F IBER -R EINFORCED C OMPOSITELAMINATES M ICROMECHANICALMODELINGOFTHINPLYEFFECTSONMICRODAMAGEIN

M ICROMECHANICAL MODELING OF THIN PLY EFFECTS ON MICRODAMAGE IN

F ^IBER -R ^EINFORCED C OMPOSITE LAMINATES

L. Di Stasio ^1,2 , Z. Ayadi ¹ , J. Varna ²

1 EEIGM, Universit´e de Lorraine, Nancy, France 2 Division of Materials Science, Lule˚a University of Technology, Lule˚a, Sweden

IMR Meeting, Saarbr¨ucken (DE), April 6-7, 2017

Thin Ply Fiber Reinforced Polymer Laminates Objectives & Approach

Micromechanical modeling Preliminary Results & Validation Conclusions & Outlook

Appendices & References

Spread Tow Technology Thin ply effect in transverse cracking

T HIN P LY FRP L AMINATES

1,2 1 2

Spread Tow Technology: Introduction

 Firstly developed for commercial use in Japan between 1995 and 1998

 In the last decade its use has been spreading, from sports’

equipments to mission-critical applications as in the Solar Impulse 2

 Only a few producers wolrdwide: NTPT (USA-CH), Oxeon (SE), Chomarat (FR), Hexcel (USA), Technomax (JP)

(a) By North Thin Ply Technology. (b) By TeXtreme.

Spread Tow Technology Thin ply effect in transverse cracking

Spread Tow Technology: Foundations

THIN PL Y LAMINA TE

TOW≈ 12/24k fibers

CONVENTIONAL LAMINA TE

1,2 1 2

Visual Definition of Transverse Cracking

(c) By Dr. R. Olsson, Swerea, SE. (d) By Prof. Dr. E. K. Gamstedt, KTH, SE.

For a visual definition of intralaminar transverse cracking.

Spread Tow Technology Thin ply effect in transverse cracking

The Thin Ply Effect

1 2 3 4 5 6 7 8

40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115

n [−]

Y

[M P a ]

In situ transverse lamina strength Y

as a function of thickness and ply orientation [0

/90

]

[±30/90

]

[±60/90

]

[90

]

Measurements of in-situ transverse strength from D. L. Flaggs & M. H. Kural, 1982 [1].

1,2 1 2

O BJECTIVES & A PPROACH

Objectives & Approach

Objectives

 Investigate the influence of volume fraction, thin ply thickness and bounding plies’

thicknesses on crack initiation

 To infere a relationship like

G ∗c = G ∗c θ debond , ∆θ debond , E _(··) , ν _(··) , G ₍₎ , VF f , t ply , t ply

t bounding plies

!

Approach

 Design and categorization of different Representative Volume Elements (RVEs)

 Automated generation of RVEs geometry and FEM model

 Finite Element Simulation (in Abaqus)

1,2 1 2

M ICROMECHANICAL MODELING

RVEs’ Design Mesh Design G c Numerical Evaluation

From macro to micro

x, i y, j

z, k

A

A

0

A A

x, i z, k

RVE

LAMINA TE AS A 3D PLA TE 2D SECTION

1,2 1 2

Representative Volume Elements (RVEs)

i, x k, z

X 2D space

X Linear elastic materials X Displacement control

X Dirichlet-type boundary conditions X Linear elastic fracture mechanics X Contact interaction

l

l

l l

u (l, z) = ¯ ε· l u (−l, z) = −¯ε · l

C ≡ (+l, +l) D ≡ (−l, +l)

B ≡ (+l, −l) A ≡ (−l, −l)

(0, R

)

(−R

, 0)

(0,−R

) (R

, 0) Ω

Ω

R

O I L

H a

Γ

Γ

Γ

Γ

θ

∆θ ∆θ

RVEs’ Design Mesh Design c Numerical Evaluation

Mesh Design and Generation

Why a good mesh is fundamental

1. Geometric discretization has a strong effect on non-linear FEM simulations

2. Damage is a process that implies changes in geometry, i.e. generation of surfaces and domain splitting 3. Fracture mechanics quantities depends on the local mesh topology and refinement

4-step procedure for mesh generation

1. The boundary is generated patching analytical parameterizations 2. The boundary is split into a set of 4 corners (c i ) and 4 edges (e i )

