Strain rate dependency of the properties of a unidirectional thermoplastic composite material

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Strain rate dependency o f the properties o f a

unidirectional thermoplastic composite material

By

Nikolaos Papadakis

A thesis submitted in partial fulfilment of the requirements

for the degree of Doctor of Philosophy in Engineering

V olum e 1 - T hesis

University o f Warwick, School o f Engineering

October 2002

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Best Copy

Available

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Abstract

This research work established the strain rate dependency of the the mechanical properties of a highly orientated glass fibre/thermoplastic composite lamina and validated a model for Computer Aided Engi neering (CAE). The mechanical properties examined for strain rate dependency were elasticity, strength and damage evolution at a ply level.

A rigourous statistical methodology were established and implemented through mechanical testing to gether with processing of the results for the development o f semi-empirical strain rate models.

Two different methods of data acquisition were considered, specifically strain measurement using videoextensometry and contacting extensometry. The resulting strain measurements were then computed. Video extensometry appeared to have clear advantages, however, scatter in the response was appreciably higher compared to the contacting extensometry. This was due to the much smaller scale of gauge length for strain measurement.

A rigourous validation methodology was further complemented through a statistical procedure and tool kit (utilising statistical tools and procedures like density distributions plot, hypothesis testing, analysis of variance). The statistical tool kit was developed to enable objective assessment of strain rate dependency and to establish the quality of a relationship (model) should one exist for the range of mechanical properties tested. Using this validation methodology, a semi empirical strain rate dependent model was validated for elasticity strength and damage evolution.

The effect of strain rate on the above mechanical properties was investigated for Plytron1 Ai. The

P lytron ™ material was supplied by Borealis as a lOOmm-wide, 0.22[mm]-deep tape, comprising aligned

continuous glass fibres in a polypropylene matrix. To manufacture a laminate, the tape was laid-up ply-by-ply into an unconsolidated stack. This stack was then consolidated using under pressure and heat according to a Warwick Manufacturing Group’s proprietary membrane-forming process [!]. For the purposes of this study, specimens were machined in accordance with ISO-527-4 from 4 different layup sequences: [0°]4, [±45°]2,, [+45°]s and [±67.5°]2„. The specimens were tested at 5, 50 and 500[min/min] crosshead displacement rates using monotonic and cyclic loading.

FYom this investigation, over the examined strain rate range, the longitudinal tensile modulus increased with strain rate, while the shear modulus and Poisson’s ratio decreased. The transverse tensile modulus did not exhibit any statistically significant difference. The shear failure stress and the longitudinal tensile failure strain and stress appeared to increase for increasing strain rate, while the shear failure strain were not strain rate dependent. The transverse tensile failure stress and strain did not exhibit any statistically significant strain rate dependency.

The characterisation parameters of the damage evolution were based on the global composite ply model for composites in the framework of continuum damage mechanics (CDM). This model was developed by Ladeveze et al. (.’), [It], for thermosetting composites. It was established that shear damage evolution of the thermoplastic materials exhibits different behaviour compared to thermosets. It also was established that the rate of shear damage evolution decreases with increasing strain rate and that the point that shear damage initiates increases with increasing strain rate.

All testing was conducted with INSTRON 4505 universal testing machine instrumented with a 100[kN] load cell. Contacting extensometry and videoextensometry has been examined as data acquisition meth ods. It was established in this work that contacting extensometry provided data with less scatter, however the videoextensometry exhibited significant advantages.

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Having established and validated semi-empirical rate dependent models, for the characterisation parame ters to service the CDM models, CAE models were established and validated using a well known explicit FE numerical simulation. To maintain rigour, the validation methodology employed new metrics to en able objective comparison between FE and experimental results. These metrics are Pearson correlation coefficient and correlation range ratio. The comparison of experimental to FEM results revealed that the available models predict adequately well the stiffness of laminates as expected.

The onset of failure is predicted at significantly lower strains compared to the experimental results (depended on layup - usually 30% of the total failure strain). The premature failure is attributed to the failure criterion implementation at ply level and/or the definition of the boundary conditions.

______________________ Strain rate effects on G FRTP properties____________________________

Keywords : strain rate, characterisation, shear damage, mechanical properties, glass/thermoplastic

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Dedication

This is dedicated to G. Tagaras, for being an inspiration during my academic life. And to my grandfather Nikos I. Papadakis, who never got to see the completed work.

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Strain rate effects on GFRTP properties

There is surely nothing other than the single purpose o f the present moment. A man’s whole life is a succession of moment after moment. If one fully understands the present moment, there will

be nothing else to do, and nothing left to pursue.

The end is important in all things.

Yamamoto Tsunetomo HAGAKURE Japan, September 1716

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Acknowledgments

I would like to thank my supervisors Dr Paul W ood and Dr Mark Pharaoh for their support and guidance during my studies and in the preparation o f this research thesis. M y acknowledgements also go to Neil Reynolds whose support and help throughout the PhD was invaluable. Working together with these people on the C R A C T A C project has been a privilege.

I would also like to thank the technicians (especially Walter Cosgriph and R ob Bromley) in the A TC for doing such a good job cutting the specimens I’ve requested. (I am sorry they were so many).

A big thank you goes to every friend that helped me have a fun time in Warwick while completing this PhD. I cannot thank all o f you in this section, but you know who you are.

Most o f all I would like to acknowledge the contribution o f my parents and grandparents to the completion of this work. Thanks for teaching me the value o f hard work and persistence and for supporting my effort all these years, although I know that y o u ’ve missed me as much as I did. To my only brother, Tony, thank you for listening and supporting me when I had those bad days.

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Declaration

I declare that all the work described in this report was undertaken by myself (unless otherwise acknowledged in the text) and that none o f the work has been previously submitted for any academic degree. All sources of quoted information have been acknowledged by means o f references.

