno com um baixo grau de defeitos

e capaz de manter o seu elevado efeito como reforco em nanocompositos.

Palavras-chave: Elementos Finitos, Grafeno, Nanocomposito, Propriedades Mecanicas,

Defeitos Atomicos

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Abstract

Graphene is a state-of-art material, known by its unique mechanical, thermal, optical and electrical

properties. For this reason, the technical and scientific communities have been exploring its immensity

of applications in the last few decades. Presently, the excellent mechanical behaviour of graphene has

leading it to be employed in composite materials with notable capability of improvement. In this disser-

tation, a two-level finite element analysis is developed to address the behaviour of composite material

reinforced with graphene. First, the mechanical properties of pristine graphene are extracted through a

refined finite element model that simulates its nanostructure, and where covalent bonds are regarded

as structural elements. Then, the atomistic representation of graphene is assembled into a suitable

representative volume element to extract the elastic moduli of nanocomposite material. The final part of

this document investigates the effects in the elastic behaviour of nanocomposite when atomic defects

are introduced in graphene. The present approach is able to reproduce the elastic properties of pristine

graphene in agreement with other methodologies, revealing also that orthotropic behaviour earlier pre-

dicted gives rise to isotropy for square sheets with larger dimensions. Additionally, the representative

model of nanocomposite shows that increasing mechanical enhancement is achieved when higher adhe-

sion is considered between graphene and surrounding matrix. Ultimately, it was identified that graphene

with low defect content is capable to maintain its superior reinforcing effect in nanocomposites.

Keywords: Finite Elements, Graphene, Nanocomposite, Mechanical Properties, Atomic De-

fects

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

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

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Introduction 1

1.1 Graphene and its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Organization of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Mechanical Behaviour of Graphene and its Nanocomposites – Literature Review 7

2.1 Pristine Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Graphene Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Graphene-based Polymer Nanocomposites . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Graphene-based Metal Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Structure-Property Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Structural Defects in Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Finite Element Modelling of Graphene and Graphene-based Nanocomposite 25

3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Force Field, Covalent and Non-Covalent Bonds in Graphene . . . . . . . . . . . . 25

3.1.2 Matrix and Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.3 Structural Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Pristine Graphene Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.2 Boundary and Displacement Conditions . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Nanocomposite Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2 Boundary and Displacement Conditions . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Results 47

4.1 Elastic Properties of Pristine Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

ix

4.1.1 Presentation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.2 Calculation of Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1.3 Model Validation and Discussion of Results . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Elastic Behaviour of Graphene-based Nanocomposite . . . . . . . . . . . . . . . . . . . . 59

4.2.1 Presentation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.2 Calculation of Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.3 Model Validation and Discussion of Results . . . . . . . . . . . . . . . . . . . . . . 64

5 Influence of Defective Graphene in Nanocomposites 69

5.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Parametric Study on Defect Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Conclusions and Future Work 77

Bibliography 81

A Single-Phase Materials 91

A.1 Interrelations among the 2D and 3D elastic moduli . . . . . . . . . . . . . . . . . . . . . . 91

A.2 Two-dimensionality elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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List of Tables

3.1 MFF constants used in graphene FE model. . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Equivalent carbon-carbon bond materials and geometrical properties used in graphene

FE model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Parameters for the interatomic force of Lennard-Jones implemented in element COMBIN39. 40

3.4 Materials and geometrical parameters for the RVE of graphene-based nanocomposite. . . 40

4.1 Displacements applied in the 24 A and 70 A size sheets. . . . . . . . . . . . . . . . . . . . 49

4.2 Corners coordinates before and after shear test obtained for the 24 A and 70 A size sheets. 49

4.3 Reaction forces, transverse displacements and nodal forces obtained for the 24 A size sheet. 50

4.4 Reaction forces, transverse displacements and nodal forces obtained for the 70 A size sheet. 50

4.5 Elastic properties predicted for graphene using the present approach and other corre-

sponding results available in literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6 Dimensions, number of nodes and elements for several graphene sheets used to investi-

gate the bulk values on elastic properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Comparison of stiffness enhancement obtained in graphene-based epoxy nanocompos-

ites between the proposed RVE and other predictions in literature. . . . . . . . . . . . . . 63

4.8 Volume mesh discretizations used in the RVE of nanocomposite for different volume frac-

tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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List of Figures

1.1 (a) Graphene nanostructure imaged by Scanning Tunnelling Microscopy (STM), reported

from [3]. (b) Scanning Electron Microscopy (SEM) image of vein graphite (Asbury c© Car-

bons 2016). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Proposed structure for GO membranes (Graphenea c© 2016). . . . . . . . . . . . . . . . . 2

2.1 RVE for the (a) molecular, (b) truss and (c) continnum models used in [21]. . . . . . . . . 8

2.2 (a) Schematic of nano-indentation on suspended graphene membrane and (b) AFM im-

age of a fractured membrane. Adapted from [30]. . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Chirality of single-layered graphene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Representation of the FE model for the honeycomb cell of graphene sheet with non-linear

springs. Adapted from [36]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Exfoliation mechanism of graphite into graphene sheets with supercritical CO2 fluid. Adapted

from [41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Atomistic morphology of graphene-based epoxy nanocomposites. Adapted from [72]. . . 14

2.7 High-resolution TEM image showing a graphene nanoplatelet with an interplanar thick-

ness of 0.34 nm embedded in Mg matrix. Adapted from [78]. . . . . . . . . . . . . . . . . 16

2.8 Schematic representation of three morphological states in graphene-based nanocompos-

ites. Adapted from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 TEM images illustrating the morphological differences in composites with PU matrix rein-

forced with different fillers and obtained through different processing methods. Adapted

from [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.10 High-resolution TEM image sequences for SW defect: (a) unperturbed lattice before ap-

pearance of the defect, (b) SW defect, (c) same image with atomic configuration superim-

posed and (d) relaxation to unperturbed lattice after 4s. Adapted from [92]. . . . . . . . . 20

2.11 (a) STM topograh of graphene grown on Ir surface (gray), where the arrows point out

edge dislocations at two domains boundary. (b) STM topograh of graphene across the

substrate showing (c) a defective line network of C rows with two edge dislocations. (d)

STM topograph of two coalesced graphene flakes forming a coeherent graphene island.

Adapted from [93]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.12 Variations of the (a) ultimate strength and (b) fracture strain with respect to defect cover-

age φ for defective graphene sheets under tensile tests along the armchair direction. The

three regions I, II, III in (b) correspond to the three stages – degrading, saturating, and

improving – of fracture strain variation with respect to φ. Adapted from [98]. . . . . . . . . 21

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2.13 Measured mechanical properties, (a) 2D elastic modulus and (b) breaking load of de-

fective graphene, as a function of the Raman parameters – I(D)/I(G) and I(2D)/I(G) –

measured at increasing plasma times. Adapted from [100]. . . . . . . . . . . . . . . . . . 22

2.14 Designed 5–8–5 defect arrangements – from (a) to (g) – in graphene membranes, with ’S’

as the nearest neighbour distance parameter. Adapted from [101]. . . . . . . . . . . . . . 23

3.1 Geometry, shape functions (u) and coordinate system of LINK180. Adapted from ANSYS c©

Mechanical APDL Element Reference guide. . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Geometry, shape functions (u,v,w,θx) and coordinate system of BEAM4. Adapted from

ANSYS c© Mechanical APDL Element Reference guide. . . . . . . . . . . . . . . . . . . . 28

3.3 Schematic of (a) an unit hexagon of carbon atoms and (b) geometrical properties of the

carbon-carbon bond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Square graphene sheet with 24 A per side and built with LINK180 element – image adapted

from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Square graphene sheet with 70 A per side and built with BEAM4 element – image adapted

from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Boundary and displacement conditions for uniaxial tensile test in the zigzag direction (X-

axis) exemplified in the 24 A size sheet – image adapted from ANSYS c©. . . . . . . . . . . 33

3.7 Boundary and displacement conditions for uniaxial tensile test in the armchair direction

(Y-axis) exemplified in the 24 A size sheet – image adapted from ANSYS c©. . . . . . . . . 34

3.8 Boundary and displacement conditions for shear test (XY-plane) exemplified in the 24 A

size sheet – image adapted from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.9 Boundary and displacement conditions for biaxial tensile test (XY-plane) exemplified in

the 24 A size sheet – image adapted from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . 36

3.10 Geometry, shape functions (u,v,w) and coordinate system of SOLID185. Adapted from

ANSYS c© Mechanical APDL Element Reference guide. . . . . . . . . . . . . . . . . . . . 37

3.11 RVE selected for the FE model of graphene-based nanocomposite. . . . . . . . . . . . . . 38

3.12 Example of interatomic potential U(r) and interatomic force F(r) curves expected for carbon-

CH2 interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.13 Geometry, coordinate system and force-deflection curve of COMBIN39. Adapted from

ANSYS c© Mechanical APDL Element Reference guide. . . . . . . . . . . . . . . . . . . . 40

3.14 Example of volume meshing implemented for the RVE of graphene-based nanocomposite

– images adapted from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.15 Boundary and displacement conditions for uniaxial tensile test in the zigzag direction ex-

emplified in the RVE (Front view) – image adapted from ANSYS c©. . . . . . . . . . . . . . 42

3.16 Boundary and displacement conditions for uniaxial tensile test in the armchair direction

exemplified in the RVE (Front view) – image adapted from ANSYS c©. . . . . . . . . . . . . 43

3.17 Boundary and displacement conditions for shear test exemplified in the RVE (Front view)

– image adapted from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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3.18 Boundary and displacement conditions for biaxial tensile test exemplified in the RVE

(Front view) – image adapted from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Initial and deformed shapes of graphene sheet: (a) Uniaxial tensile test in zigzag direction,

(b) Uniaxial test in armchair direction for 24 A size sheet – images adapted from ANSYS c©. 47

4.2 Initial and deformed shapes of graphene sheet: (a) Shear test, (b) Biaxial tensile test for

24 A size sheet – images adapted from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Representation of initial and deformed shapes of graphene sheet: (a) Uniaxial tensile

test in zigzag direction (red) and in armchair direction (green), (b,d) Shear test and shear

angles, respectively, (c) Biaxial tensile test. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Evolution of Young’s modulus in zigzag (X-axis) and armchair (Y-axis) directions, in re-

spect to the lattice diagonal length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Evolution of Poisson’s ratios in zigzag (X-axis) and armchair (Y-axis) directions, in respect

to the lattice diagonal length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6 Evolution of shear modulus in respect to the lattice diagonal length. . . . . . . . . . . . . 59

4.7 Evolution of bulk modulus in respect to the lattice diagonal length. . . . . . . . . . . . . . 59

4.8 Initial and deformed shapes of the RVE for: (a) Uniaxial tensile test in zigzag direction,

(b) Uniaxial test in armchair direction, (c) Shear test and (d) Biaxial tensile test – images

adapted from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.9 Comparison on normalized Young’s modulus in zigzag direction Ex/EM versus volume

fraction Vf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.10 Comparison on normalized shear modulus Gxy/GM versus volume fraction Vf . . . . . . . 62

4.11 Results obtained from the parametric study on various interface conditions and versus

volume fraction Vf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.12 Results obtained from the parametric study on various interface conditions and versus

volume fraction Vf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.13 Results of convergence study of volume mesh on the RVE focusing on Young’s modulus

along zigzag direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1 Introduction of a single-vacancy defect (red dot) in the FE model of DG nanostructure –

images from ANSYS c©. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 Examples of DG sheets with multiple SV type defects introduced – images from ANSYS c©. 71

5.3 Normalized mean values of elastic moduli related to defect concentration DC (%) and for

various volume fractions Vf (%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4 Standard deviations on elastic moduli related to volume fraction Vf (%) and for various

defect concentration DC (%). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Evolution on normalized Young’s modulus of DG from [97] and the presented normal-

ized mean values of Young’s modulus for the DG-based nanocomposite, related to defect

concentration DC (%) and for various volume fractions Vf (%). . . . . . . . . . . . . . . . 74

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5.6 Evolution on normalized shear modulus of DG from [96] and the presented normalized

mean values of shear modulus for the DG-based nanocomposite, related to defect con-

centration DC (%) and for various volume fractions Vf (%). . . . . . . . . . . . . . . . . . 75

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Nomenclature

Greek symbols

α Distance for van der Waals interaction.

∆ Area expansion ratio in biaxial tensile test.

ε Tensile strain.

γ Shear strain.

ν Poisson’s ratio.

ψ Well-depth of van der Waals interaction.

σ Tensile stress.

τ Shear stress.

θ Average stress in biaxial tensile test.

Roman symbols

A Cross-sectional area.

a Transverse displacement on zigzag direction.

b Transverse displacement on armchair direction.

c Axial displacement on armchair direction.

D Element cross-sectional diameter.

d Axial displacement on zigzag direction.

Ex Young’s modulus along zigzag direction.

Ey Young’s modulus along armchair direction.

F Nodal force.

F(r) Interatomic force of Lennard-Jones.

Gxy In-plane shear modulus.

I Area moment of inertia.

J Polar moment of inertia.

xvii

k Equivalent molecular force field constant.

Kxy In-plane bulk modulus.

L Element length.

R Nodal reaction force.

r Relative distance between neighbour atoms or molecules.

U(r) Interatomic potential energy of Lennard-Jones.

V Volume.

Subscripts

Ai Initial sheet area.

Ebeam Equivalent Young’s modulus for beam element (BEAM4).

Elink Equivalent Young’s modulus for link element (LINK180).

Gbeam Equivalent shear modulus for beam element (BEAM4).

Hi Initial sheet height.

Li Initial sheet width.

Vf Reinforcement volume fraction in composite material.

Superscripts

b Bond-angle variation interaction.

C Composite material.

G Single graphene sheet.

I Interface in composite material.

M Matrix in composite material.

s Bond stretching interaction.

t Bond torsion interaction.

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Acronyms

AFM – Atomic Force Microscope.

AIREBO – Adaptive Intermolecular Reactive Empirical Bond Order.

AMBER – Assisted Model Building with Energy Refinement.

APDL – Ansys Parametric Design Language.

CNT – Carbon Nanotube.

COMPASS – Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies.

CVD – Chemical Vapor Deposition.

DV – Double Vacancy (defect).

FE – Finite Element.

FEA – Finite Element Analysis.

GMNC – Graphene-based Metal Nanocomposites.

GNP – Graphene Nanoplatelets.

GO – Graphene Oxide.

GPNC – Graphene-based Polymer Nanocomposites.

MD – Molecular Dynamics.

MFF – Molecular Force Field.

MM – Molecular Mechanics.

ROM – Rule of Mixtures.

RVE – Representative Volume Element.

SEM – Scanning Electron Microscopy.

STM – Scanning Tunneling Microscopy.

SV – Single Vacancy (defect).

xix

SW – Stone-Wales (defect).

TBMD – Tight-Bending Molecular Dynamics.

TEM – Transmission Electron Microscopy.

vdW – van der Waals.

xx

Chapter 1

Introduction

1.1 Graphene and its applications

Graphene, the well-publicized and now famous two-dimensional carbon allotrope, is the most versa-

tile material ever discovered on Earth. Spite of being only an atom-thick layer of carbon atoms, it was

found to be one of the strongest materials ever tested. Its chemical structure is a repeating pattern of

hexagons in honeycomb, where each carbon atom is covalently bonded to its nearest neighbours by sp2

hybridizations. Before entering in more details, one must first contextualize its discovery to gain better

understanding about graphene outstanding properties.

It has been claimed that graphene was firstly discussed in 1946 by P. Wallace [1]. At that time,

while the theoretical physicist was aiming to understand the three-dimensional honeycomb structure of

carbon atoms in graphite, he developed a ”virtual” two-dimensional analog – later called graphene – from

which the properties of graphite could be extracted. P. Wallace showed that, if one could stack enough

sheets of graphene on top of each other, it would eventually congregate to bulk graphite. Both chemical

structures can be seen on Figure 1.1. It was just in 2004 that a group in the University of Manchester,

using the inverse process – called as mechanical exfoliation –, has successfully isolated single-layers of

graphene by softly fraying a graphite crystal on an oxidized silicon wafer, as claimed in Novoselov et al.

[2].

2 nm

0.5 nm

(a)

5 μm

(b)

Figure 1.1: (a) Graphene nanostructure imaged by Scanning Tunnelling Microscopy (STM), reportedfrom [3]. (b) Scanning Electron Microscopy (SEM) image of vein graphite (Asbury c© Carbons 2016).

For the last 60 years, a vast net of scientists, researchers and industries have been extensively

1

exploring and unveiling the potential of graphene in present and future technologies.

At the moment, the energy field is being enhanced with the superlative behaviour in electrical car-

rying transport of graphene, with energy losses reduced or even absent, as reported by Yan et al. [4].

Supercapacitors and batteries with higher lifespan and power supply, while being lighter, smaller and

with a faster charging time, are the closest technologies expected to be commercially available within

the next 5 to 10 years (Graphenea 2016). Synthesis of graphene aerogel with high surface area and

electrical conductivity has been reported in Worsley et al. [5], that sees straight usage in energetic appli-

cations. Another future aim is being investigated (University of Manchester 2016) in the growing market

of renewable energies such as wind and solar power. Since graphene has greater flexibility, larger band

of light wavelength and higher light absorption, it may be used in current electrical grids and photovoltaic

cells as a more efficient and potentially cheaper option instead of the materials used presently.

Likewise, one specific area where one will soon begin to notice employment of graphene on a com-

mercial scale is optoelectronics, such as touchscreens, liquid crystal displays and organic light emitting

diodes, as reported by Bonaccorso et al. [6]. A material must be able to transmit more than 90% of

light and also offer electrical conduction to be usable in optoelectronic applications. Besides being able

to optically transmit up to 97.7% of light, graphene is an almost completely transparent material. In that

sence, the applicability of highly conductive and transparent graphene films into flexible, stretchable and

foldable electronics has been investigated in Kim et al. [7].

However, the creation of nano-electronic devices from pristine graphene has to deal with the absence

of band gap. In other words, the electrical current that passes it can not be switched off to save energy.

Several groups of researchers have been able to overcome this drawback with ingenious solutions, such

as the investigation on lithographically patterned graphene reported by Han et al. [8] and adaptable

band gap in strained graphene described in Ni et al. [9]. Kumar et al. [10] has also solved this problem

through the development of oxygen clustering on graphene oxide (GO). This component is a processable

precursor for the production of graphene-based materials and it is obtained by chemical exfoliation of

graphite, resulting in a molecular nanostructure similar to Figure 1.2.

Figure 1.2: Proposed structure for GO membranes (Graphenea c© 2016).

Another valuable property of graphene is its capacity to be completely impermeable to most liquids

and gases, while it allows water to pass through it. As found by Bunch et al. [11] and Liu et al. [12],

graphene membranes can be effectively used in water filtration and desalination systems by control-

2

ling its inter-layer distance. Moreover, it has been proved to stop helium, which is the hardest gas to

block leakage. The same procedure could revolutionize the removal of harmful greenhouse effect gases

released into the atmosphere by power stations. Alternatively, the usage of graphene membranes as

coatings on food and pharmaceutical packaging can stop the transfer of water and oxygen, thus increas-

ing its durability and quality. Expanded graphite has began to be used in harvest of petroleum products

spilled on the water surface, by taking advantage of its high adsorption capacity, as debated by Lutfullin

et al. [13].

As for biomedical engineering, it is certainly a field where graphene will become a vital part in the

future. The development of fast and efficient bio-electric sensors, with the ability to monitor glucose

and haemoglobin levels, cholesterol and even DNA sequencing, may be upgraded with the introduction

of this nanomaterial. Also, it could be used as a platform for drug delivery, tissue and orthopaedic

engineering and biological agents. However, current estimations suggest that it would not be seen in

none of these applications until 2030, as we still need to test its biocompatibility and validate its true

benefits.