3. Interior nodes are created applying transfinite interpolation using multi-dimensional linear Lagrangian interpolants P ₁ (x, p _j ) =

n X

j=1 p _j

n Y

k =1 k6=j x − x _k x _j − x k

P ₂ (x, y , p _j , q _j ) = P ₁ (x, p _j ) ⊗ P ₁ (y , q _j )

r (ξ, η) = P ₁ (ξ, e ₂ , e ₄ ) + P ₁ (η, e ₁ , e ₃ ) − P ₂ (ξ, η, c ₁ , c ₂ , c ₃ , c ₄ ) 4. The mesh is smoothed applying elliptic mesh generation

g ¹¹ r _,ξξ + 2g ¹² r _,ξη + g ²² r _,ηη = 0

1,2 1 2

Angular discretization

x, i z, k

O Ω _f

Ω _m

δ [

]

N

Angular discretization at fiber/matrix interface: δ = ³⁶⁰ _{4N α} ^◦ .

RVEs’ Design Mesh Design c Numerical Evaluation

Virtual Crack Closure Technique (VCCT)

x, i z, k

O

∆a

a ∆a

crack closed

∆a crack closed

∆u

∆w

Z

Z

X

X

G _I = Z _C ∆w _C

2B∆a G _II = X _C ∆u _C

2B∆a ⇐⇒ *DEBOND in Abaqus

1,2 1 2

J-integral evaluation

x, i z, k

O

∆a

a ∆a

crack closed

∆a crack closed

ni

Γi nnumi

Γnumi

J _i = lim

ε→0

Z

Γ ε



W (Γ) n _i − n j σ jk

∂u k (Γ, x i )

∂x i



d Γ ⇐⇒ *CONTOUR INTEGRAL in Abaqus

Crack Shape G c Computation Performances

V ALIDATION

1,2 1 2

Numerical Crack Shape

∆θ = 15 ^◦ , δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

Crack Shape c Computation Performances

Numerical Crack Shape

∆θ = 100 ^◦ , δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

1,2 1 2

VCCT Computation of Energy Release Rates

−180−170−160−150−140−130−120−110−100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0102030405060708090 100 110 120 130 140 150 160 170 180 0

5· 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

θ [

] G

Jm2



Energy release rate G

vs debond angle θ

G

G

G

δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

Crack Shape G c Computation Performances

Numerical performances

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

∆θ [

]

T ime [h ]

Simulation time as a function of debond angular semi-aperture ∆θ Total CPU Time Wallclock Time

δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

1,2 1 2

C ONCLUSIONS

Conclusions & Outlook

Conclusions

 2D micromechanical models have been developed to investigate crack initiation in thin ply laminates

 A numerical procedure has been devised and implemented to automatize the creation of FEM models

 Analyses for VF f → 0 (matrix dominated RVE) conducted to validate the model with respect to previous literature

Outlook

 Investigate the dependence on VF f , t ply , _t ^{t ply}

bounding plies and different material systems

 Study numerical performances with respect to model’s parameters

 Repeat for different RVEs and compare

1,2 1 2

A PPENDICES & R EFERENCES

Appendices Stresses References

Spread Tow Technology: Implications

 Strong reduction in ply’s thickness and weight

 Reduction in laminate’s thickness and weight

 Higher fiber volume fraction and more homogeneous fiber distribution

 Ply thickness to fiber diameter ratio decreases of at least 1 order of magnitude, from > 100 to ≤ 10

 Increased load at damage onset and increased ultimate strength, in particular for transverse cracking

1,2 1 2

RVEs: Variations on a Theme

i, x k, z

O

Ωf Rf

Γ1 (0, Rf )

(−Rf , 0)

(0,−Rf ) (Rf , 0)

Ωm C≡ (+l, +l) D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

ll ll

Γ3 θ

∆θ

∆θ I L

H a Γ4

Γ2 w (x, l) = w (l, l)

w (x,−l) = w (l, −l)

u (l, z) = ¯ε· l

u (−l, z) = −¯εx · l

i, x k, z

O

Ωf Rf

Γ1 (0, Rf )

(−Rf , 0)

(0,−Rf ) (Rf , 0)