N. Papadakis October 2002

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II

APPENDICES

i

A Review O f Classical Laminate Theory (C L T ) 1 A .l Introduction To Laminate Analysis... 2

A .1.1 Hooke's Law For Orthotropic Materials... 3

A. 1.2 Hooke’s Law For Transversely Isotropic M a te r ia ls ... 3

A.2 Lamina C haracterisation ... 4

A .2.1 Hooke's Law For A 2D Unidirectional L a m in a ... 4

A .2.2 Hooke's Law For A 2D Angle L a m in a ... 5

A .2.3 In-Plane Loading And Bending Of A Lam ina... 7

A.3 Laminate Analysis ... 9

A .3.1 Classical Laminate Theory For A Laminate... 10

A. 3.2 Reverse Laminate Theory... 12

B Failure theories 14 B. 0.3 Tsai-Wu Theory ... 16

C Programming Scripts 19

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________________________ Strain rate effects on G FR TP properties____________________________

C .l Processing O f Raw Data... 19

C.1.1 Mntlab... 19

C .l. 1.1 Calculations For [0°],4 la m in a t e ... 19

C .l .1.2 Calculations For [± 4 5 0]2„ la m in a t e ... 21

C .l. 1.3 Calculations For [45°]g la m in a te... 26

C .l. 1.4 Calculations For [±67.50]o, la m in a t e ... 29

C .l .2 Matlab Auxiliary Scripts... 34

C .l.2.1 Bisection Script For Calculation O f The Plasticity Exponent . . . 34

C. 1.2.2 Computation O f Individual Laminate Stiffness... 35

C. 1.2.3 Calculation O f Laminate Stiffness Based On Individual Plies . . . 35

C. 1.2.4 File With Properties D a t a s e t ... 36

C. 1.2.5 Function Used By Bisection M e t h o d ... 37

C .l.2.6 Initialisation File For [0°]i lam in ate... 37

C. 1.2.7 Initialisation File For [±45°]2il lam inate... 38

C .l.2.8 Initialisation File For [+45°]* la m in a te ... 39

C .l.2.9 Initialisation File For [±67.5°]* la m in a te ... 41

C .l.2.10 Strain Rate Selection F i l e ... 42

C .l.2.11 Function For Transformation O f Angles... 43

C. 1.2.12 Matrix Trimming Function... 43

C.1.3 Video Extensometry Specific Scripts ... 44

C .l.3.1 C yclic Loading Weedout F un ction ... 44

C .l.3.2 Read Video Extensometry File F u n c t io n ... 47

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C. 1.3.3 Video Extensometry Filtering F u n ctio n ... 48

C.1.4 lnstron Contacting Extensorneter Specific S c r i p t s ... 51

C .1.4.1 Cyclic Loading Weedout F u n ction ... 51

C .1.4.2 Read lnstron Contacting Extensometer File F u n c tio n ... 54

C.1.4.3 Instron Contacting Extensometer Filtering Function ... 55

C . 1.4.4 Conversion Instron Data To Neutral Format F un ction... 58

C.2 Statistical Processing - I I ... 58

C.2.1 Listing For Generic F u n ctio n s... 59

C.2.2 Listing For Young Modulus Obtained From [Oo]., L a m in a te ... 64

C.2.3 Listing For Shear Strength Obtained From [±45o]2, L a m in a t e ... 69

C.2.4 Listing For Transverse Strain Obtained From [+45o]s L a m in a t e ... 73

C . 2.5 Listing For Critical Transverse Damage Limit Obtained From [±45o]2. Lam inate ... 77

C. 3 Experimental Vs. FE Comparison- Visual Basic For Applications... 83

D S ta tistica l P r o c e s s in g o f E x p e rim e n ta l R e su lts 95 D . l Properties Obtained From [0°].j Test... 96

I). 1.1 Strain Rate... 96

D . l .2 Strain Rate Effects On E la s t ic it y ... 97

D . l .2.1 Poisson’s ratio v\ < ... 97

D .l.3 Strain Hate Effects On S tren gth ... 102

D .1.3.1 Longitudinal Tensile Failure Strain f u ... 102

D. 1.3.2 Longitudinal Tensile Failure Stress < T n ... 108 Strain rate effects on GFRTP properties

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D.1.4 Strain Energy Density Up To F a ilu r e ... 114

D.2 Properties Obtained From [±45°]i Test... 120

D.2.1 Shear Strain Rate... 120

D.2.2 Strain Rate Effects On E la s t i c it y ... 121

D.2.2.1 Shear Modulus G ' u ... 121

D.2.3 Strain Rate Effects On Strength... 127

D.2.3.1 Shear Failure Strain 7 1 2 ... 127

D.2.3.2 Shear Failure Stress T\->... 132

D.2.4 Strain Rate Effects On Damage Evolution... 137

D.2.4.1 Initial Shear Damage Limit Value V o ... 137

D.2.4.2 Critical Shear Damage limit Value Y ,... 143

D.2.4.3 Critical Shear Damage Limit Value Y u ... 148

D.3 Properties Obtained From [+45°]s Test... 154

D.3.1 Transverse Strain Rate... 154

D.3.2 Strain Rate Effects O11 E la s t i c it y ... 155

D.3.2.1 Transverse Tensile Modulus E -n ... 155

D.3.3 Strain Rate Effects On Strength... 161

D .3.3.1 Transverse Failure Strain £•_>■_> ... 161

D.3.3.2 Transverse Tensile Failure Stress 0 ^ 2... 166

D.3.3.3 Coupling Factor Between Plastic And Shear Strains ,l 2 ... 171

D.4 Properties Obtained From [±67.5°] i Tost... 176

D.4.1 Strain Rate... 176 Strain rate effects on GFETP properties

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___________________ Strain rate effects on GFRTP properties____________________________ D.4.1.1 Transverse Strain Rate... 176 D.4.1.2 Shear Strain Rate... 177 D.4.2 Strain Rate Effects On Transverse Damage E v o lu tio n ... 179 D.4.2.1 Initial Transverse Damage Limit V’J ... 179 D.4.2.2 Critical Transverse Damage Limit V ' ' ... 186 D.4.2.3 Brittle Transverse Damage Limit Y g ... 192 D.4.3 Coupling Factor Between Transverse And Shear Damage b ... 198

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List of Figures

1.1 Categorisation o f the reinforcement type o f composite materials... 2 1.2 Differences of the molecular chains between a thermoset and a thermoplastic ((•!,

P-73])... 8 1.3 Example o f damage evolution as a function o f strain... 11

2.1 Comparison o f factors o f amplification vs strain for different analytical strain rate law... 16 2.2 Schematic o f an Instrumented Falling Weight (source llamouda and Hashmi] >])

testing configuration (left hand side) and the Bramuzzo (source Bramuzzo [ti]) con figuration (right hand side)... 24 2.3 Common Charpy and Izod test geometries], ]... 26 2.1 Generic representation o! a compressive Split Hopkinson pressure bar apparatus [S]. 28 2.5 Schematic drawing o f the gas-gun testing configuration (source Delfosse [it]... 31 2.(i Schematic drawing o f the external explosive pressurisation testing configuration

