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Caractérisation mécanique des matériaux composites àpartir d’un essai de traction biaxiale intégrant l’effet des

chemins de déformationAlaa Kobeissi

To cite this version:Alaa Kobeissi. Caractérisation mécanique des matériaux composites à partir d’un essai de tractionbiaxiale intégrant l’effet des chemins de déformation. Autre. INSA de Rennes; Université Libanaise.Faculté de génie, 2019. Français. �NNT : 2019ISAR0021�. �tel-02891797�

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THESE DE DOCTORAT DE

L’UNIVERSITE LIBANAISE

L’INSA RENNES

COMUE UNIVERSITE BRETAGNE LOIRE

ECOLE DOCTORALE N° 602 Sciences pour l'Ingénieur Spécialité : « Génie Mécanique »

« Caractérisation mécanique des matériaux composites à partir d’un essai de traction biaxiale intégrant l’effet des chemins de déformation » Thèse présentée et soutenue à « Rennes », le « 20/12/2019 » Unité de recherche : Laboratoire de Génie Civil et Génie Mécanique (LGCGM), INSA Rennes Centre de Recherche Scientifique en Ingénierie (CRSI), Université Libanaise Thèse N° : 19ISAR 38 / D19 - 38

Par

« Alaa KOBEISSI»

Rapporteurs avant soutenance :

Nathalie GODIN Maître de Conférences, HDR, Laboratoire MATEIS, INSA Lyon

Fréderic LACHAUD Professeur, ISAE Supaéro,

Toulouse

Composition du Jury :

Nathalie GODIN Maître de Conférences, HDR, Lab. MATEIS, INSA Lyon, Rapporteur

Vincent KERYVIN Professeur, IRDL, Université de

Bretagne Sud, Lorient, Examinateur

Fréderic LACHAUD Professeur, ISAE Supaéro, Toulouse,

Rapporteur Evelyne TOUSSAINT Professeur, Université Clermont

Auvergne, Clermont Ferrand, Examinateur (Présidente)

Lionel LEOTOING Maître de Conférences, HDR,

LGCGM, INSA Rennes, Co-directeur de thèse

Pierre RAHME Professeur, Faculté de Génie, Université Libanaise, Co-directeur de thèse

Intitulé de la thèse :

Caractérisation Mécanique des Matériaux Composites à Partir

d’un Essai de Traction Biaxiale Intégrant l’effet des Chemins

de Déformation

Mechanical Characterization of Composite Materials

Subjected to Biaxial Tensile Test Integrating the Effect of

Strain Paths

Alaa KOBEISSI

En partenariat avec :

Document protégé par les droits d’auteur

Acknowledgment

First, I would like to express my sincere gratitude to my research supervisors, Dr. Lionel

Leotoing, Dr. Pierre Rahme, and Dr. Dominique Guines, for their continuous support during

my thesis, their patience, motivation, and immense knowledge. Their guidance helped me

during my whole work and writing of this thesis. Thank you so much for your inspiration and

encouragement.

I would also like to thank all the colleagues at the Laboratory of Civil and Mechanical

Engineering (LGCGM) for their friendship and encouragement. I’d like to express my special

acknowledgment to the master students “Queen Tannous” and “Elias Estephan” for their

cooperation. I’m also grateful to the engineers in the laboratory for the specimen preparation

and technical support.

Last but not the least, I would like to thank my family: my parents, my brothers, and my sister

for supporting me spiritually throughout writing this thesis and my life in general.

Résumé

Au cours des dernières années, les matériaux composites à fibres continues ont été largement

utilisés dans la plupart des applications d’ingénierie en raison de leurs propriétés mécaniques

spécifiques élevées. Bien que les composites présentent de nombreux avantages par rapport aux

autres matériaux, leur fort comportement mécanique anisotrope rend la caractérisation de ces

matériaux plus compliquée lorsqu'ils sont soumis à des conditions de chargement multiaxiales.

Les essais uniaxiaux sont principalement utilisés pour la caractérisation des matériaux

composites en raison de leur simplicité et de leur faible coût. Cependant, ces essais ne sont pas

adaptés pour caractériser le comportement du matériau soumis à un état de contrainte

multiaxiale Pour mieux caractériser leur comportement, des essais biaxiaux ont été développés.

Parmi les différentes techniques de traction biaxiale, l’essai de traction sur éprouvettes

cruciformes est l'une des méthodes les plus connues. Les avantages de cette méthode par rapport

aux autres techniques biaxiales sont la possibilité de contrôler l’état de contrainte biaxiale dans

la zone centrale de l’éprouvette en chargeant chaque bras de manière indépendante. Cependant,

la difficulté de cette méthode réside dans le choix de l’éprouvette cruciforme qui doit présenter

une rupture dans la zone centrale. Pour atteindre cet objectif, une réduction d'épaisseur dans la

zone centrale est nécessaire. Dans la littérature, deux méthodes ont été utilisées pour réduire

l’épaisseur de l’éprouvette dans la zone centrale. La première méthode consiste à mettre en

sandwich le composite étudié entre deux couches de composite avec un trou au centre afin de

former une éprouvette d'épaisseur réduite complète. La seconde consiste à enlever par usinage

un nombre spécifique de plis de la zone centrale tandis que les plis restants au centre définissent

le composite étudié.

Dans cette étude, une nouvelle éprouvette cruciforme est définie pour les essais de traction

biaxiale d'un tissu de composite verre/époxy. Sur la base des études précédentes, l’idée d’usiner

la zone centrale du composite est évitée et un nouveau concept est proposé en collant des talons

en aluminium des deux côtés de la plaque composite à épaisseur constante. La réduction

d'épaisseur est réalisée uniquement par les talons en aluminium afin d'éviter l'usinage du

composite. Les dimensions finales de l’éprouvette cruciforme sont définies par une étude

paramétrique numérique tout en respectant certaines contraintes expérimentales telles que la

capacité de la machine de traction biaxiale. Une validation expérimentale est réalisée sur deux

composites verre / époxy de différentes épaisseurs (0,5 et 1 mm). Les essais ont été réalisés

pour différentes conditions de chargement biaxial (de l’uniaxial à l’équi-biaxial). La

distribution des déformations dans la zone centrale de l’éprouvette est obtenue à l'aide de la

technique de corrélation d'image (DIC). Les contraintes sont calculées en fonction des

propriétés du matériau et des déformations expérimentales à l'aide des équations de la loi de

comportement du composite. L'enveloppe à la rupture est tracée et comparée aux prédictions

de trois critères à la rupture (Max Stress, Max Strain et Norris).

Enfin, une ouverture de ce travail sur l’étude de la résistance de composites fissurés sous

differentes charges biaxiales est présentée.

Abstract Over the past years, continuous fiber composite materials have become extensively used in

most engineering applications due to their high specific mechanical properties. These materials

have many benefits compared to other materials. Although, their strong anisotropic mechanical

behavior makes the characterization of those materials complicated when they are subjected to

multi-axial loading conditions. Uniaxial tests are mostly used for the characterization of

composite materials because of their simplicity and low cost. However, these tests are not

adapted for the characterization of the behavior of these materials subjected to multiaxial stress

state. For a better characterization of their behavior, biaxial tests are used.

Among different biaxial testing techniques, the in-plane biaxial testing of cruciform specimen

is one of the most known methods. The benefits of this method compared to the other biaxial

techniques is the ability of controlling the biaxial stress state in the central zone of the specimen

by independently loading each arm. However, the main difficulty of this method is the design

of a cruciform specimen that fails in the central area. In order to achieve this, a thickness

reduction in the central zone is required. In the literature, two methods have been used in order

to reduce the thickness of the specimen in the central zone. The first method is by sandwiching

the studied composite between two layers of composite with a hole in the center in order to

form a complete reduced thickness specimen. The second is by milling a specific number of

plies from the central zone while the remaining plies in the central thickness define the studied

composite.

In this thesis, a new cruciform specimen is designed for the biaxial tensile testing of plain-

weave glass/epoxy composite. Based on the previous studies, the idea of milling the composite

central zone is avoided. A new concept is proposed by gluing aluminum tabs with a central hole

on both sides of the constant thickness composite laminate. The thickness reduction is made

only in the aluminum tabs in order to avoid machining in the composite. The final dimensions

of the cruciform specimen are defined by numerical parametric study while respecting some

experimental constraints such as biaxial tensile machine capacity. An experimental validation

is performed on two plain-weave glass/epoxy composites with different thicknesses (0.5 and 1

mm). The tests are performed under different biaxial loading conditions (from uniaxial to equi-

biaxial stress state). The strain distribution in the central zone of the specimen are obtained

using the Digital Image Correlation (DIC) technique. The stresses are calculated based on the

constitutive law equations using the material properties and the experimental strains. The failure

envelop is generated and compared to the predictions of three failure criteria (Max Stress, Max

Strain, and Norris distortional energy).

As final part of this thesis, an opening of this work towards the study of the resistance of notched

composites under different biaxial loadings is presented.

Table of Contents

General Introduction ............................................................................................................... 1

Chapter 1. State of the Art ................................................................................................... 3

1.1. Introduction ................................................................................................................. 4

1.2. Mechanical characterization of composite materials .................................................. 4

1.2.1. Tension and Compression .................................................................................... 4

1.2.2. Shear test .............................................................................................................. 5

1.2.3. Bending or Flexural test ....................................................................................... 7

1.2.4. Other purposes to perform uniaxial tests ............................................................. 7

1.3. Biaxial testing methods ............................................................................................... 8

1.3.1. Biaxial testing techniques .................................................................................... 9

1.3.2. Choice of an adapted biaxial testing technique .................................................. 11

1.3.3. Biaxial testing machines on cruciform specimens ............................................. 12

1.4. In-plane biaxial test on cruciform specimens ............................................................ 14

1.4.1. Metal cruciform specimen ................................................................................. 14

1.4.2. Composite cruciform specimen ......................................................................... 18

1.5. Conclusion ................................................................................................................. 29

Chapter 2. Design of the Cruciform Specimen ................................................................. 31

2.1. Introduction ............................................................................................................... 32

2.2. Selection of the cruciform shape ............................................................................... 33

2.2.1. Selected shapes .................................................................................................. 33

2.2.2. Numerical comparison ....................................................................................... 36

2.3. Effect of geometrical parameters .............................................................................. 40

2.3.1. External parameters ........................................................................................... 41

2.3.2. Tab geometry in the gage section ...................................................................... 42

2.4. Effect of aluminum grade .......................................................................................... 48

2.5. Final shape and behavior under different biaxial ratios ............................................ 52

2.6. Conclusion ................................................................................................................. 54

Chapter 3. Biaxial Experimental Characterization of Failure ....................................... 57

3.1. Introduction ............................................................................................................... 58

3.2. Uniaxial characterization of composite materials ..................................................... 58

3.2.1. Materials ............................................................................................................ 58

3.2.2. Strain measurement techniques.......................................................................... 59

3.2.3. Test results and material properties ................................................................... 61

3.2.4. Uniaxial tests under different strain rates .......................................................... 63

3.3. Biaxial test setup ....................................................................................................... 65

3.3.1. Biaxial testing machine ...................................................................................... 65

3.3.2. Cruciform specimen preparation........................................................................ 66

3.3.3. Experimental parameters ................................................................................... 69

3.4. Strain measurement ................................................................................................... 69

3.4.1. Effect of DIC parameters ................................................................................... 69

3.4.2. Experimental verification of the effect of aluminum grade ............................... 71

3.4.3. Strain results for M9.6 composite ...................................................................... 72

3.4.4. Strain results for M21 composite ....................................................................... 75

3.4.5. Strain path .......................................................................................................... 80

3.5. Stress calculation ....................................................................................................... 81

3.5.1. Stress calculation methods ................................................................................. 81

3.5.2. Stress results for M9.6 composite ...................................................................... 82

3.5.3. Stress results for M21 composite ....................................................................... 85

3.6. Conclusion ................................................................................................................. 89

Chapter 4. Influence of Loading Paths on Failure Limits ............................................... 91

4.1. Introduction ............................................................................................................... 92

4.2. Comparison between numerical and experimental results ........................................ 92

4.2.1. Cohesive elements modeling ............................................................................. 92

4.2.2. Boundary conditions .......................................................................................... 93

4.2.3. Numerical model results .................................................................................... 94

4.2.4. Strain path .......................................................................................................... 98

4.3. Failure of composite materials .................................................................................. 98

4.3.1. Introduction ........................................................................................................ 98

4.3.2. Failure criteria selected .................................................................................... 101

4.3.3. Failure envelop of M21 composite .................................................................. 102

4.4. Biaxial testing of Notched composite ..................................................................... 105

4.4.1. Introduction ...................................................................................................... 105

4.4.2. Notched specimen preparation ......................................................................... 106

4.4.3. Biaxial test results ............................................................................................ 107

4.5. Conclusion ............................................................................................................... 114

Conclusion and Perspectives ............................................................................................... 115

References .......................................................................................................................... 117

Appendix .......................................................................................................................... 125

Figures

Figure 1.1 (a) Tension and (b) Compression of flat specimens [2] ........................................... 5

Figure 1.2 Conventional shear tests: (a) torsion of tubes, (b) Iosipescu, (c) double notched,

and (d) off-axis [2] ..................................................................................................................... 6

Figure 1.3 Three point bending test [2] ..................................................................................... 7

Figure 1.4 Shokrieh failed specimens under different tensile rates: (a) 0.0017 s-1, (b) 0.55 s-1,

(c) 5.6 s-1, (d) 46 s-1 and (e) 85 s-1[12] ....................................................................................... 8

Figure 1.5 Biaxial testing techniques: (a) Biaxial flexure testing of plate [21], (b) Bulge test

[24], (c) off-axis test [2], (d) Iosipescu test [34], (e) Biaxial bending moment on flat

specimens [27], (f) axial/ torsion of tubes [35], (g) axial/pressure of tubes [36], and (h)

cruciform specimen [37] .......................................................................................................... 11

Figure 1.6 Biaxial testing machines with one actuator developed by Bhatnagar [40] ........... 12

Figure 1.7 Biaxial testing machines with two actuators developed by: (a) Hoferlin [41] and

(b) Lin [42] ............................................................................................................................... 13

Figure 1.8 Biaxial testing machines with four actuators developed by: (a) Makinde/Green

[44] and (b) Gower [45] ........................................................................................................... 14

Figure 1.9 Tri-axial testing machines developed by: (a) Welsh [46] and (b) ASTREE [50] .. 14

Figure 1.10 Methods used in the design of cruciform specimen [55] ..................................... 15

Figure 1.11 Cruciform specimen proposed by Hayhurst [57] ................................................. 15

Figure 1.12 Specimen with two thickness reductions [59] ...................................................... 16

Figure 1.13 Cruciform specimen without slits [37] ................................................................. 16

Figure 1.14 Cruciform proposed by (a) Yu [60] and (b) Zhang [61] ...................................... 17

Figure 1.15 Zidane optimized specimen geometry [46] .......................................................... 17

Figure 1.16 Liu optimal specimen ........................................................................................... 18

Figure 1.17 Composite cruciform specimens with slits developed by: (a) Youssef [63]and (b)

Ohtake [64] .............................................................................................................................. 19

Figure 1.18 Cruciform specimen presented by: (a) Hopgood [67], (b) Williamson [65], and

(c) Kumazawa [68] .................................................................................................................. 20

Figure 1.19 Damage evolution in the cruciform specimen [68] .............................................. 21

Figure 1.20 Specimens tested by Escarpita [70]: (a) constant thickness, (b) square central

zone, and (c) rohmboidal central zone ..................................................................................... 21

Figure 1.21 Cruciform specimens with: aluminum shims (Type I) and G10 tabs (Type II) [71]

.................................................................................................................................................. 22

Figure 1.22 Failure of all three specimen types [71] ............................................................... 22

Figure 1.23 Four specimen shapes studied by Correa et al. [72] ............................................. 23

Figure 1.24 3D geometrical model proposed by Correa [72] .................................................. 23

Figure 1.25 Cruciform geometry proposed by Welsh and Adams [47] ................................... 24

Figure 1.26 Through- thickness dimensions of Smits specimen [75] ..................................... 26

Figure 1.27 First principal strain results for the four different geometries studied by Smits et

al. [75] ...................................................................................................................................... 26

Figure 1.28 Strain variation with respect to the position in the central zone [75] ................... 26

Figure 1.29 Experimental (top) and numerical (bottom) strain field results under different

loading conditions. ................................................................................................................... 27

Figure 1.30 𝜀𝑥 strain distribution of the optimized form (a) and the commonly tested case (b)

.................................................................................................................................................. 28

Figure 1.31 Deformation results for specimens with normal and reversed tabbing ................ 28

Figure 1.32 Premature failure of composites fabricated with tabbing method ........................ 29

Figure 1.33 Crack initiation in the milled layers for (a) x and (b) y-direction loading ........... 29

Figure 2.1 Fibers orientation with respect to x and y directions .............................................. 33

Figure 2.2 Cruciform specimen parts disassembled ................................................................ 34

Figure 2.3 Cruciform specimen: shape A ................................................................................ 35

Figure 2.4 Cruciform specimen: shape B ................................................................................ 35

Figure 2.5 Cruciform specimen: shape C ................................................................................ 35

Figure 2.6 True stress- plastic strain curve for AA2017 alloy ................................................. 36

Figure 2.7 Boundary Conditions for the numerical modeling ................................................. 37

Figure 2.8 Mesh and number of elements for each specimen.................................................. 37

Figure 2.9 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) for three specimens ............................... 38

Figure 2.10 Normalized equivalent strain distribution for three specimens ............................ 39

Figure 2.11 Variation of equivalent stress with respect to the normalized distance ................ 39

Figure 2.12 Variation of the normalized equivalent strain with respect to the normalized

distance .................................................................................................................................... 39

Figure 2.13 Evolution of tensile loading with respect to the applied displacement ................ 40

Figure 2.14 Geometrical parameters for the design of the cruciform specimen ...................... 41

Figure 2.15 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) in the composite for different fillet

corners ...................................................................................................................................... 42

Figure 2.16 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) for different central zone radii ............ 43

Figure 2.17 𝜎𝑥𝑥 (𝑀𝑃𝑎) distribution for different central zone radii ....................................... 44

Figure 2.18 Variation of 𝜎𝑥𝑥 with respect to the x-direction arm normalized distance using

different central zone radii ....................................................................................................... 44

Figure 2.19 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) for different outer tapered radii .......... 45

Figure 2.20 Aluminum tabs with 0 and 1 mm ending thickness ............................................. 46

Figure 2.21 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) in the composite for (a) 0 mm and (b) 1

mm ending thickness................................................................................................................ 46

Figure 2.22 Linear and curved thickness reduction in aluminum tab ...................................... 47

Figure 2.23 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) in the composite for (a) curved and (b)

linear aluminum thickness reduction shape ............................................................................. 47

Figure 2.24 Comparison between 𝜎𝑥𝑥 results in the composite using curved and linear

thickness reduction transition .................................................................................................. 48

Figure 2.25 True stress- plastic strain curve for AA5086 and AA2017 alloys ........................ 49

Figure 2.26 𝜎𝑒𝑞 (𝑀𝑃𝑎) in composite using (a) AA2017 and (b) AA5086 aluminum tabs ..... 50

Figure 2.27 Comparison between 𝜎𝑥𝑥 (MPa) results in the composite using different

aluminum alloy tabs ................................................................................................................. 50

Figure 2.28 Plastic Strain (𝑃𝐸𝑥𝑥 ) in (a) AA2017 and (b) AA5086 aluminum tabs ................ 50

Figure 2.29 𝜎𝑒𝑞 (𝑀𝑃𝑎) in composite using AA5086 with a thickness of (a) 5.5mm and (b)

3mm ......................................................................................................................................... 51

Figure 2.30 Comparison between 𝜎𝑥𝑥 (MPa) results in the composite using different

aluminum thicknesses .............................................................................................................. 51

Figure 2.31 Load vs. displacement using the new geometrical parameters of the specimen .. 52

Figure 2.32 Cruciform specimen shape for the biaxial characterization test ........................... 53

Figure 2.33 𝜎𝑥𝑥 (𝑀𝑃𝑎) in composite under different biaxial displacement ratios .................. 54

Figure 2.34 𝜀𝑥𝑥 in composite under different biaxial displacement ratios............................... 54

Figure 2.35 Major versus minor strains for different biaxial ratios ......................................... 54

Figure 3.1 Digital Image Correlation (DIC) principle ............................................................. 60

Figure 3.2 Digital Image Correlation (DIC) parameters .......................................................... 61

Figure 3.3 M9.6 uniaxial specimens ........................................................................................ 62

Figure 3.4 M21 uniaxial specimen........................................................................................... 62

Figure 3.5 Strain results for uniaxial test of M21 composite using extensometer and DIC

techniques ................................................................................................................................ 63

Figure 3.6 Digital image correlation for specimen tested under a strain rate of 0.25 s-1 ......... 64

Figure 3.7 Uniaxial specimen tested under different strain rates............................................. 65

Figure 3.8 Elastic properties of the composite subjected to uniaxial loading for different strain

rates .......................................................................................................................................... 65

Figure 3.9 Schema of the biaxial testing machine in LGCGM ................................................ 66

Figure 3.10 Biaxial testing machine including the camera and the tested specimen ............... 66

Figure 3.11 Double-lap joint shear test specimen .................................................................... 67

Figure 3.12 Evolution of shear stresses in the adhesive with and without aluminum shot

peening surface treatment ........................................................................................................ 67

Figure 3.13 Composite and aluminum tabs cruciform shapes before assembly ...................... 68

Figure 3.14 Assembled cruciform specimen for the biaxial test ............................................. 68

Figure 3.15 Small and large strain calculation zones .............................................................. 70

Figure 3.16 Comparison between large and small strain calculation zone results under

equibiaxial test ......................................................................................................................... 70

Figure 3.17 Strain results using different DIC parameters under equibiaxial test ................... 71

Figure 3.18 Composite strain at the central point of the specimen for AA2017 and AA5086 72

Figure 3.19 Variation of R with respect to the imposed displacement .................................... 72

Figure 3.20 M9.6 cruciform specimen failure under 1/free biaxial ratio ................................. 73

Figure 3.21 Evolution of the loads and strains with respect to applied displacement for M9.6

composite subjected to 1/free, 1/2, and 1/1 biaxial ratios ........................................................ 74

Figure 3.22 𝜀𝑥 and 𝜀𝑦 strains distribution in the central zone of the M9.6 composite subjected

to 1/2 biaxial ratio .................................................................................................................... 75

Figure 3.23M21 cruciform failure under (a) free/1, (b) 1/2, and (c) 1/1.4 biaxial ratios......... 75

Figure 3.24 Evolution of the loads and strains with respect to applied displacement for M21

composite subjected to free/1, 1/2, and 1/1 biaxial ratios ........................................................ 77

