Composites Forming Analysis

Remko Akkermanwww.utwente.nl/ctw/ptr.akkerman@utwente.nl

26th September 2013

Introduction

Scope Mechanisms Constitutive Models Implementation

2

Freedom of Design

The sky is the limit? Limits in FORMABILITY Which, why, where & how?

3

Composite Life line

What is a material, what is a structure? What is a Forming Process? Micro is close to Meso is close to Macro...

4

fibre resin

prepreg

lightweight part

lightweight structure

structural deformationand failure

impregnation

production

assembly

inservice loading

autoclaveRTMhot pressingrubber moulding... low density

high stiffnesshigh strengthusually thin walledgood corrosion resistance

CFRPcontinuous fibrereinforced plastic

glasscarbon...

thermoplasticthermoset

residual stressesproduct distortions

mechanically induced stressescrack initiation & crack growth

Impregnation& consolidation quality

joining,welding & bonding environmental

loading

Composite life line

After life

recycling

5

processsettings

product properties

fibreorientation

fibre/matrixproperties

compositeproperties

productgeometry

Interrelations:Processing, Properties & Performance

6

Forming Processes

Consolidation

Drape (pre-forming)

Press Forming Compression Molding ....

7

Forming Mechanisms

8

Forming Mechanisms

9

Deformation Limits

“Form ability” Low resistance to shear & bending

High anisotropy Negligible fibre extension Low compressive “strength”

(fibre buckling)

10

Formability Analysis...

From deformation mechanisms ... to material characterisation ... to constitutive modelling ... to process modelling ... and formability prediction

11

Material Characterisation

Intra-ply shear

(a) Picture frame. (b) Bias extension.

0

100

200

300

400

500

0 10 20 30 40 50 60 70

Shear Angle (Deg)

No

rma

lise

d S

he

ar

Fo

rce

(N

/m) 160 oC

180 oC

12

Material Characterisation

Bi-axial response

0

50

100

150

200

250

0 0.2 0.4 0.6 0.8

Strain (%)

Load

(N/y

arn)

Freein the second

direction

k=0.5k=1

k=2

Yarn

T22

T22

T11T11

T11T11 > T22

T22 < T11T22 = 0

T11; T22= 0T11

T22 = 0T22

Crimp leads to non-linear behaviour depending on the warp/weft strain ratio

13

Material Characterisation

Ply/tool and Ply/ply Friction

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.0 0.5 1.0 1.5 2.0 2.5 3.0Normal Pressure (MPa)

Sh

ea

r S

tre

ss

(M

Pa

)

0.5mm/s 180C 0.5mm/s 200C0.5mm/s 220C 0.8mm/s 180C0.8mm/s 200C 0.8mm/s 220C1.2mm/s 180C 1.2mm/s 200C1.2mm/s 220C

Tool/ply friction (glass/PP)Shear stress vs pressure.

14

Continuum Mechanics

RECAP:

Continuum Mechanics =

Balance equations+ Material ‘Laws’+ Formalism

15

Continuum Mechanics

Balance Equations Conservation of mass Conservation of energy Conservation of momentum

Material ‘Laws’ Constitutive equations,

relating forces & fluxes

Formalism Scalars, vectors, tensors Deformation theories

16

Balance Equations

Conservation of mass

Conservation of momentum

Conservation of energy (1st Law)

17

Constitutive Equations

Relations between Fluxes(transport of an extensive quantity)

e.g.

and Forces(gradient of an intensive quantity)

e.g.

or, indeed, betweenstresses and strains / strain rates

e.g.

18

Formalism

Scalars:

e.g.

Vectors:

e.g.

Tensors:

e.g.

19

Formalism

Single contraction,

Dyadic product,

20

Composites Forming Processesbalance equations

Viscous & elastic forces dominant (low Reynolds number) Neglect inertia:

Neglect also body forces: Stress equilibrium

Neglect cooling during forming(at least initially)

21

Composites Forming Processesconstitutive equations

Matrix response:Viscous visco-elastic elasticlow modulus, O(1 MPa)

Fibre response: Elastichigh modulus, O(100 GPa)

Prepreg/laminate response:Elastic/high modulus - in fibre directionVisco-elastic/low modulus - transverse dir.

22

Composites Forming Processesconstitutive equations

Concluding:

Very high anisotropy

Large rotations & deformations possibleexcept in the fibre direction

23

woven fabric ud ply

Reinforcement structures… some terminology

Unidirectional Biaxial (weft & warp) Triaxial ….

24

Textiles: Woven Fabrics

plain 3x1 twill 2x2 twill 5H satin

war

p

fill

12

25

Fibre Directions

unit vectors a, b

deformation gradient F rate of deformation D

ab

26

Fibre Directions

a'b'ab

deformation

'

'

a F a

b F b

27

Constitutive Equationsdefinition of strain

Strain definition:

Frame of reference: Which “l” ?Total Lagrange or Updated Lagrange?

Differential calculus:

28

Δ 𝑙𝑙

𝜖=𝜕𝑢𝜕 𝑥

Constitutive Equationsdefinition of strain

3D Strain definition:

Good for linear elasticity

But does it work for Composites Forming?