(source llamouda and Hnshmi[5]... 33 2.7 Comparison o f the Strain rale range o f different dynamic testing techniques. . . . 34

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2.8 Percentage o f broken glass fibres vs. real stress for Weibull distribution with pa rameters a = .31 and fi = .6 (taken from X ia’s work)... 37

3.1 Vacuum former configuration... 84 3.2 ISO-527-4 1994, T ype IB specim en... 85 3.3 Marked Video extensometry specimen... 89 3.4 Failure location partitions for the ASTM dogbone... 90 3.5 Flowchart o f the calculation dependencies o f the procedure to obtain the Ladeveze

parameters from the [O0]^ laminate specimens... 91 3.0 Flowchart o f the calculation dependencies o f the procedure to obtain the Ladeveze

parameters from the [±45°]j., laminate specimens... 93 3.7 Flowchart o f the calculation dependencies o f the procedure to obtain the Ladeveze

parameters from the [-+-45®]« laminate specimens... 94 3.8 Flowchart o f the calculation dependencies o f the procedure to obtain the Ladeveze

parameters from the [±67°]-2, laminate specimens... 96 3.9 Instron raw data processing m ethodology flowchart... 98 3.10 Instron raw data processing methodology flowchart... 99 3.11 Filtering m ethodology flowchart... 100 3.12 Analysis o f results m ethodology... 101

1.1 Typical failed unidirectional specimens at different strain rates - 5(top), 50(middle) Strain rate effects on GFRTP properties

and 500(bottom )[m m /m in] crosshead displacement rate... 104

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4.2 Comparison o f representative stress vs. longitudinal and transverse strain curves of [0"].i laminated specimens at 5, 50 and 500 [m m/m in] crosshead displacement rates as obtained using the Contacting Extensometers. The longitudinal strain (along the testing direction) are positive, whilst transverse strains are negative... 105 4..'$ Failed [±45°)2,test specimen and magnification o f the failure surface... 106 4.4 Comparison o f representative stress vs. longitudinal and transverse strain curves

o f [±45°]os laminated specimens at 5, 50 and 500[mm/min] crosshead displacement rates as obtained using the Contacting Extensometers... 107 4.5 Typical failure o f a [4-45°]^ uniaxially loaded dogbone specimen... 108 4.6 Comparison o f representative stress vs. longitudinal and transverse strain curves of

[+45]* at 5, 50 and 500 [mm/min] crosshead displacement, rates as obtained using the Contacting Extensometers... 109 4.7 Typical failure o f a [±67.5°]i„ uniaxially loaded dogbone specimen... 110 4.8 Comparison o f representative stress vs. longitudinal and transverse strain curves at

5, 50 and 500 [mm/min]crosshead displacement rates as obtained using the con tacting extensometers... I l l 4.9 Comparison o f representative st ress vs. longitudinal and transverse strain curves o f

[0°]4 laminated specimens at 5, 50 and 500 [nun/min] cross head displacement rates as obtained from videooxtonsometry apparatus... 112 4.10 Comparison o f representative stress vs. longitudinal and transverse st rain curves ol

[±45°]2, laminated specimens at 5, 50 and 500 [mm/min] crosshead displacement rates as obtained from videooxtonsometry apparatus... 113

Strain rate effects on G F R T P properties

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_________________________Strain rate effects on G F R TP properties____________________________ 4.11 Comparison o f representative stress vs. longitudinal and transverse strain curves of

[+45°)s laminated specimens at 5, 50 and 500 [mm/min] crosshead displacement rates as obtained from videoextensometry apparatus... 114 4.12 Comparison o f representative stress vs. longitudinal and transverse strain curves of

[±G7.5°]-2, laminated specimens at 5, 50 and 500 [mm/min] crosshead displacement rates as obtained front videoextensometry apparatus... 115 4.13 Unidirectional Plytron tensile longit udinal Young’s Modulus vs. logarithm o f strain

rate... 115 4.14 Longitudinal tensile modulus vs. logarithm o f strain rate... 138 4.15 Density plots o f the longitudinal tensile modulus at a) all displacement rates, and

b),e) and d) at each different crosshead displacement rate separately... 141 4.16 Various curve fitted models to experimental data... 145

4.17 Transverse tensile modulus vs. logarithm o f strain rate as obtained from the tensile testing o f a [-4-45]« laminate... 146 4.18 Shear modulus vs. logarithm of shear strain rate as obtained from the tensile testing

o f a [±45]2a laminate... 148 4.19 Major Poisson’s ratio vs. strain rate logarithm... 149 4.20 Longitudinal tensile failure strain vs. strain rate logarithm... 151 1.21 Longitudinal tensile failure stress vs. strain rate logarithm... 153 4.22 Transverse tensile failure strain vs. strain rate logarithm... 155 1.23 transverse tensile failure stress vs. strain rate logarithm... 157 4.24 Shear failure strain o f vs. shear strain rate logarit hm... 159

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4.25 Shear failure stress vs. shear strain rate logarithm... 160 4.26 Master Shear Damage Law Graph for thermoplastic... 162 4.27 Initial shear damage limit value vs. shear strain rate logarithm... 164 4.28 Critical shear damage limit value vs. shear strain rate logarithm... 166 4.29 Elementary shear damage limit value vs. shear strain rate logarithm... 168 4.30 Initial transverse damage limit vs. logarithm o f strain rate as obtained from the

tensile testing o f a [± 6 7 ]j, laminate... 170 4.31 Critical transverse damage limit vs. strain rate logarithm... 172 4.32 Brittle transverse damage limit vs. strain rate logarithm... 174 1.33 Coupling factor between plastic and shear strains vs. strain rate logarithm ... 176 4.34 Coupling factor between transverse and shear damage vs. transverse strain rate

logarithm... 177

5.1 Representation o f Finite element model with the nodes involved on t he grip bound ary conditions... 184 5.2 Nodes used to compute the longitudinal strain, transverse strain and longitudinal

force... 184 5.3 Internal energy vs. total displacement o f deformed specimen... 186

6 .1 FEM st less vs. st rain curves o f [O' ] | specimen at 5. 50 and 50(1 [nun/tuin] erosshead displacement rate mechanical properties... 192 0.2 Principal st ress contour plot o f [0"]| specimen prior to first element elimination at

different crosshead displacement rates (presented from top to bottom 5. 50 and 500[mni/miri] crosshead displacement r a t e s ) ... 193

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6.3 FEM plot o f [0°] i laminate after the first element elimination at different erosshead