Figure 3.25 𝜀𝑥 and 𝜀𝑦 strains distribution in the central zone of the M21 composite subjected

to 1/1 biaxial ratio (test01) ....................................................................................................... 78


composite subjected to 1/3, 1/1.4, and 1/1.2 biaxial ratios ...................................................... 79

Figure 3.27 Strain path plot for M9.6 composite ..................................................................... 80

Figure 3.28 Strain path plot for M21 composite ...................................................................... 81

Figure 3.29 Equivalent section stress calculation principle ..................................................... 82

Figure 3.30 Evolution of the stresses 𝜎𝑥 and 𝜎𝑦 with respect to applied displacement Ux (mm)

for M9.6 composite subjected to 1/free,1/2, and 1/1 biaxial ratios .......................................... 83

Figure 3.31 Evolution the calculated equivalent sections Aequ,x and Aequ,y for M9.6 composite

subjected to 1/1 biaxial displacement ratio .............................................................................. 84

Figure 3.32 Evolution of the stresses 𝜎𝑥 and 𝜎𝑦 with respect to applied displacement for M21

composite subjected to free/1, 1/2, and 1/1 biaxial ratios ........................................................ 86


composite subjected to 1/3, 1/1.4, and 1/1.2 biaxial ratios ...................................................... 87

Figure 3.34 Evolution the calculated equivalent sections Aequ,x and Aequ,y for M21 composite

subjected to 1/1 biaxial ratio .................................................................................................... 88

Figure 4.1 One-quarter of the cruciform specimen with adhesive modeling .......................... 93

Figure 4.2 Boundary conditions for the cruciform specimen modeling .................................. 93

Figure 4.3 Major stress distribution 𝜎𝑦 (𝑀𝑃𝑎) in composite: (a) with adhesive modeling and

(b) without adhesive modeling under free/1 biaxial ratio ........................................................ 94

Figure 4.4 Shear stress distribution 𝜎𝑦𝑧 (𝑀𝑃𝑎) in the adhesive joint ..................................... 94

Figure 4.6 Comparison between numerical and experimental strain results for M21 composite

subjected to free/1 biaxial ratio ................................................................................................ 96

Figure 4.7 Shear stresses 𝜎𝑦𝑧 (MPa) in the adhesive when a displacement of 0.65 mm is

reached in M21 composite subjected to free/1 biaxial ratio .................................................... 96

Figure 4.8 Evolution of numerical and experimental strains in the composite and the

numerical equivalent plastic strain (PEEQ) in the aluminum tabs for biaxial ratio free/1 ...... 96


subjected to 1/2, 1/3, and 1/1 biaxial ratios ............................................................................. 97

Figure 4.10 Evolution of major and minor strains of M21 composite under different strain

paths ......................................................................................................................................... 98

Figure 4.11 Comparison between the predicted and measured biaxial final failure stresses

for(0°/ ± 45°/90°) AS4/3501-6 laminates [83] ........................................................................ 99

Figure 4.12 Results of AIAA failure criteria survey [88] ...................................................... 101

Figure 4.13 Experimental failure envelop of M21 composite ............................................... 102

Figure 4.14 Experimental and different non interactive criteria failure envelops of M21

composite ............................................................................................................................... 103

Figure 4.15 Experimental and Norris failure criterion envelops of M21 composite ............. 104

Figure 4.16 Experimental and selected failure criteria envelops of M21 composite ............. 105

Figure 4.17 ABS guide for notch generation in the center of the cruciform specimen ......... 107

Figure 4.18 Assembled cruciform notched specimen for biaxial test .................................... 107

Figure 4.19 Failed cruciform notched specimens under different biaxial tensile ratio ......... 108

Figure 4.20 Load Fy (N) results for notched and un-notched specimen under biaxial free/1

ratio ........................................................................................................................................ 109

Figure 4.21 Load (a) difference and (b) ratio between notched and un-notched specimen

under biaxial free/1 ratio ........................................................................................................ 109

Figure 4.22 𝜀𝑥 strain fields of notched specimen under biaxial free/1 ratio .......................... 110

Figure 4.23 𝜀𝑦 strain fields of notched specimen under biaxial free/1 ratio .......................... 110

Figure 4.24 Loads results (a) Fx and (b) Fy for notched and un-notched specimen under

biaxial 1/2 ratio ...................................................................................................................... 111

Figure 4.25 X-direction load (a) difference and (b) ratio between notched and un-notched

specimen under biaxial 1/2 ratio ............................................................................................ 111

Figure 4.26 Y-direction load (a) difference and (b) ratio between notched and un-notched

specimen under biaxial 1/2 ratio ............................................................................................ 111

Figure 4.27 𝜀𝑥 strain fields of notched specimen under biaxial 1/2 ratio .............................. 112

Figure 4.28 𝜀𝑦 strain fields of notched specimen under biaxial 1/2 ratio .............................. 112

Figure 4.29 Loads results (a) Fx and (b) Fy for notched and un-notched specimen under

biaxial 1/1 ratio ...................................................................................................................... 113

Figure 4.30 𝜀𝑥 strain fields of notched specimen under biaxial 1/1 ratio .............................. 113

Figure 4.31 𝜀𝑦 strain fields of notched specimen under biaxial 1/1 ratio .............................. 114

Tables

Table 2.1 M9.6GF/200T2/G plain-weave material properties ................................................. 36

Table 2.2 Yield and ultimate strengths for aluminum alloy AA2017 and AA5086 ................ 48

Table 2.3 Final geometrical parameters for the cruciform specimen ...................................... 52

Table 3.1 DIC parameters for the uniaxial test of M21 composite .......................................... 62

Table 3.2 Averaged material properties for both M9.6 and M21 composites ......................... 63

Table 3.3 Experimental parameters for uniaxial tests under different strain rates .................. 64

Table 3.4 Experimental parameters for the biaxial test ........................................................... 69

Table 3.5 Load and strain results at the onset of failure of M9.6 composite subjected to

different biaxial ratios .............................................................................................................. 74

Table 3.6 Load and strain results at the onset of failure of M21 composite subjected to biaxial

ratios (free/1, 1/2, and 1/1) ....................................................................................................... 78

Table 3.7 Load and strain results at the onset of failure of M21 composite subjected to 1/3,

1/1.4, and 1/1.2 biaxial ratios ................................................................................................... 80

Table 3.8 Stress results at the onset of failure of M9.6 composite subjected to 1/free, 1/2, and

1/1 biaxial displacement .......................................................................................................... 84

Table 3.9 Stress results at the onset of failure of M21 composite subjected to free/1, 1/2, 1/1,

1/3, 1/1.2, and 1/1.4 biaxial ratios ............................................................................................ 89

Notations

𝜎𝑖𝑗 Components of stress tensor

𝜎𝑖 Components of principal stresses

𝜎𝑒𝑞 Equivalent stress

𝜀𝑖𝑗 Components of strain tensor

𝜀𝑖 Components of principal strains

𝜀𝑒𝑞 Equivalent strain

𝑃𝐸𝑖𝑗 Components of the plastic strain tensor

𝜀̇ Strain rate

𝐴𝑒𝑞𝑢,𝑥 Equivalent section in x-direction

𝐴𝑒𝑞𝑢,𝑦 Equivalent section in y-direction

U Displacement

𝜗 Poisson ratio

E Young modulus

G Shear modulus

V Velocity

F Load

PEEQ Equivalent plastic strain

AA Aluminum Alloy

DIC Digital Image Correlation

General Introduction

1


Over the past decades, composites have become a key element in all mechanical parts. Their

excellent mechanical performance with low specific weight explain why these materials are

important in applications in different industries like aeronautics, transportation, sports, and

health care. Although composites present many benefits compared to traditional materials,

mechanical characterization and computation of composite structures are not straightforward

due to the complexity of these materials. This is particularly true in the case of continuous

fiber composites consisting of two phases (fibers and matrix) and of several laminae stacked

in different directions leading to strong anisotropic mechanical behavior.

Conventional uniaxial tests are usually used in order to identify the material properties of the

composite. However, these methods are not capable to identify the real mechanical behavior

of composites subjected to multi-axial stress states in engineering applications. Thus, biaxial

tests can be performed. Among the different biaxial testing methods, the in-plane biaxial test

on cruciform specimens is one of the most interesting methods because of its simplicity in

generating different biaxial loading combinations by applying loads on both arms of the flat

cruciform specimen. At the same time, the design of the cruciform specimen is the most

difficult task in this type of biaxial testing since a homogeneous and maximum stress state

must be obtained in the central area of the specimen (area subjected to biaxial stress state).

Different studies were performed by researchers in order to obtain a reliable composite

cruciform specimen for the biaxial testing. The high number of defined cruciform shapes

prove that there is no-well adapted shape for the biaxial testing. The objective of this thesis

begins with a definition of a new cruciform specimen for the biaxial characterization of plain-

weave glass/epoxy composite. The goal behind the test is to obtain the failure stresses of this

woven composite when it’s subjected to different biaxial loading conditions. The results of

this work are presented into four different chapters as below:

In Chapter 1, a complete literature review regarding the characterization of composite

materials is presented. The chapter begins by citing conventional uniaxial characterization

tests, then introduces different biaxial testing techniques leading to the in-plane biaxial

testing of cruciform specimens. Since the difficulty of this test is the design of the specimen,

different metallic and composite cruciform shapes will be presented according to previous

researchers as opening to the next chapter.


2

Chapter 2 deals with the design of the cruciform specimen for the biaxial test. First, a

numerical comparison between three different cruciform shapes inspired from the literature is

performed in order to select one shape that fulfill the requirements for a successful biaxial

test such as homogeneous and maximum stress state in the central region. According to

different experimental constraints (i.e. biaxial machine capacity and strain measurement

technique), a parametric study is performed on each geometrical parameter in order to re-

dimension the specimen while respecting the previous constraints. A final part in this chapter

is to verify the strain path and failure of the specimen in the central zone for different biaxial

loading conditions

Chapter 3 presents the complete experimental phase of this study which is an experimental

verification of the failure in the central zone of the designed cruciform specimen using two

different plain-weave glass/epoxy composites with different thicknesses. The experimental

phase begins with uniaxial characterization of the studied composites in order to obtain the

material properties such as Young modulus, shear modulus, and Poisson ratio. Later on, the

biaxial study begins with a description of the biaxial testing machine developed at INSA de

Rennes. The strains are measured using the Digital Image Correlation (DIC) method, and the

stresses are calculated based on the measured strains and the material properties using the

constitutive law equations. The results are obtained for different biaxial loading conditions

(varying from uniaxial to equi-biaxial loading states).

Chapter 4 shows the analysis of the results obtained from the previous chapters where a

comparison between the numerical and the experimental results is performed in order to show

the accuracy of the numerical model. Moreover, the strain path curves are plotted based on

the evolution of the minor and major strains in the central zone of the composite, the curves

can validate the biaxial state in the composite for different loading conditions. Also, the

traction/traction quadrant of the failure envelop is generated and compared with three

different interactive and non-interactive failure criteria in order to show the accuracy of each

criterion compared to the experimental stresses. The final section of this chapter is an

opening for a perspective work regarding the effect of impact on the behavior of the

composite. A cruciform crack is generated in the central zone of the specimen which is

supposed to represent the effect of an impact on the composite. Different biaxial tests are

performed in order to show the effect of the impact on the behavior of the composite under

different biaxial loading conditions.

The final part of the study is a conclusion summarizing all the results obtained from this

study with some perspectives about the future works.

Chapter 1. State of the Art

3



4

1.1. Introduction

This chapter presents a state of art related to the studied subjects in this thesis. First,

conventional uniaxial tests for the composite characterization are cited. However, in order to

represent the real behavior of the material subjected to multi-axial stress state during real life

applications, an overview of the biaxial testing techniques is presented. The advantages of the

in-plane biaxial test on flat cruciform specimens over the other testing methods are discussed.

Finally, a brief review of the biaxial testing machines and the different cruciform specimen

shapes found in the literature are presented.

1.2. Mechanical characterization of composite materials

Over the past decades, composites are widely used in many industries such as aeronautic,

aerospace, automotive, boating, because of their low specific weight, good mechanical

properties and corrosion resistance. Composite materials are by nature highly anisotropic.

The material properties of composite materials are most often determined by means of

uniaxial tensile tests. The characterization of the anisotropic behavior requires numerous

uniaxial tests to be determined [1]. In the following, the mostly used conventional tests to

characterize the mechanical constants of composite materials are presented based on the type

of loading.

1.2.1. Tension and Compression

1.2.1.1 Tensile testing of flat specimens

The stresses are calculated based on the loaded section and the load. The failure mode in this

type of tests can vary according to the angle between the loading direction and the fiber

angles. For example, for unidirectional fibers the failure mode can change from the rupture of

fibers (loads are parallel to the fibers direction) towards a failure perpendicular to the fibers

(transverse test) [2].

Standards have been defined the specimen for the tensile characterization test (ASTM D3039

and ISO 527). For high strength composites, it is necessary to use tabs (such as glass

reinforced composites, aluminum or veneer) bonded on the clamp sections in order to avoid

the failure of the material in the grips. The bonded area must be large enough to reach the

failure of the composite before the adhesive shear failure. Figure 1.1-a shows the shape of the

specimen with the material properties identified based on this type of test.


5

1.2.1.2 Compressive testing of flat specimens

Different approaches have been defined on the way of loading the specimen for the

compression test. Loading the specimen with only compressive load was not efficient

because of the non-uniform stress distribution which leads to premature failure. The

specimen must be stable, especially in the gage section, to prevent the buckling problem. In

1975, another way has been defined (ASTM D3410-16) for the compression tests on flat

specimens by applying shear load using the grips in a special fixture. Some studies were

carried out in order to compare the compressive strength of the material obtained from both

ways of loading. It was concluded that loading the specimen at the end gave higher

compressive strength results compared to the one loaded by shear [3]. A third method was

also proposed as the combination of the normal forces at the end of the specimen with the

shear forces applied along with the faces of the specimen. The identified material parameters

are shown in Figure 1.1-b.

Figure 1.1 (a) Tension and (b) Compression of flat specimens [2]

1.2.2. Shear test

There is no well-adapted testing method for measuring shear properties especially the shear

strength. Nevertheless, the main tests found in the literature are presented hereafter.

1.2.2.1 Torsion of tubes

In this test, both shear modulus and strength can be measured. An applied moment on the

tubular specimen generates tangential stresses around the circumferences and the length of

the specimen. In this method, the shear strength (𝜏𝑥𝑧𝑢 ) and shear modulus (𝐺𝑥𝑧) of the material

are obtained as shown in Figure 1.2-a. However, it was stated that tubular composites are

expensive and difficult to fabricate and moreover doesn’t well represent the component in

real life applications [3].


6

1.2.2.2 Iosipescu shear test

This method is one of the most common methods for direct shear testing where a double v-

notched specimen is loaded by two opposite forces to create a pure shear ([4]–[6]). In this

method, the shear strength (𝜏𝑥𝑦𝑢 ) and shear modulus (𝐺𝑥𝑦) of the material are obtained as state

shown in Figure 1.2-b. The method was initially developed for isotropic materials and then

adopted to composite materials by Adams and Walrath [7]. It was noted that a shear stress

concentration was generated at the root of each notch. This stress concentration was reduced

by changing the notch angle and rounding the bottom of each notch [5].

1.2.2.3 Double-notched shear test

This method was primarily used because of the simplicity of the specimen fabrication and the

use of the same fixture for all the tests. A defined specimen (according to ASTM D 3846-08

[8]) with two notches is loaded in a compressive state, the shear strength (𝜏𝑥𝑧𝑢 ) is obtained by

dividing the compressive load with the shear loaded zone (Figure 1.2-c). Premature failure

could occur in this type of specimens because of the stress concentrations. In this test, shear

modulus cannot be calculated because shear strains cannot be measured [3].

1.2.2.4 45º laminate shear test

A simple rectangular specimen with a fiber orientation of 45º with respect to the applied load

direction, as shown in Figure 1.2-d, is simply used [9]. The applied load induces a shear

stress state parallel to the fibers direction. The shear modulus and strength are calculated

based on the amount of the applied load and the measured strains using classical laminate

theory.

Figure 1.2 Conventional shear tests: (a) torsion of tubes, (b) Iosipescu, (c) double notched, and (d)

off-axis [2]


7

1.2.3. Bending or Flexural test

Bending tests are also one of the popular testing methods because of the simplicity of both

test and specimen shape. It was first used on isotropic metals. For the composite materials,

bending tests still have problems caused by the non-uniformity of the stress distribution in the

specimen. Bending of the beam is usually subjected to a combination of all shear, tension and

compression states (tension on the top surface, compression on the lower surface and

interlaminar shear in the middle) [3]. The most known type is the three point bending test

(Figure 1.3). The bending shear modulus, Young modulus, and the strengths can be obtained

from this test. The bending stresses distribution is in a triangular form where the stresses

increase linearly from the ends of the beam towards the middle where maximum bending

stresses are reached.

Figure 1.3 Three point bending test [2]

1.2.4. Other purposes to perform uniaxial tests

The previously presented conventional tests were also used by different researchers in order

to characterize the effect of the strain rate and temperature on the behavior of the composite.

Hsiao and Daniel [10] studied the effect of strain rate on the strength and modulus of

unidirectional and cross-ply carbon/epoxy composite specimens in compressive and shear

tests. The results show an increase of the transverse compressive strength and modulus with

the increase of the strain rate while the ultimate strain shows no strain rate effect. However,

the longitudinal compressive modulus increases slightly with the increase of strain rate while

the strength and ultimate strains show a significant increase up to 79% and 74%, respectively.

Considering the shear test, off-axis compression tests have been made for two different

angles (30° and 45°), both tests show an increase of the shear strength with the increase of the

strain rate.

Gilat et al.[11] shows the effect of the variation of the strain rate on the material stiffness and

strength of IM7/977-2 carbon/epoxy composite with different layups (90°, 10°, 45° and

[±45°]s. Gilat used a hydraulic machine for low and moderate strain rates and a Hopkinson

bar for the high strain rates. He shows that increasing the strain rate leads to a stiffer material,

and he observed high maximum stresses (associated with deformations) at high strain rates.


8

Shokrieh and Omidi [12] studied the behavior (tensile strength, failure strain and absorbed

energy) of unidirectional glass/epoxy composites under quasi-static and dynamic strain rates.

In order to make the fracture in the center of the uniaxial specimen, a thickness reduction was

made by gluing woven glass/epoxy tabs on the extremities. It was shown that the longitudinal

lamina properties increase with the increase of the strain rate. According to his experimental

results, the tensile strength, tensile modulus, strain to failure and absorbed failure energy

increase with of 52%, 12%, 10% and 53%, respectively. A visual inspection of the failed

specimens shows that the strain rate has an effect on the failure modes of the composite

where a high strain rate test causes a failure through the whole gage section while at low

strain rates, the failure was in a small portion of the surface (Figure 1.4). Also, Shokrieh and

Omidi [13] made a study on the mechanical shear properties of glass/epoxy composites by

performing off-axis uniaxial tensile tests on ±45° laminates. The stress-strain data obtained in

the off-axis are transformed to the on-axis coordinate system using transformation equations.

The results show an increase of the shear strength of approximately 37% with the increase of

the strain rate (from quasi-static to dynamic shear test). While, the shear modulus and shear

strain at failure decrease with the increase of the strain rate.

Figure 1.4 Shokrieh failed specimens under different tensile rates: (a) 0.0017 s-1, (b) 0.55 s-1, (c) 5.6 s-

1, (d) 46 s-1 and (e) 85 s-1[12]

1.3. Biaxial testing methods

In real life, composite materials are often subjected to complex loadings in which two or

three-dimensional stress state exists. For an efficient design of composites, tests must

represent these real loading conditions. The goals behind performing biaxial test will be

presented briefly based on the literature. The first purpose of biaxial testing is to determine

the ultimate stresses of the material which represent the stresses experienced during actual

service conditions. The ultimate stresses obtained will generate the biaxial failure envelop of


9

the material which is important to validate and clarify the precision of some failure theories

[14]. However, the biaxial tests are not only used to validate failure criterion, another purpose

to apply biaxial tests is the study of crack. Rashedi et al. [15] studied the notch sensitivity on

cross-ply GFRP laminates. Also, biaxial testing is an interesting way to identify the material

properties (Young modulus, shear modulus, Poisson ratios …) by performing a single test

instead of multiple uniaxial tests [16]. Moreover, Fuji and Lin [17] studied the fatigue

behavior of plain woven glass composite subjected to biaxial tension/torsion loading for

different biaxial stress state.

1.3.1. Biaxial testing techniques

In the literature, different biaxial tests have been developed in order to study the behavior of

metals and composites. Researchers had shown different ways to introduce a biaxial stress

state in the material ([18], [19]). It can be divided into two main categories: i) the first one is

by directly introducing the biaxial stress state using independent loading systems (e.g.

combined axial loading with torsion or pressure on tubular specimens, tests on cruciform

specimens using independent actuators), and ii) the second one is by generating a biaxial

stress in a non-direct way using a specific geometry or special fixture (e.g. Iosipescu test,

bluge test, off-axis tests) [20]. In this section, different biaxial testing methods will be

presented and discussed separately.

1.3.1.1 Biaxial flexure testing of plate

A circular or rectangular specimen is held on his edge and loaded at the center by an indenter

which is a circular bar. The biaxial stress state is generated but it’s not uniform and makes the

interpretation of the stress results complicated [21]. Zhang and Gibson [22] used an elliptical

loading ring instead of the central pin loading leading to a homogeneous stress state. The

stress calculation and the biaxial failure remains an issue (Figure 1.5-a).

1.3.1.2 Bulge test

It consists of deforming plates ([23], [24]) using hydraulic pressure. This test has an

advantage over flexure testing since it eliminates the friction between the indenter and the

tested plate. However, it contains different disadvantages such as only tension-tension can be

investigated, only thin specimens can be used to avoid the through thickness stresses due to

gripping, and non-homogeneous stress fields are obtained (Figure 1.5-b).


10

1.3.1.3 The off-axis method

Where a single uniaxial test is performed on a composite specimen rotated with a specific

angle to create a biaxial stress state. A problem of bending or shear could occur in this test

due to the fixed clamps. This method is generally used for the unidirectional composite

laminates and it has limited biaxial stress state (Figure 1.5-c) [18].

1.3.1.4 The Iosipescu test

It is frequently used for shear testing of materials. It was initially suggested for metallic

materials and then adopted to composite materials by Adams and Walrath [7]. Loading the

unidirectional composite specimen along the 0° represents a pure shear but loading it at other

angles between ±45° introduces a biaxial stress state. Pierron [25] showed that the

interpretation of failure in term of shear strength is erroneous since the stress distribution is

not homogeneous and failure occurs at the notch tip (Figure 1.5-d).

