Micro Mesoscale

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Micro‐ and Meso‐scale modelling

6/11/2013 1CompositesIves De Baere and Joris Degrieck – 2013‐2014

Part Design

• Stiffness• Strength• Damage tolerance

F ti

Micro‐meso‐macro homogenization methods

Motivation

• Fatigue • Static

Experimental


Wednesday, 06 November 2013

Numerical

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Textile compositesMotivation

Micro‐meso‐macro homogenization methods

UD composite

Fibre

Plain weave Twill weave


Satin weave 2D braided composite

Knitted composites etc..

Analytical Micro‐mechanical homogenization


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• if the reinforcement consists of particles, elliptical inclusions or short fibres, homogenization methods exist for prediction of elastic moduli and thermal expansion coefficients

Micro‐mechanical homogenization of particle‐reinforced composites

• the most well‐known homogenization method is the Mori‐Tanaka method

• commercial implementations exist, amongst others, by e‐XstreamEngineering (Digimat software)

• mainly applied for short‐fibre composites


• analytical homogenization methods for spherical and elliptical inclusions fail for continuous fibres (much larger aspect ratios)

• analytical Rules of Mixture (ROM) are used instead

Micro‐mechanical homogenization of UD fibre‐reinforced composites

• analytical Rules of Mixture (ROM) are used instead


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With volume fraction of the fibers

Physical properites: density

There is a direct link between the volume fraction and the mass fraction

fv

1m f vv v v

volume fraction of the matrix

volume fraction of the voids

. (1 )

f

f f mf f f

ff f v

m

Mm v v

M v v v

f

mv

vv


Remark: in literature, fibre volume fraction is often noted with capital Vf, to avoid mistaking it with the Possion’s ratio

.(1 )

m m mm m

ff v m

m

M vm v

M v v v

Representative volume element of unidirectional ply

stress taken by fibres and matrices average applied stress

Analytical Rules of Mixture (ROM) for UD composites

• Equilibrium in axial direction:

matrix

fibre

homogeneousmatrix


1 1 11 11 11

1

f f m m

f f m m

F W E W E W E

E V E V E

• Rule of Mixture requires elastic properties of fibre and matrix !

( W, Wf and Wm = surface areas, V, Vf and Vm = volume fractions !! )

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Puck’s formulas are frequently used to estimate 4 (out of 5) elastic properties of the unidirectional ply. Puck’s formulas assume an isotropic matrix and an isotropic fibre (not true for carbon fibre !!!)

E G i t i fib ti


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12

2

221.25

(1 )

(1 )

1 0.85

(1 )

f f f m

f f f m

fm

mf f

f

E V E V E

V V

VE E

EV V

E

Ef, Gf en f = isotropic fibre properties,

Em, Gm en m = isotropic matrix properties,

Vf = 1 – Vm = fibre volume fraction

0

2(1 )f

ff

m

EG

EE


121.25

1 0.6

(1 )

f

mm

f ff

VG G

GV V

G

2

2(1 )

1m

mm

mm

m

E

GE

121 E

The formula by Foye provides an estimate of the fifth elastic property of a unidirectional ply


123

2 12

1

1(1 )

1

m m

f f f m

m m m

EE

V VE

E


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Other popular formulas for micro‐mechanical homogenization are the ones proposed by Chamis





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Finite Element Micro‐mechanical homogenization


YARN(carbon)

COMPOSITE[MPa]

Finite element micro‐mechanical homogenization

MATRIX [MPa]


MATRIX(epoxy)

[MPa]

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Implementation of Periodic Boundary Conditions (PBC) through node‐coupling constraints

Loading case: uniaxial strain xx


Constraint equations: Δx

X1X2

oad g case u a a st a xx

N1N2


q

Δy = y2 – y1 = 0

Δz = z2 – z1 = 0ɛy, ɛz, γxy, γxy, γxy = 0

N2

Degrees of freedom (3) of each pair of nodes belonging to opposite faces of

Implementation of Periodic Boundary Conditions (PBC) through node‐coupling constraints


Constraint equations:

Degrees of freedom (3) of each pair of nodes belonging to opposite faces of the RVE are constrained kinematically

h d ff d l d

0

0

xu x

v

w


with Δu, Δv, Δw = difference in displacement in every directionΔx, Δy, Δz = difference in coordinates in every directionɛx, ɛy, ɛz, γxy, γxy, γxy = far‐field applied strains

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ij ijVdV

Homogenised stresses ijHomogenised strains ijOne model for every loading:

Model input Analysis output

Homogenised elastic constants Cij

ijij

j

or

P

S

1. 11 nonzero

2. 22 nonzero

3. 33 nonzero

4. 12 nonzero

5. 23 nonzero

6. 13 nonzero



48elements

4,400elements

36,320elements

4,410elements


[MPa] [MPa] [MPa][MPa]

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ORIGINAL MODEL ‘PERFECT BONDING’ ‘WEAK BONDING’


yarnmatrixno cohesive layer

yarnmatrixstiff cohesive layer

yarnmatrixsoft cohesive layer


E = 1000 MPa G1 = 1000 MPa G2 = 1000 MPa

E = 10E6 MPa G1 = 10E6 MPa G2 = 10E6 MPa

perfect bonding by definition

3D periodic boundary conditions

εy


γxy


εy = γxy = 1%

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[MPa] [MPa][MPa]




[MPa] [MPa][MPa]

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Finite Element Meso‐mechanical homogenization


1. Geometry

2 Mesh

1 2

Finite element meso‐mechanical homogenization

2. Mesh

3. Material Properties

4. Boundary conditions

3

5

4

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6. Homogenisation

5. Finite Element calculation6

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FE -Abaqus ***Micro-CT Catia V5 ***

Finite element meso‐mechanical homogenization

PBC Creator

Homogenization (Chamis)

FE l l ti Homogenized Properties

Local Material Orientations



FE calculations

Macro homogenization

Homogenized Properties

Composite material under study (Ten Cate)• 5‐harness satin weave carbon fabric‐reinforced PPS (Vf = 50 %)• [(0°,90°)]4s stacking sequence: 8 layers of

(0° 90°) fabric

Illustration of procedure for textile composite

(0 ,90 ) fabric• Composite plates were hot pressed

310 °C and 10 bar• CETEX : 6 tons in Airbus A380


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7.4mm



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Warp yarn

Weft yarn

One ply


Yarn crimp

Macro-scale

Meso-scale


Micro-scale


RVE Yarn width Yarn thicknessYarn spacing

• Weave pattern - 5 HS Yarn spacing, mm - 1.48

• Unit cell width, mm - 7.4 Unit cell thickness, mm - 0.319


, ,

• Number of fibres, K - 3 Width of yarn, mm - 1.32

•Thickness of the yarn, mm - 0.156

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Meshing arbitrary 3D yarns with correct local material orientations





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DONE Cohesive elements – applications

3D periodic boundary conditions



εx = 0,5%

MethodE11

[GPa]

E22

[GPa]

E33

[GPa]

23

G12

[MPa]

G13

[MPa]

G23

[MPa]


[GPa] [GPa] [GPa] [MPa] [MPa] [MPa]

Periodic BCs & volume averaging

56.49 56.41 10.53 0.08 0.41 0.41 4280 3048 3045

Experiment 57.0 57.0 - 0.05 - - 4175 - -



This sheet is property of Stefan Jacques and shall not be copied or disclosed to a third party without a written authorization

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Max strain Min strainAverage strainIllustration of procedure for textile composite


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Micro Mesoscale

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