Strain rate effects on GFRTP properties______________________

0.4 0.5 0.0 0.7 0.8 Of) O . H l

displacement rates (presented from top to bottom 5, 50 and 5(X)[mm/min] crosshead displacement rates)... 194 FEM stress vs. strain curves of [±45°]-js specimen at 5, 50 and 500 [mm/min] crosshead displacement, rate mechanical properties... 195 Comparison o f FEM Stress vs. strain curve's (± 4 5 0]2s Elytron specimen at 500 [mm/min] crosshead displacement rate mechanical properties with and without implementation of failure criterion... 196 Principal st ress contour plot of [±45°]-_>a specimen prior to first element elimination at different erosshead displacement rates (presented from top to bottom 5, 50 and 500[m m/min] erosshead displacement r a t e s ) ... 197 FEM plot o f [±45°]-2„ Plytron specimen after the first element elimination at differ ent crosshead displacement rates (presented from top to bottom 5, 50 and 500[inm/miir] crosshcad displacement rates)... 198 FEM stress vs. strain curves o f [+45°]* specimen at 5, 50 arid 500 [nun/tnin] crosshead displacement rate mechanical properties... 199 Principal st ress contour plot of [+45°]* specimen prior to lirst element elimination at different erosshead displacement rates (presented from top to bottom 5, 50 and 500[nmi/inin] crosshead displacement r a t e s ) ... 200 FEM plot o f [+45°]» Plvtron specimen after the lirst element elimination at different crosshead displacement rates (presented from top to bottom 5, 50 and 500[nnn/inhi] crosshead displacement rates)... 201

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fi. 11 FEM stress vs. strain curves o f [±67°]-j, specimen at 5, 50 and 500 [nim/inin] crosshead displacement rate mechanical properties... 202 6.12 Principal stress contour plot o f [±67°]2S specimen prior to first element elimination

at different crosshead displacement rates (presented from top to bottom 5, 50 and 500[mm/min] crosshead displacement r a t e s ) ... 203 0.13 FEM plot o f [±G7°]2, Plytron specimen after t in- first dem ent elimination at differ

ent crosshead displacement, rates (presented from top to bottom 5, 50 and 500[mrn/min] crosshead displacement rates)... 204

7.1 Correlation methodology... 207 7.2 Issues concerning the Pearson coefficient. - Scaling and translation... 208 7.3 How a linearly spaced stress vector is used on the experimental and numerical

simulation data to obtain data suitable for computation o f the Pearson correlation coefficient... 209 7.1 Comparison o f the Pearson correlation coefficient for different curves... 210 7.5 Correlation range ratio explanation...210 7.6 Longitudinal Pearson Correlation Coefficient, results for different stacking sequences

and different, crosshead displacement, rates... 226 7.7 Transverse Pearson Correlation Coefficient results for different stacking sequences

and different crosshead displacement rates... 227 7.8 Longitudinal Correlation Mange Ratio results for different stacking sequences and

different crosshead displacement rates... 228

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7.9 Transverse Correlation Range Ratio results for different stacking sequences and Strain rate effects on G FR TP properties

different crosshead displacement rates... 230

8.1 Comparison o f three typical Master shear law curves at 5, 50 and 500[mm/inin] crosshead displacement rate as obtained from the mechanical testing o f [±45°]2* laminate... 234 8.2 Top: Experimental failure o f O ’ ] i laminate at 500[rmn/min] crosshead displacement

rate. Middle: principal stress state contour plot obtained by an FEM analysis. Bottom: FEM plot o f a specimen with eliminated elements (failure onset)... 238 8.3 Detail o f the shear failure < ibserved at tensile test ing o f O’ ],j laminate at 500[inm/min]

crosshead displacement rate... 239 8.4 Top: Experimental failure o f [±45°]2, laminate at 500[mm/min] crosshcad displace

ment rate. Middle: principal stress state contour plot obtained by tin FEM analysis, Bottom: FEM plot o f a specimen with eliminated elements (failure o n s e t ) ... 240 8.5 Top: Experimental failure o f [+450]# laminate at 5lX)[mm/min] crosshead displace

ment rate. Middle: principal st ress state contour plot obtained by an FEM analysis, Bottom: FEM plot o f a specimen with eliminated elements (failure onset) ... 241 8.0 Top: Experimental failure o f [±fi7 ']r, laminate at 500[mm/min] crosshead displace

ment. rate, Middle: principal stress state contour plot obtained by an FEM analysis. Bottom: FEM plot o f a specimen with eliminated elements (failure! o n s e t ) ... 243 8.7 Comparison o f experimental vs FEM predicted stress vs. strain curves for [0°]i

laminate at 5[mm/min] crosshcad displacement rate... 244

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8.8 Comparison o f experimental vs FEM predicted stress vs. strain curves for [±45°]2, laminate at. 5[mni/min] crosshead displacement rate...246 8.9 Comparison <>f experimental vs FEM predicted stress vs. strain curves for [±45”]2,

laminate at 5[mni/min] crosshead displacement rate - without a failure criterion. . 246 8.10 Comparison o f experimental vs FEM predicted stress vs. strain curves for [+45°]«

laminate at 5[mm/min] crosshead displacement rate...248 8.11 Comparison o f experimental vs FEM predicted stress vs. st rain curves for [±G7°]2»

laminate at 5[mm/min] crosshead displacement rate...248

A .l Local (material) and global (testing) axes o f an angle lamina... 5 A .2 Example o f the change o f stiffness parameters with fibre angle orientation... 7 A .3 Comparison o f macro vs. micro computational levels [10]... 10

D. l Longitudinal tensile modulus vs. logarithm o f strain rate... 96 D.2 Poisson’s ratio vs. logarithm o f strain rate... 97

D.3 Poisson’s ratio vs. strain rate logarithm... 98 D.4 Density plots o f Poisson’s ratio at a) all displacement rates, and b),e) and d) at

each different crosshead displacement rate separately,... 100 D.5 Longitudinal tensile failure strain vs. logarithm of strain rate... 103 D.6 Longitudinal tensile failure strain vs. strain rate logarithm... 103 D.7 Density plots o f longitudinal tensile failure strain at a) all displacement rates, and

1 »).(■) and d) at each different cross head displacement rate separately... 106 I).8 Longitudinal tensile failure stress vs. logarithm of strain rate... 108 D.9 Longitudinal tensile failure stress vs. strain rate logarithm... 109

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L).l() Density plots o f longitudinal tensile failure stress at a) all displacement rates, and