1.3.1.5 Biaxial bending moments on flat specimen

By applying a three-point bending test on a parallelepiped bar, a biaxial stress state is

introduced in the specimen. This method is still limited to test metals and unidirectional

composites and the stress generated is heterogeneous (Figure 1.5-e) ([26], [27]).

1.3.1.6 Combined axial loading with simple torsion on tubular specimen

This method has been used to investigate the biaxial behavior of metals in 1940’s and then

composites in 1960’s. The biaxial state is created by rotating the principal directions 45° with

respect to the cylinder axis, two perpendicular normal stresses in the principal directions

exist. Biaxial extensometer is used to measure the axial and rotary displacements during the

test (Figure 1.5-f) ([28], [29]).

1.3.1.7 Combined axial loading with internal pressure on tubular specimen

Most of researchers switched from axial/torsion to axial/internal pressure because of the

limited amount of biaxial ratio in the first one. Tubular form avoids the edge effect which

was a problem in the coupon specimens ([30], [31]). However, tubular specimens have

drawbacks such as stress concentration near the end of the grips and premature failure occurs

there. Also in compression tests, buckling occurs in thin walled tubes. Increasing the

thickness of these tubes would solve the problem of buckling but makes it difficult to

determine the stresses in the specimen because of the generation of radial stresses (through

thickness caused by internal/external pressure) (Figure 1.5-g).


11

1.3.1.8 In-plane biaxial loading on flat cruciform specimen

This method consists of applying two in-plane perpendicular loads on the specimen to create

a biaxial stress state (Figure 1.5-h). This method was used first time in 1960’s to study the

biaxial behavior of metals [32] and started on composites in 1980’s. First in-plane biaxial

tests were performed on rectangular specimens. The shape was quickly changed to cruciform

to avoid stress concentration near the grips and lead to a failure in the center. The biaxial

stress state is directly controlled by the machine. The drawback of this method is the design

of the specimen which requires a stress concentration in the biaxially loaded central zone of

the specimen. Moreover, this method requires specific equipment that does not exist in every

lab [33].

Figure 1.5 Biaxial testing techniques: (a) Biaxial flexure testing of plate [21], (b) Bulge test [24], (c)

off-axis test [2], (d) Iosipescu test [34], (e) Biaxial bending moment on flat specimens [27], (f) axial/

torsion of tubes [35], (g) axial/pressure of tubes [36], and (h) cruciform specimen [37]

1.3.2. Choice of an adapted biaxial testing technique

According to the previously mentioned biaxial testing methods, problems such as stress

heterogeneity, limitation of the biaxial test on specific composite layups (unidirectional) and

on specific biaxial stress ratio are shown.

In order to select the appropriate biaxial method some requirements must be taken into

account:

1- Failure must occur in the biaxial zone.

2- Stress must be homogeneous for an accurate determination of stress and strain.


12

3- Test must cover a wide range of stress state in order to obtain the complete failure

envelope.

The only two methods that fulfill the requirements are tests on tubular specimens and test on

cruciform specimens. Different researchers show that the biaxial test on tubular specimens

produces a dominant effect of through thickness stresses which limits the characterization of

composite materials for tubular applications (e.g. pipes, vessels...) ([38], [39]). However,

since most of the engineering applications require the use of flat composites (e.g. aeronautics,

automotive…), the in-plane biaxial test is a well-adapted characterization method. Another

advantage of the in-plane biaxial test is the simplicity of varying the biaxial stress ratio in the

tested material according to the well-controlled displacements imposed on the two axes of the

cruciform specimen.

As conclusion, different biaxial test techniques exist in order to study the behavior of

materials. Regardless the design problem of cruciform specimen, the in-plane biaxial test is

one of the most appropriate method.

1.3.3. Biaxial testing machines on cruciform specimens

The in-plane biaxial testing machines can be classified into three categories: (i) machines

with one actuator, (ii) machine with two actuators, and (iii) machine with four actuators. The

machines with one actuator are developed in a way that introducing a single loading leads to

a biaxial stress state in the cruciform specimen. This type of machines needs a very specific

fixture. Bhatnagar et al. [40] developed a biaxial testing fixture using a single loading

actuator for polymer materials (Figure 1.6). The mechanical configuration of the machine

allows a single load cell to introduce different biaxial testing ratios in order to generate any

combination of σ1 – σ2 stress space. The goal behind using a single loading actuator is the

low cost compared to the multi-actuator machines. An experimental validation was made

using the machine on a short fiber polyamide and nanoparticle-reinforced PP thermoplastic

composites.

Figure 1.6 Biaxial testing machines with one actuator developed by Bhatnagar [40]


13

The machine with two actuators was developed by introducing another loading actuator

perpendicular to the one of a uniaxial test bench. The development of this machine was in

order to vary the biaxial stress ratio without having a special equipment. The biaxial stress

state was generated by fixing one side of the perpendicular arms and making the traction

from the other side. However, the problem of this kind of machines was the bending of the

specimen which gives undesirable stress state. In 2000, Hoferlin et al. [41] made a biaxial

testing machine from a uniaxial machine (Figure 1.7-a) with a capacity of 250 kN by adding

a removable hydraulic actuator having a capacity of 25 kN. The machine was tested on steel

cruciform specimen. Moreover, Lin and Ding [42] also added a manual actuator to a uniaxial

instron machine in order to study the plastic yielding of aluminum metals under biaxial test.

Lin and Ding mentioned in their study that the specimen bending can be accommodated

vertically using the springs shown in Figure 1.7-b while the horizontal movement isn’t

accommodated.

Figure 1.7 Biaxial testing machines with two actuators developed by: (a) Hoferlin [41] and (b) Lin

[42]

In order to have more reliable tests (avoid bending problems), the biaxial testing machine

with four hydraulic actuators was built. These machines usually contain a closed-loop control

system to balance the loads in the four arms and adjust the specimen in the center during the

test. Two types of machine frames were built in the literature, machines with horizontal

frames and machines with vertical frames. Makinde et al. [43] built a biaxial testing machine

with a horizontal frame. To ensure a maximum rigidity of the proposed frame, a 152 mm

thick cross-shaped steel slab was welded to an I-beam structure. The benefit of using a

machine with a vertical frame is the ease of inspection of both sides of the specimen.

Most of the researchers used machines with horizontal frame. Green et al. [44] used the

machine developed by Makinde (Figure 1.8) for the characterization of aluminum alloy

AA1145, the machine capacity is 250 kN in each loading arm. Gower and Shaw [45] used a

biaxial testing machine with a capacity of 50 kN for the characterization of carbon fiber-

reinforced plastic laminates under tension/tension loading. Some researchers used the triaxial


14

testing machine for the biaxial characterization tests. The triaxial testing machine (used by

Welsh et al.[46]–[49]), as shown in Figure 1.9, has a capacity of 94 kN in each loading arm,

it can generates any combination of tension or compression stresses in σ1 – σ2 – σ3 stress

space. Another triaxial testing machine was developed by a collaboration between Schenk

and LMT Cachan in France, the machine is composed of six hydraulic actuators where the in-

plane machine capacity is a 100 kN (tension or compression) while the out-of-plane loading

can reach 250 kN [50].

Figure 1.8 Biaxial testing machines with four actuators developed by: (a) Makinde/Green [44] and

(b) Gower [45]

Figure 1.9 Tri-axial testing machines developed by: (a) Welsh [46] and (b) ASTREE [50]

1.4. In-plane biaxial test on cruciform specimens

1.4.1. Metal cruciform specimen

Considering biaxial tests on metals, different cruciform specimens were proposed by several

researchers. Depending on the mechanical behavior to be identified (fatigue strength, forming

or rupture limits, hardening law, yield surface …), several flat specimen shapes have been

proposed during the last years ([51]–[53]). Hannon and Tiernan [54] presented two main

requirements for the design of the cruciform metal specimen for the characterization test:

homogeneous stress/strain distribution which allows the stress calculation and yield should


15

occurs in the central zone of the specimen. To avoid the premature failure in the arms and

reduce the effect of shear stresses, three methods were used in the design of the specimen

[55]: (i) making a curved fillet between the loaded arms, (ii) reducing the thickness of the

central zone, (iii) adding slits in the arms (Figure 1.10). Specimen could also have a

combination of these methods as shown in Figure 1.11 (combination of slits and thickness

reduction). In 1993, Demmerle and Boehler [56] has optimized the cruciform shape in order

to achieve more homogeneous stress and strain fields in the central zone of an anisotropic

specimen. The optimization procedure was taken by defining a cost function C that should be

minimal. Twelve parameters (representing the dimensions of the cruciform specimen) were

used as input in the optimization process. Optimization process should be repeated if the

thickness or the material parameters change.

Figure 1.10 Methods used in the design of cruciform specimen [55]

Figure 1.11 Cruciform specimen proposed by Hayhurst [57]

Dawicke and Pollock [58] proposed a variation of the cruciform specimen for the

characterization of the behavior of 2219-T87 aluminum alloy. The proposed specimen is

composed of series of the thickness reduction towards the central zone. The arms was the


16

thickest at 0.25 inch. The first thickness reduction is in a square form (0.125 inch thick). The

second thickness reduction is in a circular form leading to a 0.8 inch thick central zone

(Figure 1.12). Johnston et al. [59] used the same specimen to study the behavior of 2195

aluminum lithium alloy. The stresses and forces at the elastic limit were obtained on this type

of specimens. The strain distribution has been measured by gluing 52 strain gages in the

central zone.

Recently, Xiao et al. [37] defined a cruciform specimen without slits in the arms in order to

obtain the forming limit diagram of TA1 titanium alloy under different thermal conditions

(Figure 1.13). The specimen has a convex corner fillet between the arms and a thickness

reduction in the central zone. The thickness reduction was made from only one side of the

specimen. It was shown that at high temperatures, the limit strains were affected by the

orientation of the load with respect to the rolling direction.

Figure 1.12 Specimen with two thickness reductions [59]

Figure 1.13 Cruciform specimen without slits [37]

More, Yu et al. [60] developed a new specimen form in order to determine the forming limit

curve under different linear and non-linear strain paths. The optimized form has also two


17

thickness reductions (Figure 1.14-a). The study was based only on finite element method

without performing an experimental validation. Zhang and Sakane [61] presented another

specimen (Figure 1.14-b) with two thickness reductions in order to study the creep-fatigue

damage and life of type 304 stainless steel under low strain levels.

Figure 1.14 Cruciform proposed by (a) Yu [60] and (b) Zhang [61]

Furthermore, an optimal cruciform specimen of Aluminum alloy AA5086 was numerically

proposed by Zidane et al. [52] in 2010 (Figure 1.15). The proposed specimen has slits in each

arm and a radial transition between arms to reduce the load sharing between them. Also, slits

can reduce the influence of the arm size on the uniformity of stress in the central zone. Note

that, they reduced the cross section area of the arms which lead to a stress concentration at the

tip of slits. Since stress concentration would be in the slits, failure would occur in the arms

and not in the central zone if only slits exist. In order to solve this problem, the central zone

has a reduced thickness with a circular form. The reduction of thickness was made with a

radial transition. A comparison between the experimental biaxial test and the conventional

Marciniak test was made and good agreement is observed between the two forming limit

curves obtained.

Figure 1.15 Zidane optimized specimen geometry [46]


18

Liu et al. [53] studied the hardening behavior of metallic sheets under large strains. An

optimization of the specimen has been proposed. The specimen, as shown in Figure 1.16, has

a thickness reduction in the central area with slots in each arm and a notch between two arms

(since notches are more efficient than radii). An inverse method was applied to determine the

material constants for the hardening law. A comparison was made between the three yield

criteria: (i) isotropic Mises, (ii) anisotropic Hill48 and (iii) Bron and Besson. A modified

Voce’s hardening law is identified with these three yield criterions. Bron and Besson is the

criterion that gives the best agreement between the experimental and the numerical strain at

the central point of the specimen.

Figure 1.16 Liu optimal specimen

Liu et al. [62] also shows the influence of the strain rate on the mechanical behavior of the

dual phase high strength steel specimen (DP600). A formulation taking into account the strain

rate dependency on the initial yield stress and strain hardening of the material is established.

1.4.2. Composite cruciform specimen

The previous literature review on cruciform metal specimens shows that it is not

straightforward to define an appropriated cross specimen shape. For composite materials, the

study is even harder because of the different stress states which develop in each lamina,

leading to a difficult understanding of the specimen behavior. Unlike the metals, composites

are sensitive to stress concentrations even for very low strain level. So, particular attention

must be taken when designing a composite specimen to limit the stress/strain concentrations.

The slits in cruciform specimens arms are used in order to obtain a uniform stress field in the

central zone of the specimen [44]. In composite materials, a high stress concentration would

occur at the slits because of their sensitivity to milling process. However, some authors use

slits in composite cruciform. For example, Youssef [63] used slits to achieve a more uniform

stress field and to eliminate the coupling effect between arms on cross-ply carbon/epoxy


19

specimen under biaxial loading (Figure 1.17-a). In 2000, Ohtake et al. [64] also used slits on

cross-ply fiber reinforced carbon cruciform specimen under biaxial tensile test in order to

investigate the biaxial deformation behavior at high temperature (1000 and 1700° C) (Figure

1.17-b).

Figure 1.17 Composite cruciform specimens with slits developed by: (a) Youssef [63]and (b) Ohtake

[64]

The effectiveness of a successful biaxial test is related to different criteria. The most

important criterion is the ability of the test to accurately determine the needed property. There

is no single test method that has generally accepted by the composites community as being

superior to all others. Some indications of a successful test are noted:

a- The specimen gage section must be in a uniform stress state as far as possible.

b- Failure must occur in the gage section.

c- Repeatable results must be obtained.

d- Possibility of independently vary the biaxial stress components in order to generate a

complete biaxial failure envelop.

Similar to metals, another way, to improve the results and to obtain the failure in the central

zone, is to reduce its thickness and to use a curved fillet corner between the arms to avoid

load sharing. In the literature, a strong challenge was encountered in the thickness reduction

of the composite because of the sensitivity of the fibers to the machining effects. Some

authors suggested to reduce the thickness of the central zone by sandwiching different

composite plates with holes in the center on both sides of the studied composite in order to

have a low central zone thickness [65]. Other researchers reduced the composite thickness by

milling a number of plies from both sides of the specimen using a CNC machine [47]. Both

methods, sandwiching and milling, will be presented in this section. Moreover, comparison

between the failures results using both thickness reduction methods will be also discussed

[66].


20

1.4.2.1 Sandwiching method

The sandwiching method was successfully tested on large thick specimens which requires

high loading capacity machines. Using this concept, Hopgood et al.[67] obtained the failure

envelope of open-holes carbon fiber reinforced epoxy by performing biaxial tests (machine

capacity of 500 kN). The specimen was made of 2 mm thick carbon/epoxy composite

sandwiched between two layers of glass/epoxy composite (Figure 1.18-a). Williamson et al.

[65] also studied the biaxial failure of open-hole carbon/epoxy cruciform specimen using the

same specimen developed by Hopgood. The specimen was tested under different biaxial

tension and compression ratios in order to cover the whole failure envelop of the material.

The specimen thickness reduction was also made by adding two layers of 4 mm thick quasi-

isotropic glass/epoxy composite on both sides of the carbon composite (Figure 1.18-b). In

this study, different hole diameters, specimen thickness and layup orientation were

investigated. The biaxial testing machine used in this study has a capacity of 1500 kN. The

maximum load reached was 1124 kN for a 10 mm thick carbon specimen with a 6 mm hole

diameter. Kumazawa and Takatoya [68] studied the biaxial strength of a cruciform specimen

with a rounded central zone (designed by Susuki [69]). The specimen has a smooth transition

from arms to central zone (Figure 1.18-c). Glass fiber reinforced epoxy tabs were glued on

both sides of the quasi-isotropic carbon/epoxy laminates under biaxial tension/tension and

tension/compression tests. The thickness of the CFRP is 1 mm. A maximum load, around 140

kN, is obtained for the equibiaxial tensile test. The authors mentioned that the failure occurs

in the center of the specimen and damage was developed towards the GFRP tabs (Figure

1.19).

Figure 1.18 Cruciform specimen presented by: (a) Hopgood [67], (b) Williamson [65], and (c)

Kumazawa [68]


21

Figure 1.19 Damage evolution in the cruciform specimen [68]

On the other hand, Escarpita et al. [70] studied the biaxial failure strength of unidirectional

carbon fiber/8551-7 epoxy composite. The study starts with the comparison between two

shapes where the first one is made with an adapted corner between the loading arms without

thickness reduction. The second shape was prepared by sandwiching plates with a square hole

in the central zone on the studied composite. The stacking sequence of the total composite

thickness is [(0/90)3/0̅)s while the central zone is only made of 0° fibers. Based on this

comparison, a third enhanced specimen shape was defined as the optimal form; the shape has

a rhomboidal central zone form and a fillet corner between the arms. The optimal specimen

was defined to decrease the shear strains in the arms and to avoid delamination. However,

another shear strains appear on the edge of the central zone (Figure 1.20).

.

Figure 1.20 Specimens tested by Escarpita [70]: (a) constant thickness, (b) square central zone, and

(c) rohmboidal central zone

Torres and Maji [71] studied the strength of [(0/90)8]s 6.35 mm thick woven carbon/epoxy

composite. The shape of the specimen with a curved fillet between arms and a square central

gage section was based on the study of Welsh and Adams [47]. Three different specimens

were designed. The first specimen (type I) is a flat cruciform specimen with aluminum shims

added on the corners in order to make a thickness reduction towards the gage section as

shown in Figure 1.21. The type II specimen was developed by gluing 2.36 mm thick G10

glass/epoxy as tabs. The gluing process was made by placing an adhesive film (Hysol) in

between the composite and the G10 and curing it with the composite laminate. The type III


22

specimen was similar to type I with a hole in the center in order to study the influence of the

hole on the composite strength. In the type III specimen test, additional [(0/90)4(+45/-45)4]s

carbon fiber composite was studied. Results show a premature failure for the specimen with

aluminum shims around the central zone. The rupture is due to the debonding of the

aluminum plates occurring before the failure of the composite. Considering the specimen

type II, Torres mentioned that the failure occurs in the central zone and propagates towards

the left arm of the specimen and leads to an arm fracture (Figure 1.22). The comparison

between the strength results with and without hole using specimen type III shows that the

hole decreases the strength of the [0/90)8]s composite by 30% and has no effect on the

specimen with [(0/90)4(+45/-45)4]s layup which explains that the layup sequence influences

the stress concentration in the material.

Figure 1.21 Cruciform specimens with: aluminum shims (Type I) and G10 tabs (Type II) [71]

Figure 1.22 Failure of all three specimen types [71]

In 2017, Correa et al. [72] proposed a cruciform specimen for the biaxial transverse tests of

AS4/3501-6 carbon/epoxy unidirectional composite where the fiber direction is perpendicular

to the plane of the cruciform specimen. A comparison between the different cruciform shapes

is performed in order to select the one reaching a high stress level in the gage section

compared to the stress level in the arms. The four specimens evaluated are shown in Figure

1.23. The main difference between these specimens is the fillet radius between the arms. In

these specimens, the thickness reduction was made only in the arms by adding ±45° glass


23

fabric to the carbon (Figure 1.24). The potential of these 4 geometries is evaluated by means

of FE simulations using commercial software Patran/Nastran. The comparison is based on

two criteria: (i) the concentration factor, which is the ratio between the maximum value of σxx

at the free edge and the σxx value at the centre of the specimen; and (ii) the maximum stress in

the expected zone of failure of the four geometries tested the specimen D seems to be the

most effective.

Figure 1.23 Four specimen shapes studied by Correa et al. [72]

Figure 1.24 3D geometrical model proposed by Correa [72]


24

The behavior of this specimen has been improved by: i) making the thickness reduction

towards the central area and not only in the arms (Figure 1.24 shows how the thickness

reduction was just before the fillet between the arms and not directly in the central rea), ii)

using carbon tabs instead of glass while increasing the tabs thickness and iii) changing the

transition between arm and central zone to a spline instead of constant slope area.

1.4.2.2 Milling method

Considering the second method of thickness reduction, authors developed different

specimens by milling the composite top and bottom laminates. Starting with the known

Welsh cruciform specimen, Welsh and Adams [47] have proposed an optimal cruciform

specimen of IM6/3501-6 Carbon Epoxy cross-ply laminates (Figure 1.25). The specimen

composed of thirty-six laminae with a thickness of 5.1 mm is milled in the gage section to

reach a thickness of 2 mm with a [0/90]10s stacking sequence. The influence of the size of the

radius at the corner between the arms and the shape and the size of the central zone (square or

circular) is analyzed. The specimen with a large corner fillet between arms and a small square

gage section with large radius is selected based on the stress results. The σ1-σ2 stress space is

generated in order to present the failure envelop of the material. Unidirectional, cross-ply and

quasi-isotropic laminates of carbon fibers under tension/tension, tension/compression and

compression/compression are tested. Bhatnagar et al.[40] used the same specimen for fiber

reinforced plastics tensile testing.

Figure 1.25 Cruciform geometry proposed by Welsh and Adams [47]


25

In 2007, Welsh [48] compares the experimental and numerical failure predictions of both

cross-ply [0/90]s and quasi-isotropic [0/90/±45]s carbon composites biaxially loaded. The

quasi-isotropic specimen is designed using [(0/90)4(0/45/-45/90)2]s laminate configuration.

The [0/90]4 are milled away (using a CNC machine) from both sides of the specimen in order

to reach a stress concentration in the gage section. A bypass correction factor is defined as the

amount of load that bypasses the gage section (problem of load transfer between adjacent

loading arms). Specimens are loaded at a rate of 1.27 mm/min. The micromechanical failure

MCT theory (Multi-Continuum Theory), which is a modified Hashin failure criterion ([73],

[74]), is used to predict the failure of the specimen. A comparison between the numerical

failure envelop generated by the MCT failure criterion and the experimental results is

presented where a great agreement was shown for cross-ply composites while errors are

presented for quasi-isotropic, the errors were explained by the introduction of an in-plane

triaxial stress state due to the shear stresses obtained by the ±45° plies.

Smits et al. [75] compare four different glass fiber reinforced composite cruciform shapes

with a [(±45/0°)4/±45°]T layup to choose an optimal one. The thickness reduction was made

by milling (±45/0) plies from both sides of the central zone (Figure 1.26). Smits performed a

numerical study on the shape of the corner fillet between the loaded arms of the cruciform

specimen. An experimental validation (using digital image correlation) shows that the

specimen with adapted corner fillet from outside the arms towards inside gives better strain

distribution in the central zone. They showed that an increase of the area of the central zone

causes delamination before failure (Specimen D – Figure 1.27). The optimal specimen (C) is

verified by following strain field with respect to the relative position in gage section (Figure

1.28). The strain is almost constant and the failure is obtained in the central zone. The

selected shape is tested under different biaxial tensile ratios in order to obtain the failure

envelope of the material. Makris et al. [20] compare the results obtained on Smits optimized

specimen with different failure criteria. The first ply failure prediction is used. The

comparison shows that Tsai’s criterion is more accurate compared to other predictions such

as Max Stress and PUCK criterion.