Regarding aerospace applications, carbon fiber is commonly incorporated into the structure of air-

crafts as it is a strong and light material. On the other hand, graphene is predicted to be much

stronger while being also much lighter, with a specific strength about 5.0× 107 Nm/kg compared to

2.5× 106 Nm/kg of high modulus carbon fiber. The potential benefits from hybrid composites consti-

tuted with both reinforcements have been under evaluation to estimate its structural performance, as

outlined in Shen et al. [14] and Hadden et al. [15]. So far in this field, prospect of graphene applications

was mainly focused on polymer-based nanocomposites for several years, because their processing is

relatively simple. However, a growing interest is in the application of advanced metal-based nanocom-

posites for structural engineering, as reported by Kim et al. [16] and Moghadam et al. [17]. Their goal

is to provide an improved substitute for conventional heavier metals and alloys in the structure of aircraft

and thus, boosting fuel efficiency, flight range and reducing weight. The transverse impact behaviour

of graphene can also help in the development of high strength requirement applications such as body

armor for military personnel and vehicles, as outlined by Haque et al. [18].

Without forgetting the potential benefits in wear and corrosion effects, two interesting applications in

aerospace industry are described in literature related to branch of polymer nanocomposites. A recent

study by Kandanur et al. [19] probed the prevention in excessive wear of a solid lubricant commonly

used, by introducing graphene into it as reinforcement. On the other hand, Raji et al. [20] revealed a

new mixture of graphene injected into an epoxy resin, that is being develop to be applied onto specific

structural elements in the fuselage to create electrical conduction and produce heat as an anti-icing

agent.

Moreover, graphene could be used to coat the material in the aircraft exterior surface to prevent

electrical damage resulting from lightning strikes, due to its strong electrical conductivity. As well, the

same coating could be employ to measure strain rate of structural elements or temperature sensing,

notifying the cabin of any changes in the stress levels that the aircraft structure has under operation

or fire risk mitigation. It might also improve external surface finishing and reducing the drag caused by

3

painting roughness.

In brief, the present interest with graphene highlights its remarkable physical properties and the ever

growing list of potential applications these properties may offer for the nearest future.

1.2 Scope and Objectives

Due to its diversity on properties already known, graphene has been a nanomaterial subjected to

extensive scrutiny among the scientific and technical community in recent years. The technological

developments in processing and analysis methods have been helping to investigate additional applica-

tions of graphene that appear almost on daily basis, but also its nanoscopic features that distinguishes

it from other carbon allotropes. Regarding the mechanical behaviour of pristine graphene, complex

computational methods like Ab-initio and molecular dynamics (MD) have been able to predict similar

results. However, those numerical methods require intensive calculations and time consuming analy-

ses, as they depend not only on the electron cloud of each atom (Ab-initio), but also on thermodynamic

variables (MD) such as pressure, temperature, mole and energy. Moreover, that computational effort

increases exponentially when one desires to research on the micro-cluster morphology and its proper-

ties, resulting from incorporating nanoparticles in composite materials. In fact, numerical approaches

on the mechanical behaviour of graphene nanocomposites reported on literature are scarce. Also, due

to a lack in quality of some processing methods, naturally occurring imperfections and growth-induced

defects are usually present in produced graphene, deteriorating the overall reinforcing capability and not

much attention has been given to explore this field.

Accordingly, the scope of this dissertation is to study the mechanical behaviour of a graphene-based

nanocomposite using the finite element analysis (FEA) technique. In recent past, many authors have

been applying this technique to access nanoscale structures like pristine graphene, carbon nanotubes

(CNT) and others. Still, the versatility and efficiency of FEA are adequate to approach the problem

here proposed, under certain conditions. Therefore, the efforts in this thesis will be directed towards the

achievement of the following objectives:

• To develop a refined nanoscale finite element model capable of simulate the mechanical behaviour

of pristine graphene and validate it against other more rigorous methods in literature.

• To develop a consistent microscale finite element model of nanocomposite that incorporates the

graphene nanostructure earlier obtained as reinforcement.

• To access the mechanical behaviour of graphene nanocomposite and influence of several structure-

property relations on its mechanical performance.

• To validate the nanocomposite model in comparison with other similar approaches and experimen-

tal results.

• To incorporate atomic defects in graphene nanostructure and explore their effects on the effective

mechanical properties of the nanocomposite.

4

Hence, the present work seeks to provide an useful comparison between numerical and experimental

results and enables one to bypass complex calculations on chemistry or thermodynamics to obtain first

approximations for mechanical properties of graphene-based composites. Besides, to the author’s best

knowledge, the application of FEA to explore how the atomic defects at nanoscale may affect their

mechanical performance, will be here reported for the first time.

1.3 Organization of Contents

The present chapter is organized to reveal some major discoveries concerning the present and future

applications of graphene and its derivatives. For benchmarking purposes, Chapter 2 reviews the key

breakthroughs reported in literature regarding the mechanical behaviour of pristine graphene and its

defective condition, as well on polymer and metal matrices reinforcement by carbon nanofillers.

Having as basis some essential propositions and simplifications to solve the problem proposed in this

dissertation, the implementation of the multi-scale finite element model is described in detail in Chapter

3. Next, the set of elastic properties computed from the results of both finite element analyses are

presented in Chapter 4, regarding the mechanical behaviour of pristine graphene and its nanocomposite,

and are also validated and discussed against analogous results available in literature. In Chapter 5, the

study of influence of atomic defects present in graphene nanostructure is conducted to investigate their

effects in the elastic properties of nanocomposite through the finite element model earlier obtained.

Lastly, the main conclusions regarding the demonstrated results are identified and some proposals

for future developments are also given.

5

6

Chapter 2

Mechanical Behaviour of Graphene and

its Nanocomposites – Literature Review

The present chapter summarizes some of the major developments regarding the mechanical be-

haviour of graphene and its nanocomposites in the last two decades. While the last chapter is mainly

focused on applications, the present one is directed towards the state-of-art concerning applied methods

and their outcome, exposed in increasing evolution in time-line. The chapter outline is divided in three

main sections: Sections 2.1 and 2.2 probe pristine graphene and graphene-based nanocomposites,

respectively, while Section 2.3 introduces the topic of structural defects in graphene nanostructure.

2.1 Pristine Graphene

When modelling the macroscopic behaviour of nanostructured materials, one of the fundamental is-

sues that needs to be addressed is the large difference in length scale. On opposite ends of this scale

are computational chemistry and solid mechanics, both being composed of highly developed and reli-

able formulations/methods in present days. The former predicts molecular properties based on atomic-

quantum models while the latter predict the macroscopic mechanical behaviour of materials idealized as

continuous media. However, it did not always existed a model corresponding as an intermediate step in

the length scale.

In 2002, Odegard et al. [21] pioneered a methodology based on the equivalence between discrete

molecular structure and continuum models, adopting a representative truss model for a single hexago-

nal ring of graphene as shown in Figure 2.1. Due to the nature of the material and loading conditions,

the equivalent truss model was constructed with unidimensional structural elements like rods, whose

force constants were based on AMBER force field. The correspondence between molecular and con-

tinuum models was achieved by equating the equivalent truss potential energy with the strain energy

of a continuum plate model with finite thickness. From the results, the authors observed that mechan-

ical properties of honeycomb structure are dependent on the geometry and dimensions assumed, like

Young’s modulus and Poisson’s ratio, and may vary considerably. The main problem was, at that time,

no consistent data on the properties of graphene was available. So, the authors used bulk properties of

graphite for characterizing the equivalent-continuum plate.

In fact, between 2000 and 2004, very few researchers have reported the elastic properties of pristine

7

graphene. Van Lier et al. [22] reported Young’s modulus as 1.11 TPa, while Kudin et al. [23] computed

Young’s modulus and Poisson’s ratio of graphene as 1.02 TPa and 0.149 respectively, using both ab-

initio methods. Arrayo and Belytschko [24] calculated the Young’s modulus and Poisson’s ratio based on

Brenner potentials [25] using molecular mechanics (MM) as 0.694 TPa and 0.412, respectively. From

a general point of view, one could conclude that there was a wide dispersion in graphene properties

divulged in the literature.

Based on the last argument, in 2006, Reddy et al. [26] investigated the reason for this scatter by

analysing the effect of minimized or un-minimized potential energy configuration of graphene for the

determination of the elastic properties using atomistic molecular mechanics. The results showed a first

set of values - values for Young’s modulus and Poisson’s ratio around 0.7 TPa and 0.4, respectively -

resulting when an equilibrium configuration was used for the computation, based on bond-order Tersoff-

Brenner potentials [25, 27]. A second set - values around 1 TPa and 0.25, respectively - occurred when

minimization was neglected. So, the authors concluded that equilibrium and potential energy minimiza-

tion adjustments to the graphene lattice have large influence on the elastic properties obtained, being

this fact one of the major causes for the wide range in elastic properties values reported for graphene in

the literature. Furthermore, they extended the observations of Odegard et al. [21] to evaluate the influ-

ence of boundary conditions in the elastic constants computed. It became demonstrated that finite sized

graphene sheets do not behave exactly like an isotropic material, but in fact, resembling an orthotropic

material with an orthotropy degree between 0.92 and 0.99.

Bond-Angle

Variation

Bond Stretching

(a)

Bond-Angle

Variation Rod

Bond Stretching

Rod Truss Joints

X1

X2

(b)

X1

X2

Thickness

X3

(c)

Figure 2.1: RVE for the (a) molecular, (b) truss and (c) continnum models used in [21].

Meanwhile, experimental advances were made by Novoselov et al. [2], who obtained individual

single graphene sheets successfully by mechanical exfoliation, as prior refered in section 1.1. This

breakthrough allowed the scientific community to introduce new experimental technologies and to exe-

cute more intensive studies about the higher stiffness and strength of graphene, which were predicted

only from theoretical and numerical results until then. Frank et al. [28] measured the effective spring

constants of stacks of graphene sheets (less than 5 layers) using atomic force microscope (AFM) inden-

tation and extracted a Young’s modulus of 0.5 TPa. The same technique was used by Gomez-Navarro et

al. [29] to determine a value of 0.25 TPa of freely suspended graphene single-layers, obtained via chem-

8

ical reduction of GO. However, many difficulties arose in experimental determination of these quantities

because of uncertainty in the sample geometry, stress concentration at clamping points, unknown load

distribution and structural defects in graphene nanostructure.

In 2008, Lee et al. [30] obtained single-layer graphene membranes using Novoselov et al. [2] ap-

proach. After, they applied AFM indentation to acquire several force-displacement measurements for

various sets of raw material flakes, membrane diameters, displacement rates and indenter tip radius.

The experimental apparatus may be schematized as presented in Figure 2.2(a). A total of 67 sample

results were tested, yielding a statistically distinguishable Young’s modulus of 342 ± 30 N/m. Repeat-

ing the same procedures until failure of membrane, the fracture strength and fracture strain were also

obtained. An AFM image of a fractured membrane is presented in Figure 2.2(b). In the end, the au-

thors achieved a value of 42 ± 4 N/m for fracture strength at strains of approximately 25%. These results

serve until today as a benchmark for structural and mechanical applications of graphene and established

graphene as the strongest material ever measured.

One of the most intricate factors that has been leading experimental simulations to obtain misaligned

results compared to numerical ones, is the defect-free condition. For that reason, Lee et al. [30] finalized

their study by conducting a comparison between the breaking force distribution observed during failure

tests with the Weibull distribution [31], which characterizes the failure of brittle materials with random

defects. With this argument they were able to infer that the tested samples were mostly defect-free, at

least under the indenter tip.

(a) (b)

Figure 2.2: (a) Schematic of nano-indentation on suspended graphene membrane and (b) AFM imageof a fractured membrane. Adapted from [30].

Nevertheless, as analytical approaches were laborious and experimental methods were too expen-

sive or inaccessible, a propensy for structural finite element analysis (FEA) started to grow in recent

years. In 2009, Sakhaee-Pour [32] studied the elastic behaviour of pristine graphene sheets based on

energy equivalence used by Odegard et al. [21]. To model the covalent bond of honeycomb structure,

they used Euler-Bernoulli beam finite elements (FE) with properties, in terms of covalent stiffness and

force field constants, derived from another structural mechanics approach for the analysis of carbon

nanotubes (CNTs) [33]. The outcome confirmed the prediction of orthotropy made by Reddy et al. [26],

indicating the dependence of Young’s modulus, shear modulus and Poisson’s ratio on the crystalline

asymmetry. This asymmetry is called chirality and, as represented in Figure 2.3, it may be defined

9

according to the direction how a graphene sheet is rolled to form a CNT (indicated by black arrows).

In its turn, Scarpa et al. [34] added a deep beam theory with Timoshenko shear correction factor to

Sakhaee-Pour [32] work. The objective was to access the influence of additional deformation mecha-

nisms in equivalent honeycomb structure of graphene, such as bending and shear deformations. Also,

the inclusion of optimum shear correction factor allowed to identify distributions of values for geomet-

ric and material parameters of equivalent beam – cross-sectional thickness and Poisson’s ratio – via

nonlinear optimization. These parameters were found to diverge slightly depending upon the type of

mechanical loading and the choice of potential force field used. Moreover, the authors concluded that

from a structural point of view, the addition of bending and shear deformations can justify some higher

Poisson’s ratios values presented in earlier works. Later, Scarpa et al. [35] analysed the out-of-plane

bending behaviour of single layer graphene sheets with circular and rectangular shape. This work turns

out to be particularly interesting since it provides a direct comparison with experimental methods such

as AFM indentation [28, 29, 30], in order to characterize the flexural behaviour of single-layer graphene

sheets.

Armchair sheet Zigzag sheet Chiral sheet

Figure 2.3: Chirality of single-layered graphene.

Another approach on FEA was adopted in Georgantzinos et al. [36], who predicted the fracture

behaviour of single graphene sheets under high strains having as basis non-linear spring FEs. The

unit representative of the FE model assumed is shown in Figure 2.4, and other detailed aspects of this

approach are referred to the respective report cited. Still, the authors investigated the failure mechanism

of graphene under tensile and shear loading, showing from their results that graphene shows greater

anisotropy in its ultimate properties when subjected to large strains.

Apart from FEA, molecular dynamics (MD) simulations have been converging successfully to close

results, either validating values already predicted from other methods but also exploring other mechan-

ical characteristics about pristine graphene. Based on bond-order Tersoff-Brenner potentials, Ni et al.

[37] investigated the tensile loading for large strains, eventually reproducing results consistent with Geor-

gantzinos et al. [36] with respect to fracture process of pristine graphene.

Using the improved adaptive intermolecular reactive empirical bond order (AIREBO) potential [38],

that has been showing to accurately capture bond interaction between carbon atoms as well as bond

breaking and bond re-forming, H. Zhao et al. [39] inspected approximately square-shape graphene

sheets with sizes varying from 1.17 and 15.62 nm. The authors highlighted that graphene has approx-

imately isotropic elastic behaviour but only for a small tensile strain range below 0.5%, i.e. no chirality

10

dependence. Such results were in disagreement with the previous ones, placing an emphasis on the

sensitivity required that one should have when simulating accurate mechanical tests on the properties

of this nanomaterial. Moreover, it was also reported that a pristine sheet with diagonal length over 10

nm is estimated to have size-independent mechanical properties. This variation is similar to the size-

dependent Young’s modulus of a CNT, even though the size effect can be neglected when the diameter

of the CNT is larger than 0.75 nm [40].

Figure 2.4: Representation of the FE model for the honeycomb cell of graphene sheet with non-linearsprings. Adapted from [36].

The final subject to be addresed in this section is the synthesis of pristine graphene sheets. In

2015, a novel procedure for high-quality and large-quantity nanosheets was reported in Li et al. [41],

who use shear-assisted exfoliation with a supercritical fluid composed of CO2. When high-speed shear

stress is applied to CO2 fluid, as schematized in Figure 2.5, expanded graphite powder is effectively

laminated with 90% of exfoliated sheets with less than 10 layers and approximately 70% from 5 to

8 layers. Supercritical fluids have optimum characteristics, such as high permeability and diffusivity

while having low viscosity, and have been used to exfoliate others types of layered materials. The

obtained samples in [41] were then characterized by a battery of experiments, such as scanning electron

microscopy (SEM), transmission electron microscopy (TEM), AFM, Raman spectroscopy, and others.

Therefrom, the authors concluded that this synthesis process of graphene nanosheets had resulted in

higher mechanical and electrical quality than those obtained with chemically reduced graphene oxide

(GO).

11

Figure 2.5: Exfoliation mechanism of graphite into graphene sheets with supercritical CO2 fluid.Adapted from [41].

Finally, an alternative approach on pristine graphene synthesis is chemical vapour deposition (CVD)

growth on a catalyst surface. A large-scale MD simulation was perfomed by Z. Xu et al. [42] to explore

the self-aggregation of carbon atoms on a nickel surface. Throughout this numerical procedure, growth

mechanism of single layer graphene was successfully observed and formation of defects appeared to be

diluting – for higher temperatures and longer annealing time – with the assistance of catalyst particles

and thermal energetic motion, resulting in graphene sheets with low-defect content.

In sum, one may conclude that mechanical behaviour of pristine graphene has been investigated and

characterized in great extent, specially by numeric and experimental means, and their results proved that

this outstanding material has very attractive mechanical properties and also, must serve as basis in a

huge variety of applications in the near future in many areas of engineering.

2.2 Graphene Nanocomposites

One possible route to harness the list of awesome mechanical properties of graphene reported in

literature, is to incorporate graphene or one of its derivatives into a composite material. This section

will summarize some of the fundamental processing methods and recent studies about the mechan-

ical behaviour of graphene-based polymer and graphene-based metal nanocomposites, as well their

structure-property relationships.

2.2.1 Graphene-based Polymer Nanocomposites

It is well known that pristine graphene is not compatible with organic polymers unless it has being

previously reduced of functionalised, as stated in Potts el al. [43]. Also, a strong van der Waals inter-

12

action between adjacent layers of pristine graphene is responsible for formation of agglomerates and

a poor enhancement effect. In contrast, GO sheets are more chemically reactive, with various organic

radicals attached on its surface, and as a result it has attracted considerable attention as filler for poly-

mer nanocomposites. According to the literature, a diversity of processing methods have been reported

for synthesis of graphene-based polymer nanocomposites (GPNC), some of which are divided in the

following main categories: solution blending, melt mixing and in-situ polymerization.

The first one has been the most popular technique, since it only involves the solubilisation of polymer

– in common aqueous or organic solvents – and its mixing with a colloidal suspension of graphene-based

materials. The suspension achieved can be precipitaded causing the polymer chains to encapsulate the

filler under precipitation. The polymers polystyrene (PS) [44], polycarbonate (PC) [45], polyacrylonitrile

and polymethyl methacrylate (PMMA) [46], polyacrylamide [47], polyurethane (PU) [48, 49, 50], poly-

imide [51, 52] and epoxy [53, 15] have been successfully mixed with modified graphene filler through so-

lution blending, and producing GPNC with considerable mechanical enhancement. Additionally, the easy

production of aqueous filler platelets suspensions through sonication – act of applying sound energy to

agitate particles in a sample – makes solution blending a notably appealing technique for water-soluble

polymers such as polyvinyl alcohol (PVA) [54, 55], polyallylamine [56] and epoxy [57].

The crucial challenges with solution blending are minimizing residual solvents and obtaining good

dispersion properties in viscous polymeric solutions. Even though it leads to better particle dispersion

than melt mixing process, its slow solvent evaporation may induce particle re-aggregation. In addition,

the use of large quantities of solvent and its associated environmental issues have prevented this tech-

nique from large-scale production.