____________________ Strain rate effects on GFETP properties______________________

b ),c) and d ) at each different orosshead displacement rate separately... 112 D .ll Strain energy density up to failure vs. logarithm o f strain rate... 114 D.12 Strain energy density to failure vs. strain rate logarithm... 115 D.13 Density plots o f strain energy density up to failure at a) all displacement rates, and

b),c.) and d ) at each different crosshead displacement rate separately... 118 D.14 Logarithm o f shear strain rate vs. Crosshead displacement rate as obtained from

the tensile testing o f a [±45)2(, laminate... 120 D. 15 Conditional plot o f Shear modulus vs. logarithm o f shear strain rate as obtained

from the tensile testing o f a [±45]2, laminate, conditioned with respect o f data acquisition source and Failure location... 121 D.16 Shear modulus vs. logarithm o f shear strain rate as obtained from the tensile testing

o f a [±45)2, laminate... 122 D.17 Density plots o f the shear modulus o f at a) all displacement rates, and b ),c) and d)

at each different erosshead displacement rate separately... 125 D. 18 Conditional plot o f shear strain at failure vs. logarithm o f shear strain rate as

obtained from the tensile testing o f a [±45)2, laminate, conditioned with respect o f data acquisition source and failure location... 128 D.19 Shear failure strain vs. shear st rain rat».' logarithm... 129 D.20 Density plots o f shear failure strain at a) all displacement rates, and h),c) and d)

at each different crosshead displacement rate separately... 131

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D.21 Conditional plot o f Shear stress at failure vs. logarithm o f shear strain rate as

Strain rate effects on GFRTP properties

obtained from the tensile testing o f a [±45]2» laminate, conditioned with respect of data acquisition source and failure location... 132 D.22 Shear failure stress vs. shear strain rate logarithm... 133 D.23 Density plots of shear failure stress at a) all displacement rates, and b ),c) and d)

at each different crosshead displacement rate separately... 136 D.24 Conditional plot o f initial shear damage limit value vs. logarithm o f shear strain

rate as obtained from the tensile testing of a [±45]2, laminate, conditioned with respect o f data acquisition source and failure location... 138 D.25 Initial shear damage limit value vs. shear strain rate logarithm... 139 D.2C Density plots o f unitial shear damage limit value at a) all displacement rates, and

b ),c) and d) at each different crosshead displacement rate separately. ... 141 I).27 Conditional plot o f critical shear damage limit vs. logarithm o f shear strain rate as

obtained from the tensile testing o f a [±45]_>„ laminate, conditioned with respect of data acquisition source and Failure location... 143 D.28 Critical shear damage limit value vs. shear strain rate logarithm... 144 D.29 Density plots of critical shear damage limit value at a) all displacement rates, and

b),c) and d) at each different crosshead displacement rate separately... 147 I ).3() Condit ional [riot of elementary shear damage limit vs. logarithm of shear strain rate

as obtained from the tensile testing o f a [±45]2„ laminate, conditioned with respect o f data acquisition source and Failure locat ion... 149 I).31 Elementary shear damage limit value vs. shear strain rate logarithm... 150

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D.32 Density plots o f elementary shear damage limit value o f at a) all displacement rates,

Strain rate effects on G FRTP properties

and b ),c) and d) at each different crosshead displacement rate separately... 152 D.33 Logarithm o f Transverse Strain Rate vs. Crosshead displacement rate as obtained

from the tensile testing o f a [+45]« laminate... 154 D.34 Conditional plot o f transverse tensile modulus vs. logarithm o f shear strain rate as

obtained from the tensile testing o f a [+45]« laminate, conditioned with respect of data acquisition source and failure location... 156 D.35 Transverse tensile modulus vs. logarithm o f strain rate as obtained from the tensile

testing o f a [+45]« laminate... 157 D.36 Density plots o f the transverse tensile modulus at a) all displacement rates, and

b ),c) and d ) at each different crosshead displacement rate separately... 160 D.37 Conditional plot o f transverse tensile failure strain vs. logarithm o f shear strain rate

as obtained from the tensile testing o f a [+45]« laminate, conditioned with respect o f data acquisition source and failure location... 162 D.38 Transverse tensile failure strain o f vs. strain rate logarithm... 163 D.39 Density plots o f transverse tensile failure strain at a) all displacement rates, and

b),e) and d ) at each different crosshead displacement rate separately... 165 0.40 Conditional plot o f transverse tensile failure strews vs. logarithm o f shear st rain rate

as obtained from the tensile testing o f a [+45]« laminate, conditioned with respect o f data acquisition source and failure location... 167 IJ.41 Transverse tensile failure tensile stress vs. strain rate logarithm... 168

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D.42 Density plots of transverse' tensile failure stress at a) all displacement rates, and b),c) and d ) at each different erosshead displacement rate separately... 170 D.43 Conditional plot o f the coupling factor between plastic and shear strains vs. loga

rithm o f shear strain rate as obtained from the tensile testing o f a [+45]» laminate, conditioned with respect o f data acquisition source and failure location... 172 D.44 Coupling factor between plastic and shear strains vs. strain rate logarithm... 173 D.45 Density plots of coupling factor between plast ic and shear strains at a) all displace

ment rates, and b ),c) and d) at each different erosshead displacement rate' separately. 175 D.46 Logarithm of Transverse Strain Rate vs. Crosshead displacement rate as obtained

from the tensile testing o f a [±t>7]2., laminate... 177 D.47 Density plots of transverse strain rate for the different displacement rates... 178 D.48 Logarithm of Shear Strain Rate vs. Crosshead displacement rate as obtained from

the tensile testing of a [±67]2„ laminate... 178 D.49 Density plots of shear strain rate for the different displacement rates... 179 D.50 Conditional plot o f initial transverse damage limit vs. logarithm o f shear strain rate

as obtained from the tensile testing of a [± 6 7 ]j, laminate, conditioned with respect o f data acquisition source and failure location... 180 D.51 Initial transverse damage limit vs. logarithm o f strain rate as obtained from the

tensile testing o f a [±67]2„ laminate... 181 I).52 Density plots of initial transverse damage limit at a) all displacement rates, and

b),c) and d) at each different erosshead displacement rate separately...184

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D.53 Conditional plot o f critical transverse damage limit vs. logarithm o f shear strain rate as obtained from the tensile testing o f a [±G7]2., laminate, conditioned with respect o f data acquisition source and failure location... 187 D.54 Critical transverse damage limit vs. strain rate logarithm... 188 D.55 Density plots o f critical transverse damage limit at a) all displacement rates, and

b),e) and d) at each different crosshead displacement rate separately... 190 D.56 Conditional plot <>f brittle transverse damage limit vs. logarithm o f shear strain rate

as obtained from the tensile testing o f a [±t>7]2., laminate, conditioned with respect o f data acquisition source and failure location... 193 D.57 Brittle transverse damage limit vs. strain rate logarithm... 194 D.58 Density [¡lots o f brittle transverse damage limit at a) all displacement rates, and

b ),c) and d ) at each different erosshead displacement rate separately... 196 D.59 Conditional plot o f coupling factor between transverse and shear damage vs. log

arithm o f transverse strain rate as obtained from the tensile testing of a [itw ji, laminate, conditioned with respect of data acquisition source and failure location. 198 I).fit) Coupling factor between transverse and shear damage vs. strain rate logarithm. . 199 D.61 Density plots o f coupling factor between transverse and shear damage at a) all

displacement, rates, and b),e) and d) at each different, crosshead displacement, rate separately... 202