Ωm C≡ (+l, +l) D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

ll l

l

Γ3 θ

∆θ

∆θ I L

H a Γ4

Γ2 (u (x, l) = u (l, l) xl w (x, l) = w (l, l)

(u (x,−l) = u (l, −l)xl w (x,−l) = w (l, −l)

u (l, z)x = ¯ε· l

u (−l, z) = −¯εx · l

i, x k, z

l

l

l l

u (l, z) = ¯ε· l

u (−l, z) = −¯ε · l

C≡ (+l, +l) D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

G≡ (+l, +tratio· l) K≡ (−l, +tratio· l)

F≡ (+l, −tratio· l) E≡ (−l, −tratio· l)

(0, Rf)

(−Rf, 0)

(0,−Rf) (Rf, 0)

Ωf

Ωm

Ωu [0◦]

Ωb [0◦]

Rf

O I L

H a

Γ1 Γ3

Γ4

Γ2

θ

∆θ

∆θ

i, x k, z

X 2D space X Linear elastic materials X Displacement control

X Dirichlet-type boundary conditions X Linear elastic fracture mechanics

l

l l l

u (l, z) = ¯ε· l u (−l, z) = −¯ε · l

C≡ (+l, +l) D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

(0, Rf)

(−Rf, 0)

(0,−Rf) (Rf, 0) Ωf Ωm

Rf

O I L

H a

Γ1 Γ3

Γ3

Γ2 θ

∆θ∆θ

L. Di Stasio ^1,2 , Z. Ayadi ¹ , J. Varna ² IMR Meeting, Saarbr¨ucken (DE), April 6-7, 2017 26

Appendices Stresses References

RVEs: First Variation on a Theme

i, x k, z

O

Ωf Rf

Γ1 (0, Rf)

(−Rf, 0)

(0,−Rf)

(Rf, 0)

Ωm C≡ (+l, +l) D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

l

l

l l

Γ3 θ

∆θ

∆θ I L

H a Γ4

Γ2 w (x, l) = w (l, l)

w (x,−l) = w (l, −l)

u (l, z) = ¯ε· l

u (−l, z) = −¯εx· l

Isolated RVE with zero vertical displacement BC.

1,2 1 2

RVEs: Second Variation on a Theme

i, x k, z

O

Ωf Rf

Γ1 (0, Rf)

(−Rf, 0)

(0,−Rf)

(Rf, 0)

Ωm C≡ (+l, +l) D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

l

l

l l

Γ3 θ

∆θ

∆θ I L

H a Γ4

Γ2 (

u (x, l) = u (l, l)^xl w (x, l) = w (l, l)

(u (x,−l) = u (l, −l)x l w (x,−l) = w (l, −l)

u (l, z)x= ¯ε· l

u (−l, z) = −¯εx· l

Isolated RVE with homogeneous displacement BC.

Appendices Stresses References

RVEs: Third Variation on a Theme

i, x k, z

l

l

l l

u (l, z) = ¯ε· l

u (−l, z) = −¯ε · l

C≡ (+l, +l) D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

G≡ (+l, +tratio· l) K≡ (−l, +tratio· l)

F≡ (+l, −tratio· l) E≡ (−l, −tratio· l)

(0, Rf)

(−Rf, 0)

(0,−Rf)

(Rf, 0)

Ωf

Ωm

Ωu [0◦]

Ω^b[0◦] Rf

O I L

H a

Γ1 Γ3

Γ4

Γ2

θ

∆θ

∆θ

Bounded RVE.

1,2 1 2

Topological transformation

i, x k, z

α β γ δε

l

l

l

l C≡ (+l, +l)

D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

(0, Rf)

(−Rf, 0)

(0,−Rf) (Rf, 0)

Ωf

Ωm O

Γ1

I L

a Γ3

Γ2

(a)

i, x k, z

A B

C D

A

(b)

i, x k, z

A B

C D A

(c)

i, x k, z

A B C D A

E F, G

H I

J K, L M N

J

K

, L

M

N

O P, Q R S

O

P

, Q

R

S

T U, W X Y

T

U

, W

X

Y

E

F

, G

H

I

I

N

S

Y

(d)