Secondly, melt mixing combines high temperatures and shear forces to distribute nanofiller powder

in thermoplastic polymer matrices in molten state, using conventional methods like extrusion or injection

molding. In melting process, no solvent is involved and thus making it an economical and environmen-

tally friend method for mass production of GPNC. A wide range of GPNC have been manufactured with

enhanced mechanical properties through this method, based on polyamide-6 [58], polyethylene [59, 60],

PC [61], polypropylene (PP) [62] and polyphenylene sulphide [63]. However, this process is less effective

in dispersing the nanofiller due to increased viscosity, as the filler content enlarges. Another challenge

for the melt mixing is the low bulk density of graphene in dry powder form, which difficults its feed to the

extruder. Also, it is not a suitable option for matrix materials that are susceptible to thermal degradation,

as reported in H. Kim et al. [48].

Lastly, through in-situ polymerization, a graphene-based material is dispersed in solvent and mixed

with a monomer solution with a suitable reactive initiator. After the initiator is dissociated by heat or

radiation, graphene-based materials can be mixed or cross-linked with polymer chains, providing strong

interactions between the filler and the polymer via covalent bonding. Unlike solution blending or melt

mixing techinques, polymerization achieves a high level of dispersion of graphene-based filler without

solvents or prior exfoliation. Currently, it is still very time consuming to initiate mass production since

it should be performed in soluble state to elimate large quantities of residual solvents. Even so, a

large variety of GPNC has been successfully produced, based on polyaniline [64], PS [65], PMMA [66],

13

polyimide [67], PU [68] and PP [69].

More recently, some numerical studies on the mechanical behaviour of GPNC have been also em-

ployed. The phenomena of re-agglomeration occurs in graphene-based materials due to high interlayer

adhesion energy of graphene [70], resulting in stacked structures called graphite flakes, which are rarely

investigated. The mechanical properties of graphite flakes and single-layered graphene were charac-

terized using MD simulations in Tsai and Tu [71], where the authors took into consideration clarify if the

mutual influences of adjacent layers are relevant in the mechanical response. Selecting AMBER force

field for the carbon-carbon atomic interaction, it was found that single-layered graphene exhibit higher

moduli – approximately +12% – than graphite flakes. Thus, the authors suggested that to achieve better

mechanical properties in graphene-based nanocomposites, the aggregated flakes should be expanded

and effectively exfoliated into single sheets.

Moreover, graphene-based epoxy nanocomposites were modelled through MD simulations in Shiu

and Tsai [72] and characterized for three different formats of reinforcement – graphite flakes, intercalated

graphene and intercalated GO – as represented in Figure 2.6. The covalent bonding, as well as the non-

covalent interactions among the molecular chains were described by the COMPASS force field [73].

(a) Graphite flakes (b) Intercalated graphene

sheets

(c) Intercalated GO sheets

Figure 2.6: Atomistic morphology of graphene-based epoxy nanocomposites. Adapted from [72].

The results shown in [72] proved that intercalated forms of graphene yield a greater amount of epoxy

chains with high density on both sides of each sheet, as compared to graphite flakes. Since more

interfacial areas were generated in intercalated nanocomposites, it was verified that this high density

distribution of polymer provides superior enhancement in the mechanical properties of the nanocom-

posite material. Besides, from the three different morphologies addressed, it was found that the oxide

functional group present in GO is another important factor in GPNC. Due to the development of higher

interaction energy with the surrounding matrix, the latter provides a superior reinforcing effect in Young’s

modulus than unmodified pristine graphene.

Ultimately, Giannopoulos and Kallivokas [74] and Spanos et al. [75] employed FEA to investigate

the mechanical behaviour of GPNC. Due to the high dependence of graphene’s mechanical properties

in respect to its nanostructure, a multi-scale technique was established from atomistic structural me-

14

chanics approach based on FE models representing the interatomic force field in graphene. The next

level was regarding the matrix material surrounding a single graphene sheet as a continuum medium,

by implementing three-dimensional isoparametric elements, since it reduces computational cost and

model complexity. In order to approximate effectively the load transfer between the two phases, a hy-

brid interface of specific thickness was also considered. Therefrom, interfacial mechanical properties

were assumed varying according to an exponential law dependent and bounded by the surrouding ma-

terial properties of the two basic composite components, i.e. graphene and matrix. This concept has

proven its ability to model experimental findings concerning the interfacial stress and strain fields of

multi-walled CNT-polymer nanocomposites [76]. Spite of polymeric chains nearby the surface of inter-

calated graphene sheets behave differently from those in bulk matrix, as noticed earlier, it has not yet

been proved experimentally that this dependence causes anisotropy on interface mechanical properties

of graphene-based nanocomposites. Even so, the results revealed that their elastic properties are influ-

enced by the size of reinforcement and its volume fraction, as well by interface stiffness. In sum, all the

numerical studies aforementioned showed consistent results with the experimental ones and thus will

provide an important comparison with the present report.

2.2.2 Graphene-based Metal Nanocomposites

A growing interest on metal reinforcement by carbon nanofillers is leading to novel materials for a

variety of applications, as mentioned before in Chapter 1. Compared to conventional metals and alloys,

graphene-based metal nanocomposites (GMNC) are expected to have higher strength combined with

reduced weight and also enhanced wear and corrosion resistance. However, very few experiments or

studies were conducted about the mechanical behaviour on GMNC so far. For that reason, one might

not dissipate much effort on understanding all sorted methods in detail but for now only browse their

results.

Starting in 2012, graphene nanoplatelets (GNP) were combined with powdered aluminium (Al) in

order to observe the effects on mechanical strength, as reported in J. Wang et al. [77]. The tensile

strength of the GNP-Al nanocomposite was enhanced though below theoretical expectations, with a

62% enhancement obtained for only 0.3% of weight content in GNP. The authors attributed an effective

interface stress transfer as the main strengthening mechanism. However, only a very small percentage

of the matrix was affected by the GNP content because there was a large difference in size of flakes for

both materials.

Regarding other metallic elements, a successfully attempt on using GNP to reinforce a metal matrix

based on magnesium (Mg) is described in Chen et al. [78]. This method combined liquid-state ultrasonic

processing and solid-state flutter to disperse the filler uniformly into the Mg matrix. Adequate tests were

conducted to evaluate the mechanical performance of the GNP-Mg nanocomposite, revealing a 78%

increase in micro-hardness with 1.2% volume fraction in GNP, compared to the hardness of pure Mg

prepared under the same condition. High-resolution TEM was used to study the interfacial bonding,

confirming that most of it occurred through GNP encapsulation inside Mg grains, as represented in

Figure 2.7.

15

In addiction, GO functionalised with ionic functional groups was chemically bonded into copper (Cu)

matrices through molecular-level mixing, as reported in Hwang et al. [79]. The strengthening effects of

this hybrid graphene content were evaluated, with elastic modulus and the yield strength being signif-

icantly incremented over bulk copper, 1.3 and 1.8 times higher, respectively. However, to consolidate

both composite phases, the mixture underwent through spark plasma sintering, which caused a more

defective state in GO, damaging its honeycomb bonding network.

Figure 2.7: High-resolution TEM image showing a graphene nanoplatelet with an interplanar thicknessof 0.34 nm embedded in Mg matrix. Adapted from [78].

In a similar way, Jiang et al. [80] conducted a comparative study to access the influence of pristine

graphene and GO as reinforcements on Cu matrices. To improve the chemical reactivity of pristine

graphene, polymeric chains were also attached onto its surface to provide a more uniform attachment

on Cu particles. After preparation of graphene-Cu and GO-Cu nanocomposites with powder metallurgy

process, the former exhibited yield and 5% compression strengths increased up to 172 and 228 MPa,

respectively, which constitute a 90% and 81% enhancement comparing to the same properties of pure

Cu. On the other hand, the latter resulted only in yield and 5% compression strengths of 156 and 208

MPa, respectively. Based on these results, the authors verified through experimental methods that the

produced hybrid graphene provided greater reinforcing improvement because of its high crystallinity and

refined interface adhesion energy with the matrix. However, the fracture strain of both nanocomposites

decreased significantly compared to bulk Cu, transforming a ductile into brittle material in consequence

of an interface impermeable to dislocations. Since grain dislocations tend to concentrate in interfacial

regions, the crack tip can not release stress concentration by further deformation.

Finally, an important study already mentioned in Chapter 1, was conducted by Moghadam et al. [17]

about self-lubricating GMNC. Because of its good tribological properties – high wear and corrosion resis-

tance – graphite may act as solid lubricant and has been used to produce self-lubricating composites for

several applications in automotive and aerospace industries. However, when graphene-based materials

are embedded into metal matrices, it is expected that wear rate and coefficient of friction of the resulting

nanocomposite could be even more reduced. Keeping this in mind, an extensive review on applicability

of various processing developments on self-lubricating nanocomposites reinforced with carbonaceous

materials was developed in the aforementioned report.

16

2.2.3 Structure-Property Relations

Relatively limited search has been conducted to understand the microscopic structure-property re-

lationships in graphene-based nanocomposites. The mechanical behaviour enhancement observed

due to the presence of graphene is generally attributed to the its higher aspect ratio – in-plane dimen-

sions related to out-of-plane thickness – and excellent mechanical properties. Concerning all types of

graphene-based materials, the filler can exist in different forms such as stacked, intercalated or exfoli-

ated, as shown in Figure 2.8.

With increased interlayer spacing, intercalated clusters tend to have better dispersion than a stacked

phase. In exfoilated structure, the platelets have the largest interfacial contact with the host matrix.

However, because the increasing interaction with the matrix is proportionaly related to exfoliation level

and sheet size, the reinforcement may have a wavy or wrinkled structure that tend to fold itself rather

than stretch under an applied loading, as reported by Wakabayashi et al. [81]. Even so, a wrinkled

surface texture has been observed to create mechanical interlocking and effective load transfer, leading

to improved mechanical strength as reported in Srivastava et al. [82] and F. Liu et al. [83].

(a) Stacked phase (b) Intercalated phase (c) Exfoliated phase

Figure 2.8: Schematic representation of three morphological states in graphene-basednanocomposites. Adapted from [43].

In fact, nanocomposite processing methods yield different morphologies of graphene. According

to Potts et al. [43], solvent mixing and in-situ polymerization methods give a microstructure with ho-

mogeneously exfoliated and randomly oriented distribution of filler within the matrix, while melt blending

produces an intercalated and aligned distribution of graphene, as shown in Figure 2.9. As reported in the

aforementioned work, these different microstructures have serious effects in the mechanical properties

of GPNC.

Aligned with the preceding arguments, an innovative study on the effects of morphological state of

graphene on nanocomposites was outlined in Bayrak et al. [84]. The main objective was to access how

different morphological structures may influence the stiffness of graphene-based nanocomposites by

comparing experimental and theoretical approaches. Common assumptions in micromechanics, – such

as the rule of mixtures (ROM) –, consider the reinforcement as fully exfoliated and homogeneously dis-

tributed. Therefore, modified theory versions were constructed to account for intercalated morphology

instead, which has been effectively observed in most of experimental characterization studies. There-

from, the accuracy of existing micromechanical models and their upgraded versions was measured,

comparing the set of theoretical models with experimental tests. From those results, an accuracy im-

17

provement was revealed when morphological adjustments were made to existent theories and several

morphological effects, such as matrix stiffness, graphene thickness and graphene interlayer spacing,

were also explored.

(a) Unexfoliated graphite processed by melt

mixing

(b) GO processed by melt mixing

(c) GO processed by solution blending (d) GO processed by in-situ polymerization

Figure 2.9: TEM images illustrating the morphological differences in composites with PU matrixreinforced with different fillers and obtained through different processing methods. Adapted from [43].

Another important structure-property relationship is the interface adhesion, which may lead to lower

load transfer when poorly designed. AFM has been successfully employed by Kranbuehl et al. [85] to

measure the attractive forces in the interface of GO particles and PMMA/PVA polymers. The nanoparti-

cles were sandwiched between different polymeric materials possessing atomic-level smooth surfaces,

and the interfacial adhesion was determined by comparing the strength of the interfacial forces between

the nanoparticle and the top surface of the polymer to the interfacial forces between the nanoparticle

and the bottom substrate. In the same field, Raman spectroscopy has been be used to follow stress

transfer in a variety of polymeric composites reinforced with carbon–based materials such as carbon

fibres [86] and single/double-walled CNT [87, 88]. In respect to GPNC, stress-induced Raman bands

were found to shift directly with stress transfered between matrix and filler, which enables interfacial load

transfer to be monitored, as reported by Gong et al. [89]. All abovementioned techniques are likely to be

used also in GMNC in the near future, as further investigations on these type of materials are receiving

quite attention nowadays.

18

In conclusion, GPNC have successfully established themselves as materials with improved mechan-

ical behaviour and its scale production may soon become a reality. On the other hand, much knowledge

can be retained from recent achievements on GMNC while their upcoming developments seem even

more promising. In fact, graphene-based nanocomposites have a plurality of properties and also the

potential to be transformed in order to satisfy various demands that exist in science and daily life.

2.3 Structural Defects in Graphene

Structural defects in graphene have been predicted ever since it was originally obtained through

experimental means [2]. Computational, theoretical and experimental works have been implementing

serious efforts in order to gain better awareness on the various defective conditions of graphene and

their implications on the mechanical behaviour. According to L. Liu et al. [90], these conditions may be

summarized in two different groups: an imperfection without the presence of foreign atoms is referred

as of intrinsic type, while others are referred to as of extrinsic type. In terms of dimensionality, the

defects present in graphene can also be categorized as point defects (0D) – concerning vacancy and

Stone-Wales (SW) defects – and line defects (1D). As it will be demonstrated, certain defect types may

have atomistic mobility that cause lattice defects to undergo coalescence or reconstruction phenomena,

resulting in the formation of polygonal rings rather than hexagons, with sp-sp2-sp3 transitions.

Following these last arguments, in 2005 Lee et al. [91] addressed the diffusion, coalescence and

reconstruction of vacancy defects in graphene layers at high temperature regimes using tight-bending

molecular dynamics (TBMD). A single vacancy (SV) defect is formed when there is a missing lattice

atom. For the first time reported, it was numerically verified that two SVs initially apart in the lattice,

begin to move next to each other near 3000 K and eventually coalescing into a double vacancy (DV),

which is no more than a group of two missing neighbour atoms. In its turn, this DV defect reformed

into a 5-8-5 type defect, which is composed of two pentagonal rings and one octagonal central ring.

After temperature was increased to 3800 K, the same 5-8-5 defect undergone another structural trans-

formation, eventually being reconstructed into a new 555-777 defect type, which is composed by three

pentagons and three heptagons. This last defect type was found to remain stable even near the melting

temperature (∼ 4000 K) of graphene.

The first optical visualization on structural defects in graphene membranes was achieved in 2008

by an innovative aberration-corrected TEM investigation on crystalline sheets with 1-atom thickness at

an acceleration voltage of 80 kV, reported in Meyer et al. [92]. With an impressing 1-A resolution

and enhanced capability to detect and resolve every individual carbon atom in sight, visual data on the

atomistic dynamics for formation and annealing of SV and SW defects was recorded in real-time. The

SW defect type is formed due to a C-C bond rotation of 90◦ that results in a double pair of pentagons and

heptagons – known also as 55-77 defect –, as shown in Figure 2.10. One-dimensional line defects have

been observed in Coraux et al. [93], through scanning tunnelling microscopy (STM) in mono-layered

graphene produced by CVD on iridium surface. Likewise, in Lahiri et al. [94], a domain boundary has

been observed due to a lattice mismatch in graphene grown on nickel surface. Generally, line defects

19

have oblique boundaries separating two domains of different lattice orientations.

Figure 2.10: High-resolution TEM image sequences for SW defect: (a) unperturbed lattice beforeappearance of the defect, (b) SW defect, (c) same image with atomic configuration superimposed and

(d) relaxation to unperturbed lattice after 4s. Adapted from [92].

From this point forward, a series of authors have been investigating the mechanical properties for

various defective conditions of graphene. By means of ab initio methods, the effect of SV defect con-

centration on Young’s modulus of graphene was evaluated in Fedorov et al. [95] for several sheets with

approximately square shape, and the results highlighted an inversely proportional relation between the

two. Tapia et al. [96] implemented a FEA model based on Sakhaee-Pour [32] work to investigate the

influence of SV defect location, arrangement and concentration on the elastic properties. The authors

found that when an unique SV defect is imposed, the effective elastic stiffness is insignificantly dimin-

ished, independently of its position. By extending that concentration, it leads to increasing reduction with

fairly linear trend. Although it has been also verified that defect position could greatly influence shear

modulus, the authors seemed to attribute some defect arrangements in regions of large strain/stress

gradients, like constrained edges and/or boundaries where loading was imposed.

Figure 2.11: (a) STM topograh of graphene grown on Ir surface (gray), where the arrows point out edgedislocations at two domains boundary. (b) STM topograh of graphene across the substrate showing (c)

a defective line network of C rows with two edge dislocations. (d) STM topograph of two coalescedgraphene flakes forming a coeherent graphene island. Adapted from [93].

Later in 2012, a full TBMD characterization of the displacement and stress fields generated in pres-

20

ence of SV, DV, SW, 5-8-5 and 555-777 defects is described in Dettori et al. [97]. The authors proved

that the majority of the displacement and stress fields generated tend to uniformise around 10 A from

the defected region, being its spacial distribution dependent on the defect symmetry. Moreover, the

authors were able to evaluate the number density influence of some defects types in the elastic prop-

erties. The results for Young’s modulus confirmed that SW defects induced minor variation in graphene

stiffness, independently of defect density, as contrary to SV. Additionally, the stiffness degradation found

in the presence of SV or 555–777 defects is very similar. In fact, these evidences seems to verify

that bond-switch mechanisms, which drive the formation of SW defects, leave the graphene elasticity

nearly unaffected. The same is true for possible reconstruction of dangling bonds generated upon atom

removal, like the 5-8-5 and 555-777 defects.

Regarding the fracture behaviour, Xu et al. [98] unveiled a remarkable property of highly defective

graphene through MD simulations performed with AIREBO potential. Considering SV and SW defects,

the results revealed that with high defect coverage on the sheet, graphene becomes weaker but having

a super-ductile behaviour, with reduced Young’s modulus and fracture strength but an enlarged fracture

strain compared with graphene with only an unique defect. Both outcomes are depicted in Figure 2.12.

The authors stated that increasing occurrences of bond re-hybridizations in the defective sheet by SV

type enlarge the overall system length and therefore, increased the fracture strain. However, localized

geometric re-arrangement around the SW defects trapped the crack without system elongation, resulting

in saturated fracture strain under high defect coverage.

(a) (b)

Figure 2.12: Variations of the (a) ultimate strength and (b) fracture strain with respect to defectcoverage φ for defective graphene sheets under tensile tests along the armchair direction. The three

regions I, II, III in (b) correspond to the three stages – degrading, saturating, and improving – of fracturestrain variation with respect to φ. Adapted from [98].

Adopting a similar method as Xu et al. [98], a detailed investigation on the effects of the orientation

of SW defects was executed in He et al. [99]. In fact, the C-C bond rotation for a SW to form may be

perpendicular in the armchair direction or in the zigzag direction. Their results showed that defective

graphene stiffness remains almost unchanged with the orientation of this defect type, while the fracture

strength varied greatly by changing the lattice chirality and tensile direction.

On the experimental field, in 2014, Zandiatashbar et al. [100] used an analogous technique as Lee

et al. [30] but to study the influence of structural defects induced by oxygen plasma etching. Firstly, they

were able to differentiate the type and density of defects depending on the plasma exposure time via

21

Raman spectroscopy in sp3 (such as SW) and vacancy defects. Then, to quantify the mechanical prop-

erties of defective graphene, the authors used AFM indentation on various samples with different plasma

exposure times to test flexural stiffness and fracture strength while mapping their Raman spectra. As

depicted in Figure 2.13, this approach predicted two distinct evidences. On the one hand, defective

graphene stiffness was not significantly diminished, as it was capable of tolerating a considerable quan-

tity of defects. Only in severe vacancy-defect regime, when plasma began to remove large numbers

of carbon atoms, the elastic modulus suffered a larger degradation. On the contrary, breaking strength

showed greater sensibility in both defect regimes. This work provides a direct relation between the Ra-

man spectra signature of defective graphene and its mechanical properties, allowing one to choose for

a simple and non-destructive methodology to predict its elastic stiffness and breaking strength without

pass through laborious tests.