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List of Tables

1.1 Propert ies of E-Glass [11] [ 1 2 ] ... 6 1.2 Comparison of thermoplastic's and thermosets matrix phases [12]... 8 1.8 Properties of the E-glass/PP Plytron material according to manufacturer... 9

2.1 Formulation o f strain rate analytical models used by PAM-CRASH... 14 2.2 Summarising table o f characterisation capabilities o f dynamic testing machines. . 35 2.3 Parameters required for the complete delimtion o f the hi-phase ply definition. . . . 68 2 .1 Parameters that are obtained through a standardised test procedure... 69

3.1 Table with the number o f experiments for each erosshead displacement rate and stacking sequence... 86

1.1 Hypothesis test ing statist ics for t he equality o f means of longitudinal Young’s mod ulus... 116 1.2 llvpot la sis test mg for equality o f variances stal istics o f longit udinal Young’s modulus. 118 1.3 Ladeve/.c composite material model parameters as obtained from the Plytron [0°].«

laminated specimens at .r>[nnn/miu] erosshead displacement rate... 121 1.3 (continued) ... 122

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4.4 Ladeveze composite material model parameters as obtained from the Plytron [0°].* laminated specimens at 50[mm/min] crosshead displacement rate... 122 4.4 (continued) ... 123 4.4 (continued) ... 124 4.5 Ladeveze composite material model parameters as obtained from the Plytron [0°].|

laminated specimens at 500[mm/min] crosshead displacement rate... 124 4.5 (continued) ... 125 4.6 Mechanical properties as obtained from (±45)2* Plytron laminate experimental re

sults at 5[mm/min] crosshead displacement rate... 126 4.7 Mechanical properties as obtained from (±45)2, Plytron laminate experimental re

sults at 50[mm/min] crosshead displacement rate... 127 4.7 (continued) ... 128 4.8 Mechanical properties as obtained from [±45]a, Plytron laminate experimental re

sults at 5()0[imn/min] crosshead displacement, rate... 128 4.8 (continued) ... 129 4.9 Mechanical properties as obtained from [+45]a Plytron laminate experimental re

sults at 5[mm/min] crosshead displacement rate... 131 4.10 Mechanical properties as obtained from [+45],s Plytron laminate experimental re

sults at 50[mm/inin] crosshead displacement rate... 132 4 .1 1 Mechanical properties ns obtained from [+45]« Plytron laminate experimental re

sults at 500(tnrn/min] crosshead displacement rate... 133 Strain rate effects on G FR TP properties

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4.12 Mechanical properties as obtained from [±67.5°]j, Plytron laminate experimental results at 5[mm/min] crosshead displacement rate... 135 4.13 Mechanical properties as obtained from [±67.5°]2« Plytron laminate experimental

results at 50[mm/min] crossliead displacement rate... 135 4.13 (continued) ... 136 4.14 Mechanical properties as obtained from [±67.5°]2» Plytron laminate experimental

results at 500[mm/min] crosshead displacement rate...136 4.14 ( c o n t i n u e d ) ... 137 4.15 Statistics for longitudinal tensile modulus at different crosshead displacement rates

rates... 139 4.16 Hypothesis testing statistics for the equality of means of longitudinal tensile m od

ulus... 140 4.17 Hypothesis testing for equality of variances statistics of longitudinal tensile modulus.140 4. In Statistics for the Goodness-of-Fit o f longitudinal tensile modulus probability density

distribution... 143 4.19 ANOVA results for the selection of the strain rate model order for the longitudinal

tensile modulus... 144 4.20 Statistics for the transverse tensile modulus at different crosshead displacement

rates as obtained from a [-+-45]» laminate... 147 4.21 Statistics for shear modulus at different crosshem 1 displacement rates as obtained

from a [±45]j, laminate... 148 1.22 Statistics for the major Poisson's ratio at different crosshead displacement rates. . 150

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_____________________ Strain rate effects on GFRTP properties____________________________

4.23 Statistics for the failure strain at different cross head displacement r a t e s ... 152 4.24 Statistics for the longitudinal tensile failure stress at. different crosshead displace

ment rates... 154 4.25 Statistics for the transverse tensile failure strain at different crosshead displacement

r a t e s ... 156 4.26 Statistics for the transverse tensile failure stress at different crosshead displacement

r a d 's ... 158

4.27 Statistics for the shear failure strain at different crosshead displacement rates . . . 159

4.28 Statistics for the shear failure strain at different crosshead displacement raft's. . . 160

4.29 Statistics for the initial shear damage limit value at different crosshead displacement rates... 164 4.30 Statistics for the critical shear damage limit value at different crosshead displace

ment rates... 166 4.31 Statistics for the elementary shear damage limit value at different crosshead dis

placement rates... 168 4.32 Statistics for the initial transverse damage limit ¡it different crosshead displacement

rates as obtained from a [±67]a« laminate... 170 4.33 StiM ¡sties for the critical transverse damage limit at different crosshcad displacement

r a t e s ... 172 1.34 Statistics for the brittle transverse damage limit at different crosshead displacement

r a t e s ... 174

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4.35 Statistics for the coupling factor between plastic and shear strains at different crosshead displacement r a t e s ... 176 4.36 Statistics for the coupling factor between transverse and shear damage at different

crosshead displacement r a t e s ... 178

5.1 Ladeveze characterisation o f Plytron at different strain rates... 181

7.1 Quantitative metrics for the comparison between experimental and FEM results for the [0°]4 specimens at 5[mm/rnin] crosshead displacement rates... 211 7.1 (continued) ... 212 7.2 Quantitative metrics for the comparison between experimental and FEM results for

the [0°]i specimens at 50[mm/inin] crosshead displacement rates... 212 7.2 (continued) ... 213 7.2 (continued) ... 214 7.3 Quantitative metrics for the comparison between experimental and FEM results for

the [0°] i specimens at 500[inrn/min] crosshead displacement rates... 214 7.3 (continued) ... 215 7.4 Quantitative metrics for the comparison between experimental and FEM results for

the [± 4 5c']2, specimens at 5[mm/rnin] crosshead displacement rates...216 7.5 Quantitative metrics for the comparison between experimental and FEM results for

the [±45°)2, specimens at 50[nim/miuJ crosshead displacement rates... 216 7.5 (continued) ... 217 7.6 Quantitative metrics for the comparison between experimental and FEM results for

the [± 4 5 n]a, specimens at 5()0[mm/min] crosshead displacement rates... 217 Strain rate effects on G FRTP properties