26

Figure 1.26 Through- thickness dimensions of Smits specimen [75]

Figure 1.27 First principal strain results for the four different geometries studied by Smits et al. [75]

Figure 1.28 Strain variation with respect to the position in the central zone [75]


27

In 2009, Makris et al. [76] studied the failure of the cross-ply carbon epoxy cruciform

specimen under quasi-static biaxial test. The layup used in their study was [(90/0)2, (0/90)2]s

where one group of [(90/0)2] was milled away in order to reach the failure in the central

region. The digital image correlation technic (DIC) has been used for the biaxial strain

measurements. Authors used a progressive damage modeling to predict the failure of the

composite. Hashin failure criterion is used for the matrix, fiber compression and delamination

failure modes, while the Maximum Stress is adopted for the fiber tension failure. A good

correlation is obtained between experimental and numerical strain field results in the central

area (Figure 1.29).

Figure 1.29 Experimental (top) and numerical (bottom) strain field results under different loading

conditions.

Makris et al. [77] optimized the cruciform specimen shape by means of a cost function based

on two different ratios: (i) failure ratio (FR) which is the maximum deformation outside the

central zone over the maximum deformation inside the central zone; and (ii) coefficient of

variation (COF) which defines the heterogeneity of stress inside the central zone. Both ratios

should be minimal to obtain an optimal form. Two different shapes are tested, one has a

circular central zone and the other has a rectangular form. Makris et al. showed the x-

direction strain distribution (𝜀𝑥) for the specimen with square form before and after

optimization (Figure 1.30).


28

Figure 1.30 𝜀𝑥 strain distribution of the optimized form (a) and the commonly tested case (b)

1.4.2.3 Comparison between both methods

From the previously presented specimens, some studies have compared the results obtained

from the both thickness reduction methods. Ramault et al. [66] use two different composite

materials: the first is a glass/epoxy composite with a [(±45/0°)4/±45°] layup while the second

material is carbon/epoxy with a [(90/0)2 (0/90)2]s layup. The tabbing method is based on

gluing composite tabs (with a hole in the center) on both side of a composite plate (without a

hole) after doing a surface preparation. The results show that the failure stresses of the tabbed

specimen are low compared to the milled specimen. This is explained by the premature

failure in the tabbed specimen caused by the debonding of the tabs due to a failure in the

glue. Moreover, the study shows that reversing the tabs (i.e. reversing the stacking of the tabs

as shown in Figure 1.31) gives a better result since it reduces the interlaminar shear strains

between the upper ±45° lamina of the central laminate and the lower 0° lamina of the tab.

However, premature failure still occurs in the adhesive and failure occurs outside the central

zone (Figure 1.32).

Figure 1.31 Deformation results for specimens with normal and reversed tabbing


29

Figure 1.32 Premature failure of composites fabricated with tabbing method

Ramault study [66] was based on the idea of milling the material would affect the mechanical

properties of the composite due to the machining process. This could generate a stress/strain

concentration in the milling transition. This is mentioned by Lamkanfi et al. [78] who studied

the effect of geometrical discontinuities (generated by milling) on the strain distribution in

the specimen. A strain concentration is noticed between the milled central zone and the un-

milled surrounding area (Figure 1.33). The results show that milling the specimen has a major

influence on the premature failure of the composite specimen where a crack always appears

at the transition zone regardless the type of the load and the loading direction.

Figure 1.33 Crack initiation in the milled layers for (a) x and (b) y-direction loading

1.5. Conclusion

Based on the literature review, the in-plane biaxial test on cruciform specimen is adapted to

achieve the objectives of this thesis which is the characterization of the failure of composites

subjected to different biaxial loading conditions in order to obtain the failure envelop of the

material. At the same time, different biaxial testing methods existent (e.g. Bulge test, test on

tubular specimens ...). However, some of these tests cannot obtain a complete biaxial

envelop, while others cannot generate a homogeneous stress state in the material which

makes the analysis of the results more complicated. In this study, the biaxial test on cruciform


30

specimen is used and the tests are realized using a machine with four independent hydraulic

actuators to impose different loading conditions.

In addition, the literature study shows that an issue for a successful biaxial test is the design

of the cruciform specimen. A significant amount of researchers put a lot of effort in the

design of the metal cruciform specimens in order to study different mechanical behaviors

subjected to biaxial loadings such as fatigue, forming limit curves, hardening, anisotropy,

yield surfaces.

Numerous studies have also been conducted for the characterization of the mechanical

behavior of composites subjected to biaxial stress state. The sensitivity of the composites to

stress concentrations, even for very low strain level, and the strong anisotropic behavior made

their characterization more complicated. The two major difficulties in defining the cruciform

specimen are: (i) to locate the maximum stress in the central area, so that the loading paths

leading to the failure can be easily controlled; (ii) to obtain as homogeneous a stress field as

possible in this area in order to simplify the assessment of the stress level. However, a lot of

restrictions must be taken into account when designing the composite cruciform specimen.

According to the previous studies, the stress concentrations generated on the corners and the

thickness reduction transition are the main causes of premature failure in this type of tests.

The design of a composite cruciform specimen that fulfills the requirements for a successful

biaxial test remains an open question. The next chapter will present a complete numerical

study about the design of the cruciform specimen used in this thesis including the influence

of the geometrical parameters on the stress/strain results.

Chapter 2. Design of the Cruciform Specimen

31

Chapter 2. Design of the Cruciform

Specimen


32

2.1. Introduction

As mentioned in the literature part, the design of the composite cruciform specimen is not

straightforward. The choice of the shape of the specimen remains the main difficulty when

performing in-plane biaxial tensile test. The first aim of this chapter is to propose an adapted

cruciform shape for the in-plane biaxial tensile test. However, the specimens proposed in the

literature are made purely from composite materials where a thickness reduction is obtained

by either milling some plies from the central zone or sandwiching composite laminates with

holes in the center on both sides of the composite to be characterized. The literature review

presents different generic forms of cruciform specimens. In this chapter, a new concept is

numerically proposed by gluing two aluminum tabs on the top and bottom side of the

constant thickness composite. The aluminum tabs, obtained by milling, are preferred over the

composite for two reasons: (i) easy of obtaining accurate shapes for the transition zone,

between the specimen center and the arms, where a thickness reduction is needed, (ii) avoid

the risk of delamination in the tabs, when the thickness reduction in the composite is obtained

by machining. A comparison between different cruciform shapes generated by this method is

numerically investigated using Finite Element analysis in order to select the adapted shape

according to the stress and strain distributions. Based on different experimental constraints, a

study on different geometrical parameters (e.g. arm width, central zone radius, etc...) is

presented in order to obtain a final design of the specimen that respects those constraints.

Finally, the designed specimen is tested under different biaxial tensile ratios in order to

validate the failure of the composite in the central zone regardless the ratio of loads applied

on each arms.

Different limitations or constraints must be taken into consideration regarding the dimensions

of the specimen. These constraints related to the experimental setup must be integrated in the

design of the specimen. They are cited and explained:

a. Machine capacity: The most important constraint related to the specimen design is

the capacity of the machine. In this work, a machine of maximum capacity of 50 kN

for each loading arm is used (more details will be presented in Chapter 3). The failure

stresses must be obtained in the composite before reaching the maximum capacity of

the machine. The capacity of the machine is directly related to the arm cross-section

which consists of the arm width and thickness (thickness of the composite and the

aluminum tabs). Increasing one of these components would increase the load required

to reach the failure of the specimen.


33

b. Central zone size: A high-resolution camera is used to acquire a sequence of images

from the test. The images are post-treated using a Digital Image Correlation (DIC)

software in order to determine the strain fields at the specimen surface. A compromise

must be considered between the dimensions of the central zone size and the image

resolution.

c. Aluminum tabs fabrication: The last parameter that should be taken into

consideration is the thickness reduction of the aluminum tabs. A minimal thickness of

the aluminum tabs is fixed arbitrary at 1 mm during the thickness reduction in order to

obtain accurately similar aluminum tabs.

In this work, the composite 0° fibers are aligned with the x-direction as shown in Figure 2.1.

Figure 2.1 Fibers orientation with respect to x and y directions

2.2. Selection of the cruciform shape

This study will be divided into two parts. In the first one, different cruciform specimen

shapes will be presented according to the literature. In a second part, different shapes are

compared by means of a finite element numerical study performed with the commercial

ABAQUS software in order to select only the one(s) that fulfill(s) the different requirements.

2.2.1. Selected shapes

It is important to note, in this section, that the optimal specimen is selected as the one that

respects different requirements listed by Daniel et al. [79] such as: (i) failure most occurs in

the biaxial central zone for different biaxial loading ratios, and (ii) a homogeneous stress state

is required in the test section. Based on the literature, three different shapes have been

selected: Specimens A [75], B [47] and C [69]. The difference between these three shapes is

either the central zone form (circular or square) or the corner fillet between the arms.


34

In order to avoid machining the glass/epoxy composite laminate to be characterized, a

sandwich specimen solution consisting of two aluminum tabs, drilled in their central part, and

glued on each side of the constant thickness composite is chosen in this work (Figure 2.2).

To compare these different specimen shapes, the three main dimensions of the total length of

the specimen, the width of the arms and the hole diameter of the tabs are fixed at 250 mm, 30

mm, and 17 mm respectively. In this way, the influence of the shape and dimensions of the

central zone on the stress and strain distributions at the center of the specimen can be

assessed. For all these three specimens, the aluminum tabs present a tapered thickness

reduction in their central area (Figure 2.2).

Specimen A (Figure 2.3) and specimen B (Figure 2.4) have the same adapted corner fillet

between arms but different central form. Specimen A has a circular central zone while

specimen B has a squared one. Specimen C (Figure 2.5) has also a circular central zone but it

has a rounded corner between the cruciform arms. The corner fillet between the arms in all

specimens was performed in order to avoid the load sharing between the arms and redirect

the load towards the central zone. The central zone circular and square shapes are considered

because they are mostly used in the literature according to the homogeneity of the

strain/stress fields in the central zone.

Figure 2.2 Cruciform specimen parts disassembled


35

Figure 2.3 Cruciform specimen: shape A

Figure 2.4 Cruciform specimen: shape B

Figure 2.5 Cruciform specimen: shape C


36

2.2.2. Numerical comparison

In this section, a numerical comparative study using ABAQUS software has been made

between these three shapes. For the comparative study, the plain-weave glass/epoxy

composite M9.6GF/200T2/G is selected with a thickness of 1mm where M9.6GF stands for

the resin type, 200T2 represents the reinforcement and G means glass fibers. A more detailed

description will be presented in the third chapter regarding the used composites. Table 2.1

shows the M9.6GF/200T2/G material parameters obtained from simple uniaxial tests.

Table 2.1 M9.6GF/200T2/G plain-weave material properties

Material E1 (GPa) E2 (GPa) G12 (GPa) υ12 GFRP 22.8 22.8 4.1 0.14

Different aluminum grades and thicknesses were tested and will be presented later in this

chapter. For the following comparative study, the aluminum alloy AA2017 with a thickness

of 3 mm has been used. AA2017 is considered as isotropic material with a Young modulus of

𝐸 = 70 𝐺𝑃𝑎 and a Poisson ratio of 𝜈 = 0.3. The yield and ultimate stresses of this aluminum

alloy are 230 MPa and 534 MPa, respectively as shown in Figure 2.6.

Figure 2.6 True stress- plastic strain curve for AA2017 alloy

Because of the symmetry of the specimen, one-fourth of the specimen is modeled in order to

decrease the computational time (Figure 2.7). A displacement of 1 mm is applied at the

extremity of each loaded arm by coupling reference points with the loading arm, all the

degrees of freedom of the extremity of the loading arm are driven by the degree of freedom

of the reference points. The reference points are also used in order to obtain the evolution of

the applied load on each arm with respect to the displacement. For the symmetry, the edge of

the horizontal arm is blocked in the y-direction while the edge of the vertical arm is blocked

in the x-direction (Figure 2.7). For the meshing (Figure 2.8), a three-dimensional hexahedral

structured solid elements (C3D8R) have been used for both aluminum and composite mesh

type. The mesh is refined in the central zone of the specimen (area of interest). A rigid link


37

has been made between the aluminum tabs and the composite plate (without modeling the

adhesive behavior).

Figure 2.7 Boundary Conditions for the numerical modeling

Figure 2.8 Mesh and number of elements for each specimen

In order to compare the performance of all three shapes, the equivalent stress (Eq 2.1) and the

equivalent strain (Eq 2.2) are used since they take into part all the stress and strain

components in a single equation. These parameters were already used by previous researchers

to identify the zone where the failure occurs in cruciform specimens ([52], [80]). In addition,

the stress and strains in each direction will be considered later in order to analyze the results

when the specimen is subjected to different biaxial loading conditions. The equivalent

stresses and strains are presented in Eq 2.1 and Eq 2.2 respectively as:


38

𝜎𝑒𝑞 = 1

√2√(𝜎1 − 𝜎2)2 + (𝜎1 − 𝜎3)2 + (𝜎2 − 𝜎3)2

Eq 2.1

𝜀𝑒𝑞 = 1

√2(1 + 𝜐)√(𝜀1 − 𝜀2)2 + (𝜀1 − 𝜀3)2 + (𝜀2 − 𝜀3)2 Eq 2.2

The equivalent strain is normalized by dividing the equivalent strain 𝜀𝑒𝑞 with the maximum

equivalent strain value reached in each specimen. This will present a clear comparison about

the equivalent strain distribution regardless the equivalent strain level obtained in each

specimen. The results are shown when a stress level of 440 MPa is reached in each specimen.

This stress level was chosen since it fits well within the range of the uniaxial failure stresses

of this type of composites.

Figure 2.9 and Figure 2.10 shows the stress and the normalized strain distributions in each

specimen, respectively. Regarding the stress distribution, the three specimens show a

homogeneous stress state in the central zone. However, a stress concentration is obtained in

the corner between the arms of the specimens with an adapted corner fillet, for specimens A

and B. This was also clearly shown in the equivalent strain distribution where the maximum

strain is reached between the arms of these specimens. For specimen C (with a large corner

fillet), the equivalent strain is maximal in the central zone. To interpret the results, a

normalized distance has been taken using a 45° line between the central point (normalized

distance = 0) of the specimen towards the corner between the arms (normalized distance = 1)

(Figure 2.10). This shows that the maximum stress level is obtained at the corners of

specimens A and B, meanwhile the stresses are maximum only in the center of specimen C

(Figure 2.11). The strain curve, presented in Figure 2.12, shows also that only specimen C

reaches its maximum value in the centre of the specimen while for other specimens the

maximum value is exactly reached at the extremity of the corner (normalized distance equal

to 1).

Figure 2.9 Equivalent stress distribution 𝜎𝑒𝑞(𝑀𝑃𝑎) for three specimens


39

Figure 2.10 Normalized equivalent strain distribution for three specimens

Figure 2.11 Variation of equivalent stress with respect to the normalized distance

Figure 2.12 Variation of the normalized equivalent strain with respect to the normalized distance

The evolution of the load with respect to the applied displacement is plotted until a stress

level of 440 MPa is reached in each specimen, as shown in Figure 2.13. Regarding the

machine capacity (50 kN), a load of 34, 42, and 60 kN are required to obtain a stress level of

440 MPa in all three specimens A, B, and C respectively.

Normalized

distance = 1

Normalized

distance = 0


40

Among the three tested shapes, according to the previous results, the overall shape of the

specimen C is adopted in the following since it allows to obtain the maximum value and the

best homogeneity of the stress/strain field in the central zone. Nevertheless, the dimensions of

this specimen must be adapted in order to reduce the loading level and make it suitable with

the capacity of the machine.

Figure 2.13 Evolution of tensile loading with respect to the applied displacement

2.3. Effect of geometrical parameters

According to the constraints and the experimental limitations presented before, the specimen

is re-dimensioned, keeping the same generic shape (tabs with circular central hole and

rounded fillet corner) and same thicknesses since the composite thickness will remain the

same. The numerical model shape can be then defined based on two groups of parameters:

external parameters (such as H, R and L) and parameters related to the tab geometry (such as

r, D, and the shape of the thickness reduction transition) as shown in Figure 2.14. Each

parameter is cited and defined below:

- H: Width of the arms

- R: Fillet radius between the two adjacent arms

- L: length of the specimen

- D: Outer tapered radius

- r: Inner tapered radius of the aluminum tabs (or the gage section radius)


41

Figure 2.14 Geometrical parameters for the design of the cruciform specimen

2.3.1. External parameters

2.3.1.1 Effect of the specimen length “L”

The length of the arms does not have an important effect on the stress distribution in the

central zone of the specimen since arms are only used to transfer loads from actuators to the

central area of the specimen. So, based on the available working space of the machine and on

the grip devices of the specimen, a 160 mm specimen length is defined.

2.3.1.2 Effect of the arm width “H”

As mentioned at the beginning of this chapter in the experimental constraints, the width

parameter (H) is directly related to the amount of load/stress that will be transferred from the

cruciform arm towards the central zone. A smaller H means a less amount of required load in

order to reach the failure of the specimen, but at the same time, it increases the stress level in

the loading arms which increases the possibility of the failure in the arms instead of the

central zone. To stay compatible with the maximum load possible reached by the biaxial

machine (50 kN), an arm width H is considered to be 18 mm, this value reduces the amount

of load required to reach the failure stresses of the composite.

2.3.1.3 Effect of the fillet radius “R”

Regarding the arms corner fillet radius “R”, it could be noticed that this parameter has some

limitations since it is related to other parameters (D and H). The effect of the fillet radius R is

investigated using two different values of 18 and 30 mm while fixing the other geometrical


42

parameters. The value of 18 mm was taken according to the same ratio used when reducing

the width of the arm H from 30 mm to 18 mm. In addition, 18 mm corresponds to a minimal

value of this parameter, decreasing the value more than 18 mm would lead to an overlap

between the corner fillet radius (R) and the outer tapered radius (D). These two values of 18

and 30 mm were enough to obtain a clear conclusion regarding this parameter.

Figure 2.15 presents the equivalent stress distribution in the composite when a stress level of

440 MPa is reached. The figure shows that increasing the fillet radius between the arms lead

to a stress concentration in the loading arms which ensure the failure outside the area of

interest. Thus, a fillet radius of 18 mm is considered leading to a maximum stress in the

central area.

Figure 2.15 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) in the composite for different fillet corners

2.3.2. Tab geometry in the gage section

2.3.2.1 Effect of the central zone radius “r”

Considering the central radius r, a numerical study has been made for three different values of

r (5, 7.5 and 10mm) while fixing the other parameters. The choice of the radius will be based

on the requirements of the successful biaxial test, i.e. a homogeneous maximum stress

concentration in the central zone. An equibiaxial tensile test is performed by applying 1 mm

of displacement on both cruciform arms. The results are presented when a stress level of 440

MPa is reached for all three cases. Figure 2.16 shows that, for different central zone radii, the

equivalent stresses are homogeneous and maximum in the central zone. However, the figure

shows also that the stresses in the central zone, using a small radius (radius of 5 mm), are

close to the stresses in the arms. This means that a higher possibility of failure would occur in

the loading arms using this central zone radius.


43

In addition, the x-direction stresses 𝜎𝑋𝑋 (Figure 2.17) are also analyzed for all specimens

since those stresses are used to predict the failure of the composite according to different

failure criteria (will be more explained later in Chapter 3). The results show that, increasing

the radius of the central zone, leads to an increase of the difference between the stress levels

reached in the central zone and in the loading arms. At the same time, a more heterogeneous

stress state in the central zone (radius of 10 mm) is obtained.

The variation of 𝜎𝑋𝑋 from the central point of the specimen towards the extremity of the x-

direction arm is presented in Figure 2.18. A distance of zero refers to the central point of the

specimen while a distance of 1 refers to the extremity of the x-direction loaded arm. The

graph shows that, increasing the central zone radius, leads to higher difference between the

stress levels obtained in the central zone and the loading arms. However, taking into

consideration the stress homogeneity in the central zone, the results in all three specimens

show nearly a similar 𝜎𝑋𝑋 distribution where a homogeneous stress state is obtained in the

central zone with a small increase on the borders of the central area. The difference between

the stresses in the middle of the central zone and on the borders is 23, 25, and 33 MPa for

radius of 5, 7.5, and 10 mm respectively. This can show that increasing the central zone

radius would increase the heterogeneity of the stress states in the central zone but in a very

slight amount.

As conclusion, having a small central zone radius (r=5 mm) leads to a more possibility of

failure in the loading arms. Increasing the central zone radius (r=10 mm) is acceptable and

would ensure a failure in the central zone but increases slightly the heterogeneous stress state

in the central area. Thus, a radius of 7.5 mm is selected.

Figure 2.16 Equivalent stress distribution 𝜎𝑒𝑞(𝑀𝑃𝑎) for different central zone radii


44

Figure 2.17 𝜎𝑋𝑋 (𝑀𝑃𝑎) distribution for different central zone radii

Figure 2.18 Variation of 𝜎𝑋𝑋 with respect to the x-direction arm normalized distance using different

central zone radii

2.3.2.2 Effect of the outer tapered radius “D”

The influence of the outer tapered radius is investigated using two different values of d (12.5

and 17.1 mm) while fixing the width of the arm and fillet corner at 18 mm and the central

zone radius at 7.5 mm. Similarly to the fillet radius R, the value 17.1 mm was also obtained

according to the ratio used to change the value of the width from 30 to 18 mm. This

parameter has a maximum limit, increasing the parameter with an important value (more than

17.1 mm) would make it more complicated in the fabrication process for the same

overlapping problem presented previously in the effect of the fillet radius.

The equivalent stress distributions are plotted when a stress level of 440 MPa is obtained. As

shown in Figure 2.19 a radius D of 17.1 mm gives a more homogeneous stress state in the

central zone compared to a radius D of 12.5 mm. In addition, comparing the stress levels

between the central zone and the loading arms, it could be noticed that a high difference of

stress levels were obtained in the specimen with an outer tapered radius of 17.1 mm which

emphasize the failure in the central zone. Considering the radius of 5 mm, a small difference


45

between the stresses in the central zone (𝜎𝑒𝑞 = 407 𝑀𝑃𝑎) and the stresses in the loading

arms (𝜎𝑒𝑞 = 365 𝑀𝑃𝑎) which increases the risk of obtaining a failure in the loading arm.