(a) (b)

Figure 2.13: Measured mechanical properties, (a) 2D elastic modulus and (b) breaking load ofdefective graphene, as a function of the Raman parameters – I(D)/I(G) and I(2D)/I(G) – measured at

increasing plasma times. Adapted from [100].

Finally, another important type of defect that deserves atention is the 5-8-5. In their paper, S. Wang

et al. [101] employed MD simulations with the AIREBO potential to investigate on multiple 5-8-5 defects

and their influence in Young’s modulus and fracture strength and strain. Several parametric studies were

conducted to explore the nearest neighbour distance ’S’ and defect arrangement, as depicted in Figure

2.14. The results revealed that defective graphene stiffness has distinct behaviour for higher values

of ’S’, depending upon tensile direction. Along armchair direction, the values for Young’s modulus of

22

graphene with 5-8-5 defects became always smaller, while there were cases with values even higher

(about 10%) than those of pristine graphene in the zigzag direction for certain defect arrangements.

Figure 2.14: Designed 5–8–5 defect arrangements – from (a) to (g) – in graphene membranes, with ’S’as the nearest neighbour distance parameter. Adapted from [101].

In addition, the failure mechanism of defective graphene sheet was also inspected in [101]. The

fracture strength and fracture strain of defective graphene became lower than pristine graphene, but the

values of zigzag oriented were larger than those of armchair oriented. After potential minimization, some

case studies suffered out-of-plane changes from its original flat configuration to a wavy configuration,

where the first breaking bond was either shared by 5-8 rings or by 6-8 rings. This fact verified that higher

stresses are imposed onto defective bonds, as it was also observed in other works [98, 99]. Besides,

the authors noticed that defective bonds break directly and no reconstruction phenomena occurred.

This proved that the overall system could not overcome the transformation barriers at low temperature,

whereas the brittle fracture dominated the fracture process.

In sum, the relevance of defects is as delicate as an urgent subject to investigate so that the scientific

community can understand their real implications on graphene properties, mostly important mechanical,

but as well electrical or thermal ones.

23

24

Chapter 3

Finite Element Modelling of Graphene and

Graphene-based Nanocomposite

The present chapter begins with the problem definition (Section 3.1), which comprises the assump-

tions that are adopted as simplifications and /or as constraints to construct the multi-scale finite element

(FE) model and the respective finite element analysis (FEA) conducted. The first level (Section 3.2)

refers to the nanoscale structure of pristine graphene, where several formulations are tested to achieve

the most refined elastic properties of this nanomaterial. After, the second level (Section 3.3) aims the

formulation of a representative volume element of graphene nanocomposite, where the discrete nanos-

tructure of pristine graphene is embedded into a polymeric matrix.

3.1 Problem Definition

Throughout the present and following chapters, when and if necessary, some of the contents exposed

in the next section will be recalled.

3.1.1 Force Field, Covalent and Non-Covalent Bonds in Graphene

Primarily, the FE model setup involves the space-frame representation of the molecular structure of

pristine graphene. The latter was chosen over other types of graphene-based materials – such as carbon

nanotubes (CNT) or graphene oxide (GO) – since it consists in a two-dimensional and regular structure,

being easier to build its model and because its surface is not chemically reactive. So, it becomes

unnecessary to incorporate complex chemical phenomena, such as radical groups that would result in

graphene functionalisation and whose description is dependent of the chemical nature of interacting

atoms/molecules.

As referred by Odegard et al. [21], molecular mechanics (MM) of a nanostructured material describes

the forces between individual atoms, wherein the sum of the overall energetic contributions for each

interaction constitutes a molecular force field (MFF), Unsm:

Unsm =∑

Uρ +∑

Uθ +∑

Uτ +∑

Uω +∑

UvdW +∑

Uel (3.1)

and where Uρ, Uθ, Uτ and Uω are the energetic terms for carbon-carbon bond associated with bond

stretching, angle variation, torsion and inversion interactions, respectively. The non-covalent terms con-

25

sist of van der Waals UvdW and electrostatic Uel, interactions. Depending on the material and its loading

conditions, a set of experimental or numerical data is fitted into a functional form of equivalent force

constants, which can describe the potential force field with great accuracy and solely with the relative

positions of the nuclei constituting each atom. As such, this report will explore two different equivalent

force fields for comparison, which have been tested and verified also in other reports regarding carbon

allotropes: the AMBER force field described in Cornell et al. [102] and the linearised version of the

Morse force field introduced by Belytschko et al. [103].

Due to two-dimensionality of graphene, the loading conditions will be applied only parallel to its sur-

face and considering exclusively small deformations onto the nanostructure. Based on these arguments,

it is valid to neglect the covalent energetic terms corresponding to bond torsion and inversion which are

more likely to increase in an out-of-plane load and also, no cross-interactions are included. Still, since

each carbon atom is covalently bonded to its nearest neighbours through a hexagonal pattern, by com-

parison the non-covalent terms are much weaker and totally negligible.

3.1.2 Matrix and Interface

The incorporation of single-layered graphene within the polymeric matrix produces a triple-layered

nanocomposite – called sandwich-like model – by assuming that a sheet of large dimensions is embed-

ded. For this FE model and for reasons of simplicity, the matrix is regarded as a continuum medium with

isotropic elastic properties. Although a discrete molecular approach for the matrix is more accurate, this

report will avoid that kind of methodology which would take to higher modelling complexity and compu-

tational cost. Besides, some similar approaches [74, 75, 76] have been taking the same assumption,

when the reinforcement of polymer composites with carbon allotropes is studied.

In between the two basic components, an interfacial region is modelled to represent the overall in-

terfacial load transfer. Similar to matrix, the interface is assumed also as a continuum isotropic elastic

medium. Moreover, when full exfoliation of graphene is achieved, the interfacial contact with the host

matrix is the largest and it results in better stress transfer. As such, the graphene reinforcement is sup-

posed as exfoliated and uniformly dispersed within the matrix, and having its total surface area adhering

into the internal surface of matrix. Thus, the mechanical encapsulation of graphene is responsible for

interfacial load transfer in the present approach.

On the other hand, three structure-property relationships have been reported to affect critically the

mechanical behaviour of nanocomposites: volume fraction of reinforcement, interface stiffness and in-

terface thickness. Firstly, the volume fraction is defined as the volume of graphene reinforcement per unit

volume of nanocomposite, and where it will be necessary to attribute a thickness value for graphene. In

its turn, the interface stiffness has been related to the level of intefacial adhesion, which could be higher

or lower depending the nanocomposite processing method and its resulting micro-cluster morphology.

In respect to interface thickness, several authors have been used the graphitic inter-planar distance of

3.4 A in graphene-based nanocomposites, without taking in account the type of matrix and its respective

equilibrium distance with graphene surface. As such, it was preferred an interface thickness based on

the intermolecular equilibrium distance of van der Waals interaction. Accordingly, it becomes convenient

26

to consider those non-covalent bonds in the model of nanocomposite.

3.1.3 Structural Defects

The last objective of this dissertation is to study a group of samples of the FE model of graphene-

based nanocomposite, whose reinforcement nanostructure is degraded by inserting single-vacancy (SV)

type defects randomly located. Considering the total surface area of a pristine sheet, these defects will

be generated only inside an area fraction, where it is expected the stress and strain distributions are not

influenced by boundary conditions, in compliance with the Saint-Venant’s principle. Moreover, to avoid

the formation of a double-vacancy (DV) complex, the relative distance between two neighbour vacancies

must be larger than 4 A. This last assumption is based on the outcome achieved by Dettori et al. [97].

These considerations will be only necessary for Chapter 5, where the influence of structural defects is

investigated.

3.2 Pristine Graphene Finite Element Analysis

This section concerns the description and construction of the equivalent FE model of pristine graphene,

as well the boundary conditions imposed towards the computing of the elastic properties desired. Here-

after, pristine graphene will be referred only as graphene until further note.

3.2.1 Model Description

Firstly, to use the FEA technique, ANSYS c© Mechanical APDL 15.0 software was chosen. This soft-

ware includes several modules and tools that revealed themselves essential to accomplish the objectives

that this report intended. Then, a suitable FE must be selected to simulate the considered interactions

between carbon atoms as they were defined in last section 3.1. A single type of covalent bond takes

place inside the hexagonal pattern of graphene, which is a sp2 hybridization, typical of aromatic com-

pounds. Taking into consideration these last arguments, two distinct FEs may fit the task.

As a spar element, LINK180 is an uniaxial and two-node tension-compression element with three

degrees of freedom at each node - three translations about the respective nodal axis X, Y and Z. It

behaves almost like a spring or as a pin-jointed link, since no bending, torsion or shear effects are

considered on the element. However, only a restrict number of input data like cross section area and

material properties are allowed to be controlled. As referred in ANSYS c© Mechanical APDL Element

Reference guide, the axial displacement shape function implies an uniform stress in the spar, axially

loaded at its nodes and with uniform properties from end to end. LINK180 has been used only in few

reports [21, 34] because of its limitations, though it provides an useful comparison against other FE

types. The representative geometry, shape functions and coordinate system of the element LINK180

are showed in Figure 3.1.

27

Figure 3.1: Geometry, shape functions (u) and coordinate system of LINK180. Adapted from ANSYS c©

Mechanical APDL Element Reference guide.

The BEAM4 element is also uniaxial but a two-node structural beam that has six degrees of freedom

per node – three translations and three rotations about the respective nodal axis X, Y and Z. An optional

node is also available to define the element orientation, but in the present case it can be neglected

since all elements are modelled in XY-plane. The beam theory reference is based on Euler-Bernoulli,

as reported in ANSYS c© Mechanical APDL Theory Reference guide. In consequence, plane sections

remain plane and normal to the longitudinal axis after deformation since it neglects transverse shear

effects. Still, it is a more versatile element than LINK180 since it has tension, compression, torsion,

and bending capabilities, as one can verify in ANSYS c© Mechanical APDL Element Reference guide.

In terms of user control, this FE enables several types of input data that are quite convenient such as

the type and area of cross section, area moment of inertia about the two principal axis of the beam

cross section, polar moment of inertia, initial strain and so on. In fact, a diversity of authors has been

using this FE to model carbon-carbon bonds in graphene with concordant results [32, 34, 35, 96]. The

representative geometry, shape functions and coordinate system of the element BEAM4 are showed in

Figure 3.2.

Figure 3.2: Geometry, shape functions (u,v,w,θx) and coordinate system of BEAM4. Adapted fromANSYS c© Mechanical APDL Element Reference guide.

The next step is the definition of the geometrical properties of graphene nanostructure. Having

as basis an unit hexagon of carbon atoms, one may assume that nodal positions, element length L,

and element cross-sectional diameter D, are defined as schematized in Figure 3.3. The cross-section

of elements LINK180 and BEAM4 are circular shape for convenience, though this last one allows the

user to select other types. In addition, a set of parameters are needed to completely define the cross-

section of BEAM4 such as its area moment of inertia I – symmetry for circular cross-section implicates

– I = Iyy = Izz – and polar moment of inertia J . These are readily determined through the following

28

basic formulas:

I =πD4

64J =

πD4

32(3.2)

t

ac-c

Carbon atom - Node position

C-C bond - Finite element

(a)

D = t

L = a c-c

(b)

Figure 3.3: Schematic of (a) an unit hexagon of carbon atoms and (b) geometrical properties of thecarbon-carbon bond.

The forces equivalent to atomic bonds are obtained via two different formulations where the concept

of equivalent material for carbon-carbon bond is introduced. The first one was proposed by Odegard et

al. [21], where bond stretching and bond-angle variation deformations were considered to simulate the

covalent bond. Using the element LINK180, and since this FE has only stretching/compression degrees

of freedom, two elastic links are needed for each carbon-carbon bond and the equivalent Young’s moduli

for bond stretching Eslink and bond-angle variation Eblink interactions, are obtained as:

Eslink =2ksL

πD2(3.3a)

Eblink =24kb

πD2L(3.3b)

where ks and kb are the equivalent molecular force field (MFF) constants for bond stretching and bond-

angle variation, respectively. Since the stiffness matrix for element LINK180 only contemplates axial

stiffness EA, the cross-sectional areas for both links will cancel out in the solving process. Thus, the

results are independent of the respective element cross-sectional diameters and for simplicity one may

use an unitary value for D in this case.

The second formulation resembles the one proposed by Scarpa et al. [34], but neglecting Timo-

shenko shear deformation, to conform to element BEAM4, and bypassing the non-linear optimization

process. The versatility of this element allows to use only a single beam for each carbon-carbon bond.

This modified formulation comes up with equivalent bond Young’s modulus Ebeam and shear modulus

Gbeam, defined as:

Ebeam =4ksL

πD2(3.4a)

Gbeam =32ktL

πD4(3.4b)

where kt is the equivalent MFF constant for torsional interaction. With the simplification used in this case,

29

the association of strain energy with molecular force constant for bond-angle variation is not directly

defined, but an equivalent covalent deformation is associated with beam flexural rigidity EI. Also, if

one assumes an equivalent isotropic material for the carbon-carbon bond, the condition for its Poisson’

ratio ν = −1 + E/2/G with −1 < ν < 0.5 leads to the following maximum condition for the value of

cross-sectional diameter:

D <√

1 + ν

√16kt

ks(3.5)

Thus, for the second formulation, the beam diameter will be determined using the maximum value

obtained with Equation (3.5). A null Poisson’s ratio is attributed for both formulations, since the equivalent

carbon-carbon bond suffers negligible transverse dilation when is mechanically stretched, as reported

in Scarpa et al. [34].

The values for equivalent materials and remaining geometric parameters, which will be implemented

in ANSYS c© Mechanical APDL code, are obtained following Equations (3.2) and (3.5). As stated in

Section 3.1, AMBER and linearised Morse equivalent MFFs are employed here for comparison pur-

poses. Tables 3.1 and 3.2 summarize all the necessary input data for all modelling cases that are being

compared.

Table 3.1: MFF constants used in graphene FE model.

AMBER force field [102] Morse force field [103]

Parameter LINK180 BEAM4 LINK180 BEAM4 Units

ks 32.6 65.2 42.3 84.7 nN/A

kb 4.38 – 4.5 – nN A/rad2

kt – 2.78 – 2.78 nN A/rad2

Table 3.2: Equivalent carbon-carbon bond materials and geometrical properties used in graphene FEmodel.

AMBER force field [102] Morse force field [103]

Parameter LINK180 BEAM4 LINK180 BEAM4 Units

Eslink 28.8478 – 37.4313 – nN/A2

Eblink 24.0725 – 24.7320 – nN/A2

Ebeam – 169.1435 – 285.4480 nN/A2

Gbeam – 84.5718 – 142.7240 nN/A2

Poisson’s ratio, ν 0 –

Element length, L 1.39 A

Element diameter, D 1 0.8260 1 0.7247 A

Area moment of inertia, I – 2.2846× 10−2 – 1.3537× 10−2 A4

Polar moment of inertia, J – 4.5691× 10−2 – 2.7075× 10−2 A4

30

With the equivalent materials and geometry of covalent bonds defined, now one may advance with

spatial modelling of graphene nanostructure. This procedure was assisted with a supplementary tool

[104], which simulates two-dimensional graphene lattices and allows the user to save its output data

– atoms locations, interatomic bonds, lattice dimensions and chirality – into manageable files for post-

processing. The default bond length of the tool is set to be 1.39 A, which is used as the value of element

length as given in Table 3.2. This value is within the range of bond length variation found in [26].

In terms of dimensions, two square shape sheets were constructed with approximately, 24 A and 70 A

per side, respectively. These two different cases were investigated to enable a consistent validation of

the FE model of graphene, since there is a great diversity in shape and size reported in literature. In

its turn, the global coordinate system has its X-axis aligned with the zigzag direction while the Y-axis is

aligned with the armchair direction. The Z-axis is not yet used in this first model, as graphene is a two-

dimensional material, however it could be considered as if it was aligned with the graphene thickness

direction. Illustratively, Figures 3.4 and 3.5 represent some of the cases constructed and their coordinate

systems.

Y

X

Figure 3.4: Square graphene sheet with 24 A per side and built with LINK180 element – image adaptedfrom ANSYS c©.

31

Y

X

Figure 3.5: Square graphene sheet with 70 A per side and built with BEAM4 element – image adaptedfrom ANSYS c©.

3.2.2 Boundary and Displacement Conditions

At this stage, the FEA for graphene is almost complete but it is not yet solvable by any solver until

a group of specific boundary conditions and displacement conditions are applied into the sheet. These

conditions are related to the elastic properties of graphene which this report aims to determine, obtained

through the following mechanical tests:

• Uniaxial tensile test in the zigzag direction (X-axis) to compute the elastic modulus Ex and Poisson’

ratio νyx.

• Uniaxial tensile test in the armchair direction (Y-axis) to compute the elastic modulus Ey and Pois-

son’s ratio νxy

• Shear test (XY-plane) to compute the shear modulus Gxy

• Biaxial tensile test (XY-plane) to compute the in-plane bulk modulus Kxy

Since the respective boundary and displacement conditions for each test are valid for all modelling

cases described earlier, in the next pages a single example using BEAM4 will be illustrated for simpler

visualization. Figures 3.6 and 3.9 only regard the 24 A size sheet, but a similar idea is valid for the larger

32

one. In the end of this section, thirty-two distinct codes are written in ANSYS c© APDL, divided in eight

groups, per each modelling case to be tested.

Uniaxial tensile test in the zigzag direction (X-axis)

The first mechanical test applied to graphene sheet is the uniaxial tensile test aligned with the zigzag

direction (X-axis). Recalling the considerations refereed in Section 3.1, a small tensile strain of 0.01%

is simulated in all nodes in the right edge of the sheet along the X-axis, while the nodes on the left

edge are restrained from displacement in X-axis also. The opposite procedure would also be correct. To

prevent rigid-body motion, the displacement along Y-axis is fixed for two nodes, one on each right and

left edges and at mid-height. Besides, it is also necessary to restrain any residual bending behaviour

of the sheet by fixing the displacement along the Z-axis for all nodes. In Figure 3.6, is represented an

example of uniaxial tensile test in the zigzag direction. The objective of this analysis is to extract the

following values: (i) the sum of nodal reaction forces in zigzag direction∑Rx along the left edge, (ii) the

mean of nodal displacements in transverse direction∑a along the upper and lower edges. Knowing

the deformation d applied and with the remaining values extracted, the zigzag elastic modulus Ex and

respective Poisson’s ratio νyx are readily computed.

Y

X

d

Figure 3.6: Boundary and displacement conditions for uniaxial tensile test in the zigzag direction(X-axis) exemplified in the 24 A size sheet – image adapted from ANSYS c©.

33

Uniaxial tensile test in the armchair direction (Y-axis)

The second mechanical test concerns uniaxial tensile test aligned with the armchair direction (Y-

axis). A small tensile strain of 0.01% is applied in all nodes in the upper edge of the sheet along

the Y-axis, while the nodes on the lower edge are restrained from displacement in Y-axis also. The

opposite procedure would also be correct. To prevent rigid-body motion, the displacement along X-

axis is fixed for two nodes, one on each upper and lower edges and at mid-width. As in the previous

test, the displacement along the Z-axis for all nodes is fixed. An example of uniaxial tensile test in

the armchair direction is shown in Figure 3.7. The objective of this analysis is to extract the following

values: (i) the sum of nodal reaction forces in armchair direction∑Ry along the lower edge, (ii) the

average of nodal displacements in transverse direction∑b along the right and left edges. Knowing

the deformation c applied and with the remaining values obtained, the armchair elastic modulus Ey and

respective Poisson’s ratio νxy are quickly extracted.

cY

X

Figure 3.7: Boundary and displacement conditions for uniaxial tensile test in the armchair direction(Y-axis) exemplified in the 24 A size sheet – image adapted from ANSYS c©.