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7.6 (continued) ... 218 7.7 Quantitative metrics for the comparison between experimental and FEM results for

the [4-45'’ ]« specimens at 5[mm/niin] crosshead displacement rates... 219 7.8 Quantitative metrics for the comparison between experimental and FEM results for

the (+45°]s specimens at 50[inin/min] crosshead displacement rates... 220 7.8 (continued) ... 221 7.9 Quantitative metrics for the comparison between experimental and FEM results for

the [+45“]s specimens at 50()[mm/niin] crosshead displacement rates...221 7.9 (continued) ... 222 7.10 Quantitat ive metrics for t he comparison bet ween experimental and FEM results for

the [±67°]i, specimens at 5[mm/min] crosshead displacement rates... 222 7.10 (continued) ... 223 7.11 Quantitative metrics lor the comparison between experimental and FEM results for

the [±670)a., specimens at 50[tnm/min] crosshead displacement rates... 223 7.11 (continued) ... 224 7.12 Quantitative metrics for the comparison between experimental and FEM results for

th<' (±67°)2* specimens at. 50()[inm/min] crosshead displacement rates... 224 7.12 (continued) ... 225

D .l Statistics for measured strain rate at different crosshead displacement rates. . . . 96 D.2 Statistics for the Poisson’s ratio at different crosshead displacement rates... 98 11.3 Hypothesis testing statistics for ('quality of means o f Poisson's ratio... 99 D. l Hypothesis testing for equality o f variances statistics o f Poisson’s ratio... 99

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D.5 Statistics for the Goodness-of-Fit, o f Poisson’s ratio probability density function. . 101 D.6 AN OVA results for the selection o f the strain rate model order o f the Poisson’s ratio. 102 D.7 Statistics for the longitudinal tensile failure strain at different crosshead displace

ment. rates... 104 I).8 Hypot hesis testing st atist ics for equality o f means o f longitudinal tensile failure strain. 104 D.9 Hypothesis testing for equality o f variances statistics o f longitudinal tensile failure

strain... 105 D. 10 Statistics for the Goodness-of-Fit o f longitudinal tensile failure st rain probability

density distribution... 107 D .ll AN OVA results for the selection o f the strain rate model order o f the longitudinal

tensile failure strain... 108 D. 12 Statistics for the longitudinal tensile failure stress at different crosshead displace

ment rates... 109 D. 13 Hypothesis test ing statistics for equality o f means of longitudinal tensile failure stress. 110 D.l 1 Hypot hesis testing for equality o f variances statistics o f longitudinal tensile failure

stress... I l l D.15 Statistics for the Goodness-of-Fit of longitudinal tensile failure stress distribution. 112 D.lti ANOVA results for t ire selection of the strain rate model order o f the longitudinal

tensile failure stress... 113 D. 17 Statistics for the strain energy density up to failure! at different strain rate's... 115 D .l8 Hypothesis testing statistics for equality erf menus o f strain energy density to failure. 116

Strain rate effects on G FR TP properties

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D. 19 Hypot hesis testing for equality o f variances statistics o f strain energy density to failure... 117 D.20 Statistics for the Goodness-of-Fit o f strain energy density to failure distribution. . 118 D.21 ANOVA results for the selection of the strain rate model order o f the strain energy

up to failure... 119 D.22 Statistics for measured shear strain rate at different, crosshead displacement rates. 120 D.23 Statistics for shear modulus at different crosshead displacement rates as obtained

from a [± 4 5 ]2, laminate... 122 D.24 Hypothesis testing statistics for the equality o f means o f shear modulus...123 D.25 Hypothesis testing for equality of variances statistics of shear modulus... 124 D.26 Statistics for the Goodness-of-Fit o f Shear modulus probability density distribution. 126 D.27 ANOVA results for the selection o f the shear strain rate model order for the shear

modulus... 127 D.28 Statistics for the shear failure strain at different erosshead displacement rates. . . 128 D.29 Hypothesis testing statistics for equality o f means o f shear failure strain...129 D.30 Hypothesis testing for equality of variances statistics of shear failure strain... 130 D IM Statistics for the Goodness-of-Fit o f shear failure strain probability density distri

bution... 132 D.32 Statistics for the shear failure stress nt different erosshead displacement rates. . . 133 D.33 Hypot hesis test ing statistics for equality o f means o f shear failure stress... 134 1) 31 Hypothesis testing for equality o f variances statistics of shear failure stress... 135 I).35 Statistics for the Goodness-of-Fit o f the shear failure stress distribution... 137

Strain rate effects on G F R T P properties

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D.36 ANOVA results for the selection o f the shear strain rat*' model order o f the shear