Figure 2.19 Equivalent stress distribution 𝜎𝑒𝑞(𝑀𝑃𝑎) for different outer tapered radii

2.3.2.3 Effect of the aluminum thickness reduction shape

During the milling process of the aluminum tabs, the end of the aluminum tabs cannot be

accurately machined at a thickness of 0 mm. This issue is must be taken into consideration in

the numerical study when designing the cruciform specimen. The milling of the aluminum

tabs is considered to start from a thickness of 3 mm towards a minimum thickness of 1 mm

which generates similar aluminum tabs for all the cruciform specimens (Figure 2.20).

In Figure 2.21, the equivalent stress results show that in both cases, a maximum and

homogeneous stress state is still found in the central zone for both cases. However, the

difference between the stresses in the central zone and the stresses obtained in the loading

arms decreases when using an end thickness of 1 mm of the aluminum tabs.


46

Figure 2.20 Aluminum tabs with 0 and 1 mm ending thickness

Figure 2.21 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) in the composite for (a) 0 mm and (b) 1 mm

ending thickness

According to the previous results, a 1 mm of aluminum tabs thickness on the edge of the

central zone would increase the risk of having the failure outside the central zone. Based on

that conclusion, a study was performed to investigate the influence of the shape of the

thickness reduction transition in the aluminum tabs on the stress distribution in the composite.

A numerical comparison between linear and curved transition (radius of 24 mm) is

performed. The radius 24 mm is obtained when the curve is tangent to the plane parallel to

the plane of the specimen as shown in Figure 2.22 (tangent to the vertical line). Figure 2.23

shows the equivalent stress distribution (stress level of 440 MPa) in the composite when both

linear and curved thickness reduction transitions are used. It could be noticed that the

difference between the stress levels obtained in the central zone and the loading arms is

higher when using a curved thickness transition. In addition, the variation of the x-direction

Thickness of 0

mm is not easily

experimentally

feasible


47

stresses in the composite from the central point (normalized distance 0) towards the x-

direction arm extremity (normalized distance 1) using both shapes of thickness reduction is

presented in Figure 2.24. The results show that the curved transition gives a higher difference

between the stresses in the center and the loading arms which emphasize the failure of the

composite in the central area where a difference of 132 MPa is noticed for the linear shape

while 200 MPa is obtained for the curve transition.

In addition, according to these curves, the x-direction stresses are more homogeneous in the

central zone when using a curved shape transition since a less difference is obtained between

the stresses in the middle and on the border of the central zone (difference of 9 MPa)

compared to the one obtained when using a linear transition (25 MPa). Thus, a curved

transition is adapted.

Figure 2.22 Linear and curved thickness reduction in aluminum tab

Figure 2.23 Equivalent stress distribution 𝜎𝑒𝑞 (𝑀𝑃𝑎) in the composite for (a) curved and (b) linear

aluminum thickness reduction shape


48

Figure 2.24 Comparison between 𝜎𝑋𝑋 results in the composite using curved and linear thickness

reduction transition

2.4. Effect of aluminum grade

In this section, the influence of the mechanical properties of the aluminum tabs on the

behavior of the specimen is evaluated. Two aluminum alloys AA2017 and AA5086, with

very different initial yield strengths (and ultimate stress), have been used for this numerical

comparison. Table 2.2 shows the initial yield and ultimate strengths for both aluminum

grades. These grades are considered to have similar Young modulus (𝐸 = 70 𝐺𝑃𝑎) and

Poisson ratio (𝜈 = 0.3). Figure 2.25 shows the isotropic hardening behavior obtained from

uniaxial tensile tests of both aluminum alloys.

Table 2.2 Yield and ultimate strengths for aluminum alloy AA2017 and AA5086

Aluminum Yield Strength (MPa) Ultimate Strength (MPa)

AA5086 126 400

AA2017 230 534


49

Figure 2.25 True stress- plastic strain curve for AA5086 and AA2017 alloys

An equibiaxial numerical tensile test has been applied on both cruciform arms using both

aluminum alloys types. A linear thickness reduction transition is considered in this

comparison. The effect of the aluminum type on the stress distribution in the composite is

presented in Figure 2.26 when a stress level of 440 MPa is reached in the composite. The

figure shows that in both cases, the stress is homogeneous in the central zone of the

composite but at the same time, a high level of stresses is obtained in the loading arm of the

specimen using AA5086 aluminum. In addition, Figure 2.27 shows the variation of the x-

direction stresses 𝜎𝑋𝑋 is plotted with respect to the normalized distance from the middle of

the specimen towards the extremity of the x-direction loaded arm. The curves show that a

higher difference between the stresses in the central zone and the stresses in the loading arms

are obtained when using AA2017 aluminum tab.

The high stresses in arms when using AA5086 are obtained due to the earlier plasticity that

occurs in AA5086 tabs compared to AA2017 based on their yield strength. To validate this,

the plastic strains in the x-direction (𝑃𝐸𝑋𝑋) are presented (Figure 2.28) in both aluminum tabs

when a stress level of 440 MPa is reached in the composite. It can be noticed that a high

plastic deformation is obtained in the AA5086 (1.5%) compared to AA2017 (0.9%). A high

plastic deformation in the aluminum tabs will lead to higher shear stresses in the adhesive

which can cause an adhesive failure and debonding of the aluminum tabs. This will lead to an

undesirable failure in the loading arms of the composite. Based on this study, it is important

to use the aluminum alloy with the highest yield strengths (AA2017) in order to avoid the

adhesive failure before reaching the failure in the composite.


50

Figure 2.26 𝜎𝑒𝑞 (𝑀𝑃𝑎) in composite using (a) AA2017 and (b) AA5086 aluminum tabs

Figure 2.27 Comparison between 𝜎𝑋𝑋 (MPa) results in the composite using different aluminum alloy

tabs

Figure 2.28 Plastic Strain (𝑃𝐸𝑋𝑋) in (a) AA2017 and (b) AA5086 aluminum tabs


51

Another way is to improve the performance of the aluminum tabs is by increasing its

thickness, in this way, the aluminum would enter into the plastic region for a higher amount

of applied load which decrease the shear stresses in the adhesive when the failure stresses in

the composite are reached. In this part, the effect of the tab thickness on the stresses in the

central zone has been numerically studied using two different thicknesses (3 and 5.5 mm).

The thickness of 5.5 mm was chosen according to the ratio between the yield stresses of both

aluminum types. Figure 2.29 shows the equivalent stress distribution in the composite when

using AA5086 tabs with two different thicknesses. The use of a thicker aluminum tab

increases the stress level in the composite central zone compared to the stresses obtained in

the loading arms. In addition, the variation of the x-direction stresses 𝜎𝑋𝑋 is plotted with

respect to the normalized distance from the middle of the specimen towards the extremity of

the x-direction loaded arm (Figure 2.30). The curves show that a higher difference between

the stresses in the central zone and the stresses in the loading arms are obtained when using a

thicker aluminum tab (stress difference of 200 MPa) compared to the 3 mm tab (stress

difference of 80 MPa).

Figure 2.29 𝜎𝑒𝑞 (𝑀𝑃𝑎) in composite using AA5086 with a thickness of (a) 5.5mm and (b) 3mm

Figure 2.30 Comparison between 𝜎𝑋𝑋 (MPa) results in the composite using different aluminum

thicknesses


52

As conclusion, it can be retained that the choice of the aluminum grades, for the same

thickness, is based on the high level of yield stress. Meanwhile, increasing the thickness of

the aluminum tabs leads to a high level of stress concentration in the central zone of the

composite. The aluminum AA2017 with a thickness of 3 mm presents good results regarding

the stress distribution and concentration in the central zone.

2.5. Final shape and behavior under different biaxial ratios

A complete numerical study on the geometrical parameters has been performed to obtain a

final design of the specimen. The obtained parameters are presented in Table 2.3. In addition,

it was retained that using a curved aluminum thickness reduction would give better results

regarding the stress homogeneity and the maximization in the central zone of the specimen

compared to a linear transition. Figure 2.32 presents the final shape of the cruciform

specimen for the biaxial test.

Table 2.3 Final geometrical parameters for the cruciform specimen

L (mm) H (mm) R (mm) D (mm) r (mm)

160 18 18 17.1 7.5

The evolution of the load vs. applied displacement is shown in Figure 2.31 when using the

final designed specimen. The curve shows that even when 1 mm displacement is reached in

both loading directions (𝜎𝑒𝑞 = 600 𝑀𝑃𝑎 in the central area), the load reaches 40 kN which is

still under the maximum load capacity of the machine.

Figure 2.31 Load vs. displacement using the new geometrical parameters of the specimen


53

Figure 2.32 Cruciform specimen shape for the biaxial characterization test

One of the main requirements for a successful biaxial test is to obtain failure in the central

zone for different biaxial displacement ratios. The biaxial displacement ratio is defined as the

ratio 𝑈𝑋 𝑈𝑌⁄ of the displacement applied on the ends of the arms in 𝑋 and 𝑌 directions,

respectively. Taking a thickness of 3mm for the AA2017 tabs, three different tensile ratios

have been numerically studied: 1/free, 1/0.5 and 1/1. This could verify that the failure always

appears in the central zone. The distribution of the major stresses and strains are presented

respectively in Figure 2.33 and Figure 2.34 when 1 mm of displacement is reached in the x-

direction loading arm. The major stresses and strains are considered since they are an

important key to represent the failure of the composite regarding the failure envelops and the

strain paths. In another way, if the major stresses are maximum and homogeneous in the

central zone, then the failure will occurs in this zone.

For the considered biaxial ratios, the stress and strain distributions show a maximum value in

the central zone. This means that the failure should occur in this zone. The variation of minor

and major strains in the center of the composite for different biaxial ratios are presented in

Figure 2.35. It could be noticed that a linear evolution of the strains is presented for the equi-

biaxial test (1/1) and for the uniaxial test (1/free)


54

Figure 2.33 𝜎𝑋𝑋 (𝑀𝑃𝑎) in composite under different biaxial displacement ratios

Figure 2.34 𝜀𝑋𝑋 in composite under different biaxial displacement ratios

Figure 2.35 Major versus minor strains for different biaxial ratios

2.6. Conclusion

In this chapter, a cruciform shape for the biaxial characterization of the glass/epoxy

composite material is selected among three different shapes based on the literature. All the

studied specimens consist of two cruciform aluminum tabs (with holes in the center) glued on


55

both sides of the composite in order to ensure a stress maximization in the central zone of the

composite. The study is based on FE numerical investigations. According to the stress and

strain results, the selected shape is the one having homogeneous and maximum stress/strain

state in the central zone.

According to different experimental constraints, mainly the capacity of the machine, a

parametric study has been performed in order to select the geometrical parameters of the

specimen. The final shape consists of a circular central zone with a curved thickness

reduction and a fillet corner between the loading arms. The proposed cruciform specimen has

a length of 160 mm, arm width of 18 mm, central zone radius of 7.5 mm, arms corner fillet

radius of 18 mm, and an outer tapered radius of 17.1 mm.

As final step, the specimen was numerically tested under different biaxial displacement

ratios. The stress and strain results show that the failure will occur in the central zone for all

biaxial ratios. The next chapter consists of an experimental validation of the defined

specimen shape in order to validate the failure of the composite in the central part under

different biaxial tensile ratios.


56

Chapter 3. Biaxial Experimental Characterization of Failure

57

Chapter 3. Biaxial Experimental

Characterization of Failure


58

3.1. Introduction

The previous chapter goals were achieved by defining numerically, using finite element

analysis, a cruciform specimen that fails in the central area under different biaxial loading

conditions. This chapter handles the experimental phase. The chapter starts with different

uniaxial tests performed on the composite and on the adhesive in order to characterize the

composite and the adhesive, respectively. Indeed, the behavior of the adhesive subjected to

shear loading should be defined in order to improve its performance in the biaxial testing. In

addition, the effect of the strain rate on the composite mechanical properties is shown by

performing uniaxial tensile tests under quasi-static and dynamic strain rates. After, a

description is given about the experimental testing bench composed of the biaxial testing

machine and the strain measurement techniques. The specimen preparation which consists of

gluing the aluminum tabs with the composite is also briefly presented. Afterwards, the

experimental biaxial testing results such as strain determination using full field measurement

technic and stress calculation methods are presented for different biaxial tensile ratios.

Finally, a conclusion behind the experimental work is retained.

3.2. Uniaxial characterization of composite materials

3.2.1. Materials

In this thesis, two different plain-weave glass/epoxy composites were studied. The first one

“M9.6GF/200T2/G”, fabricated by C3 Technologies Company, is composed of four laminae

(thickness of each lamina is 0.25 mm) with a total thickness of 1 mm. The second composite

“M21/58%/G” was fabricated by the Laboratory “Institut Clément Ader” composed also of

four laminae (thickness of 0.125 mm) with a total thickness of 0.5 mm.

Since both composites are plain-weave glass/epoxy fabrics, each composite will be called by

his matrix code (M9.6 and M21). The global reason behind the use of different composite

thicknesses in the study was based on the biaxial results obtained for the 1mm thick

composite (M9.6) where the specimen fails ideally in the central zone for specific biaxial

ratios and some difficulties were encountered with other biaxial ratios. For this reason, the

second composite with 0.5 mm thickness (M21) was introduced in order to ensure the failure

inside the central zone of this type of specimen under different biaxial ratios without

increasing the maximum tensile force.

Different uniaxial tensile tests have been made on both composites in order to obtain their

material properties which will be used later in the study. Since the tests are in-plane tests,

only the in-plane parameters are required (𝐸1, 𝐸2, 𝐺12 𝑎𝑛𝑑 ʋ12 ). In this study, the composites


59

considered are plain-weave fabrics, so the elastic Young modulus 𝐸1 in the longitudinal

direction is equal to 𝐸2 in the transverse direction.

A uniaxial Instron testing machine with a maximum capacity of 50 kN is used for these

uniaxial tests. The Young modulus is calculated based on the stress/strain curve and the

Hooke’s Law (Eq 3.1) obtained from the longitudinal tests. The shear modulus is obtained

from the 45° tensile tests using Eq 3.2. Poisson ratio is calculated as the ratio between the

compressive and longitudinal strains determined by the strain measurements (Eq 3.3). Since

the measured strains are essential in order to obtain the material properties, different strain

measurement techniques are presented and analyzed in this part.

𝐸1 =𝜎1

𝜀1 Eq 3.1

𝐺12 =𝜎1

2 (𝜀1 − 𝜀2) Eq 3.2

ʋ12 = −𝜀2

𝜀1 Eq 3.3

3.2.2. Strain measurement techniques

3.2.2.1 Strain gages

One of the most widely used methods to measure strain is by using strain gages. The concept

is based on the change of the electrical resistance of a metallic conductor placed on the

specimen when the specimen is loaded. The change of resistance is the measure of the

deformation of the specimen. Strain gages are often used because of their high accuracy

measured strain. However, strain gages measure only the strain on a local point (gage

position) and cannot measure a strain distribution.

3.2.2.2 Extensometer

Another strain measurement method recommended by the most used ASTM tests is the

extensometer. Extensometers are used more often than strain gages in uniaxial tests because

of their simplicity, high accuracy and low cost. The extensometer is connected to the

specimen via two clips, it measures the displacement in between both clips in order to obtain

the strain evolution during the test. The extensometer has a reference length L0 which is the

initial length before starting the test. The strain calculation using the extensometer is the ratio

of the measured displacement over the reference length.


60

3.2.2.3 Digital Image Correlation

Another method to obtain the strain field in the specimen is by using the method of image

correlation. This method is a great way to measure the displacement or deformation fields on

specimen surface under any kind of loading. Since the tests are in-plane, a single camera can

be used to cover the zone where the strain calculation is needed. The process of image

correlation can be divided into three parts (Figure 3.1): first, the specimen is prepared by

applying a random speckle pattern. After that, the camera films the whole test by taking a

series of images according to a pre-defined frequency by the user. Finally, the images are

treated using specific correlation software on the computer and the strain fields are generated.

Figure 3.1 Digital Image Correlation (DIC) principle

Unlike the strain gages and the contact extensometers, DIC technique is a non-contact strain

measuring method where the strain results are not influenced by the measuring system. Also,

it is capable to calculate the strain field in a specific area instead of obtaining a local strain

measurement which is very important later in the biaxial testing.

In this thesis, the software “Correlate” developed by GOM Company in Germany is used for

the strain calculation. The strain calculation using this software is based on creating a surface

component composed of facets (squares) defined by the user. An area of calculation is

selected first where the strain field will be plotted. The accuracy of the results depends on the

image quality and the correlation parameters. The parameters related to the facets are the

facets size which defines the pixels covered by each side of the facet, and the point distance

which defines the distance between the center points of each two adjacent facets in pixels

(Figure 3.2). The choice of these parameters depends on the accuracy needed with respect to

the computation time. Increasing the facet size would increase the computation time and

decrease the accuracy of results inside the facets while increasing the point distance would

decrease the computational time and decrease the measurement point density.


61

Figure 3.2 Digital Image Correlation (DIC) parameters

The correlation software calculates the displacement of the center points between facets

according to the deformed image with respect to a reference image. The facet matching

parameter is related to the matching (or comparing) the facets over time. This parameter can

be either “against definition stage” where the software compares the patterns with respect to

the first image or “against previous stage” where the software compare the pattern with

respect to the previous stage. This option should be selected for high deformations when the

software cannot compute the displacements based on the first image in some stages. Since the

deformations in composite materials are usually low, the facet matching was selected as

against definition stage.

3.2.3. Test results and material properties

Considering the M9.6 composite, four specimens have been prepared for the uniaxial

characterization test: two tests are loaded with the fiber direction and two tests loaded on 45°

angle with respect to the fibers direction. Strain gages have been glued in the middle of each

specimen in order to determine the longitudinal and transverse strains during each test

(Figure 3.3). Static tensile tests were realized with a velocity of 0.005 mm/s. Aluminum tabs

(thickness of 2 mm) were glued on the extremity of each specimen in order to avoid the

failure on the grips. Tests applied with the fiber direction allow to determine the Young

Modulus and the Poisson ratio, while the tests loaded on 45° angle allow to calculate the

shear modulus. Material properties are calculated based on the previous equations according

to the strain gages measurement and the applied load by the machine (section 1.2.1. The

averaged results of the four tests are presented in Table 3.2.


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Figure 3.3 M9.6 uniaxial specimens

On the other hand, six specimens were prepared for the uniaxial tensile test of the M21

composite. Four tests have been performed in the fiber directions and two in 45° direction

with respect to the fibers direction. The specimens were prepared while respecting the ISO

527-4 dimensions. Aluminum tabs (with a length of 50 mm and a thickness of 3 mm) were

glued on the extremity of the specimen in order to avoid the failure in the grips. In these tests,

an extensometer was used in order to measure the strain through the loading direction. The

extensometer was used based on the ISO standards with a reference length of 50 mm.

Figure 3.4 M21 uniaxial specimen

In addition, the specimens were painted on the other side using white paint with black dots

for the Digital Image Correlation (DIC) strain measurement. In this way, the strains obtained

from the DIC method will be validated based on the extensometer, and the results obtained

from the image correlation will help to obtain the deformations in the transverse direction for

the Poisson ratio and shear modulus calculation. The idea is also to validate the DIC technic

for the composite characterization in order to use it in biaxial tensile experiments on cross

specimen where mechanical extensometer cannot be used. The following Table 3.1 shows the

parameters related to the digital image correlation technique including the image resolution

and camera frequency.

Table 3.1 DIC parameters for the uniaxial test of M21 composite

Resolution Frequency Facet size Point distance 384x761 pixels 2 image/s 32x32 pixels 32 pixels


63

The evolution of the strain in the loading direction is plotted with respect to the applied

displacement U (mm) measured by the machine. Figure 3.5 shows that using both strain

measurement techniques, the strain results fit perfectly until the applied displacement reaches

0.75 mm. After, the strain measured by DIC has a slow decrease and then the evolution

continues with the same initial slope. Meanwhile, the strain results obtained by the

extensometer is almost linear. The small decrease in the strain measured by the DIC was also

noticed by different authors previously when studying only plain-weave composite type. It

was explained as the beginning of the failure of the transverse fibers with respect to the

applied load ([49], [81]). However, the material properties are calculated based on the results

before reaching this small decrease according to the standards (deformation level between

0.0005 and 0.0025). Table 3.2 summarizes the averaged material properties obtained for this

composite.

Figure 3.5 Strain results for uniaxial test of M21 composite using extensometer and DIC techniques

Table 3.2 Averaged material properties for both M9.6 and M21 composites

Material E1 (GPa) E2 (GPa) G12 (GPa) υ12 M9.6 22.8 22.8 4.1 0.14

M21 18.8 18.8 2.1 0.12

3.2.4. Uniaxial tests under different strain rates

The previous uniaxial tensile tests were performed under a low traction velocity (strain rate

𝜀̇ of 8x10-5 s-1) which is very low compared to the traction velocity used in the biaxial tests.

For this reason, different uniaxial tests are performed on the M9.6 composite in order to

investigate the effect of the strain rate on the behavior of the material.

A uniaxial dog-bone specimen, with a gage length of 10 mm, was designed using finite

element analysis in order to obtain a failure in the center of the specimen and avoid the

failure on the grips. The evolution of the strain is measured using the Digital Image


64

Correlation method. The facet size and point distance are considered to be 32x32 pixels and

32 pixels respectively. Figure 3.6 shows an example of the image treated in the DIC software

in order to obtain the strain measurements.

Figure 3.6 Digital image correlation for specimen tested under a strain rate of 0.25 s-1

Having a gauge length of 10 mm, the used head velocity is 2.5, 250, and 500 mm/s, in order

to achieve strain rates of 0.25, 25, and 50 s-1, respectively. Based on the traction velocity,

different camera acquisition frequency are selected. Table 3.3 shows the different

experimental parameters for each strain rate. Some of these tests were performed twice in

order to verify the repeatability of the results. Figure 3.7 shows an example of a failed

specimen with dimensions.

Table 3.3 Experimental parameters for uniaxial tests under different strain rates

Strain rate �̇� 0.25 s-1 25 s-1 50 s-1

Head velocity 2.5 mm/s 250 mm/s 500 mm/s

Force acquisition

frequency 10 kHz 100 kHz 100 kHz

Camera acquisition

frequency 1kHz 25 kHz 25 kHz

Image Resolution 384 x 464 256 x 304 256 x 304

The Young modulus and shear modulus were obtained for each strain rate (logarithmic scale).

Figure 3.8 shows that both material parameters were not significantly affected by the

variation of the strain rate. These results show that this material is non-sensitive to the

variation of the strain rate in both quasi-static and intermediate (dynamic) strain rates where a


65

small difference is noticed between the results obtained for each test. That means, the

specimen can be tested later at higher strain rates than the one reached previously in the

uniaxial characterization test (8x10-5 s-1).