34

Shear test (XY-plane)

The third mechanical test involves a shear test (XY-plane), where a small shear strain of 0.04%

is applied in all nodes along each edge of the sheet. In contrast to latter tests, rigid-body motion is

already restrained in this test and there is no need to fix additional in-plane displacement. Moreover,

the displacement along the Z-axis for all nodes of the sheet is fixed for the same reason as mentioned

before. Figure 3.8 represents an example of shear test implemented. The objective of this analysis is to

extract the following values: (i) the sum of nodal forces in armchair direction∑Fy along the left edge,

(ii) the sum of nodal forces in zigzag direction∑Fx along the lower edge. The procedure would also be

correct if the respective opposite sides are used in these calculations. Including the shear deformation

γ applied and the remaining values extracted, the shear modulus Gxy is easily computed.

c- c

Y

X

d

- d

Figure 3.8: Boundary and displacement conditions for shear test (XY-plane) exemplified in the 24 A sizesheet – image adapted from ANSYS c©.

35

Biaxial tensile test (XY-plane)

The fourth mechanical test is a biaxial tensile test (XY-plane). In this case, a small tensile strain of

0.01% is applied in all nodes along each edge of the sheet. As for the shear test, rigid-body motion

is already restrained in this test and there is no need to fix additional in-plane displacement. Also, the

displacement along the Z-axis for all nodes of the sheet is restrained. An example of biaxial tensile test

is depicted in Figure 3.9. The objective of this analysis is to extract the following values: (i) the sum of

nodal forces in zigzag direction∑Fx along the left edge, (ii) the sum of nodal forces in armchair direction∑

Fy along the lower edge. The procedure would also be correct if the respective opposite sides are

used in these calculations. Concerning the biaxial deformation (d, c) applied and the remaining values

measured, the in-plane bulk modulus Kxy is promptly extracted.

c

- c

d - d

Y

X

Figure 3.9: Boundary and displacement conditions for biaxial tensile test (XY-plane) exemplified in the24 A size sheet – image adapted from ANSYS c©.

36

3.3 Nanocomposite Finite Element Analysis

This section regards the description and construction of the equivalent FE model corresponding to

the graphene-based nanocomposite, besides the respective boundary conditions established for obtain-

ing the elastic properties of the composite material.

3.3.1 Model Description

The FE model for nanocomposite is also analysed in ANSYS c© Mechanical APDL 15.0 software.

First, to model the molecular structure of polymeric matrix, a suitable FE has to be chosen. Keeping

in mind the assumptions mentioned in Section 3.1, it occurs obviously to use a three-dimensional solid

element, like SOLID185. This isoparametric element has eight nodes, having three degrees of freedom

at each node: translations in the nodal X, Y, and Z directions. Although it is a hexahedron by default,

it allows the user to shape it also in prismatic and tetrahedral options. However, those other options

would require a more complicated volume meshing. As referred in ANSYS c© Mechanical APDL Element

Reference guide, the material properties are assumed as orthotropic by default. Since both matrix and

interface are here assumed as isotropic materials, only Young’s modulus and Poisson’s ratio are input

and SOLID185 will assume the same properties in all material directions. The representative geometry,

shape functions and coordinate system of the element this FE are shown in Figure 3.10.

Figure 3.10: Geometry, shape functions (u,v,w) and coordinate system of SOLID185. Adapted fromANSYS c© Mechanical APDL Element Reference guide.

Now, the geometrical properties of the FE model of graphene-based nanocomposite may be de-

fined, starting with the selection of an appropriate representative volume element (RVE). The RVE will

be assembled as a triple-layered composite material (denoted as C), assuming that a single sheet of

graphene (denoted as G) is positioned within the matrix (denoted as M) as reinforcement, as depicted

in Figure 3.11. For convenience, the transversal dimensions (along XY-plane) of the matrix volume will

be the same as those of graphene FE model. Also, both adjacent regions of matrix material are consid-

ered to have the same finite value of thickness, wherein the RVE total volume VC is dependent on the

37

reinforcement volume fraction Vf as it follows:

Vf =V G

V C(3.6)

where V G is the volume of graphene filler. In turn, an interface (denoted as I) is considered between the

two basic components, whose volume is already included into the matrix for obtaining the last equation.

To study the effects of interfacial stiffness, the interface volume is modelled separately from the matrix

though. Seeing that transversal size is uniform along RVE thickness, it follows that Equation (3.6) can

be rewritten as:

Vf =tG

tC(3.7)

where tG and tC are the thicknesses of graphene and RVE, respectively. However, graphene has no real

thickness, as it was mentioned earlier. From a physical point of view, its thickness may be defined as

the normal distance – from the atomistic structure of graphene – wherein interfacial interactions occur.

Usually, it is assumed that the inside surface of matrix is located at the same position as the outside

surface of graphene, giving an interface thickness tI equal to half thickness of graphene. Since in the

present RVE, matrix is covering graphene in both sides, it is reasonable to assume both interface and

graphene have the same thickness. Thus, the total interface region takes place from z = 0+ to z = +tI

and from z = −tI to z = 0−. It remains only to define the interface thickness, which in turn will fix the

value of graphene thickness.

zLx

0.5tC

0.5tC

xy

Ly

tI

Matrix (M)

Graphene (G)

Interface (I)

tI

z

Figure 3.11: RVE selected for the FE model of graphene-based nanocomposite.

As referred in Section 3.1, apart from mechanical interlocking, van der Waals (vdW) interactions

should also form a mechanism of stress transfer between matrix and graphene reinforcement in the

present approach. Thus, it was preferred to use a value for interface thickness equal to the equilibrium

vdW distance between a carbon atom and a common polymer chain CH2. This value may be reached

38

using the well-known 12-6 potential of Lennard-Jones, reported also in Meguid et al. [105] and which is

defined as:

U(r) = 4α

[(ψ

r

)12

−(ψ

r

)6]

(3.8)

where α is the potential well depth, ψ is the distance where potential U(r) is zero and r is the actual

interatomic distance. Usually, only interacting atoms within Lennard–Jones cut-off distance of 2.5ψ are

considered, since at this distance the vdW forces are negligible. From Equation (3.8), one can obtain

the first derivative of potential U(r), deriving to the following expression for vdW interaction force:

F (r) = −24α

[2

(ψ12

r13

)−(ψ6

r7

)](3.9)

After some manipulation of Equation (3.8), it can be verified that its absolute minimum is rmin = 21/6ψ,

which is the vdW equilibrium distance. If the value for rmin is substituted in Equation (3.9), one would

obtain a zero force value. The curves obtained for interatomic potential and its derivative, interatomic

force F(r), are similar to those exemplified in Figure 3.12.

-0.05

-0.03

-0.01

0.01

0.03

0.05

3 4 5 6 7 8

-0.05

-0.03

-0.01

0.01

0.03

0.05

Len

nard

-Jon

es p

oten

tial

, U

Interatomic distance, r (Å)

Inte

rato

mic

For

ce, F

(n

N)

U(r) F(r)

rmin

Figure 3.12: Example of interatomic potential U(r) and interatomic force F(r) curves expected forcarbon-CH2 interaction.

Equation (3.9) enables one to implement a structural finite element to represent such non-covalent

force along the interface,, though it can be observed that its expression is non-linear. Thereby, an addi-

tional FE must be selected, which has to have non-linear force-deflection capability. The choice is to use

COMBIN39, an uniaxial spring element defined by two nodes and a generalized force-deflection curve.

As referred in ANSYS c© Mechanical APDL Element Reference guide, this element has longitudinal or

torsional capabilities in one, two, or three dimensional applications. For the present case, it will be used

the longitudinal option, which turns it into an uniaxial tension-compression element with three degrees

of freedom at each node: translations in the nodal X, Y, and Z directions. In that sense, the line joining

the nodes defines the direction of the force. Finally, the force-deflection curve should be input such that

deflections are increasing from the third (compression) to the first (tension) quadrants. To gain better

understanding of what has been described, one should visualize Figure 3.13.

Though the number of vdW interactions for each atom is governed by the separation distance 2.5ψ,

for simplicity reasons, only vdW interactions from a carbon atom facing directly a nodal position of the

inside surface of matrix are considered here. Thus, two equivalent non-linear springs are implemented

39

per carbon atom in graphene nanostructure, one for each side out of graphene surface.

Figure 3.13: Geometry, coordinate system and force-deflection curve of COMBIN39. Adapted fromANSYS c© Mechanical APDL Element Reference guide.

With the RVE geometry characterized, one may now attribute values for the equivalent materials,

geometrical parameters and non-covalents bonds, which will be implemented in ANSYS c© Mechanical

APDL code as an extension of the FEA for graphene. Initially, both matrix and interface have same

mechanical properties, which correspond to those of common epoxy. The values for parameters in the

potential of Lennard-Jones – Equations (3.8) and (3.9) – are obtained from [106] and the force-deflection

curve obtained is interpolated with 20 data points to be input in ANSYS c©. For convenience, Tables 3.3

and 3.4 summarize all the necessary input data for the RVE described until this point.

Table 3.3: Parameters for the interatomic force of Lennard-Jones implemented in element COMBIN39.

12-6 Lennard-Jones Potential [106] Units

Interaction

Carbon-CH2

ψ 0.4492 kJ/mol

α 3.4 A

rmin 3.8163 A

Table 3.4: Materials and geometrical parameters for the RVE of graphene-based nanocomposite.

RVE Units

Matrix (M)EM 3 GPa

νM 0.340 –

Interface (I)

EI 3 GPa

νI 0.340 –

tI 3.8613 A

Transversal

Dimensions

Lx 103.5247 A

Ly 102.8600 A

The final step in the construction of the RVE is to characterize its volume mesh (see Figure 3.14).

This is an important stage because the mesh refinement can influence the stress-strain fields and the

overall load transfer. For instance, graphene and interface are connected at plane z = 0 and line

elements (BEAM4) and solid elements (SOLID185) must have coincident nodal positions. In the present

40

approach, it is presumed that use of regular rectangular divisions is most suitable, in respect to in-plane

discretization of the volume mesh, as depicted in Figure 3.14(a). These divisions are based on the

interatomic distances of the nanostructure of graphene, and this kind of coupling may be applied since

both element types are three dimensional while having common degrees of freedom per node.

(a) Front view (XY-plane) (b) Right view (YZ-plane)

(c) Isometric view

Figure 3.14: Example of volume meshing implemented for the RVE of graphene-based nanocomposite– images adapted from ANSYS c©.

In respect to other interactions, appropriate nodal coupling takes place between elements COMBIN39-

BEAM4 at plane z = 0 and between elements SOLID185-COMBIN39-SOLID185 at plane z = +tI . In

turn, volume mesh discretization along thickness (Z-axis) is dependent on total volume of RVE, which

depends also on volume fraction of reinforcement. For this condition, it was decided to use uniform

divisions within the interface, from z = 0+ to z = +tI , whereas for the matrix an expansive growth of

the thickness of solid elements is applied, from z = +tI to z = +tC . Figure 3.14(b) shows the lateral

view of RVE for better understanding. The other half of the RVE is meshed symmetrically. An example

of volume mesh obtained for the RVE is presented in Figure 3.14(c).

41

3.3.2 Boundary and Displacement Conditions

With the RVE conveniently assembled and meshed, now the respective boundary conditions and

displacement conditions have to be applied in order to obtain the elastic properties desired for the

nanocomposite. Since the mechanical tests are equivalent, these conditions bear similitude with those

applied in graphene model (as described in Section 3.2).

Uniaxial tensile test in the zigzag direction (X-axis)

As before, the first mechanical test applied to the RVE is the uniaxial tensile test aligned with the

zigzag direction (X-axis) of graphene. A small tensile strain of 0.01% is simulated along the X-axis in

all nodes on the right face (plane x = Lx) while the respective nodes at the left face (plane x = 0) are

restrained from displacement in X-axis also. The opposite procedure would also be correct. To prevent

rigid-body motion, one node in the center of the left face is constrained from displacements in Y-axis,

Z-axis and rotations about X-axis. In Figure 3.15, is represented an example of uniaxial tensile test in

the zigzag direction. The objective of this analysis is to extract the sum of nodal reaction forces in zigzag

direction∑Rx along the left face. Regarding the deformation applied d, the zigzag elastic modulus Ex

is easily acquired.

Y

X

d

Figure 3.15: Boundary and displacement conditions for uniaxial tensile test in the zigzag directionexemplified in the RVE (Front view) – image adapted from ANSYS c©.

42

Uniaxial tensile test in the armchair direction (Y-axis)

In respect to the uniaxial tensile test aligned with the armchair direction (Y-axis) of graphene, a small

tensile strain of 0.01% along the Y-axis is applied in all nodes on the upper face whereas all nodes on the

lower face are restrained from displacement in Y-axis. The opposite procedure would also be correct. To

prevent rigid-body motion, one node in the center of the lower face is constrained from displacements in

X-axis, Z-axis and rotations about Y-axis. Depicted in Figure 3.16, is an example of uniaxial tensile test

in the armchair direction. The objective of this analysis is to extract the sum of nodal reaction forces in

armchair direction∑Ry along the lower face. Knowing the deformation c applied, the armchair elastic

modulus Ey is readily evaluated.

cY

X

Figure 3.16: Boundary and displacement conditions for uniaxial tensile test in the armchair directionexemplified in the RVE (Front view) – image adapted from ANSYS c©.

43

Shear test (XY-plane)

For shear test (XY-plane), a small shear strain of 0.04% is applied in nodes spread along right, left,

upper and lower faces. In contrast to graphene model, rigid-body motion is yet not totally restrained

in this test. To prevent it, displacement along Z-axis is avoided for each of the central nodes of those

former faces. Figure 3.17 represents an example of shear test implemented. The objective of this

analysis is to extract the following values: (i) the sum of nodal forces in armchair direction∑Fy along

the left face, (ii) the sum of nodal forces in zigzag direction∑Fx along the lower face. The procedure

would also be correct if the respective opposite faces are used in these calculations. With the value for

shear deformation γ and the remaining values extracted, the in-plane shear modulus Gxy is promptly

determined.

c- c

d

- d

Y

X

Figure 3.17: Boundary and displacement conditions for shear test exemplified in the RVE (Front view) –image adapted from ANSYS c©.

44

Biaxial tensile test (XY-plane)

Finally, regarding the biaxial tensile test (XY-plane), a small tensile strain of 0.01% is applied in the

same faces as shear strain. Also, rigid-body motion needs to be restrained in this test, where similar

constrains for motion along the Z-axis as for shear test are implemented. An example for biaxial tensile

test is shown in Figure 3.18. The objective of this analysis is to extract the following values: (i) the

sum of nodal forces in zigzag direction∑Fx along the left face, (ii) the sum of nodal forces in armchair

direction∑Fy along the lower face. The procedure would also be correct if the respective opposite

faces are used in these calculations. Knowing the biaxial deformation (d, c) applied and the remaining

values computed, the in-plane bulk modulus Kxy is quickly obtained.

c

- c

d- d

Y

X

Figure 3.18: Boundary and displacement conditions for biaxial tensile test exemplified in the RVE (Frontview) – image adapted from ANSYS c©.

In the end of this section, four distinct codes are written in ANSYS c© APDL, one per each mechanical

test.

45

46

Chapter 4

Results

The present chapter exhibits the results obtained from the multi-scale finite element (FE) model

described in Chapter 3, from where the respective elastic properties are readily determined after being

post-processed. Subsequently, their validation and discussion is conducted by comparing with similar

results found in the literature review.

4.1 Elastic Properties of Pristine Graphene

The results obtained from the FEA concerning pristine graphene are presented next. Since all me-

chanical tests conducted were static linear simulations, the sparse direct solving process on ANSYS c©

Mechanical APDL was adopted here.

4.1.1 Presentation of Results

To confirm that all the mechanical tests run properly, it is important to visually inspect the plot for the

deformed shape for each case. The next images in Figures 4.1 and 4.2 represent the deformed shape

obtained for the 24 A size sheet, with the FE formulation related to BEAM4, though the other modelling

cases would lead to similar results.

(a) (b)

Figure 4.1: Initial and deformed shapes of graphene sheet: (a) Uniaxial tensile test in zigzag direction,(b) Uniaxial test in armchair direction for 24 A size sheet – images adapted from ANSYS c©.

47

(a) (b)

Figure 4.2: Initial and deformed shapes of graphene sheet: (a) Shear test, (b) Biaxial tensile test for24 A size sheet – images adapted from ANSYS c©.

In comparison, the deformed shape for each mechanical test can be represented qualitatively as

shown in Figure 4.3. This figure show the relevant displacements, deformed sheet size and positions of

displaced nodes needed to calculate the elastic properties of graphene.

a

d

cb

(a)

1’

1

2

2’

3’

3

4’

4

(b)

Hf

Lf

(c)

1’

1

2

2’

3’

3

4’

4

γ

β

α

(d)

Figure 4.3: Representation of initial and deformed shapes of graphene sheet: (a) Uniaxial tensile test inzigzag direction (red) and in armchair direction (green), (b,d) Shear test and shear angles, respectively,

(c) Biaxial tensile test.

48

A black square is equivalent to graphene sheet before any deformation takes place. First, the red

and green rectangles (see Figure 4.3(a)) are representations of the deformation after the uniaxial tests

are performed, in the zigzag direction (X-axis) and armchair direction (Y-axis), respectively. Succes-

sively, the orange diamond (see Figure 4.3(b)) shows the shear deformation (XY-plane) with its corners

numbered for post-processing. Finally, the blue square (see Figure 4.3(c)) exhibits the final dimensions

resulting from biaxial tensile test (XY-plane). It can observed the similarity between the mechanical tests

(Figures 4.1 and 4.2) performed and its representations ( Figures 4.3(a) to 4.3(c)), thus indicating their

modelling was the correct.

The objective values of each mechanical test are presented in the next tables. The displacements

applied in all tests are presented in Table 4.1, while the undeformed and deformed coordinates obtained

due to shear deformation are presented in Table 4.2 to facilitate the post-processing. The remaining

values include the reactions forces, transverse displacements and nodal forces, which are shown in

Tables 4.3 and 4.4 concerning the 24 A and 70 A size sheets, respectively.

Table 4.1: Displacements applied in the 24 A and 70 A size sheets.

Sheet size

Mechanical testDisplacement

applied24 A 70 A Units

Tensile test

on X-axisd 2.4076× 10−3 6.9819× 10−3 A

Tensile test

on Y-axisc 2.3630× 10−3 6.9500× 10−3 A

Shear test

c

(transverse along X-axis)2.3630× 10−3 6.9500× 10−3 A

d

(transverse along Y-axis)2.4076× 10−3 6.9819× 10−3 A

Biaxial tensile test

d

(along X-axis)2.4076× 10−3 6.9819× 10−3 A

c

(along Y-axis)2.3630× 10−3 6.9500× 10−3 A

Table 4.2: Corners coordinates before and after shear test obtained for the 24 A and 70 A size sheets.