___________________ Strain rate effects on GFRTP properties______________________

failure stress... 137 D.37 Statistics for the initial shear damage limit value at different crosshead displacement

rates... 138 D.38 Hypothesis testing statistics for equality o f means of init ial shear damage limit value. 140 D.39 Hypothesis testing for equality o f variances statistics o f Poisson’s ratio initial shear

damage limit value... 140 D.40 Statistics for the Goodness-of-Fit o f the initial shear damage limit probability den

sity function... 142 D. 11 ANOVA results for the selection o f the shear strain rate model order o f the Initial

shear damage lim it value... 143 D. 12 Statistics for the critical shear damage limit value at different crosshead displace

ment rate«... 144 D.43 Hypothesis testing statistics for equality o f means of critical shear damage limit

value... 145 D. 14 Hypothesis testing for equality o f variances statistics o f critical shear damage limit

value... 146 D.45 Statistics for the Goodness-of-Fit o f the critical shear damage limit probability

density function... 148 D.46 ANOVA results for the selection o f the shear strain rate model order o f the Critical

shear damage limit value... 148

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D.47 Statistics for the elementary shear damage limit value at different crosshead dis placement rates... 149 D.48 Hypothesis testing statistics for equality o f means o f elementary shear damage limit

value... 151 D.49 Hypothesis testing for equality o f variances statistics o f elementary shear damage

limit value... 151 D.50 Statistics for the Goodness-of-Fit o f the elementary shear damage limit probability

density function... 153 D.51 ANOVA results for the selection o f the shear strain rate model order o f the elemen

tary shear damage limit value... 154 D.52 Statistics for measured strain rate at different crosshead displacement rates. . . . 155 D.53 Statistics for the transverse tensile modulus at different crosshead displacement

rates as obtained from a [+45]a laminate... 157 D.51 Hypothesis testing statistics for the equality o f averages o f transverse tensile m od

ulus... 158 D.55 Hypothesis testing for equality o f variances statistics of transverse tensile modulus. 159 D.56 Statistics for the Goodness-of-Fit o f Transverse tensile modulus probability density

distribution... 160 I).57 Statistics for the transverse tensile failure strain at different crosshead displacement

rates... 163 1) 58 Hypothesis testing statistics for equality o f means o f transverse tensile failure strain. 163

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D.59 Hypothesis testing for equality o f variances statistics o f transverse tensile failure

strain... 164 D.60 Statistics for the Goodness-of-Fit o f transverse tensile failure strain probability

density distribution... 166 D.61 Statistics for the transverse tensile failure stress at different crosshead displacement

rates... 168 D.62 Hypothesis testing statistics for equality o f means o f transverse tensile failure stress. 169 D.63 Hypothesis testiirg for equality o f variances statistics of transverse tensile failure

stress... 169 U.64 Statistics for the Goodness-of-Fit of transverse tensile failure stress probability den

sity distribution... 171 D.65 Statistics for the coupling factor between plastic and shear strains at. different

crosshead displacement rates... 173 D.66 Hypothesis testing statistics for equality o f means o f transverse tensile failure stress. 173 1) 67 Hypothesis testing for equality o f variances statistics o f coupling factor between

plastic and shear strains... 174 D.68 Statistics for the Goodness-of-Fit o f coupling factor between plastic and shear

strains probability density distribution... 176 D.69 Statistics for transverse strain rate at different crossbead displacement rates. . . . 177 D.70 Statistics for shear strain rate at different crosshead displacement rate's... 179 D.71 Statistics for t he initial transverse damage limit at different crosshead displacement

rates as obtained front a [±67j2, laminate... 181

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0 .7 2 Hypothesis testing statistics for the equality o f averages o f the initial transverse damage limit... 182 D.73 Hypothesis testing for equality o f variances statistics o f the initial transverse damage

limit... 183 0 .7 4 Statistics for the Goodness-of-Fit o f initial transverse damage limit probability den

sity distribution... 184 D.75 ANOVA results for the selection of the strain rate model order for the initial trans

verse damage limit... 186 D.76 Statistics for the critical transverse damage limit at different crosshead displacement

rates... 186 0.77 Hypothesis testing statistics for equality o f means o f critical transverse damage limit. 188 0 .7 s Hypothesis testing for equality of variances statistics o f critical transverse damage

limit... 189 0 .7 9 Statistics for the Goodness-of-Fit o f critical transverse damage limit probability

density distribution... 191 0 .8 0 ANOVA results for the selection of the strain rate model order o f the critical trans

verse damage limit... 192 0.81 Statistics for the brittle transverse damage limit at different crosshead displacement

rates... 192 0.82 Hypothesis testing statistics for equality o f means of brittle transverse damage limit.194 0 .8 3 Hypothesis testing for equality of variances statistics o f brittle transverse damage

limit... 195

______________ Strain rate effects on GFRTP properties____________________________

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D.84 Statistics for the Goodness-of-Fit o f brittle transverse damage limit probability

_______________________ Strain rate effects on G F R T P properties______________________

density distribution... 196 D.85 ANQVA results for the selection o f the strain rate model order o f the brittle trans

verse damage limit... 197 D.86 Statistics for the coupling factor between transverse and shear damage at different

crosshead displacement rates... 199 D.87 Hypothesis testing statistics for equality o f means o f coupling factor between trans

verse and shear damage... 200 11.88 Hypothesis testing for equality o f variances statistics o f coupling factor between

transverse and shear damage... 201 D.89 Statistics for the Goodness-of-Fit o f coupling factor between transverse and shear

damage probability density distribution... 202 !).!)() ANOVA results for the selection o f the strain rate model order o f the coupling factor

between transverse and shear damage... 203

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Abbreviations

A T C : Advanced Technology Centre;

C A E : Computer Aided Engineering;

C C D : Charge Coupled Device;

FE : Finite Elements;

c d f : Cumulative density function;

C D R : Crosshead displacement rate;

C R R : Correlation Range Ratio;

D IL A : Dynamic Interface Loading Apparatus;

G F R P : Glass fibre reinforced composite;

G F R T P : Glass fibre reinforced thermoplastic composite;

F R P : Fibre reinforced plastic;

F E : Finite Element;

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Ins : Instron;

LPCC : Longitudinal Pearson’s Correlation Coefficient;

LCRR : Longitudinal Correlation Range Ratio;

PCC : Pearson’s Correlation Coefficient;

p d f : Probability density function;

RVE : Representative volume element;

T P C C : Transverse Pearson’s Correlation Coefficient;

T C R R : Transverse Correlation Range Ratio;

U D : Uni-directional;

V E : Video-extensometry;

_______________Strain rate effects on G F R T P properties

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Notation

• General ply properties:

pUD Mass density; tUD Ply thickness; Vf : Fibre volume fraction; A : Cross-sectional area;

• Elasticity properties:

E{q : Tensile fibre Young’s modulus; E /o : Compressive fibre Y oung’s Modulus;

Ell t o> E2 2 101 £33 ( 0 : M atrix phase tensile stiffness modulii; EJ'1,.0, £ ’ScO> £33,0 : M atrix phase compression stiffness modulii ;

G?i t 0 , G i'id), G2330 : M atrix phase shear modulii;

^i2,c,oi 0, G S,c,o : M atrix phase shear modulii;

"¡211 "¡3i> " S i '■ Matrix phase Poisson’s ratio under tensile loading; 1/J5 , i/J5c, i/g r : Matrix phase Poisson’s ratio under compressive loading;