29

𝑑𝑢𝑥

𝑑𝑥 𝜖𝑥𝑥=𝜕𝑢𝑥

𝜕𝑥

Constitutive Equationsdefinition of strain

Rigid rotation:

Often non-zero axial strain

Except for the “average configuration”

30

𝑑𝑋 𝜖𝑥𝑥=𝜕𝑢𝑥

𝜕 𝑋𝑑𝑢𝑥

𝑑𝑢𝑦𝑑𝑥 𝜖𝑥𝑦=12

𝜕𝑢𝑦

𝜕 𝑋

Constitutive Equationsdefinition of strain

Average configuration:

But in which direction does the stress act?

Should be in the Final Configuration!

(considering the high anisotropy)

31

𝑑𝑋 𝜖𝑥𝑥=𝜕𝑢𝑥

𝜕𝑥 12

=0

𝑑𝑥

INCONSISTENCY

Constitutive Equationsdefinition of strain

Result (tensile test simulation, E1/E2=105):

Exact strain definition required

32

Constitutive Equationsdefinition of strain

Large deformation theory

Deformation gradient:

and also:

33

Constitutive Equationsdefinition of strain

The usual polar decomposition:

(R orthogonal, V & U symmetric)maintains an orthogonal basis

which is usually wrong!

34

Constitutive Equationsdefinition of strain

Solution: multiplicative split(R orthogonal, G non-symmetric), knowing such that and hence leading to

as the scalar fibre strain ϵ in direction a

with

35

Continuum model

• Now directional properties f (a,b)

21 2

with , ,i i D D D

p

I II III

σ 1 D D

• Recall incompressible isotropic viscous fluids:

36

Continuum model

Inextensibility: 0

0

a D a

b D b

or introduce

A aa

B bb

leads to : 0

: 0

A D

B D

37

Continuum model

Incompressibility: : 0tr D 1 D

Combine with

A Bp s s σ 1 A B τ leads to

: 0

: 0

A D

B D

38

Continuum model

extra stress t , ,τ τ a b D

Form-invariance under rigid rotations:isotropic function of its arguments

Assume linearity, leads to:

1 2

3 4

2 2 2

2 2 ,

T T

τ D a b D A D D A B D D B

C D D C C D D C

C abwith

39

Continuum model

Fabric Reinforced Fluid (FRF) model

1 2

3 4

2 2 2

2 2 ,

T T

τ D a b D A D D A B D D B

C D D C C D D C

Can be simplified by symmetry considerations (sense of a, b, fabric symmetry)

40

Constitutive Modelling

1. Continuum mechanics2. Alternative: Discrete approach

(resin + fibre + structure)

for instance using mesoscopic modelling

...m a b σ σ σ σ

41

Mesoscopic modelling

Composite property prediction from mesostructure

Shear response from FE model

42

Mesoscopic modelling

Composite property prediction from mesostructure

Biaxial 2D3D

Triaxial 2D KnitMultiaxial 2D (NCF)

TexGen, WiseTex, etc

43

Implementation issues

Accuracy especially concerning fibre directions

Consistent tangent (as above) Shear locking

(due to large stiffness differences)

44

Shear Locking

Linear triangle (N1, N2, N3)

,

,

x x x x

y y y y

u x y a x b y c

u x y a x b y c

Linear strains & rotations

xx x

yy y

y xxy y x

y xxy y x

ua

xu

by

u ua b

x y

u ua b

x y

45

Shear Locking

Fibres in x and y direction (inextensibility)

Eliminate rigid body displacements

0

0x x

y y

a

b

0,0 0 0

0,0 0 0

x x

y y

u c

u c

46

Shear Locking

N1 in the origin (0,0)

Remaining d.o.f.s

x

y

N1

N3

N2

1,

21

,2

x x xy xy

y y xy xy

u x y b y y

u x y a x x

47

Shear Locking

Suppress a single node Ni (i=2,3)

Shear locking !

, 0 0

, 0 0

x i i x

y i i y

u x y b

u x y a

Unless: xi=0 or yi=0 (i=2,3)

Edge coincides with fibre direction!

48

Shear Locking

Result of locking: Far too high stiffness Spurious wrinkles Incorrect deformations

Example: bias extension

49

Shear Locking

Aligned vs unaligned mesh (quads)

50

Shear Locking

Aligned vs unaligned mesh (triangles)Force vs Displacement

51

UD laminates:

52

Process ModellingINCLUDE RELEVANT DEFORMATION MECHANISMS

Intra-ply shear Inter-ply shear Laminate bending

Process Modelling

Reduction of trial & error

draping trimming spring-backdraping trimming spring-back

Production process simulation of wing leading edge stiffeners

Benchmarking

experiments+ analysis

+ modelling

53

Recap: Formability Analysis of Composites

Very high anisotropy Highly Sensitive to Fibre Directions –

use exact (non linearised) strain definition Shear Locking for non-aligned meshes ‘Stiff systems’ – Consistent Tangent Operators to

prevent divergence

54

Composites Forming Processesnumerical aspects

In summary:

Very high anisotropy Highly Sensitive to Fibre Directions –

use exact (non linearised) strain definition Shear Locking for non-aligned meshes ‘Stiff systems’ – Consistent Tangent Operators to

prevent divergence

55