Sheet size Sheet size

Corner 24 A 70 A Corner 24 A 70 A Units

1 (0 ; 0) (0 ; 0) 1’ (−0.0024 ; −0.0024) (−0.0070 ; −0.0070) A

2 (0 ; 23.6300) (0 ; 69.5000) 2’ (0.0024 ; 23.6276) (0.0070 ; 69.4931) A

3 (24.0755 ; 23.6300) (69.8190 ; 69.5000) 3’ (24.0779 ; 23.6324) (69.8260 ; 69.5070) A

4 (24.0755 ; 0) (69.8190 ; 0) 4’ (24.0731 ; 0.0024) (69.8120 ; 0.0070) A

49

It is important to note that the shear strain γ was calculated taking in consideration a qualitative

deformed shape as represented in Figure 4.3(d). For a pure shear test, the following relation is valid:

γ = α+ β (4.1)

Knowing the corners’ coordinates at the end of deformation from Table 4.2, simply remains compute the

sum of shear angles through Equation (4.1) to verify the shear strain applied.

Table 4.3: Reaction forces, transverse displacements and nodal forces obtained for the 24 A size sheet.

AMBER force field [102] Morse force field [103]

Mechanical

test

Result

computedLINK180 BEAM4 LINK180 BEAM4 Units

Tensile test

on X-axis

∑Rx 5.4751× 10−2 6.2323× 10−2 5.7477× 10−2 7.0120× 10−2 nN∑a −5.2396× 10−4 −4.6762× 10−4 −5.5301× 10−4 −5.6202× 10−4 A

Tensile test

on Y-axis

∑Ry 5.2177× 10−2 5.8430× 10−2 5.4962× 10−2 6.6068× 10−2 nN∑b −4.8762× 10−4 −4.3355× 10−4 −5.1292× 10−4 −5.1983× 10−4 A

Shear test

∑Fx 6.6516× 10−2 7.5756× 10−2 6.8339× 10−2 7.9254× 10−2 nN∑Fy 6.9447× 10−2 8.1343× 10−2 7.1977× 10−2 8.7261× 10−2 nN

Biaxial

tensile test

∑Fx 1.9014× 10−1 2.0009× 10−1 2.0874× 10−1 2.5917× 10−1 nN∑Fy 1.8271× 10−1 1.8962× 10−1 2.0053× 10−1 2.4590× 10−1 nN

Table 4.4: Reaction forces, transverse displacements and nodal forces obtained for the 70 A size sheet.

AMBER force field [102] Morse force field [103]

Mechanical

test

Result

computedLINK180 BEAM4 LINK180 BEAM4 Units

Tensile test

on X-axis

∑Rx 1.4542× 10−1 1.6379× 10−1 1.5224× 10−1 1.8312× 10−1 nN∑a −1.5778× 10−3 −1.4097× 10−3 −1.6653× 10−3 −1.6982× 10−3 A

Tensile test

on Y-axis

∑Ry 1.4278× 10−1 1.5981× 10−1 1.4972× 10−1 1.7917× 10−1 nN∑b −1.5336× 10−3 −1.3678× 10−3 −1.6165× 10−3 −1.6451× 10−3 A

Shear test

∑Fx 1.9090× 10−1 2.1991× 10−1 1.9613× 10−1 2.3071× 10−1 nN∑Fy 1.9400× 10−1 2.2544× 10−1 1.9991× 10−1 2.3853× 10−1 nN

Biaxial

tensile test

∑Fx 5.2526× 10−1 5.4421× 10−1 5.7628× 10−1 7.0622× 10−1 nN∑Fy 5.1718× 10−1 5.3324× 10−1 5.6736× 10−1 6.9233× 10−1 nN

4.1.2 Calculation of Elastic Properties

In this subsection, the elastic properties from the FE model of graphene are determined, having

in mind the set of objective values determined earlier. Nonetheless, one must remember that since

graphene is a two-dimensional nanostructure, it has no real thickness. In that sense, the elastic moduli

here demonstrated are defined in terms of nN/nm – denominated as surface elastic moduli. However,

50

it is regular to use the interlayer thickness of graphite of 3.4 A to compare with the results from other

authors, which are obtained commonly in units of GPa (or TPa). This value comes from the interlayer

adhesion energy, as reported recently by J. Zhao et al. [70]. Hence, to determine the bulk elastic moduli,

one must divide the respective surface stress by the thickness considered.

Next, to determine the elastic properties, first one must recall a set of equations to realize this task:

• Elastic modulus Ex and Poisson’s ratio νyx (uniaxial tensile test in the zigzag direction,

X-axis)

Ex =σxεx

εx =d

Liσx =

∑RxHi

(4.2a)

νyx = − εyεx

εy =2∑a

Hi(4.2b)

In Equation (4.2a), σx and εx are the axial stress and axial strain on X-direction, respectively. Also, εy

and νyx account for the transverse strain and respective Poisson’s ratio, as listed in Equation (4.2b).

• Elastic modulus Ey and Poisson’s ratio νxy (uniaxial tensile test in the armchair direction,

Y-axis)

Ey =σyεy

εy =c

Hiσy =

∑RyLi

(4.3a)

νxy = −εxεy

εx =2∑b

Li(4.3b)

Similarly, in Equation (4.3a), σy and εy are the axial stress and axial strain on Y-direction, respectively.

As well for Equation (4.3b), εx matches transverse strain and νxy stands for the respective Poisson’s

ratio.

• Shear modulus Gxy (shear test, XY-plane)

Gxy =τxyγ

τxy =

∑Fx

2Li+

∑Fy

2Hi(4.4)

Regarding Equation (4.4), τxy denote in-plane shear stress due to shear strain γ. Because the sheet

dimensions are not exactly equal, one must compute it as an average stress as indicated.

• Bulk modulus Kxy (biaxial tensile test, XY-plane)

Kxy =θ

24θ

2=σx + σy

2=

∑Fx

2Hi+

∑Fy

2Li4 =

4AAi

(4.5a)

4A = Af −Ai Af = (Li + 2d)(Hi + 2c) (4.5b)

Lastly, the Equation (4.5a) stand for mean in-plane bulk stress θ – just as for the previous test – and

area expansion4, which is computed taking into consideration the final area Af for the deformed sheet

through Equation (4.5b). Applying the latter expressions abovementioned, the set of elastic properties

computed for graphene is now exhibited in Table 4.5.

51

Table 4.5: Elastic properties predicted for graphene using the present approach and other correspondingresults available in literature.

PresentEx

(nN/nm)

Ey

(nN/nm)νyx νxy

Gxy

(nN/nm)

Kxy

(nN/nm)MethodSize

SheetFE Type MFF

24 A

LINK180AMBER 231.7 216.7 0.443 0.405 71.3 195.4

FEA

Morse 243.2 228.3 0.468 0.426 73.5 214.5

BEAM4AMBER 263.7 242.7 0.396 0.360 82.3 204.3

Morse 296.7 274.4 0.476 0.432 87.3 264.7

70 A

LINK180AMBER 209.2 204.5 0.454 0.439 69.1 187.0

Morse 219.1 214.4 0.479 0.463 71.1 205.2

BEAM4AMBER 235.7 228.9 0.406 0.392 79.9 193.3

Morse 263.5 256.0 0.489 0.471 84.2 250.9

Oth

erau

thor

s

Lee et al. [30] 335 – – – Experimental

Arroyo and

Belytschko [24]236.0 0.412 – –

MM

Reddy et al. [26] 277.4 228.1 0.520 0.428 60.9 –

Sakhaee-Pour [32] 354.3 353.6 – 75.0∗ –

FEAScarpa et al. [34]

517.0 342 0.523 0.509 76.0 –

546.0 408.0 0.577 0.551 97 –

Van Lier et al. [22] 377.4 – – –Ab-initio

Kudin et al. [23] 345 0.149 – –

H. Zhao et al. [39] 343.1 0.210 – –

MDJ. Zhao et al. [70] 291.2 327.8 0.143 0.157 84.6∗ –

Tsai and Tu [71] 310.1 0.261 121.7 –

Kalosakas et al. [107] 320 0.22 84.0 200.0

Superscript * – Mean of results for simple shear tests along each chiral direction.

4.1.3 Model Validation and Discussion of Results

In possession of all results for the elastic properties of graphene, the proposed FE model of graphene

is validated in comparison with other methods available in the literature to choose the most efficient

and accurate FE formulation and MFF. Then, one must outline several important conclusions about the

results determined through the present approach.

Model Validation

In order to validate the FE model of graphene, all elastic moduli are converted to units of nN/nm to

facilitate their comparison. It must be noticed that some works of other authors have predicted values

for Young’s modulus and Poisson’s ratio without discriminating the chirality of graphene, as well as

52

conducting simple shear tests rather than pure shear.

Still, it should be referred that in the literature there are several methods, like Ab-initio and molecular

dynamics (MD) simulations, which require intensive computational effort and time consuming analyses.

In fact, the latter methods should predict more accurate results than the ones obtained by FEA, since

they take into account many other complex parameters involved at atomic-scale, which are neglected

in this more simplistic approach. Therefore, the FE model here described has to be considered as

an approximation. However, the application of very complex methods is limited in terms of simulation

scalability and that is the main reason why FEA has been applied to larger structures at nanoscale to

evaluate their mechanical behaviour.

Firstly, from a global point of view, LINK180 and BEAM4 formulations have predicted lower values

for Young’s moduli than the majority present in the literature. Compared to Ab-initio and MD simulations,

this fact may be justified by the choice of MFF constants compared to those methods that use sophis-

ticated potential functions, and which have a major weight in graphene’s elastic stiffness, as referred

before. Besides, the present approach assumes a finite-size sheet while other methods more rigorous

are often conducted with periodic boundary conditions, resulting in larger stiffness. Even so, it can be

observed that BEAM4 approximates better than LINK180 the results determined with Ab-initio and MD

approaches, particularly for the smaller sheet size. Regarding the experimental field, Lee et al. [30]

found a value for Young’s modulus of mono-layered graphene close to the highest value predicted by

BEAM4 formulation. In respect to atomistic methods such as molecular mechanics (MM), their outcome

is in good agreement with the present results. Concerning other results based in FEA, as obtained by

Sakhaee-Pour [32] and Scarpa et al. [34], the values of Young’s modulus are considerably distanced

from those here presented. Both authors also used FE models based on structural beams but with

different material properties and geometrical characteristics for equivalent bonds, resulting in dissimilar

results. Lastly, one may observe that Morse force field provides Young’s moduli more adjusted to the

ones in literature.

Regarding Poisson’s ratio, it may be observed in Table 4.5 that the present results are largely dis-

tanced from results determined from Ab-initio and MD computations, while showing a good agreement

with predictions from MM and FEA. The latter methods provide ratios’ values near 0.5 or even above,

as the case of Scarpa et al. [34] and Reddy et al. [26]. It is known that a 3D elastic and isotropic

material has its Poisson’s ratio within interval [−1; 0.5], but because graphene behaves as a 2D mate-

rial, it is necessary to establish the relationship with 3D-elasticity to understand the implications. This

correspondence was deduced by Eischen and Torquato [108] as described in Appendix A, resulting sev-

eral expressions that relate the 2D elastic properties with 3D ones whether the conditions occurring are

plane-strain (Equations (A.3) and (A.6)) or plane-stress (Equations (A.7) and (A.10)). Because the load-

ing conditions here applied on graphene surface are displacements and not forces, the corresponding

state of 3D-elasticity is plane-strain.

To understand what is being stated, one might assume for instance a value for ν(2D) = 0.45 as the

average of values for Poisson’s ratio presented in Table 4.5 and then substituting it in Equation (A.4). A

value for ν(3D) = 0.31 is obtained, which would represent the actual value considering that graphene has

53

real thickness. When the latter value is compared with others reported in the literature (see Table 4.5),

one may conclude that better agreement is achieved with results achieved through Ab-initio and MD

simulations. For most of those methods, the loading conditions comprise applied forces and thus, the

state of 3D-elasticity is plane-stress and Equation (A.8) shows that ν(3D) = ν(2D). In fact, one may verify

that for a maximum value of ν(3D) = 0.5, the maximum value for Poisson’s ratio for a 2D material using

Equation (A.4) is ν(2D) = 1, which confirms that is possible to obtain a Poisson’s ratio larger than 0.5

for a 2D material. In respect to Young’s modulus, it becomes obvious that when the value of ν(3D) is

substituted in Equation (A.3), the value of E(3D) will always be lower than E(2D).

In respect to shear modulus, some results have been reported in literature but the most is related with

simple shear rather than pure shear. Even so, all modelling cases present in Table 4.5 produced values

for shear modulus in decent accordance with other methods referenced in the literature, wherein BEAM4

agrees better with results from MD simulations. Furthermore, it can be verified that 2D and 3D shear

moduli are exactly equal through Equation (A.6), confirming that in-plane shear does not depend on

graphene thickness. Regarding the in-plane bulk modulus of graphene, it was only possible to found in

theliterature the work of Kalosakas et al. [107]. Still, their work proves great agreement with the present

approach, though one has obtained fairly large values when using Morse force field. As the case of

Young’s modulus, one may also notice that K(3D) is always lower than K(2D) through Equation (A.5).

In sum, it may be concluded that the FE model employed to predict the elastic properties of graphene

in the present report was successfully implemented and proved itself as a capable approximation in

respect to other most accurate and rigorous methods. Moreover, BEAM4 is considered slightly more

refined, since it provides considerably better results in conformity with the literature available and it allows

more geometrical characterization of covalent bonds than LINK180. Moreover, if one desires to extent

the present report to evaluate other mechanical properties such as yield strength or failure, BEAM4

quickly permits to account for other types of mechanical deformations (non-linear) without increasing the

model complexity. Therefore, the formulation for BEAM4 elements embedded with Morse as equivalent

MFF has been chosen as the most appropriate to simulate graphene.

Finite Element Type

An important remark from the present approach concerns the differentiation in the results because

of the FE types used, as one may verify in Table 4.5. As both formulations depicted the nanostructure

of graphene assuming different equivalent space-frames, the association of structural strain energies

with MFF constants was distinct, as the characterization of equivalent bond too. Nevertheless, though

BEAM4 has contributed on the increase of model complexity – by multiplying the number of degrees of

freedom from 3n per node to 6n per node –, the molecular deformation within the graphene nanostruc-

ture can be more conveniently described. Particularly in those cases where additional loading conditions

are applied, other types of atomic interactions that not only bond stretching and bond-angle variation may

appear as non-negligible energetic terms.

On the one hand, if one examines the results for elastic moduli and ignoring for now, the effects

of size sheet and equivalent MFF, it is readily verified that greater values are obtained when using

54

BEAM4 formulation rather than using LINK180. However, one would anticipate the reverse scenario

since accounting for more deformation mechanisms leads to a decrease in stiffness, as found in [34].

A more detailed comparison confirms that the axial stiffness EA of LINK180 – around 23 nN – is much

lower than that of BEAM4 – around 91 nN – which is related to bond stretching deformation. In fact,

the latter covalent interaction has the larger contribution on the MFF when only small deformations are

applied. In that sense, the reason for the significant differences in elastic moduli when using distinct FE

types becomes clear.

Molecular Force Field and Size Sheet

The second aspect that one should address is the effects of different MFF constants upon the com-

puted elastic properties. By observing Table 3.1, it is clear that Morse always gives a stronger MFF

constant equivalent to bond stretching interaction. Recalling that this is the predominant atomic interac-

tion in the conditions here tested, one may justify the evidence of higher values for elastic moduli – Ex,

Ey, Gxy and Kxy – being computed with Morse rather than with AMBER, as it can be seen in Table 4.5.

In respect to Young’s modulus, the most significant difference between MFFs is around 14%, while 13%

and 30% are obtained for shear modulus and bulk modulus, respectively.

However, the increase in stiffness of graphene is followed also by a significant increase in Poisson’s

ratio. In other words, as the graphene nanostructure becomes more rigid in the axial direction – direction

of pull – it also suffers larger transverse deformation. Due to the special orthotropy of structural hon-

eycombs, one needs to compare the stretching-bending stiffness ratio, which is the ratio of stretching

stiffness per bending stiffness of the equivalent honeycomb cell. In respect to BEAM4 formulaton, the

bond-angle variation stiffness was not defined. However, as referred in [34], one may obtain an equiva-

lent expression for bending force constant associated to the Euler–Bernoulli beam flexural deformation

Kb as:

Kb =F

L4θ(4.6)

where 4θ = F LD2EI . Considering Equation (3.4a) for equivalent Young’s modulus and substituting for

Equation (4.6), after some algebraic manipulations, the stretching-bending stiffness ratio may be defined

as:Ks

Kb=

8L

D(4.7)

Substituting the proper values from Table 3.2 in Equation (4.7), it is obtained a ratio Ks/Kb of 13.7

and 15.3 when considering AMBER or Morse, respectively. These values confirm that for the same

tensile deformation applied, Morse provides less honeycomb bending stiffness and in consequence,

more transverse deformation. For BEAM4 formulation, the most considerable contrast in Poisson’s ratio

for different MFFs is around 20%. So, one may conclude that related to BEAM4, the MFFs have a

significant impact on transverse contraction of graphene.

For LINK180 formulation, a similar analysis could be performed. For an equivalent honeycomb cell

as presented in Figure 2.1(b), the values of bond stretching stiffness and bond-angle variation stiffness

presented in Table 3.1 are readily used to compute the equivalent stretching-bending stiffness ratio.

55

Once again, higher transverse deformation occurs for Morse, because the values obtained for ratio

Ks/Kb are of 7.4 and 9.4 when using AMBER and Morse, respectively. Even so, the biggest difference

(around 6%) in Poisson’s ratio between MFFs is less for LINK180 than for BEAM4.

A third aspect that needs proper attention is the influence of sheet size. As one can observe in

Table 4.5, the results for the 24 A and 70 A size sheets are fairly dissimilar. While the graphene sheet

extends, all elastic moduli diminish while both Poisson’s ratios tend to increase. In the literature, it

was found only a single study about the effects of size-dependent elastic properties, reported by H.

Zhao et al. [39]. In their work, Young’s modulus and Poisson’s ratio on graphene nanoribbons were

observed as converging to bulk values while the lattice diagonal length increased above 10 nm. Since

the present report intends to investigate the elastic behaviour of a graphene-based nanocomposite,

one may consider that it is more suitable the embedding of a reinforcement agent whose mechanical

performance is size-independent. For that reason, the last part of this section will address the study on

the influence of size-dependent properties of graphene.

Orthotropic Behaviour

From the results presented in Table 4.5 , one may also observe distinct values for Young’s modu-

lus along zigzag and armchair directions. This evidence agrees well with other reports, because even

though different size and shapes – rectangular, square or circular – have been tested, each case pro-

vided dissimilar results depending on graphene’s chirality. Regarding Poisson’s ratio, since this elastic

property becomes smaller when the sheet suffers tensile deformation along the zigzag edges rather than

along the armchair edges, one can conclude that graphene chirality also affects its transverse deforma-

tion. The maximum difference in Young’s moduli and Poisson’s ratios when comparing both crystalline

directions is about 9% and 10%, respectively, regarding the 24 A size sheet. Still, those differences

decrease when the bigger sheet is considered, converging to values around 3% and 4%, respectively.

Before going into details about the in-plane shear modulus and in-plane bulk modulus, it is useful

to remember that because all mechanical test conducted involved displacements imposed parallel to

graphene surface (remember Section 3.1), the FE model behaves actually as a two-dimensional (2D)

medium rather than three-dimensional (3D). Having basis in two-dimensional elasticity relations for an

isotropic material, as deduced by Eischen and Torquato [108] and described in Appendix A, the relations

for in-plane shear modulus and in-plane bulk-modulus are readily obtained. However, these 2D elasticity

relations do not refer with either plane-strain or plane-stress elasticity. Such specifications only arise

when one desires to make contact with 3D-elasticity, as referred previously.