Figure 3.7 Uniaxial specimen tested under different strain rates

Figure 3.8 Elastic properties of the composite subjected to uniaxial loading for different strain rates

3.3. Biaxial test setup

3.3.1. Biaxial testing machine

According to the state of art, different in-plane biaxial testing machines were developed by

laboratories and companies. The biaxial testing machine, developed at the Laboratory of Civil

and Mechanical Engineering (LGCGM) of INSA de Rennes, consists of four independent

servo-hydraulic actuators which allow static and dynamic tests (Figure 3.9). The machine is

only limited to apply tension/tension tests by controlling each loading arm displacement.

Each actuator can produce loads up to 50 kN and a maximum loading speed of 2 m/s. The

device allows imposing different tensile velocities along both axes of the cruciform

specimen. For force measurement, load sensors are placed in each loading axis, as shown in

the figure. The machine is also provided with a cooling and heating system (range from -60

to 150 °C) in order to study the effect of temperature on the material behavior under biaxial

testing. However, this system will not be used in this thesis. To obtain the deformations


66

during the test, a high speed camera (Photron FASTCAM-APX RS), with a maximum

frequency of 3000 images per second and a maximum resolution of 1024x1024 pixels, is

placed on the top of the specimen as shown in Figure 3.10.

Figure 3.9 Schema of the biaxial testing machine in LGCGM

Figure 3.10 Biaxial testing machine including the camera and the tested specimen

3.3.2. Cruciform specimen preparation

Before going into the biaxial testing of the specimen, an important step that would affect the

results is the specimen preparation since the specimen is composed from different materials

glued together. A complete study was made in order to improve the adhesive strength and

decrease the possibility of adhesive failure in the biaxial test. The biaxial failure of the

composite before reaching the failure of the adhesive requires a strong multi-material


67

adhesive for this kind of tests. AEROBOND 7300 (Datasheet in Appendix) is a

recommended epoxy adhesive to be used between aluminum and glass composite plates.

In order to improve the adhesive performance, a comparative study has been made between

two uniaxial tests under different conditions. Two double-lap joint tests were performed by

gluing three aluminum plates for uniaxial testing. One of these tests was realized after

treating the aluminum glued surfaces by shot peening. In the other test, the aluminum plates

were only cleaned with acetone. Both tests were exactly performed under the same conditions

with a traction velocity of 0.004 mm/sec. The tests are performed according to the ASTM

D3528 standard for double lap shear testing of adhesive with an adhesive length of 16 mm.

Figure 3.11 shows the double-lap joint specimen shape with dimensions. The measured shear

stresses are calculated as the ratio of the applied load by the adhesion area. According to the

results in Figure 3.12, it is clear that the treatment of the surface shows an important

improvement in the shear performance of the adhesive. The surface treatment increases the

shear strength of the cohesive from 8.1 MPa to 16.8 MPa. This shows that the surface

treatment has an important effect on the shear strength and it must be considered in the

biaxial tensile test.

Figure 3.11 Double-lap joint shear test specimen

Figure 3.12 Evolution of shear stresses in the adhesive with and without aluminum shot peening

surface treatment

For the biaxial test, a 400x600 mm composite plates were provided by the manufacturers.

The composite plates were cut using water jet technique into cruciform shapes (Figure 3.13).

For the M21 composite, the plate includes peel plies on top and bottom of the composite,

these peel plies are removed before the gluing process in order to have similar composite top


68

and bottom surface states. This way provides a good composite surface state for the adhesion

process. Considering the aluminum tabs, aluminum plates of thickness of 3 mm were cut into

cruciform specimen shape and a thickness reduction with hole in the central zone were made

using a milling machine.

For the last step, aluminum tabs and composite are glued together by hand spreading. The

adhesive, mixed with the microbeads with an amount of 1% of the adhesive weight, provides

a continuous adhesive thickness of 0.1 mm. An adhesive thickness of 0.1 mm provides a

higher shear strength compared to different thicknesses, this was recommended by the

manufacturer and was also presented in different researches. A 3D printed plastic pins were

fabricated and placed into the aluminum and composite arms holes during the gluing process

in order to keep the tabs and the composite aligned together (Figure 3.14). The full specimen

was set for 48 hours at room temperature under pressure for the adhesive curing.

Figure 3.13 Composite and aluminum tabs cruciform shapes before assembly

Figure 3.14 Assembled cruciform specimen for the biaxial test


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3.3.3. Experimental parameters

The tension in each arm is defined by the traction velocity (in mm/s) up to the failure.

Specimens were tested under different biaxial displacement ratios. The biaxial displacement

ratio is the ratio between the displacement applied on the x-direction loading arm and the

displacement applied on the y-direction loading arm (𝑈𝑥/𝑈𝑦). For example, a biaxial

displacement ratio of 1/2 means that the applied displacement in the Y-direction is 2 times

equal to the displacement in the X-direction arm. The M9.6 composite is tested only under

three different biaxial displacement ratios (1/free, 1/2 and 1/1/) while the M21 composite is

tested under six different biaxial displacement ratios (free/1, 1/2, 1/1, 1/1.4, 1/3 and 1/1.2).

For M21, the biaxial ratios free/1, 1/2, and 1/1 are tested twice in order to validate the

repeatability of the results. A traction velocity of 0.1 mm/s is imposed on the x-direction of

each arm, while the velocity in the y-direction vary in order to have the desired biaxial tensile

ratios. Since the tests are static, an acquisition frequency of 50 images per second is sufficient

for the camera. For the two load sensors (one in each direction), a frequency of 500 Hz is

considered. The acquisition of these three signals (camera and load sensors) are

synchronized. Table 3.4 summarizes the experimental parameters introduced in the biaxial

machine and the camera for the biaxial tests.

Table 3.4 Experimental parameters for the biaxial test

Force acquisition

frequency

Camera acquisition

frequency

Image

resolution

Traction

velocity (Vx)

Traction

velocity

(Vy)

500 Hz 50 Hz 832x896

pixels 0.1 mm/s

Depends on

the biaxial

ratio

3.4. Strain measurement

In this section, the strain results for each glass/epoxy composite based on the digital image

correlation technique are discussed separately. The strain fields in the central zone and the

failure strains in both x and y directions are presented for each biaxial displacement ratio.

3.4.1. Effect of DIC parameters

Before going into the strain results of each biaxial test, the effect of the digital image

correlation parameters must be investigated. The study is divided into two steps, the first one

is a study on the size of the strain calculation area which is defined in the digital image

correlation technique and the second one is a study on the effect of the facet size and point

distance on the strain results. For this study, the equibiaxial test on the M9.6 composite is

considered.


70

Two surface components (strain calculation zones) are evaluated (Figure 3.15). The first zone

is the large one with an area of 6.7x6.8 mm2 (321x323 pixels). The second zone is the small

one with an area of 3.58x3 mm2 (170x142 pixels). In this study, a facet size of 32x32 pixels

and a point distance of 32 pixels are considered. The average of strain field results in the

central zone is considered as the strain in the central area (average of 50 and 20 calculation

points in the surface component of large and small zones respectively). A comparison

between the calculated strains in both zones is shown in Figure 3.16. The figure shows a

similar results for both cases where a very small difference is noticed between the strains in

the y-direction (difference of 0.1%). Thus, both large and small zone can be taken as strain

calculation zone. In addition, the similarity of the strain results proves the homogeneous

strain field in the central zone which will be discussed later.

Figure 3.15 Small and large strain calculation zones

Figure 3.16 Comparison between large and small strain calculation zone results under equibiaxial

test


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For the same equibiaxial test, the effect of the facet size and point distance is presented. Two

different sets of parameters were taken into consideration: 32x32 pixels with a distance of 32

pixels and 24x24 pixels with a distance of 24 pixels. For this comparison, a similar large

surface component zone is considered. Figure 3.17 shows the average measured strain in the

central zone using each set of parameters. Similar results were obtained using both

parameters. In both cases, the computation time was nearly similar. However, it was

recommended by the DIC software developers to use a facet size large enough so that the

computation of the facet is always possible. Thus, 32 pixels facets were considered.

Figure 3.17 Strain results using different DIC parameters under equibiaxial test

3.4.2. Experimental verification of the effect of aluminum grade

Experimental validation of previous numerical investigations is performed between both

aluminum alloys AA2017 and AA5086. Because of the difference between the aluminum

alloy yield stresses and in order to decrease the possibility of failure in the arms for the

specimen with AA5086 tabs (as mentioned in the previous chapter), a thicker AA5086 tab

has been used for the experimental comparison (thickness of 5.5 mm). Two static equi-biaxial

tests (velocity of 0.1 mm/s) have been made using both aluminum tabs in order to validate the

selection one of these tabs for the biaxial characterization test. The comparison is made by

applying a similar amount of displacement in each test on each loading arm. Since the test is

symmetric, the strain in both x and y directions are equal. Figure 3.18 shows the experimental

evolution of the strain in composite with respect to the applied displacement using both

aluminum tabs. For the same amount of displacement, the composite reaches a high strain

level when using AA2017 tabs (1.13%) compared to AA5086 (0.56%). Moreover, the

experimentally measured load is introduced into the numerical model as boundary conditions

in order to validate the influence of the aluminum tabs on the composite. A variable R has

been defined as the ratio between the strain in the center of the composite using AA5086 and

AA2017 (Eq 3.4). A ratio R is less than one meaning that the strain in the center using

AA2017 is higher than the strain using AA5086.


72

𝑅 = 𝜀𝑐𝑜𝑚𝑝 (𝐴𝐴5086)

𝜀𝑐𝑜𝑚𝑝 (𝐴𝐴2017) Eq 3.4

Figure 3.19 shows the variation of experimental and numerical R values with respect to the

applied displacement. The results show that in both numerical and experimental tests, the

strain in the central zone using AA2017 is higher than the one using AA5086. A high

deformation in the composite under the same amount of displacement imposed means higher

possibility of composite failure before reaching the adhesive failure. According to these

results, AA2017 is chosen for the complete composite biaxial tests.

Figure 3.18 Composite strain at the central point of the specimen for AA2017 and AA5086

Figure 3.19 Variation of R with respect to the imposed displacement

3.4.3. Strain results for M9.6 composite

The M9.6 glass/epoxy composite with a thickness of 1mm is tested under three biaxial

displacement ratios (1/free, 1/1 and 1/2). In each test, the strain distribution in the central

zone is presented. Since the biaxial displacement ratio is defined with respect to the x-

direction applied displacement, both load and strain curves are presented with respect to the

displacement in this direction. Figure 3.20 shows an example of a failure specimen under

1/free biaxial tensile ratio.


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Figure 3.20 M9.6 cruciform specimen failure under 1/free biaxial ratio

Figure 3.21 shows the evolution of the loads applied by the machine (collected by the loading

sensors) and the measured strains until the composite failure with respect to the applied

displacement (Ux) for all three biaxial ratios 1/free, 1/2, and 1/1 respectively.

Considering the load curves, a non-linear evolution is noticed. This can be explained as the

beginning of the plastic phase in the aluminum tabs. Moreover, for the biaxial ratio 1/2, a

small decrease in the loading curves is presented when a displacement of 0.9 mm is reached

in the x direction. This decrease is caused by the crack initiation of the adhesive joint

between the aluminum tabs and the composite. However, the traction continues until the

failure of the composite. The evolution of the load for the biaxial ratio 1/1 shows the equi-

biaxial state of the test where both loads Fx and Fy reach a level of 40 kN at the moment of

failure. However, a small de-synchronization is noticed at the beginning of the test. This can

be explained due to the geometrical accuracy of the biaxial setup (e.g. specimen fabrication

and biaxial machine tolerances).

For the strain measurement, the strains in both x and y directions are the average of the

calculated strains in the central zone of the composite. For the biaxial ratio 1/free, the results

show the compression behavior of the material in the transverse direction (y-direction) where

a negative value of the ε𝑦 deformations are obtained due to Poisson effect. It could be noticed

that for all biaxial ratios, the composite fails at a nearly similar major strain level of 3 %.

Table 3.5 summarizes the failure loads and strains under all three biaxial tensile ratios.


74

Biaxial

ratio Load evolution Strain evolution

1/free

1/2

1/1

Figure 3.21 Evolution of the loads and strains with respect to applied displacement for M9.6

composite subjected to 1/free, 1/2, and 1/1 biaxial ratios

Table 3.5 Load and strain results at the onset of failure of M9.6 composite subjected to different

biaxial ratios

Biaxial ratio 𝛆𝒙 (%) 𝛆𝒚 (%) 𝐅𝒙 (kN) 𝐅𝒚 (kN)

1/free 3.1 -0.9 34.8 -

1/2 1.46 3.1 40 31.5

1/1 2.7 3.2 41.6 42.2

To obtain more reliable information about the behavior of the material during the test, the

strain distributions are presented in both directions. Since in all tests the strains obtained are

homogeneous and to avoid repeatability, only one test was considered (biaxial ratio 1/2).

The strain distributions are similar during the whole test, thus only the strain fields are

presented when 80% of the failure loads are reached (equivalent to 0.94 mm of displacement


75

in the x direction). Figure 3.22 shows the homogeneous state in both 𝜀𝑥 and ε𝑦 distributions

where a small scatter is obtained in both strain fields.

Figure 3.22 𝜀𝑥 and 𝜀𝑦 strains distribution in the central zone of the M9.6 composite subjected to 1/2

biaxial ratio

3.4.4. Strain results for M21 composite

As mentioned before, the M21 woven glass/epoxy composite with a thickness of 0.5 mm is

tested under six different biaxial ratios (free/1, 1/2, 1/1, 1/1.4, 1/3 and 1/1.2). To validate the

repeatability of the results, the biaxial ratios free/1, 1/2, and 1/1 are doubled. Compared to the

previous composite, the traction in the biaxial test “free/1” (uniaxial test) is in the y direction

instead of the x direction. However, since in both cases the composite is a plain-weave fabric,

the results of tests subjected to a free/1 biaxial ratio should be similar to the test under 1/free

ratio. This section will be divided into two parts where the first includes the repeated tests

and the second contains the rest. Figure 3.23 shows an example of some M21 composite

failed specimens.

Figure 3.23M21 cruciform failure under (a) free/1, (b) 1/2, and (c) 1/1.4 biaxial ratios


76

3.4.4.1 Results for biaxial ratios free/1, 1/2, and 1/1

Similarly to the previous composite, the evolution of the load and strains are presented until

the failure of the composite in Figure 3.24.

The biaxial ratio free/1 is analyzed first, the evolution of loads and strains are presented with

respect to the displacement in the y direction (traction direction). The figure shows that in

both repeated tests (Test01 and Test02), the composite fails at the same loading level of 28

kN. However, a small different behavior was noticed in both curves, this can be due to the

fact that both aluminum and composite used were not cut out from the same plate for both

tests. The strain evolution also is plotted in the same figure, a comparison between both tests

shows the repeatability of the results where the strain evolution in both cases are similar and

both specimens fail at the same amount of strain level (𝜀𝑥 around -0.8% and ε𝑦 around 2.9%).

As noticed in the previous composite, a compression state is also obtained in the transverse

direction to the applied load which is caused by the Poisson effect of the material.

For the biaxial ratio 1/2, the loading figure shows the identical behavior of the material

response during both tests where similar shape of the curves is obtained for both tests.

However, the curves show more a linear state in the x direction and a non-linear state

evolution in the y direction, this can be due to the high displacement applied in the y direction

compared to x direction, the plastic behavior of the aluminum tabs and the adhesive joint

would affect more the slope of the curves in the y direction. The evolution of the strains in

the central zone of the specimen is also plotted, the figure shows the strains at failure for both

tests where ε𝑥 measured at failure is 2.9 % and 3.2 % while ε𝑦 is 0.7 % and 0.8%

respectively.

Considering the equi-biaxial test, the effect of the geometrical accuracy of the specimen can

be clearly noticed in the loading curves where a desynchronization was noticed between the

Fx and Fy loading curves in the Test01 while for Test02 the test was perfectly in equibiaxial

state. In addition, a decrease of the loading curves in both tests was presented during the

biaxial test due to the adhesive crack initiation. However, the equibiaxial tensile test

continues until the failure of the composite in the central zone as shown in the loading curves.

The variation of the strains in the central zone of the specimen for both tests is also presented.

The curves show that in both tests, a similar amount of strain decrease is obtained due to the

adhesive crack initiation for both tests (a decrease of 0.5% of 𝜀𝑥 and ε𝑦 strains). However, the

strains at the onset of failure of the composite are similar for both equibiaxial tests (strain

level of 2.3%).


77

Biaxial


free/1

1/2

1/1


composite subjected to free/1, 1/2, and 1/1 biaxial ratios

Figure 3.25 shows the strain fields obtained for one equibiaxial test when 80% of the failure

load is reached (equivalent to a displacement Ux =0.96 mm for Test01). The figure shows the

homogeneous strain field in the central zone of the composite. Table 3.6 shows the failure

loads and strains in the composite material for all the biaxial tests presented in this section.


78

Figure 3.25 𝜀𝑥 and 𝜀𝑦 strains distribution in the central zone of the M21 composite subjected to 1/1

biaxial ratio (test01)

Table 3.6 Load and strain results at the onset of failure of M21 composite subjected to biaxial ratios

(free/1, 1/2, and 1/1)


free/1 (Test01) -0.7 2.8 - 27.6

free/1 (Test02) -0.9 2.99 - 28.3

1/2 (Test01) 2.9 0.7 35 25.6

1/2 (Test02) 3.2 0.8 33 22.5

1/1 (Test01) 2.35 2.2 33.3 33

1/1 (Test02) 2.32 2.36 34.5 34.7

3.4.4.2 Results for biaxial ratios 1/3, 1/1.4, and 1/1.2

In order to obtain more results about the behavior of the composite under biaxial tensile state,

different biaxial ratios were studied. The biaxial ratios were selected based on the previous

results where a biaxial ratio of 1/3 would give more information about the behavior of the

composite in between the two biaxial ratios free/1 and 1/2. While the biaxial ratios 1/1.2 and

1/1.4 will give more detailed results about the behavior of the composite between the biaxial

ratio of 1/2 and the equibiaxial test (1/1).

Figure 3.26 shows the evolution of the loads and strains until the failure of composite for all

three biaxial ratios. For the biaxial ratio 1/3, the same conclusion of the previous biaxial ratio

1/2 is obtained where the evolution of load in the y direction is curved while in the x direction

is linear due to the plasticity of the aluminum tabs in the y direction. In addition, an adhesive

crack initiation was also occurred for both biaxial ratios 1/1.2 and 1/1.4 but the traction

continues until the failure of the composite. A brief conclusion can be retained from these

results, when both loads Fx and Fy are both nearly dominating in the test (e.g. biaxial ratios

1/1.2, 1/1.4 and 1/1) the adhesive tends to crack before the composite failure but the traction


79

continues until the composite failure. Considering the tests where one load is dominating with

respect to the other (e.g. Biaxial ratio of free/1 and 1/3), the composite completely fails

before the adhesive. The strains are also measured for all three tests. The figure shows that

having a higher difference between the biaxial applied displacements in x and y directions

(Biaxial ratio 1/3) is more similar to the uniaxial test on the cruciform specimen where a very

small displacement is noticed in the non-dominating direction (𝜀𝑥 = -0.003). While biaxial

test 1/1.2 is closer to the equibiaxial test. Table 3.7 summarizes all the failure loads and

strains for the different biaxial ratios.

Biaxial


1/3

1/1.4

1/1.2


composite subjected to 1/3, 1/1.4, and 1/1.2 biaxial ratios


80

Table 3.7 Load and strain results at the onset of failure of M21 composite subjected to 1/3, 1/1.4, and

1/1.2 biaxial ratios


1/3 -0.3 2.97 12.6 30.5

1/1.4 1.5 2.75 28.6 32.8

1/1.2 1.6 2.5 30.6 31.8

3.4.5. Strain path

The strain results obtained for each biaxial tensile ratio are presented together in the strain

path plot. The strain path plot is defined based on the variation of the major strain with

respect to the minor strain. For each biaxial displacement ratio, one curve is plotted which

shows the evolution of the strains from the beginning of the test towards the failure of the

material. Figure 3.27 and Figure 3.28 show the experimental strain paths obtained for both

M9.6 and M21 composites, respectively. First, considering the common biaxial test ratios

1/1, 1/2, and 1/free for both composite materials, it can be noticed that both materials have a

similar variation of strains where a linear variation is observed for the biaxial ratio 1/1 in both

cases. Moreover, the effect of the aluminum plasticity on the strain evolution is clearly seen

on the uniaxial tests of both composites where a non-linear behavior is observed when the

major strain reaches 1.5 % and 1.3 % for M9.6 and M21 respectively.

Figure 3.27 Strain path plot for M9.6 composite


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Figure 3.28 Strain path plot for M21 composite

3.5. Stress calculation

3.5.1. Stress calculation methods

The calculation of the stresses in a simple uniaxial coupon specimen is usually obtained by

dividing the applied load on the cross-sectional area of the composite. However, the stress

calculation in a cruciform specimen is more difficult because of the complex shape of the

specimen. Since the specimen has a curved fillet between the loading arms, the applied loads

on each arm are not directly transferred to the central zone and a load sharing between the

arms occurs. The amount of the load which goes through the central zone is not easily

obtained. However, different methods were proposed in order to calculate the stresses in the

composite with and without relying on the measured strains. In this thesis, only two methods

will be presented and discussed: (i) constitutive laws and (ii) equivalent section method.

3.5.1.1 Constitutive laws equation

The constitutive equations (also known as generalized Hook’s law) consists of calculating the

stresses in the composite according to measured strains and the material properties (obtained

from the simple uniaxial tests). This method was used by different researchers in the biaxial

testing of metallic and composite cruciform specimens. The following equation (Eq 3.5)

shows the relation between the stresses and the strains in both x and y directions.

𝜎𝑥,𝑒𝑥𝑝 = (𝜀𝑥 + 𝜗𝑥𝑦𝜀𝑦)𝐸𝑥

1 − 𝜗𝑥𝑦2 𝜎𝑦,𝑒𝑥𝑝 =

(𝜀𝑦 + 𝜗𝑥𝑦𝜀𝑥)𝐸𝑦

1 − 𝜗𝑥𝑦2 Eq 3.5


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3.5.1.2 Equivalent section method

The second method for the stress calculation is the equivalent section method. This method is

based on the idea that the amount of the transmitted load towards the central zone is always

the same with respect to the applied on each loading arm. The equivalent section, which is the

section subjected to the load including the load sharing between the arms, is constant for

every biaxial tensile ratio for the same cruciform shape.