Appendices Stresses References

Mesh parameters

i, x k, z

l

l

l

l C≡ (+l, +l)

D ≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

E F G H

P N

N

N

N

N

E≡ (−f

· R

, +f

· R

) F≡ f

R

(− cos 45

, sin 45

)

G≡ R

(− cos 45

, sin 45

)

H≡ (R

+ f

(l− R

)) (− cos 45

, sin 45

) (0, R

)

(−R

, 0)

(0,−R

)

(R

, 0)

Ω

Ω

O

Γ

I L

a

Γ

Γ

i, x k, z

l

l

l

l C≡ (+l, +l)

D≡ (−l, +l)

B≡ (+l, −l) A≡ (−l, −l)

E F G H

P Nα Nβ Nγ Nδ Nε

E≡ (−f1· Rf, +f1· Rf) F≡ f2Rf(− cos 45^◦, sin 45◦) G≡ Rf(− cos 45^◦, sin 45◦) H≡ (Rf+ f3(l− Rf)) (− cos 45^◦, sin 45◦)

(0, Rf)

(−Rf, 0)

(0,−Rf) (Rf, 0)

Ωf

Ωm O

Γ1

I L

a Γ3

Γ2 O≡ (+l, +tratio· l) K≡ (−l, +tratio· l)

N≡ (+l, −tratio· l) Q≡ (−l, −tratio· l)

Nζ

Nζ

1,2 1 2

Finite Element Model in Abaqus

Method

ABAQUS/STD static analysis + VCCT + J-integral.

Type

Static, i.e. no inertial effects. Relaxation until equilibrium.

Elements CPE4/CPE8 Interface

Tied surface constraint & contact mechanics Input variables

R f , V f , material properties, interface properties.

Control variables θ, ∆θ, ¯ε x . Output variables

Stress field, crack tip stress, stress intensity factors, energy release rates, a.

Appendices Stresses References

VCCT Computation of Mode Ratio

−180−170−160−150−140−130−120−110−100−90−80−70−60−50−40−30−20−100 102030405060708090100110120130140150160170180

0 5· 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

ψ [◦] G(··)GTOT[−]

Mode ratio vs debond angle θ

GI GI+GGIIII GI+GII

δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

1,2 1 2

J-integral Computation of Energy Release Rates

−180−170−160−150−140−130−120−110−100−90−80−70−60−50−40−30−20−100 102030405060708090100110120130140150160170180

0 5· 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

ψ [◦] JJm2

Contour integral J vs debond angle θ

Contour 1 Contour 2 Contour 3 Contour 4 Contour 5 Contour 6 Contour 7 Contour 8 Contour 9 Contour 10

δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

Appendices Stresses References

Stresses at fiber/matrix interface

−180−170−160−150−140−130−120−110−100−90−80−70−60−50−40−30−20−100 102030405060708090100110120130140150160170180

−40

−30

−20

−10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210

ψ [◦] σ(··),τ(··)[MPa]

Stress components at fiber/matrix interface on matrix surface

σnn τnψ τnz τnψ^f

∆θ = 5 ^◦ , δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

1,2 1 2

σ xx along circumferential sections

−150 −100 −50 0 50 100 150

ρ [µm]

20 30 40 50 60 70

σ

[M P a ]

Stress distribution σ xx along circumferential sections r [µm] = 0.09 r [µm] = 0.18 r [µm] = 0.27 r [µm] = 0.36 r [µm] = 0.45 r [µm] = 0.55 r [µm] = 0.64 r [µm] = 0.73 r [µm] = 0.82 r [µm] = 0.91 r [µm] = 3.68 r [µm] = 6.37 r [µm] = 9.05 r [µm] = 11.74 r [µm] = 14.42 r [µm] = 17.1 r [µm] = 19.79 r [µm] = 22.47 r [µm] = 25.16 r [µm] = 27.84

∆θ = 5 ^◦ , δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

Appendices Stresses References

σ zz along circumferential sections

−150 −100 −50 0 50 100 150

ρ [µm]

−20

−15

−10

−5 0

σ

[M P a ]