Having the proper expressions for in-plane shear modulus (Equation (A.11)) and in-plane bulk mod-

ulus (Equation (A.13)), now one can evaluate the orthotropy effects in these moduli by comparing the

numerical results in Table 4.5 with those that one would obtain from the assumption of 2D isotropic

material using the latter expressions. The process is to substitute the values of Young’s modulus and

Poisson’s ratio in those expressions for each chiral direction of graphene, i.e. along zigzag and armchair

directions. In respect to shear modulus, when considering the 24 A size sheet, the numerical shear

modulus has an average difference of around 13% comparing to the one obtained by Equation (A.11)

56

along zigzag direction, while an average difference of 8% is computed along armchair direction. Doing

an analogous analysis concerning the 70 A size sheet, smaller average differences of about 4% and 3%

are obtained in the zigzag and armchair directions, respectively.

The orthotropic behaviour influence is also evidenced in bulk modulus. An average difference around

7% for zigzag and armchair directions is obtained between the numerical result and the outcome from

Equation (A.13), when comparing for the 24 A size sheet. In contrast, lower average differences of 3%

and 2% are verified in zigzag and armchair directions, respectively, for the 70 A size sheet.

In sum, the FE model implemented is able to confirm that elastic properties of graphene are governed

by chirality effects, thus leading to orthotropic behaviour. On the one hand, a similar behaviour has

been verified in other carbon allotropes. In their report, J. Zhao et al. [70] also observed orthotropy

on supergraphene, cyclicgraphene and graphyne. On the other hand, the results here presented also

demonstrate that as the sheet size increases, graphene may approach to an isotropic behaviour when

under small deformations, as predicted in H. Zhao et al. [39].

Study on the Influence of Sheet Dimensions

To complete this section, it is important to inquire the behaviour of size-dependent properties on

graphene nanostructure. As such, several square-sheets are dimensioned with increasing size, resem-

bling the study reported in H. Zhao et al. [39], and employing on ANSYS c© Mechanical APDL the FE

model previously chosen. It must be noted that this study is not a common structural mesh convergence

study, since the number of FEs per equivalent honeycomb cell is maintained whereas the pristine sheet

dimensions are increased. The interval of lattice diagonal lengths comprises values between [34; 258] A.

The set of dimensions, number of nodes and elements can be consulted at Table 4.6.

Table 4.6: Dimensions, number of nodes and elements for several graphene sheets used to investigatethe bulk values on elastic properties.

SheetWidth

(A)

Height

(A)

Diagonal length

(A)

No. of

nodes

No. of

elements

1 24.0755 23.6300 33.7344 252 356

2 40.9284 40.3100 57.4459 700 1013

3 69.8190 69.5000 98.5136 2006 2946

4 103.5247 102.8600 145.9368 4350 6432

5 139.6379 140.3900 198.0103 7956 11 808

6 182.9738 182.0900 258.1399 13 464 20 032

As one may verify in Figure 4.4, as the lattice diagonal length increases, both Young’s moduli show a

rapid decreasing trend and then slowly converging. The bulk values obtained are around 253 nN/nm and

251 nN/nm for zigzag and armchair directions, respectively, with a difference of only 1% when compared

to the maximum of nearly 9% verified in the results earlier presented in Table 4.5. As shown in Fig-

ure 4.5, both Poisson’s ratios increase but at different rates, since νyx does not enlarge as significantly

as νxy. The bulk values predicted are about 0.492 and 0.485, for each respectively, with a difference of

57

1.5% (compared to the maximum value of 10% earlier). These results also demonstrate that the stiffer

crystalline direction always provides the major Poisson’s ratio.

Regarding in-plane shear modulus and in-plane bulk modulus, both decrease with similar behaviour

as Young’s modulus, spite of the absolute reduction in shear modulus being fairly minor, as it may

be noticed in Figures 4.6 and 4.7. The corresponding bulk values predicted are about 83 nN/nm and

247 nN/nm, respectively. Once again, if one compares the numerical bulk values with those obtained

with ”isotropic” assumption (through Equations (A.11) and (A.13)), the maximum differences are only

about 2% and 1% for each case, respectively. Eventually, the facts abovementioned corroborate the

hypothesis of isotropic behaviour of graphene for greater sheet sizes.

240

250

260

270

280

290

300

0 50 100 150 200 250 300

s

ulu

doM s'g

nuoY ecafruS

(nN

/nm

)

EX

EY

Diagonal Length (Å)

Figure 4.4: Evolution of Young’s modulus in zigzag (X-axis) and armchair (Y-axis) directions, in respectto the lattice diagonal length.

0.42

0.44

0.46

0.48

0.50

0 50 100 150 200 250 300

Poi

sson

's r

atio

νXY

νYX

Diagonal Length (Å)

Figure 4.5: Evolution of Poisson’s ratios in zigzag (X-axis) and armchair (Y-axis) directions, in respectto the lattice diagonal length.

58

83

84

85

86

87

88

0 50 100 150 200 250 300 suludo

M raehS ecafruS(n

N/n

m)

Diagonal Length (Å)

Figure 4.6: Evolution of shear modulus in respect to the lattice diagonal length.

245

250

255

260

265

270

0 50 100 150 200 250 300

su lud oM klu

B ec afr uS(n

N/n

m)

Diagonal Length (Å)

Figure 4.7: Evolution of bulk modulus in respect to the lattice diagonal length.

Last but not least, it must be noted that spite of graphene bulk values have been successfully de-

termined, a huge number of FEs comprises the graphene models with diagonal lengths close or above

200 A. In that sense, the graphene sheet obtained with a diagonal length around 150 A (sheet 4 in Ta-

ble 4.6) is used as reinforcing agent in the RVE of nanocomposite, thus minimizing the computational

cost and guaranteeing enough size-independent properties.

4.2 Elastic Behaviour of Graphene-based Nanocomposite

The results obtained from the FEA regarding graphene-based nanocomposite were determined using

the method of Newton-Raphson for solving process on ANSYS c© Mechanical APDL, due to the nature

of these non-linear structural analyses. The outcome is presented next.

4.2.1 Presentation of Results

The deformed shapes for each mechanical test applied to the RVE are presented in Figure 4.8. Only

in-plane deformations (XY-plane) are illustrated but one must recall that due to tridimensionality, the

RVE suffers additional deformation along Z-axis. This fact has to do with Poisson’s ratio effect in elastic

regime.

59

(a) (b)

(c) (d)

Figure 4.8: Initial and deformed shapes of the RVE for: (a) Uniaxial tensile test in zigzag direction, (b)Uniaxial test in armchair direction, (c) Shear test and (d) Biaxial tensile test – images adapted from

ANSYS c©.

Comparing the deformed shapes from Figure 4.8 to the qualitative representations earlier described

in Figure 4.3, it can be assumed that modelling of each test was performed correctly. So, one may

proceed to the extraction of objective values.

Now, two parametric studies are set for the FE model of nanocomposite to access the importance

of several structure-property relations (see Section 3.1). Firstly, the effects of graphene volume fraction

in the elastic properties of the RVE are evaluated. For this study, different volume mesh discretizations

are used accordingly to each volume fraction. Then, using the same discretization, a secondary study

concerning the influence of interface stiffness is targeted. The influence of interface thickness on the

mechanical behaviour will not be addressed, though it was assumed constant for simplicity. Due to the

amount of data to be processed, the collection of objective values will not be presented hereafter.

60

4.2.2 Calculation of Elastic Properties

To compute the elastic properties from the RVE, Equations (4.1) and (4.5b) are applied considering

that all elastic moduli must contemplate their stresses in units of force per area by dividing the respective

surface stress by the total thickness tC of the model. Then, the respective properties are conveniently

calculated in units of GPa. In its turn, the rule of mixtures (ROM) is also used for qualitative comparison,

and it is expressed as:

pijC = pij

GVf + pijM (1− Vf ) (4.8)

where pijG and pij

M are the mechanical properties of graphene and epoxy, respectively, while pijC is

the one predicted for the RVE of nanocomposite. Although this theoretical approach is more suitable

for composite materials with long and unidirectional fibres, in this case it assumes equal strain for both

phases and it should provide a good approximation to the mechanical tests implemented in this work.

To apply Equation (4.8), the mechanical properties obtained from the FE model of graphene must also

consider the thickness assumed as tG = 3.8613 A. The respective values are Ex = 676.6GPa and

νyx = 0.491 for the zigzag direction (X-axis), Ey = 665.0GPa and νxy = 0.478 for the armchair direction

(Y-axis), while Gxy = 218.7GPa and Kxy = 652.0GPa are obtained for the in-plane shear and bulk

moduli, respectively.

For the parametric study in respect to volume fraction, the proposed RVE has an interface stiffness

equal to the matrix one, or by other words, the nanocomposite has no interface region. The results

obtained are shown in Figures 4.9 and 4.10.

0

5

10

15

20

25

30

35

40

0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

EX/E

M

Vf

Proposed RVE

ROM

Giannopoulos et al. [74] (No Interface)

Giannopoulos et al. [74] (Hybrid Interface)

Spanos et al. [75] (Hybrid Interface)

Figure 4.9: Comparison on normalized Young’s modulus in zigzag direction Ex/EM versus volumefraction Vf .

61

0

10

20

30

40

50

60

0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

GX

Y/G

M

Vf

Proposed RVE

ROM

Giannopoulos et al. [74] (No Interface)

Giannopoulos et al. [74] (Hybrid Interface)

Figure 4.10: Comparison on normalized shear modulus Gxy/GM versus volume fraction Vf .

While the previous study only assumes EI = EM , the parametric study concerning the influence of

interface stiffness takes also into account the cases where EI = 2 × EM and EI = 4 × EM , shown in

the following Figures 4.11 and 4.12.

0

5

10

15

20

25

0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

EX/E

M

Vf

EI = E

M

EI = 2xE

M

EI = 4xE

M

(a) Normalized Young’s modulus in zigzag direction, Ex/EM

0

5

10

15

20

25

0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

EY/E

M

Vf

EI = E

M

EI = 2xE

M

EI = 4xE

M

ROM

(b) Normalized Young’s modulus in armchair direction, Ey/EM

Figure 4.11: Results obtained from the parametric study on various interface conditions and versusvolume fraction Vf .

62

0

5

10

15

20

25

0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

GX

Y/G

M

Vf

EI = E

M

EI = 2xE

M

EI = 4xE

M

ROM

(a) Normalized shear modulus Gxy/GM

0

5

10

15

20

25

30

35

0.0% 2.0% 4.0% 6.0% 8.0% 10.0%

KX

Y/K

M

Vf

EI = E

M

EI = 2xE

M

EI = 4xE

M

ROM

(b) Normalized bulk modulus Kxy/KM

Figure 4.12: Results obtained from the parametric study on various interface conditions and versusvolume fraction Vf .

It is also important to compare the accuracy of the present approach with actual experiments avail-

able in the literature, which usually report the amount of graphene content in weight percentage wt (%)

rather than volume fraction. For the RVE of graphene-based nanocomposite assembled, the weight per-

centage was computed considering the total number of carbon atoms in the pristine sheet, the atomic

mass of carbon – equal to 1.9943× 10−26 kg – and the typical density value of epoxy resin – 1200 kg/m3.

Next, several predictions are demonstrated in Table 4.7.

Table 4.7: Comparison of stiffness enhancement obtained in graphene-based epoxy nanocompositesbetween the proposed RVE and other predictions in literature.

Graphene ContentNotes

wt (%) Stiff. Enhanc. (%)

Present 0.1013.5 EI = EM

13.9 EI = 4× EM

Numerical[74] 0.10 16.7 Hybrid Interface

[75] 0.10 18.5 Hybrid Interface

Experimental[53] 0.10

13.3 Reduced Graphene

18.0 Functionalised Graphene

[57] 0.10 14.4 Graphene Nanoplatelets

63

4.2.3 Model Validation and Discussion of Results

Having determined the array of elastic properties for the graphene-based nanocomposite, the val-

idation of the FE model needs to be established. However, firstly is necessary to demonstrate the

convergence study performed to guarantee that all solutions previously shown were mesh-independent.

Therefrom, it is mandatory to debate some aspects that emerged from the parametric studies performed.

Convergence Study of Volume Mesh

Remembering the volume mesh discretization explained earlier (see Figure 3.14), several meshes

are obtained with increasing refinement for each value of volume fraction of graphene reinforcement.

The mesh refinement is higher in the region where high stress concentration is expected, such as the

graphene-interface border. Besides, a 20%-30% element growth along RVE thickness for the matrix

was used to minimize computational effort. Table 4.8 summarizes the discretizations involved, number

of nodes and elements adopted throughout the refinement procedure.

Table 4.8: Volume mesh discretizations used in the RVE of nanocomposite for different volume fractions.

Discretization 1 Interface Matrix

Vf No. of solid elements No. of nodes No. of solid elements No. of nodes

0.02 50 912 77 778 127 280 155 556

0.04 50 912 77 778 152 736 181 428

0.06 50 912 77 778 178 192 207 408

0.08 50 912 77 778 203 648 233 334

0.10 50 912 77 778 254 560 285 186

Discretization 2 Interface Matrix

Vf No. of solid elements No. of nodes No. of solid elements No. of nodes

0.02 101 824 129 630 152 736 181 482

0.04 101 824 129 630 178 192 207 408

0.06 101 824 129 630 229 104 259 260

0.08 101 824 129 630 254 560 285 186

0.10 101 824 129 630 381 840 414 816

Discretization 3 Interface Matrix

Vf No. of solid elements No. of nodes No. of solid elements No. of nodes

0.02 203 648 233 334 254 560 285 186

0.04 203 648 233 334 305 472 337 038

0.06 203 648 233 334 356 384 388 890

0.08 203 648 233 334 381 840 414 816

0.10 203 648 233 334 509 120 544 446

The convergence study was conducted to evaluate the values of Young’s modulus along the zigzag

direction and shear modulus of the nanocomposite for the range of volume fractions considered previ-

64

ously. In Figure 4.13, one may verify the convergence results for the axial modulus. Since the results

converged for bulk values immediately after the second case, one may conclude that the choice of

volume mesh discretization is adequate for capturing the strain-stress fields occurring within the RVE.

Therefore, the first discretization case was implemented for obtaining accurate results while a minimal

computational effort is achieved.

0

10

20

30

40

50

60

70

80

2.0% 4.0% 6.0% 8.0% 10.0%

Ex(G

Pa)

Vf

Discretization 1

Discretization 2

Discretization 3

Figure 4.13: Results of convergence study of volume mesh on the RVE focusing on Young’s modulusalong zigzag direction.

Model Validation

In order to validate the FE model of graphene-based nanocomposite, its accuracy and efficiency

must be compared with other similar ones, as well some experimental results available in the literature.

For this reason, it was decided to attach Table 4.7 with several predictions on the stiffness enhancement

obtained in graphene-based epoxy nanocomposites, and whose experimental methods used different

graphene-based materials.

Starting with a general examination of Figures 4.9 and 4.10, one may verify that present results are

almost exact with the theoretical approach from ROM for both moduli. This was expected since both

approaches have concordant assumptions. Still, it would not have been possible to solve the problem

here defined with the limitations from ROM, since the latter does not account for atomistic effects present

in the nanostructure of graphene. Even so, the results from ROM on the in-plane bulk modulus (see

Figure 4.12(b) for instance) also show good agreement with the present approach. Turning the attention

back to Figures 4.9 and 4.10, similar FEA approaches from other authors reveal linear enhancement on

elastic properties of the nanocomposite with progressive increments of volume fraction as well. When

no interface is considered, i.e. matrix and interface have the same stiffness, numerical predictions

from Giannopoulos et al. [74] resemble on the ones provided by the proposed RVE. However, it is still

verified an appreciable deviation between those values, which is related with differences in the FE model

implemented by the authors to simulate graphene nanostructure. Consequently, it is expected to occur

different reinforcing effects. The deviation aggravates when hybrid interface is assumed, and this fact

will be discussed in the following.

While the present approach admits an interfacial region with isotropic mechanical behaviour but low

65

adhesion modulus – in other words, low interface stiffness –, other works based on FEA [74, 75] adopted

a hybrid interface. As it was explained in Section 2.2, that methodology assumes that interfacial mechan-

ical properties are dependent and bounded by the respective ones of the two phases that surround it,

in this case graphene and epoxy resin. Obviously, such formulation provides a larger stiffening effect in

interfacial region than the prediction of the proposed RVE, because while there are sections which have

similar properties to matrix, closer to the reinforcement these properties resemble those of graphene.

Although the use of homogeneous properties for interface is more simplistic, from a physical point

of view, the polymer high density distribution that occurs near to graphene surface may be represented

by a volume of material with small thickness but distinct properties from the bulk matrix. For higher

volume fractions, when the distance between two successive filler particles is smaller or a larger sheet

is embedded, it is obvious that a greater portion of matrix will be also transformed, being expected that

interface thickness increases considerably. Even so, the adoption of constant stiffness across interface

is fairly applicable and allows further adjustments in conformity with the desired level of adhesion or

optimal load transfer. Moreover, in situations where graphene reinforcement has no strong covalent

bonds with matrix, the interfacial adhesion may derive only from non-bonded interactions, such as van

der Waals interactions, or the mechanical interlocking due to polymeric chain roughness and wrapping.

Both contributions are sufficiently represented in the proposed RVE, though their validity is restrained to

small deformations.

By comparing the values of stiffening enhancement obtained from experimental results described in

Table 4.7, it can be confirmed that a good correlation from the proposed RVE is achieved when some

experimental nanocomposites produced have lower interfacial adhesion, as those which employed less

chemically reactive forms of graphene-based materials – nanoplatelets [53] – or more degraded ones

in terms of crystalline nanosctructure – reduced graphene [57]. On the other hand, the employment of

hybrid interface leads to results more concordant with experiments where higher interfacial adhesion is

established – as functionalised forms of graphene [53] –, resulting in better load transfer and increasing

reinforcing effect.

Despite of several limitations in the present approach that do not allow to scale this analysis to access

other structure-property relations that arise in graphene nanocomposite, the proposed FE model has the

efficiency and practicability required to evaluate the mechanical behaviour of graphene nanocomposites

with fairly accurate results, highlighting the excellent enhancement that this outstanding nanomaterial

may provide.

Study on the Influence of Structure-Property Relations

As it is noticed in Figures 4.9 and 4.10, there is a direct correlation of volume fraction with the

enhanced mechanical behaviour of the graphene-based nanocomposite. On the proposed RVE, the

stiffening effect grows proportional upon the volume fraction of graphene for both elastic moduli. When

compared to bulk matrix properties, this enhancement is consequence of two characteristics: (i) the

superior mechanical properties of graphene and (ii) mechanical coupling in common nodal positions,

that considers no yielding or slip occurring between graphene surface and epoxy chains.

66

On the other hand, the linear trend of enhancement may be explained by several factors. First, the

mechanical tests implemented consider only small deformations, from which one can infer that graphene

and matrix behaved as linear elastic materials. Spite of non-linear vdW interactions being modelled be-

tween both phases, the force-deflection results verified that those interactions remain restrained in their

linear portion after deformation takes place (see Figure 3.12). Besides, the proposed RVE is constructed

assuming that graphene sheets are uniformly dispersed within the matrix and thus, neglecting the de-

grading effects of graphene re-agglomeration [71]. For a nanocomposite with high fiber volume fraction,

Wei et al. [57] deduced two possible microstructures in addition to that where particles are randomly dis-

persed. In the first case, the resin is completely entrapped within the filler aggregates or clusters, while

in the second type the contrary occurs. Such nanocomposites would consist of regions with randomly

dispersed fillers, but with each region containing a different volume fraction. Therefore, it is expected

that some simplifications used here are not valid at higher values of graphene content.

Regarding Figures 4.11 and 4.12, the second parametric study demonstrate that for the same

graphene concentration, as the interface stiffness increases, the interfacial load transfer is more effi-

cient. Although the improvement achieved is very reduced for the values inspected here, if one admits

a stiffer interface than those here assumed, it is expected to obtain a sharper slope in the mechanical

enhancement. In its turn, the assumption of vdW interactions turns out to be less significant for the

interfacial adhesion when higher interface stiffness is considered in the present conditions. However,

the influence of non-linear vdW stiffness on the interfacial behaviour would be more important in con-

ditions of higher strain applied. In sum, the proposed RVE proves that the efficiency of load transfer in

nanocomposites is highly dependent from a suitable interface adhesion between matrix and graphene

reinforcement.