To explain this method in details, first the strains are measured in the central zone of the

specimen using strain gages or DIC for one single biaxial test. The stresses are calculated

using the constitutive laws and the equivalent section is defined as the ratio of the applied

load over the calculated stresses in each direction. Thus, the stresses for each different biaxial

ratio can be calculated as the ratio between the applied loads on each arm divided by the

equivalent section as shown in Eq 3.6. Figure 3.29 shows a scheme of the concept of

equivalent section stress calculation method.

𝜎𝑥,𝑒𝑥𝑝 = 𝐹𝑥

𝐴𝑒𝑞𝑢,𝑥 𝜎𝑦,𝑒𝑥𝑝 =

𝐹𝑦

𝐴𝑒𝑞𝑢,𝑦 Eq 3.6

Figure 3.29 Equivalent section stress calculation principle

3.5.2. Stress results for M9.6 composite

The evolution of the stresses with respect to the applied displacement for each biaxial ratio

are investigated. First, the constitutive method was used in order to analyze the composite

behavior and to obtain the biaxial stresses at the onset of failure for each test. Using the

previously calculated material parameters, Figure 3.30 shows the variation of both 𝜎𝑥 and σ𝑦

stresses with respect to the applied displacement Ux for all three biaxial ratios 1/free, 1/2, and

1/1, respectively. It can be concluded from these figures that for this type of composites, the


83

stresses at failure increase when the material is subjected to biaxial stress state (𝜎𝑥= 840

MPa) compared to the uniaxial state (𝜎𝑥= 700 MPa). Considering the 1/free ratio, a negative

value of σ𝑦 (-100 MPa) was obtained according to the equations, this can be explained based

on the structure of the specimen (Aluminum tabs, adhesive, and shape) an extra compression

occurs which generates a compressive stresses in the central zone of the composite.

Biaxial ratio Stress evolution

1/free

1/2

1/1

Figure 3.30 Evolution of the stresses 𝜎𝑥 and 𝜎𝑦 with respect to applied displacement Ux (mm) for

M9.6 composite subjected to 1/free,1/2, and 1/1 biaxial ratios


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The calculated stresses obtained from the equi-biaxial tensile test are used in order to

investigate the variation of the equivalent section with respect to the applied displacement.

The ratio of the applied load in each direction over the calculated stresses are plotted in

Figure 3.31. It can be seen that the equivalent section at the beginning of the test changes

with an important amount. This is due to the low value of the stresses at the beginning of the

test. However, the curves show that the equivalent section obtained at the moment of the

composite failure are Aequ,x = 55 mm2 and Aequ,y = 51 mm2.

Figure 3.31 Evolution the calculated equivalent sections Aequ,x and Aequ,y for M9.6 composite subjected

to 1/1 biaxial displacement ratio

Since in these tests, the stresses at failure are the most important, the failure stresses for both

biaxial ratios 1/free and 1/2 are calculated using the obtained equivalent sections and the

failure loads. Table 3.8 shows the failure stresses of each biaxial test using both constitutive

laws and equivalent section method. A disadvantage of the equivalent section is shown in the

table for the biaxial ratio 1/free where the generated stresses in the transverse direction σ𝑦

(due to the specimen structure) cannot be calculated since there is no load in that direction.

The table also shows that a small difference was noticed between both results except for the

minor stresses σ𝑥 obtained in the biaxial ratio 1/2 where an error of 29% is presented between

both methods. However, this method fits well with the major stresses obtained using the

constitutive equations. The equivalent section calculation method will be presented more in

details for the M21 composite because of the high amount of tests performed using that

composite.

Table 3.8 Stress results at the onset of failure of M9.6 composite subjected to 1/free, 1/2, and 1/1

biaxial displacement

Constitutive laws Equivalent section

Biaxial ratio 𝛔𝒙 (MPa) 𝛔𝒚 (MPa) 𝛔𝒙 (MPa) 𝛔𝒚 (MPa)

1/free 696 -107 633 -

1/1 740 839 740 839

1/2 441 774 572 784


85

3.5.3. Stress results for M21 composite

Figure 3.32 and Figure 3.33 shows the stress results for all the repeated biaxial tests (free/1,

1/2, 1/1) and single tests (1/3, 1/1.4 and 1/1.2) respectively. According to the failure stresses

for all the biaxial ratios, it can be noticed that the major stress (𝜎𝑦) at failure is around 550

MPa for all the biaxial ratios. In addition, the fact that negative stresses are obtained again in

the uniaxial test on the cruciform specimen proves again that the reason is the specimen itself.

Applying a small displacement in the perpendicular direction will compensate this extra

compression as shown in the biaxial ratio of 1/3 where the minor stresses are zero (𝜎𝑥).

Results can show the different stress state for each biaxial ratio.


86


free/1

1/2

1/1


composite subjected to free/1, 1/2, and 1/1 biaxial ratios


87


1/3

1/1.4

1/1.2


composite subjected to 1/3, 1/1.4, and 1/1.2 biaxial ratios

The calculation of the stresses is also investigated using the equivalent section method. The

equibiaxial test has been considered in order to calculate the equivalent section in both x and

y directions. The evolution of the equivalent sections is also plotted with respect to the

applied displacement in Figure 3.34. The curve shows two important change of slope where

the first is at a displacement of 0.4mm, this change can be explained as when the traction

begins, the stresses in the composite are low since the load is still compensated by the

aluminum tabs, thus an increase of the load will be presented while the stresses in the


88

composite are still low, this will make a sudden increase in the equivalent section value

calculated. The second change of slope is at a displacement of 1.15 mm, which is related to

the adhesive crack initiation which was previously explained. However, both curves show

that at the moment of failure, the equivalent section is equal which means the same amount of

load is transmitted towards the central zone. This can show again the equi-biaxial state of the

test.

According to the curves, the equivalent sections at the onset of failure of the composite are

Aequ,x = 61.5 mm2 and Aequ,y = 63 mm2. The calculated sections are used in order to calculate

the failure stresses of the composite subjected to different biaxial displacement ratios.

Figure 3.34 Evolution the calculated equivalent sections Aequ,x and Aequ,y for M21 composite subjected

to 1/1 biaxial ratio

Table 3.9 shows the failure stresses obtained from both constitutive equations and equivalent

section methods (only one test is chosen between the repeated tests since they already gave

similar results).

The analysis of these results will be composed of two parts according to both major and

minor stresses. For the major stresses, it can be seen that using the equivalent section method

gives a good results that correlates well with the results obtained from the constitutive

equations, this is due to the domination of the load in the major direction compared to the

minor direction. The major stresses are always around the level of 500 MPa using both stress

calculation methods which prove that the equivalent section method works well on the major

direction.

For the minor stresses, a high scatter was presented between both methods (e.g. biaxial ratio

1/3 where the calculated minor stresses are 4 and 206 MPa using constitutive equations and

equivalent section respectively). The stresses obtained using the equivalent section are higher

than the one obtained from constitutive equation which shows that the estimation of the

equivalent section based on a single equi-biaxial test is not sufficient.


89

To decrease the errors between the minor stresses calculated from the equivalent section and

the constitutive laws, an updated relation integrating the biaxial ratio (Eq 3.7) seems more

appropriated to evaluate the minor stresses using the applied load and the equivalent section

as shown in Table 3.9.

(𝜎𝑥)𝑢𝑝𝑑𝑎𝑡𝑒𝑑 = 1

𝑟

𝐹𝑥

𝐴𝑒𝑞𝑢,𝑥 Eq 3.7

Table 3.9 Stress results at the onset of failure of M21 composite subjected to free/1, 1/2, 1/1, 1/3,

1/1.2, and 1/1.4 biaxial ratios

Constitutive laws Equivalent section Updated Equivalent Section

Biaxial ratio 𝛔𝒙 (MPa) 𝛔𝒚 (MPa) 𝛔𝒙 (MPa) 𝛔𝒚 (MPa) 𝛔𝒙 (MPa) 𝛔𝒚 (MPa)

free/1 -73 518 - 451 - 451

1/1 497 470 497 470 497 470

1/2 218 572 420 555 210 555

1/3 4 560.5 206 484 68 484

1/1.2 370 507 497 504 399 504

1/1.4 346.5 560 465 522 333 522

3.6. Conclusion

In this chapter, the experimental part of this thesis is presented in details. At first, uniaxial

tests were performed in order to obtain the correct material mechanical properties. A

comparison between the use of extensometer and the digital image correlation technique

shows a good agreement and validate the use of the correlation technique in the biaxial tests.

This method has an important advantage over other strain measurement techniques by

measuring the strain fields in the whole central zone.

After adapting reliable digital image correlation parameters, the experimental biaxial results

were presented. The measurement of the strains in the composite central zone is required in

order to analyze the behavior of the composite without relying on the stresses (which is not

simply obtained experimentally). A study was performed on two plain-weave composites

M9.6 and M21 subjected to different biaxial displacement conditions. According to the strain

results, a homogenous strain state was obtained in the central zone for both composites and

the failure strains were measured under different biaxial ratios.

For the stress calculation, the equivalent section method shows good results concerning the

major stresses at failure in the composite according only to the load sensors. However,

obtaining the minor stresses is more complicated using this method especially when the

minor stress is very low compared to the major stress. On the other hand, the constitutive

laws equations were used also in order to calculate the experimental stresses according to the

material parameters and the measured strains. In the next chapter, the obtained stress and

strain results will be compared to the results obtained from the numerical model. In addition,


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the strain path and failure envelop of both composites will be presented and compared with

different failure criteria.

Chapter 4. Potential and Applications of the Biaxial Tensile Test

91

Chapter 4. Influence of Loading Paths

on Failure Limits

Chapter 4. Influence of Loading Paths on Failure Limits

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4.1. Introduction

In the following chapter, first, a comparison between numerical and experimental results is

performed. Thereafter, the experimental failure envelop of the studied composite is

generated, according to the stress results obtained in the previous chapter, and is compared to

different interactive and non-interactive failure criteria. Finally, an introduction to the

perspectives of this thesis is shown in the last part of this chapter. Different notched

cruciform specimens are tested in order to investigate the effect of the biaxial loads on the

opening of the notches.

4.2. Comparison between numerical and experimental results

As mentioned previously in the introduction, both numerical and experimental approaches are

compared together, through the measured and calculated strains. In the first part of this

section, the modeling of the adhesive joint between the aluminum tabs and the composites is

introduced into the numerical model in order to have a more reliable model and to obtain

more information about the interface. After, a comparison between both numerical and

experimental results are presented. Finally, both numerical and experimental strain paths are

generated and discussed.

4.2.1. Cohesive elements modeling

A more reliable numerical model is defined by including a modeling of the adhesive joint

between the aluminum tabs and the composite into the previous FE model proposed in

Chapter 2. A 3D model of the adhesive layer is defined by the use of 8 nodes cohesive

elements (COH3D8) available in the Abaqus software environment (Figure 4.1). The

adhesive used in this thesis, AEROBOND 7300, is considered as isotropic material with a

Young modulus of 2 GPa and shear modulus of 770 MPa, as given by the supplier. It should

be noted that the rheological behavior model associated with the cohesive elements used to

model the adhesive joint does not consider any damage effect or failure criteria. A thickness

of 0.1 mm is considered to model the adhesive film between the aluminum tabs and the

composite which is similar to the adhesive thickness in the experimental phase. A tie

constraint is defined between the cohesive and its surrounding surfaces (Figure 4.1).


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Figure 4.1 One-quarter of the cruciform specimen with adhesive modeling

4.2.2. Boundary conditions

The comparison between the experimental and numerical strain results will be presented in

the following section. The numerical model main goals are just to define the specimen shape.

The experimental loading curves (obtained from the machine sensors) are introduced as

boundary conditions in the numerical model. The load is preferred over the applied

displacement because in the cruciform specimen the displacement sensors are far from the

specimen arms (as shown in Chapter 3) and there is no direct information about the

displacement applied at the extremity of the loading arms of the specimen while the applied

load are directly transmitted to the arms and accurately obtained via the loading sensors. In

ABAQUS, the measured load is introduced as a concentrated force applied on the reference

point which is coupled with each loading arm. The applied loading is then a uniformly

distributed pressure on the arm edges. Since one-fourth of the specimen is considered, half of

the load is applied on the both loading arms (Figure 4.2).

Figure 4.2 Boundary conditions for the cruciform specimen modeling

Figure 4.3 shows the major stress distribution (σ𝑦) in the composite for both cases, with and

without cohesive elements, for a free/1 biaxial loading ratio at the moment of failure of the

composite. Similar results were obtained with and without adhesive modeling where the

adhesive does not affect the composite strain (or stress) level in the central part of the

cruciform specimen since damage is not considered in the model of cohesive elements.

Nevertheless, these cohesive elements allow to obtain the stress state at the interface between


94

the tabs and the composite, in particular the shear stresses (𝜎𝑦𝑧) as shown in Figure 4.4. Note

that, the high shear stress level obtained at the moment of the composite failure is reached

numerically due to the missing of damage modeling.

Figure 4.3 Major stress distribution 𝜎𝑦 (𝑀𝑃𝑎) in composite: (a) with adhesive modeling and (b)

without adhesive modeling under free/1 biaxial ratio

Figure 4.4 Shear stress distribution 𝜎𝑦𝑧 (𝑀𝑃𝑎) in the adhesive joint

4.2.3. Numerical model results

4.2.3.1 Strain results

The comparison is presented only for the M21 composite. Since both stress and strain

evolution show a similar behavior, only strain results are presented. First, the results of the

biaxial ratio free/1 are presented and discussed. Figure 4.5 shows the evolution of strain with

respect to the applied displacement Uy (mm). It could be noticed that at the beginning of the

test, a similar behavior between the numerical and experimental results is observed during the

purely elastic behavior of the cruciform specimen at a low level of deformations. However, a

change in the slope of the major numerical strain (ε𝑦) curve is observed (at Uy = 0.55 mm).


95

This is due to the fact that when the load starts to increase with a significant amount, the

plastic deformation tends to increase in the aluminum tabs which generates a high shear stress

level in the adhesive joint leading to the appearance of damage in the adhesive. Since this

damage is not taken into consideration in the numerical model, the load transfer towards the

composite central zone tends to be higher numerically compared to the experimental results.

Figure 4.6 shows the shear stresses σyz in the adhesive joint when a displacement of 0.55 mm

is reached. It could be seen that a shear level of 34 MPa is reached at the extremity of the

central area, this shear stress level is concentrated at the extremity of the central zone, this

stress level is still comparable to the shear strength of the adhesive (16.8 MPa) obtained in

the experimental double lap shear test of the adhesive (previously shown in Chapter 3).

However, it should be mentioned that the shear obtained numerically is a local shear result

while the shear obtained experimentally is averaged along the adhesive zone.

Considering the minor strain (ε𝑥), a compressive strain is shown numerically from the

beginning of the test (which is the case of an ideal adhesion between the tabs and the

composite). From an experimental point of view, negative strains are also obtained even if

some deviation is observed between the evolutions of the numerical and experimental strains.

These differences may result from the difficulty of calculating such low levels of strain by

DIC technic and/or from imprecise modelling of the adhesive's behavior, particularly in

compression.

It is difficult to measure that low level of strain experimentally using the image correlation. In

addition, the compressive behavior of the adhesive is not evaluated experimentally and the

high difference between both numerical and experimental curves show that the adhesive is

not transmitting the compressive load generated by the shape of the specimen (because of the

aluminum compression) towards to the composite.

To obtain more information about the aluminum tabs, the evolution of the equivalent plastic

strain (PEEQ) in the aluminum tabs with respect to the applied displacement is plotted. The

PEEQ is taken at the element where the plastic strain is maximal as shown in Figure 4.7. The

curve shows that the plastic strain tends to increase early (Uy=0.25 mm), this early plasticity

would increase the shear stresses in the adhesive joint until its failure initiation at higher

applied displacement (Uy=0.55 mm) as shown previously.


96


subjected to free/1 biaxial ratio

Figure 4.6 Shear stresses 𝜎𝑦𝑧(MPa) in the adhesive when a displacement of 0.65 mm is reached in

M21 composite subjected to free/1 biaxial ratio

Figure 4.7 Evolution of numerical and experimental strains in the composite and the numerical

equivalent plastic strain (PEEQ) in the aluminum tabs for biaxial ratio free/1

Among the other biaxial ratios, since similar conclusions can be drawn for all, only the

results of the three ratios 1/2, 1/3, and 1/1 are presented in Figure 1.9. The

experimental/numerical comparison is carried out until the initiation of the first observed

rupture (in the adhesive) as explained in chapter 3, section 3.4.4. Overall, the trends observed

from the numerical simulations are consistent with the experimental results, particularly for

the 1/2 and 1/3 ratios. The aluminum plasticity effect is clearly seen in the numerical model


97

based on the change of the slope of the major strain curves (e.g. at displacement of 0.45 and

0.75 mm for biaxial ratios 1/2 and 1/1 respectively).

Biaxial ratio Strain evolution

1/2

1/3

1/1


subjected to 1/2, 1/3, and 1/1 biaxial ratios


98

4.2.4. Strain path

For each biaxial displacement ratio, the strain path defined by the variation of the major strain

with respect to the minor strain is plotted up to the failure. Based on the M21 composite

results, both numerical and experimental strain paths are plotted as shown in Figure 4.9.

A relatively good agreement is observed between the experimental and simulated strain paths

for the ratios 1/1, 1/1.2 and 1/1.4, although the simulations underestimate the strain levels.

For the others ratios (free/1, 1/3 and 1/2), good trends are observed even if the low level of

the minor strain leads to greater deviations, particularly at the beginning of the tests.

Regarding the uniaxial results (free/1 ratio), it could be noticed that after reaching a major

deformation of 1%, the experimental and numerical behavior of the composite is similar

based on the slope of both curves.

Figure 4.9 Evolution of major and minor strains of M21 composite under different strain paths

4.3. Failure of composite materials

4.3.1. Introduction

Over the past years, different studies were presented in order to understand and predict the

failure of composite materials. The goal behind these studies was to generate a reliable failure

theory that predicts well the behavior of composite materials subjected to different loading

conditions. The numerous number of different approaches confirm that the failure prediction


99

is still an interesting issue. The validation of any failure theory requires a comparison with

experimental data under a wide range of loadings [82].

In 2004, the first world-wide failure exercise (WWFE-I) [83] was done where 19 different

failure theories were presented and compared with experimental tests under 2D stress state.

As results, some of the theories were modified and improved for the first time after 40 years.

However, it was concluded that there is a lack in the failure prediction of composite materials

under multiaxial stress state. In addition, some failure theories were only validated for certain

materials, specific layup, and stress state. Figure 4.10 shows an example of failure envelops

generated by different failure theories. It can be noticed that some failure criteria give the

same results in some zones (traction/traction and compression/compression) but at the same

time, a difference between the failure stresses generated by the criteria is presented in other

zones (traction/compression and compression/traction zone). At the same time, the biaxial

failure envelops generated by Tsai, Wolfe, and Rotem show a high scatter between the stress

results which validate the problems of generating a reliable failure theory for composites.

Another important notice is the dispersion of the experimental results for the same loading

condition, this idea shows the missing of reliable experimental methods to obtain the failure

stresses (for example, the results obtained between the biaxial ratios SR 1:0 and 2:1, also the

failure stresses obtained for the ratio 0:-1).

Based on this conclusion, the second world-wide failure exercise (WWFE-II) was performed;

the study focuses on the failure characterization of composites under tri-axial stress state.

However, the same conclusion was obtained and only a few theories could give acceptable

correlation (within ±50%) with test data for 75% of the test cases [84].

Figure 4.10 Comparison between the predicted and measured biaxial final failure stresses for(0°/ ±

45°/90°) AS4/3501-6 laminates [83]


100

Previous experiments were investigated in order to generate complete failure envelops of

different types of composite materials. The generation of a complete biaxial failure envelop

requires testing the specimen through all combinations of tension and compression as shown

previously in Figure 4.10 where SR is the Stress Ratio between the x and y directions.

The failure criteria allows the designers to evaluate the strength of the material. For

composite materials, understanding the failure is not even straightforward because of the

different failure modes (fiber failure, matrix failure, fiber-matrix debonding … etc.). In

addition, the use of laminates made the study even more complicated since the complete

failure of the material is usually reached after a progressive failure process that begins from a

single lamina failure (first ply failure FPF). To be more specific about the materials used in

this thesis, woven laminates are more complex because of the weave geometry where a

different arrangement of fibers affect the stress concentrations in the laminate.

The failure criteria, in general, can be divided into two main categories: (i) interactive and (ii)

non-interactive failure theories. The first one takes into account the interaction between the

stresses in different directions and the influence of each one of the other (example: Hill, Tsai-

Hill [85], Norris distortional energy [86]) by using a single high order equation including all

stress (or strain) components. The non-interactive criteria defines the failure by comparing

each stress (or strain) component with its corresponding strength. It doesn’t take the

interaction between stresses into account (example: Max stress and Max strain criteria) [87].

Some criteria also separate the failure modes of fibers and matrix in both tension and

compression loading state, e.g. Hashin failure criterion. The failure of the composite is

defined when either fiber or matrix failure equation is validated. Note that most of the

interactive failure criteria were only defined for unidirectional composites. The

American Institute of Aeronautics and Astronautics (AIAA) failure criteria survey [88] shows

that 80% of the respondents used the four failure criteria: Max strain, Max stress, Hill-Tsai

and Tsai-Wu. It was also shown that the Maximum Strain criterion is the mostly used one

among the others (Figure 4.11).


101

Figure 4.11 Results of AIAA failure criteria survey [88]

4.3.2. Failure criteria selected

In this work, only the envelope at rupture of the tensile/tensile quadrant of the M21 material

is generated using the failure experimental stresses since only biaxial tensile ratios can be

imposed. Considering the non-interactive criteria, both maximum stress (Eq.4.1 where

σ𝑢,𝑥 and σ𝑢,𝑦 represents the ultimate stresses of the composite in both x and y directions

respectively) and maximum strain (Eq.4.2) were selected to plot failure contours. Since the

failure envelop is presented in the stress space, the maximum strain failure criterion is also

considered in this same stress space as expressed by the Eq.4.3 by means of the compliance

matrix (S𝑖𝑗). The interactive failure criteria that include the compressive strength of the

material, e.g. Tsai-Wu, cannot be plotted because of the lack of experimental data. Tsai-Hill

failure criterion is usually only defined for unidirectional composites. Applying Tsai-Hill

failure equation on the studied plain-weave fabric leads to Maximum stress failure criterion.