Stress distribution σ zz along circumferential sections r [µm] = 0.09 r [µm] = 0.18 r [µm] = 0.27 r [µm] = 0.36 r [µm] = 0.45 r [µm] = 0.55 r [µm] = 0.64 r [µm] = 0.73 r [µm] = 0.82 r [µm] = 0.91 r [µm] = 3.68 r [µm] = 6.37 r [µm] = 9.05 r [µm] = 11.74 r [µm] = 14.42 r [µm] = 17.1 r [µm] = 19.79 r [µm] = 22.47 r [µm] = 25.16 r [µm] = 27.84

∆θ = 5 ^◦ , δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

1,2 1 2

τ xz along circumferential sections

−150 −100 −50 0 50 100 150

ρ [µm]

−15

−10

−5 0 5 10 15

σ

[M P a ]

Stress distribution σ xz along circumferential sections r [µm] = 0.09 r [µm] = 0.18 r [µm] = 0.27 r [µm] = 0.36 r [µm] = 0.45 r [µm] = 0.55 r [µm] = 0.64 r [µm] = 0.73 r [µm] = 0.82 r [µm] = 0.91 r [µm] = 3.68 r [µm] = 6.37 r [µm] = 9.05 r [µm] = 11.74 r [µm] = 14.42 r [µm] = 17.1 r [µm] = 19.79 r [µm] = 22.47 r [µm] = 25.16 r [µm] = 27.84

∆θ = 5 ^◦ , δ = 0.4 ^◦ , VF f = 0.001, _R ^l _f ≈ 28

Appendices Stresses References

References

Donald L. Flaggs, Murat H. Kural; Experimental

Determination of the In Situ Transverse Lamina Strength in Graphite/Epoxy Laminates. Journal of Composite Materials, vol. 16, n. 2, 1982.

Parvizi A., Bailey J.E; On multiple transverse cracking in glass fibre epoxy cross-ply laminates. Journal of

Materials Science, 1978; 13:2131-2136.

1,2 1 2

References

Miguel Herr´aez, Diego Mora, Fernando Naya, Claudio S.

Lopes, Carlos Gonz´alez, Javier LLorca; Transverse cracking of cross-ply laminates: A computational micromechanics perspective. Composites Science and Technology, 2015; 110:196-204.

Luis Pablo Canal, Carlos Gonz´alez, Javier Segurado, Javier LLorca; Intraply fracture of fiber-reinforced composites: Microscopic mechanisms and modeling.

Composites Science and Technology, 2012;

72(11):1223-1232.

Appendices Stresses References

References

Stephen W. Tsai; Thin ply composites. JEC Magazine 18, 2005.

Znedek P. Bazant; Size Effect Theory and its Application to Fracture of Fiber Composites and Sandwich Plates. in Continuum Damage Mechanics of Materials and

Structures, eds. O. Allix and F. Hild, 2002.

Robin Amacher, Wayne Smith, Clemens Dransfeld, John Botsis, Jo¨el Cugnoni; Thin Ply: from Size-Effect

Characterization to Real Life Design CAMX 2014, 2014 Ralf Cuntze; The World-Wide-Failure-Exercises -I and - II for UD-materials.

1,2 1 2

References

Pinho, S. T. and Pimenta, S.; Size Effects on the Strength and Toughness of Fibre-Reinforced Composites.

Pedro P. Camanho, Carlos G. D´avila, Silvestre T. Pinho,

Lorenzo Iannucci, Paul Robinson; Prediction of in situ

strengths and matrix cracking in composites under

transverse tension and in-plane shear. Composites Part

A: Applied Science and Manufacturing, vol. 37, n. 2,

2006.

Appendices Stresses References

References

P.P. Camanho, P. Maim´ı, C.G. D´avila; Prediction of size effects in notched laminates using continuum damage mechanics. Composites Science and Technology, vol. 67, n. 13, 2007.

J. A. Nairn; The Initiation and Growth of Delaminations Induced by Matrix Microcracks in Laminated Composites.

International Journal of Fracture, vol. 57, 1992.

Joel Cugnoni , Robin Amacher, John Botsis; Thin ply technology advantages. An overview of the TPT-TECA project. 2014.

1,2 1 2

References

Donald L. Flaggs, Murat H. Kural; Experimental

Determination of the In Situ Transverse Lamina Strength

in Graphite/Epoxy Laminates. Journal of Composite

Materials, vol. 16, n. 2, 1982.