67

68

Chapter 5

Influence of Defective Graphene in

Nanocomposites

Several studies have been investigating various types of defects that are known to appear in graphene

nanostructure, but also their intricacy in the mechanical behaviour of the nanomaterial. However, to the

author’s best knowledge, there are no references in the literature of any report that had quantified the

effects of defective graphene on the mechanical properties of nanocomposites. For this reason, this

chapter pioneers the exploration of that influence, specifically how the defect content percentage affects

the overall elastic properties of the nanocomposite material. Hereafter, pristine graphene and defective

graphene will be referred as PG and DG, respectively, for reading clarity. Moreover, the finite element

analysis (FEA) accomplished for PG-based nanocomposite in Section 3.3 will be named as reference

FEA (Ref-FEA).

5.1 Model Description

The most common structural defect that is referred in literature is the vacancy-type, usually intro-

duced in graphene-based materials produced via mechanical exfoliation or chemical reduction [43].

Having as basis the Ref-FEA, it is necessary to account for minor adjustments to simulate this type of

atomic defect in the DG nanostructure.

Knowing that a single-vacancy (SV) defect is introduced when a carbon atom is removed from the

graphene lattice, also the three covalent bonds which connect the removed atom to its neighbours must

be neglected. In terms of modelling, the process of defect introduction is performed by removing the

respective node, coincident with the atom to be withdrawn, and the structural beam elements that are

linked to it. Figure 5.1 depicts what has been explained in the last sentence.

The whole procedure was accomplished through an external script assembled in MATLAB c© 2015b.

This script executed specific commands on the output data obtained from the previously mentioned tool

[104] to rearrange the set of atoms locations and interatomic bonds in accordance to each defect allo-

cated in graphene nanostructure. One may repeat the latter process as many times as the total number

of defects desired, though non-physical lattice deconstruction might happen due to erroneous defect

coalescence, when a highly defective condition is considered. Besides, as stated in Section 3.1, the for-

mation of a double-vacancy (DV) complex is more susceptible when the relative distance between two

69

neighbour vacancies is lower than 4 A. To avoid these events, the latter value is assumed as minimum

distance between neighbour SVs.

After the FE model of DG is substituted over the former PG one, the remaining geometry, material

properties and spatial modelling are almost the same as those described on the Ref-FEA. Only the

non-linear springs representing van der Waals (vdW) interface interactions are neglected in each defect

position.

Y

X X

Y

Figure 5.1: Introduction of a single-vacancy defect (red dot) in the FE model of DG nanostructure –images from ANSYS c©.

It is must be also mentioned that for this study, all mechanical tests performed – uniaxial tensile test

along zigzag direction, shear test and biaxial tensile test – and calculation of the elastic properties –

Young’s modulus along zigzag direction, shear modulus and bulk modulus – are executed analogously

to what has been described in Chapter 3.

5.2 Parametric Study on Defect Coverage

As referred earlier, it was decided to simulate the randomness of defect distribution rather than a

set of predefined locations as conducted by several other authors. With this option, one may expect to

reflect more adequately actual experiments and processing methods where graphene-based materials

are thermally or chemically derived, which tend to have a degraded crystalline nanostructure. A suitable

quantification of the defective condition of graphene is the defect concentration percentage DC (%),

defined as:

DC(%) =No. of Removed Atoms

Total No. of Atoms(5.1)

70

Next, using equation 5.1, five distinct values of DC are investigated – SV (0.023% DC) and multiple

SVs (0.25% DC, 0.5% DC, 1% DC and 2% DC) defects – and each concerning to a group of of five

DG nanostructures samples, which have defects randomly located. Some examples for the multiple SVs

degraded sample sheets are depicted in Figure 5.2.

Y

X

(a) 0.25% DC

Y

X

(b) 0.5% DC

Y

X

(c) 1% DC

Y

X

(d) 2% DC

Figure 5.2: Examples of DG sheets with multiple SV type defects introduced – images from ANSYS c©.

A total of 375 simulations in ANSYS c© APDL were conducted, divided in five main sets - one for each

defect concentration percentage – and each set consisting of seventy-five different cases with respect

to five different volume fractions and three mechanical tests. Hereafter, the mean values for the elastic

properties are obtained from each group of samples in several conditions of volume fraction. For each

volume fraction, the results obtained are properly normalized using the respective value determined in

the Ref-FEA, as presented in Figure 5.3.

71

0.75

0.80

0.85

0.90

0.95

1.00

2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

N

orm

aliz

ed M

ean Y

oung’s

Modulu

s

0.023 % DC (SV) 0.25 % DC 0.5 % DC 1 % DC 2 % DC

(a) Young’s modulus along zigzag direction.

0.75

0.80

0.85

0.90

0.95

1.00

2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

N

orm

aliz

ed M

ean

Sh

ear

Modulu

s

0.023 % DC (SV) 0.25 % DC 0.5 % DC 1 % DC 2 % DC

(b) In-plane shear modulus.

0.75

0.80

0.85

0.90

0.95

1.00

2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

N

orm

aliz

ed M

ean

Bu

lkM

odulu

s

0.023 % DC (SV) 0.25 % DC 0.5 % DC 1 % DC 2 % DC

(c) In-plane bulk modulus.

Figure 5.3: Normalized mean values of elastic moduli related to defect concentration DC (%) and forvarious volume fractions Vf (%).

Additionally, the standard deviation on each elastic property for all the groups of is computed to

explore the randomness effect, as it is usually used to quantify the amount of variation or dispersion of

a set of data values. An useful property of standard deviation is that it is expressed in the same units as

the data. The outcome is shown in Figure 5.4.

72

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0% 0.5% 1.0% 1.5% 2.0%Sta

ndar

d D

evia

tion

on

You

ng’s

Mod

ulus

(G

Pa)

DC

2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

(a) Young’s modulus along zigzag direction.

0.00

0.01

0.02

0.03

0.04

0.05

0.0% 0.5% 1.0% 1.5% 2.0%

S

tand

ard

Dev

iati

on o

n S

hear

Mod

ulus

(G

Pa)

DC

2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

(b) In-plane shear modulus.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.0% 0.5% 1.0% 1.5% 2.0%

Sta

ndar

d D

evia

tion

on

Bul

kM

odul

us (

GP

a)

DC

2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

(c) In-plane bulk modulus.

Figure 5.4: Standard deviations on elastic moduli related to volume fraction Vf (%) and for variousdefect concentration DC (%).

5.3 Discussion of Results

Firstly, when one observes Figure 5.3, the major remark is that the mechanical properties of graphene-

based nanocomposite are indeed affected by the defective condition of the reinforcing agent. Based on

literature results about structural defects in graphene, explored in Chapter 2, Fedorov et al. [95], Dettori

et al. [97] and Tapia et al. [96] have confirmed that Young’s modulus of DG shows a monotonic decrease

as SV defect density grows.

73

0.95

0.96

0.97

0.98

0.99

1.00

0.10% 0.20% 0.30% 0.40% 0.50%

sul

ud

oM s'

gn

uo

Y d

ezi l

amr

oN

Defect Content, DC

Dettori et al. [97] 2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

Figure 5.5: Evolution on normalized Young’s modulus of DG from [97] and the presented normalizedmean values of Young’s modulus for the DG-based nanocomposite, related to defect concentration DC

(%) and for various volume fractions Vf (%).

Regarding Figure 5.3(a), for the same volume fraction, it is predicted a progressive deterioration of

effective Young’s modulus for the nanocomposite while multiple SV concentration increases. Because

the enhancement in performance of the nanocomposite is mostly dependent on the mechanical proper-

ties of the reinforcement, it is expected that both effects of defective condition and stiffening effect are

also conjugated. This evidence may be visualized in Figure 5.5. It can be also noticed in Figure 5.3(a),

that the smallest influence in effective Young’s modulus of nanocomposite happens when an unique

SV defect (0.023% DC) is introduced. Moreover, the reduction on effective Young’s modulus is always

larger for 2% DC, indicating that this defective condition of DG largely affects the nanocomposite tensile

performance.

In respect to shear modulus, the results exhibited in Figure 5.3(b) also show a decreasing trend with

defect concentration. This behaviour is consistent with the influence of number of defects reported in

Tapia et al. [96], whose comparison is demonstrated in Figure 5.6. As for Young’s modulus, the minor

and major aggravations in effective shear modulus occur for 0.023% DC and for 2% DC. Concerning

the influence of defects in bulk modulus, no studies were found available in literature, though one may

expect that this property varies with graphene’s defective condition as well. Actually, from the results

depicted in Figure 5.3(c), it seems that bulk modulus of nanocomposite is greatly influenced by the

defective condition of the reinforcement, when compared to the other two moduli.

Another interesting finding from Figure 5.3 is the repercussion of volume fraction in the mechanical

performance of the DG-based nanocomposite. The present results demonstrate that for the same level

of defectiveness, it occurs some worsening of elastic properties of the nanocomposite while the volume

fraction of DG grows. However, this worsening is dependently on the vacancy-defect concentration. For

small defect coverage – between 0.023% DC and 0.5% DC –, an increase in volume fraction tends to

maintain constant the effective elastic properties. On the other hand, for larger defect coverage – from

1% DC to 2% DC –, the degradation of mechanical performance seems to have greater sensitivity on

74

increasing volume fraction between 2% Vf and 4% Vf , rather than increasing volume fraction from 6%

Vf to 10% Vf .

Until this point, one may conclude that for a low-level of defect coverage, the presence of SV impu-

rities in DG only affects a marginal reduction in the elastic properties of the nanocomposite. However,

that reduction increases significantly when it occurs a higher diminution in absolute mass density on

the lattice of DG. Besides, it is predicted that for a lower sensitivity of effective elastic properties upon

variation in defect coverage, one should design the nanocomposite at higher filler volume fractions.

0.94

0.95

0.96

0.97

0.98

0.99

1.00

0.05% 0.25% 0.45% 0.65% 0.85%

sul

ud

oM rae

hS

dezilamr

oN

Defect Content, DC

Tapia et al. [96] 2 % Vf

4 % Vf

6 % Vf

8 % Vf

10 % Vf

Figure 5.6: Evolution on normalized shear modulus of DG from [96] and the presented normalizedmean values of shear modulus for the DG-based nanocomposite, related to defect concentration DC

(%) and for various volume fractions Vf (%).

Concerning the impact of randomness in the results obtained, as shown in Figure 5.4, it becomes

clear that this effect does not own a critical preponderance in the dispersion of the elastic properties,

at least quantitatively. When an arbitrary defect distribution is assumed, the largest standard deviation

is less than 1 GPa for all moduli. Specifically for the case when an unique SV is adopted, the results

of standard deviation clearly show for all elastic properties of the nanocomposite that, not only are not

sensitive of the defect location, but are also not susceptible to vary when increasing the volume fraction.

Still, it becomes clear that for the remaining defect concentrations explored in Figure 5.4, the absolute

deviations tend to increase for greater volume fractions.

On the other hand, while the defect coverage enlarges over the DG surface, for the same volume

fraction, the standard deviation tends to increase but non-monotonically. In fact, one may find slight

reductions in standard deviation that occur for Young’s modulus, between 0.25% DC to 0.5% DC in

Figure 5.4(a), and bulk modulus, from between 0.25% DC to 0.5% DC and from 1% DC to 2% DC in

Figure 5.4(c). These small variations may be explained by several factors. It can be due to a certain

level of similarity in the samples used, though it does not seem obvious, since in that case one would

expect a common behaviour in standard deviation between each elastic property. Instead, it can be a

result of mutual interaction between stress fields generated by each isolated defect. As Dettori et al. [97]

found, these fields generally normalize only above 10 A from the defect position. Therefore, by mutually

75

offering similar/opposite (in sign) stress fields along the approaching direction, the mutual defect–defect

interactions and their distribution may affect each elastic property in particular ways, providing more or

less deviation from its mean.

76

Chapter 6

Conclusions and Future Work

The present dissertation had the main objective of studying the mechanical behaviour of a graphene-

based nanocomposite. Since this has been a cutting edge topic in recent years, as such, Chapter 1

introduced an overview on the latest discoveries about graphene and a brief description of the vastness

of applications of this carbon allotrope due to its excellent mechanical, thermal, electrical and optical

properties. In its turn, Chapter 2 reviewed the most important breakthroughs reported in the literature

in the last few decades, concerning the mechanical properties of graphene and its nanocomposites,

and examined various structural defects usually present in graphene, how they may be formed and their

influence in its mechanical performance. The newest calculation methods and latest developments in

experiments that have been applied in this area were also explored.

In Chapter 3, the multi-scale computational model based on finite element analysis (FEA), developed

to describe the mechanical behaviour of graphene-based nanocomposite, was accomplished. In the

first section, the problem was defined through some adequate assumptions and constraints, which were

carefully discussed to allow an efficient but accurate modelling. The second section was dedicated to

construct the nanoscale equivalent model based on pristine graphene (PG) molecular structure and

where the finite element (FE) type and interatomic forces dependence on molecular force field (MFF)

were addressed in detail. In order to extract the elastic properties of PG (Young’s modulus, Poisson’s

ratio, shear modulus and bulk modulus), different mechanical tests were performed with proper boundary

conditions. The third and last section conducted the embedding of PG equivalent model into epoxy

resin through a tridimensional representative volume model (RVE) of composite material. Both phases

were connected through a homogeneous interface of specific thickness and van der Waals interactions

were also accounted by Lennard-Jones interatomic potential. After boundary conditions were imposed,

several structure-property relations were tested to investigate their effects in elastic behaviour of the

RVE of nanocomposite.

The abovementioned modelling process allowed the examination of two distinct set of results on

Chapter 4. Firstly, the elastic properties of PG obtained were discussed and validated against analogous

results obtained by other authors. Secondly, the parametric studies conducted about the graphene

nanocomposite were intended to test its efficiency and effectiveness, by validating it in comparison with

other similar numerical approaches and experimental results reported in literature.

In Chapter 5, the defective state on defective graphene (DG) molecular nanostructure was adopted

and attributed in the RVE nanocomposite already obtained, by substituting the defective sheet (DG) over

77

the pristine one (PG). Then, several parametric studies were performed to explore the influence of atomic

defects at nanoscale in the mechanical performance of the nanocomposite. Eventually, this chapter

presents the conclusions and main discoveries found throughout this work, which can be enumerated

as follows:

• The implementation of the proposed FE model to simulate the mechanical behaviour of PG proved

to be simple but fairly accurate. Its orthotropic behaviour, predicted by other authors, was observed

and also verified that mechanical properties are considerably influenced by the MFF constants

assumed, allowing to refine the results in accordance to more rigorous methods.

• The study of size effects confirmed that elastic properties of PG converge to bulk values, while the

diagonal length of square-shape sheets increases. However, by substituting those bulk values in

appropriate isotropic relations, it was also proven that a large sheet of PG behaves almost as an

isotropic material rather than orthotropic.

• The FE model developed to study the mechanical behaviour of the PG-based nanocomposite

was compared with other similar numerical models and some experimental results reported in

literature. It was verified that by assuming the homogeneous interface described, the proposed

model is compliant with the stiffness enhancement occurring in experiments when the reinforcing

phase, i.e. graphene, and the matrix are not covalently bonded.

• The influence of PG volume fraction in the nanocomposite material proves that the former can

increase significantly the mechanical properties of pure epoxy. Besides, the design of higher inter-

face stiffness leads to increasing adhesion between graphene and matrix due to a more effective

stress transfer between both phases.

• The effective elastic properties of the nanocomposite material are worsened by the presence of

single-vacancy (SV) type defects in the reinforcing agent. For a lower number of defects there is a

marginal reduction, but the degrading effect increases upon the defective condition. The influence

of randomness in defects location/distribution has not resulted in an appreciable dispersion of

effective elastic properties.

Since there are no similar reports in literature so that one can directly validate the results achieved

in Chapter 5, some observations were identified by making the correlation between the influence of

structural defects on graphene and on graphene-based nanocomposites. Qualitatively, the aggravation

of mechanical performance with the presence of SV defects in both cases was verified as quite similar,

by comparing the present approach with Fedorov et al. [95], Dettori et al. [97] and Tapia et al. [96] works.

This fact demonstrates that numerical methods on graphene-based nanocomposites have to have the

ability to describe, coherently, the mechanical behaviour of DG in order to obtain accurate results with

recent experiments.

Ultimately, after the presentation of this work, still some improvements can be accomplished in the

FE models described, and other interesting future developments as well. These include:

78

• Include more accurate MFF constants in graphene equivalent bonds, which can be obtained

through the parametrisation of more recent potential functions, as the recently achieved AIREBO

potential.

• Introduce non-linear constitutive relations in equivalent carbon-carbon bond properties to evaluate

the strength and toughness of graphene.

• Development of similar models with randomly oriented graphene sheets to test the possibility of

distinct reinforcing effects on nanocomposites.

• Conduct a numerical pull-out test on graphene-based nanocomposite to evaluate graphene sheet

debonding and obtain interfacial axial and shear stresses distributions throughout the process.

• Address other types of atomic defects, such as Stone-Wales (SW), 5-8-5 or 555-777 and relate

their location/distribution with the mechanical performance of graphene-based nanocomposites.

79

80

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90

Appendix A

Single-Phase Materials

A.1 Interrelations among the 2D and 3D elastic moduli

Consider a d-dimensional, linear, isotropic homogeneous material with bulk, shear, Young’s modulus,

and Poisson’s ratio denoted by K(d), G(d), E(d), and ν(d), respectively. For such a material, the d-

dimensional strain-stress relations were developed by Eischen and Torquato [108] and are expressed

as:

εij =1

E(d)

[(1 + ν(d)

)σij − ν(d)σkkδij

], i, j, k = 1, ..., d (A.1)

Similarly, the d-dimensional stress-strain relations are given by:

σij =(K(d) − 2G(d)

)εijδij + 2G(d)εij , i, j, k = 1, ..., d (A.2)

To connect the 3D moduli to the 2D moduli, one may assume either plane-strain or plane-stress elasticity.

For plane-strain elasticity, we take ε11 = ε12 = ε13 = 0 in the relation (A.1) with d = 3. If we compare this

simplified 3D expression to relation (A.1) with d = 2, we find the interrelations:

E(2) =E(3)

(1− ν(3))(1 + ν(3))(A.3)

ν(2) =ν(3)

1− ν(3)(A.4)

Similarly, by comparing equation (A.2) with d = 3 under plane-strain conditions and equation (A.2) with

d = 2, it can be found that:

K(2) = K(3) +G(3)/3 (A.5)

G(2) = G(3) (A.6)

For plane-stress elasticity, one must take σ11 = σ12 = σ13 = 0 in the expression (A.1) with d = 3.

Comparing this simplified 3D relation to relation (A.1) with d = 2 gives:

E(2) = E(3) (A.7)

ν(2) = ν(3) (A.8)

91

Finally, comparing equation (A.2) with d = 3 under plane-stress conditions and equation (A.2) with d = 2,

yields:

K(2) =9K(3)G(3)

3K(3) + 4G(3)(A.9)

G(2) = G(3) (A.10)

A.2 Two-dimensionality elasticity

To define the elastic moduli for two-dimensional media, one must compare expressions (A.1) and

(A.2) with d = 2, resulting in the following interrelations:

G(2) =E(2)

2(1 + ν(2)

) (A.11)

ν(2) =K(2) −G(2)

K(2) +G(2)(A.12)

Clearly, there are only two independent moduli. Replacing equation (A.11) in equation (A.12) and solving

this with respect to K(2), the following definition is obtained for the 2D bulk modulus:

K(2) =E(2)

2(1− ν(2)

) (A.13)

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