σ𝑥 < σ𝑢,𝑥 or σ𝑦 < σ𝑢,𝑦 Eq.4.1

ε𝑥 < 𝜀𝑢,𝑥 or 𝜀𝑦 < ε𝑢,𝑦 Eq.4.2

σ𝑥 − 𝜗𝑥𝑦σ𝑦 < σ𝑢,𝑥 or σ𝑦 − 𝜗𝑦𝑥σ𝑥 < σ𝑢,𝑦 Eq.4.3

An interactive failure criterion was proposed by Charles Norris [86] for orthotropic materials

subjected to combined stresses. This criterion was validated for different orthotropic

materials such as plywood and glass composite and is given in equation Eq.4.4 for a plane

stress state where σ𝑢,𝑥𝑦 represents the shear strength of the composite.

(σ𝑥

σ𝑢,𝑥)

2

−σ𝑥σ𝑦

σ𝑢,𝑥σ𝑢,𝑦+ (

σ𝑦

σ𝑢,𝑦)

2

+ (σ𝑥𝑦

σ𝑢,𝑥𝑦)

2

= 1 Eq.4.4


102

4.3.3. Failure envelop of M21 composite

According to the experimental stresses, the failure envelop of M21 composite is plotted.

Since the test is symmetric, the failure contour is symmetric with respect to the 45° line (σ𝑦 =

σ𝑥 line) and the results obtained from a test of a biaxial ratio 1/2 are similar to the results

obtained from test 2/1. Figure 4.12 shows the failure contour of M21 composite obtained

from the experimental failure stresses including the repeated tests.

Figure 4.12 Experimental failure envelop of M21 composite

In order to present clear results, the composite failure envelop is compared first with the non-

interactive failure criteria (Max Stress and Max Strain) then with interactive criterion

(Norris).

Regarding for example the Maximum Stress criterion, this criterion depends simply on the

ultimate stresses of the material. It could be calibrated based on one single test. Figure 4.13

shows two Maximum Stress failure envelops plotted based on three different failure stresses :

σ𝑢,𝑥 = 518 MPa based on the free/1 test, σ𝑢,𝑥 = 560 MPa based on the 1/3 biaxial test, and

σ𝑢,𝑥 = 503 MPa based on 1/1. It could be noticed that, calibrating the Max Stress criterion

based on the free/1 test increases the errors with respect to the results from the biaxial ratios

1/2 (error of 9%) and 1/1.4 (error of 7.5%). However, the Max Stress criterion, calibrated

using the equi-biaxial stress results (Max Stress_1/1), is the most conservative since it fits

well the biaxial failure stress results (ratio 1/1.2) and underestimates the other ratios (free/1,

1/3 and 1/2).


103

More, if the same criterion is calibrated using the 1/3 biaxial, an inverse results were obtained

where the errors decrease with biaxial ratios 1/2 and 1/1.4. The errors increase with respect to

the results obtained from the biaxial ratios 1/1.2 (error of 10%) and 1/1 (error of 11%).

However, it’s better to adapt the criterion that underestimates some experimental points (Max

Stress_1/1) since in that case, the composite material during application would not reach the

real experimental failure stresses making the material always in a safe zone.

Considering the Max strain criterion, the Max Strain failure envelop based on the free/1 ratio

is plotted. It is clearly seen in the figure that this criterion cannot estimate the behavior of the

composite subjected to different biaxial loading conditions. Even when calibrating this

criterion according to different biaxial testing results, important errors will be noticed

(calibrating the criterion based on the equi-biaxial test would give nearly a similar envelop of

the free/1 since both have same major ultimate stress levels of 500 MPa)

Figure 4.13 Experimental and different non interactive criteria failure envelops of M21 composite

Similarly to the Maximum Stress criterion, the Norris distortional energy can also be

calibrated according to different biaxial tests in order to have the less amount of error with

the experimental results. Three ultimate major stress values are considered: i) from the

uniaxial test (518 MPa as shown in Norris_free/1 curve), ii) from the equi-biaxial test (503

MPa as shown in Norris_1/1 curve), and iii) from the biaxial ratio 1/3 (560.5 MPa as shown

in Norris_1/3).

According to these curves, it can be concluded that, when calibrating this failure criterion

according to the equi-biaxial test stress results, this fits well with the experimental results for

the biaxial ratios 1/2, 1/1.2,1/1.4, and 1/1. However, the error increases with the experimental

free/1 and 1/3 results where the criterion underestimates the failure results obtained on 1/3


104

biaxial ratio with an error of 10%. Meanwhile, calibrating the criterion according to the free/1

stress results would decrease the error with 1/3 biaxial ratio to 7% but increase the errors with

the experimental results obtained from 1/2, 1/1.2, 1/1.4 and 1/1 results.

In addition, it could be seen on the experimental failure envelop a difference of the results

obtained from the repeated tests (biaxial ratio 1/2) where the major stress reaches 636 MPa in

one test while the other reaches 572 MPa. Calibrating the Norris criterion on the major stress

level obtained from the 1/3 biaxial test (560.5 MPa as shown in Norris_1/3 curve) fits well

with the high stress level of 636 MPa reached in the biaxial test 1/2. This would keep the

material in the safe zone if this level of stresses is applied, but it highly overestimates the

other biaxial ratios at the same time.

According to these results, it can be noticed that the Norris distortional energy failure

criterion calibrated based on the equi-biaxial stress results (ratio 1/1) predicts well

experimental failure stresses but overestimates the failure stresses for the biaxial ratio 1/1.2.

Figure 4.14 Experimental and Norris failure criterion envelops of M21 composite

As conclusion, each criterion can be calibrated according to single or multiple tests in order

to have a less amount of errors with the experimental results and predict well the failure of

the composite. However, it was proved that Norris failure criterion predicts well the behavior

of the composite subjected to different biaxial tensile ratios by performing a single equi-

biaxial tensile test on the material and calibrating the failure criterion based on the failure

stresses obtained from that test. So, if the material is subjected to different biaxial stress state

during the engineering application, Norris distortional energy could be adapted.


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Figure 4.15 Experimental and selected failure criteria envelops of M21 composite

4.4. Biaxial testing of Notched composite

4.4.1. Introduction

Previous studies were proposed regarding the crack growth problems in fracture mechanics of

materials under different crack propagation modes. The study of multiple cracks propagated

from a local failure initiation is an example of an impacted stressed panels. Thus, a generated

cruciform crack would represent the impact on the material as mentioned by previous

researchers [89], [90]. Li [91] also shows how Vickers indentation hardness test (which is a

powerful technique to measure the mechanical properties of materials) generates a cruciform

crack in the material. Moreover, the study of the cruciform crack propagation in aerospace

engineering structures is important for the sake of safety of these structures where invisible

micro-cracks may grow slowly as mentioned by Mishra et al. [92].

Regarding the biaxial tests on composite cruciform specimens, different studies were

performed on cruciform notched specimens in order to analyze the effect of the crack on the

behavior of the composite. Rashedi et al. [15] studied the effect of the existence of a crack

glass fiber composite laminate under different biaxial loading conditions. The study was

covering two crack lengths (40 and 45 mm) with two different orientations (0° and 45°). A

numerical comparison between four different cruciform geometries is performed in order to

choose the shape that generates the most uniform far field stresses. A cruciform shaped

geometry with corner fillet radius of 25 mm is chosen. It is shown that the failure stresses of

the notched specimen under biaxial loading conditions is higher compared to uniaxial loading

case. In addition, the study shows that a higher crack length would decrease the failure


106

stresses of the material, and the cruciform specimen with a 45° crack reaches higher stress

level until the failure compared to the one with 0° crack for different biaxial loading ratios

(1/0, 1/0.3, 1/0.6, and 1/0.8). Moreno et al. [93] used the eXtended Finite Element Method

(XFEM) in order to simulate the crack initiation and propagation in chopped glass reinforced

composite with quasi-isotropic homogeneous elastic behavior. The study covered a wide

range of biaxial loading ratios (0.5/1, 1/1, 1/1.5, and 1/-1). The comparison between

numerical and experimental results show a good agreement regarding the crack initiation,

propagation path and velocity, and the stress levels. However, up to now there is no work

regarding the study of the cruciform crack effect in a cruciform specimen subjected to

multiple biaxial loading conditions.

In this section, six cruciform specimens were prepared with a cruciform notch inside the

central zone of the composite in order to study the behavior of the notched composite

subjected to different biaxial stress states. The composite M21 with a 0.5 mm thickness is

used in this study. The idea of having a cruciform notch in the composite specimen is similar

to the previous literature part where the specimen including the cruciform crack is considered

as a representation of an impacted composite subjected to multi-axial stress state. First, the

specimen preparation are shown regarding the method used in order to have a similar defined

crack. After, the biaxial testing results are presented and analyzed.

4.4.2. Notched specimen preparation

In order to prepare the notched specimens, a specific device for holding the specimen and

positioning a drilling guide in the center of the cruciform specimen is manufactured by 3D

printing of ABS (Acrylonitrile-Butadiene-Styrene) plastic material (shown in Figure 4.16).

The cruciform notch is achieved by punching the central area of the composite specimen with

a sharp metallic blade guided through the ABS guide. A cruciform notch with 8 mm total

length in each direction is achieved, for recall, the central zone has a diameter of 15 mm.

After, the aluminum tabs are glued on both sides of the composite using the same adhesive

used previously in order to obtain the final cruciform specimen. Figure 4.17 shows an

example of the final form of the notched specimen before biaxial testing.


107

Figure 4.16 ABS guide for notch generation in the center of the cruciform specimen

Figure 4.17 Assembled cruciform notched specimen for biaxial test

4.4.3. Biaxial test results

As shown previously in Figure 4.17, the specimen is painted in order to obtain the strain

distribution around the crack during the biaxial test. Three biaxial tensile testing ratios are

studied free/1, 1/2, and 1/1. Each test is performed twice in order to verify the repeatability of

the results. In this part, the evolution of the loads with respect to the applied displacement is

presented for each notch and un-notched specimen and for each biaxial loading condition. In

addition, the strain fields are presented for each test of the notched specimens in order to

show the strain distribution around the notches and the propagation of the cruciform crack

based on the type of loading. For the DIC parameters, small size of facets (16 pixels × 16

pixels) and small distances (16 pixels) are used. The repeatability of the tests is shown


108

through the loading curves. Only one test is considered to present the strain field results for

each biaxial loading condition. Figure 4.18 shows an example of failed notched cruciform

specimens under different biaxial loading conditions. Note that, the results are still under

analysis and the presented results are only an introduction for a perspective work.

Figure 4.18 Failed cruciform notched specimens under different biaxial tensile ratio

4.4.3.1 Uniaxial test

The first comparison that can be made is related to the evolution of the load during the test.

The applied load in the uniaxial test on the cruciform specimen with and without the notch is

presented in Figure 4.19. The curves are plotted until failure of the un-notched composite.

Two conclusions can be directly extracted from these curves. First, the existence of the notch

affects clearly the rigidity of the whole specimen where a difference between the loading

curves is noticed after 0.8 mm of applied displacement. In addition, the repeatability of the

results is clearly verified when comparing the two tests with notches. Both tests present

exactly the same loading curves.

To quantify the difference between the results, the ratio and the difference between the loads

obtained from the un-notched and notched biaxial tests are plotted in Figure 4.20. A

difference of 2.8 kN is obtained at the moment of failure of the composite. This is due to the

effect of the crack propagation during the test which leads to a reduced resistant central

section compared to the un-notched one. This can also be concluded regarding the ratio

between the loads where a ratio of 1.1 is observed. In addition, both comparisons show a

similar evolution of the difference and the ratio with respect to the un-notched specimen

when the load starts to be more important and significant (displacement of 0.8 mm) which

shows the repeatability of the results.


109

Figure 4.19 Load Fy (N) results for notched and un-notched specimen under biaxial free/1 ratio

Figure 4.20 Load (a) difference and (b) ratio between notched and un-notched specimen under

biaxial free/1 ratio

Since similar results were obtained for both repeated tests, only one of the notched tests,

Test01, is analyzed. The ε𝑥 and ε𝑦 strain fields are obtained (at two different applied

displacement of 0.3 and 0.78 mm) using the digital image correlation method and presented

in Figure 4.21 and Figure 4.22, respectively. The figures show globally the low ε𝑥 strain level

which is perpendicular to the loading direction, also it could be noticed that a compressive

ε𝑥 is obtained at the y-direction crack tip since the traction is parallel to the direction of this

crack. The crack propagation starts when a displacement of 0.78 mm is reached (this was

already shown previously in the loading curves), Figure 4.22 shows the ε𝑦 strain

concentration at the x-direction crack tip at the moment of the crack opening where a strain

level of 2.5 % is reached.


110

Figure 4.21 𝜀𝑥 strain fields of notched specimen under biaxial free/1 ratio

Figure 4.22 𝜀𝑦 strain fields of notched specimen under biaxial free/1 ratio

4.4.3.2 Biaxial test (1/2)

Considering the biaxial ratio 1/2, the loading curves are plotted for both un-notched and

notched specimens. The results show that the notch has more impact on the y-direction

loading arm (which is the dominating load in this test) compared to the x-direction loading. A

higher difference is noticed between the un-notched and notched y-direction loading curves

compared to the one obtained for the x-direction as shown in Figure 4.23. To quantify this

effect of adding a cruciform crack, the difference and the ratio between the loads are

presented in Figure 4.24 and Figure 4.25, for the x and y directions respectively. From these

results, it can be seen that the difference between the load in the y-direction increases during

the tensile test (the difference reaches 1.8 kN). This is due to the crack opening in the same

direction of the load which leads to a decrease in the global rigidity of the specimen.


111

Figure 4.23 Loads results (a) Fx and (b) Fy for notched and un-notched specimen under biaxial 1/2

ratio

Figure 4.24 X-direction load (a) difference and (b) ratio between notched and un-notched specimen

under biaxial 1/2 ratio

Figure 4.25 Y-direction load (a) difference and (b) ratio between notched and un-notched specimen

under biaxial 1/2 ratio


112

The ε𝑥 and ε𝑦 strain fields are also presented for this test in order to present the evolution of

the strain distribution during the crack propagation (as shown in Figure 4.26 and Figure

4.27). A displacement Ux of 0.5 mm refers to the moment of crack opening. The figures

show again the high strain concentration around the crack tip which is due to the crack

opening and closing. Moreover, it could be noticed that based on the type of loading, and

since the y-direction load is dominating, the ε𝑦 strain is maximum at the border of horizontal

crack and the crack opening will occurs there.

Figure 4.26 𝜀𝑥 strain fields of notched specimen under biaxial 1/2 ratio

Figure 4.27 𝜀𝑦 strain fields of notched specimen under biaxial 1/2 ratio

4.4.3.3 Biaxial test (1/1)

The evolution of the load for both notched and un-notched specimens under an equi-biaxial

tensile test are presented in Figure 4.28. Contrarily to the previous biaxial ratios, the

existence of the cruciform crack did not show a significant effect on the loading evolution

compared to the un-notched loading results. In addition, a small drop of the loading curves


113

are presented in all three curves in both x and y directions. For the un-notched specimen, this

drop is already seen before and explained by the adhesive crack initiation. For the notched

specimens, the drop happened exactly when the crack propagation reaches the extremity of

the central zone which means when the crack propagates towards the glued zone.

Figure 4.29 and Figure 4.30 show the strain distribution in the central zone of the notched

specimen subject to equi-biaxial loading conditions. When a displacement of 1.06 mm is

reached in each loading arm, the crack starts to propagate. The figures show clearly the

maximization of the strains in both x and y directions around the crack tip. The same

maximum strain level of ε𝑥 on the border of the vertical crack and ε𝑦 on the horizontal crack

shows the equi-biaxial state at the moment of crack opening.

Figure 4.28 Loads results (a) Fx and (b) Fy for notched and un-notched specimen under biaxial 1/1

ratio

Figure 4.29 𝜀𝑥 strain fields of notched specimen under biaxial 1/1 ratio


114

Figure 4.30 𝜀𝑦 strain fields of notched specimen under biaxial 1/1 ratio

4.5. Conclusion

This chapter discussed the comparison between the numerical and experimental stress and

strain results obtained from different biaxial tensile ratios. To obtain more information about

the interface between the aluminum tabs and the composite, the adhesive is introduced into

the numerical model without modeling its damage criterion. The results show that even

without modeling the adhesive damage, the numerical model still gives acceptable results. In

addition, the numerical and experimental major and minor strain results are plotted for M21

composite where a similar behavior is noticed for different strain paths.

According to the obtained stresses, the traction/traction quadrant of the failure envelop is also

plotted and compared to different failure criteria. The generated failure envelop is compared

to three different failure criteria (Max Stress, Max Strain and Norris distortional energy). The

results show that calibrating the Maximum Stress failure criterion based on an equi-biaxial

tensile test predicts well the failure of the material for different loading conditions. Norris

distortional energy criterion is also in a relatively good agreement with experimental results

when calibrated on the equi-biaxial tensile test (biaxial ratio 1/1)

The final part of this chapter consists of an introduction for perspective works of this thesis

regarding biaxial tensile tests of cruciform notched specimens subjected to different biaxial

tensile ratios. The loading curves clearly show the stiffness changes of the whole specimen

for the biaxial ratio free/1 and 1/2 due to the notches in the center of the composite. The

strain fields are also plotted for this kind of specimens under different biaxial loading

conditions.

Conclusion and Perspectives

115


The main objective of this work is achieved by defining a cruciform shape for the in-plane

biaxial testing of composite materials. A dedicated shape is designed by the finite element

method to obtain the failure stresses in the central area of the specimen under different biaxial

loading ratios. The cruciform specimen is composed of two aluminum tabs glued on both

sides of the studied composite in order to avoid thickness reduction of the composite by

machining. Based on a comparative study between different shapes, the final shape of the

cruciform specimen exhibits a thickness reduction in the aluminum tabs and a circular central

zone (composite zone) with corner fillet between the loading arms. This shape generates a

maximum and homogeneous stress state in the central area of the composite.

The designed cruciform shape is validated by experiments on two plain-weave glass/epoxy

composites with two different thicknesses under different biaxial tensile ratios. Digital Image

Correlation (DIC) technique is applied in the biaxial test in order to obtain the experimental

strain fields in the central zone where the failure is successfully observed. As expected, a

homogeneous strain distribution is obtained in the central zone. The stresses are calculated

based on the measured strains and the material properties of the composite materials using the

constitutive law equations.

The predictive and experimental strain paths (from in-plane minor and major strains) for all

the biaxial tensile ratios show that very different strain states can be reached in the composite.

The traction/traction quadrant of the failure envelop is obtained based on the failure stresses.

The experimental failure envelop is compared with the predictions of three different failure

criteria (Max Stress, Max Strain and Norris distortional energy). It is shown that for a

composite part subjected to different biaxial stress states, the Norris failure criterion is well

adapted to predict the failure of the composite material.

According to this work, several perspectives may be envisaged in order to take better

advantage of the in-plane biaxial tensile test.

The first tests on specimens with cruciform crack in the central zone show that the

loading biaxial ratio has an influence on the difference of measured maximum forces

between cracked and non-cracked specimens. If a cruciform crack is representative of

the defect induced by an impact on a composite sheet, tests with different loading


116

conditions and crack orientations could be helpful to calibrate failure criteria

associated to impacted sheets. In a more general framework, the test could be used to

evaluate the efficiency of a repair method by comparing the bearing of non-cracked

and repaired specimens under different loading ratios.

Thanks to the technical specifications of the biaxial machine, which can be associated

to an air flow generator, in-plane biaxial tests can be performed on a wide range of

temperature (from -60°C to 150°C) and strain rate. Then, the actual operating

conditions of the composite part can be reproduced during the test in order to fully

characterize the composite material for these conditions. New experimental database

will be provided in order to enhance the reliability of failure criteria on a wide range

of operating conditions.

117

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Appendix

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127

Titre : Caractérisation mécanique des matériaux composites à partir d’un essai de traction biaxiale intégrant l’effet des

chemins de déformation

Mots clés: Composites, traction biaxiale, éprouvette cruciforme, corrélation des images, chemin de déformation

Résumé : Les matériaux composites à fibres

continues sont largement utilisés dans la plupart des

applications d’ingénierie en raison de leurs propriétés

mécaniques spécifiques élevées. Les essais uniaxiaux

sont principalement utilisés pour la caractérisation de

ces matériaux en raison de leur simplicité et de leur

faible coût. Cependant, ces essais ne sont pas adaptés

pour caractériser le comportement du matériau

soumis à un état de contrainte multiaxiale. Pour

mieux caractériser leur comportement, des essais

biaxiaux ont été développés. Parmi les différentes

techniques de traction biaxiale, l’essai de traction sur

éprouvettes cruciformes est l'une des méthodes les

plus connues. Cependant, la difficulté de cette

méthode réside dans le choix de la forme de

l’éprouvette qui doit présenter une rupture au centre.

Dans cette étude, une nouvelle éprouvette cruciforme

est définie pour des essais de traction biaxiale d'un

composite verre/époxy. Les dimensions finales de

l’éprouvette cruciforme sont définies par une étude

paramétrique numérique tout en respectant certaines

contraintes expérimentales telles que la capacité de la

machine de traction biaxiale. Une validation

expérimentale est réalisée sur deux tissus de

composites verre/époxy de différentes épaisseurs. Les

essais ont été réalisés pour différentes conditions de

chargement biaxial. La distribution des déformations

dans la zone centrale de l’éprouvette est obtenue à

l'aide de la technique de corrélation d'image (DIC).

L’enveloppe à la rupture est tracée et comparée aux

prédictions de trois critères à la rupture (Max Stress,

Max Strain, et Norris).

Title : Mechanical characterization of composite materials subjected to biaxial tensile test integrating the effect of strain

paths

Keywords : Composites, biaxial test, cruciform specimen, digital image correlation, strain path, failure envelop

Abstract: Continuous fiber composite materials

have become extensively used in most engineering

applications due to their high specific mechanical

properties. Uniaxial tests are mostly used for the

characterization of composite materials because of

their simplicity and low cost. However, these tests

are not adapted for the characterization of the

behavior of these materials subjected to multiaxial

stress state. For a better characterization of their

behavior, biaxial tests were developed. Among

different biaxial testing techniques, the in-plane

biaxial testing of cruciform specimen is one of the

most known methods. However, the main difficulty

of this method is the design of a cruciform specimen

that fails in the center.

In this work, a new cruciform specimen is designed

for the biaxial tensile testing of plain-weave

glass/epoxy composite. The final dimensions of the

specimen are defined by a numerical parametric

study while respecting some experimental constraints

such as biaxial tensile machine capacity. An

experimental validation is performed on two plain-

weave glass/epoxy composites with different

thicknesses. The tests were performed under different

biaxial loading conditions. The strain distribution in

the central zone of the specimen are obtained using

the Digital Image Correlation (DIC) technique. The

failure envelop is generated and compared to the

predictions of three failure criteria (Max Stress, Max

Strain, and Norris).

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Caractérisation mécanique des matériaux composites à