1. Composites Forming Analysis
Remko Akkerman
26th September 2013

2. Introduction
Scope
Mechanisms
Constitutive Models
Implementation

3. Freedom of Design
The sky is the limit?
Limits in FORMABILITY
Which, why, where & how?

4. 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...

5. Composite life line After life residual stresses product distortions
Impregnation & consolidation quality
recycling
joining, welding & bonding
environmental loading
mechanically induced stresses
crack initiation & crack growth
After life

6. Interrelations: Processing, Properties & Performance
process settings
product
properties
fibre orientation
fibre/matrix properties
composite
geometry
hypothese, experiment, conclusie

7. Forming Processes
Consolidation
Drape (pre-forming)
Press Forming
Compression Molding
....

8. Forming Mechanisms

9. Forming Mechanisms

10. Deformation Limits
“Form ability”
Low resistance to shear & bending
High anisotropy
Negligible fibre extension
Low compressive “strength” (fibre buckling)

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

12. Material Characterisation
Intra-ply shear
(a) Picture frame. (b) Bias extension.

13. Material Characterisation
Bi-axial response
Crimp leads to non-linear behaviour depending on the warp/weft strain ratio

14. Material Characterisation
Ply/tool and Ply/ply Friction
Tool/ply friction (glass/PP)
Shear stress vs pressure.

15. Continuum Mechanics
RECAP: Continuum Mechanics = Balance equations + Material ‘Laws’ + Formalism

16. 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

17. Balance Equations
Conservation of mass
𝜌 =−𝜌𝛁∙𝒗
Conservation of momentum
𝜌 𝒗 =𝛁∙𝝈+𝜌𝒃
Conservation of energy (1st Law)
𝜌 𝑢 =𝝈:𝑫−𝛁∙𝒒

18. Constitutive Equations
Relations between Fluxes (transport of an extensive quantity)
e.g. 𝑞 and 𝑣
and Forces (gradient of an intensive quantity)
e.g. 𝛁𝑇 and 𝑝 𝑣
or, indeed, between
stresses and strains / strain rates
e.g. 𝜎=𝐸𝜖 and 𝜎=𝜂 𝛾

19. Formalism
Scalars:
e.g. 𝑢,𝑇,𝜌, 𝑝
Vectors:
e.g. 𝒒,𝒗
Tensors:
e.g. 𝑫,𝝈

20. Formalism
Single contraction,
𝒂∙𝒃 ↔ 𝑎 𝑖 𝑏 𝑖
𝑨∙𝑩 ↔ 𝐴 𝑖𝑗 𝐵 𝑗𝑘
Dyadic product,
𝒂𝒃 ↔ 𝑎 𝑖 𝑏 𝑗
Double contraction,
𝑨:𝑩 ↔ 𝐴 𝑖𝑗 𝐵 𝑗𝑖

21. Composites Forming Processes balance equations
Viscous & elastic forces dominant (low Reynolds number) Neglect inertia:
𝛁∙𝝈+𝜌𝒃=𝜌 𝒗 =𝟎
Neglect also body forces: Stress equilibrium
𝛁∙𝝈=𝟎
Neglect cooling during forming (at least initially)

22. Composites Forming Processes constitutive equations
Matrix response: Viscous ⋯ visco-elastic ⋯ elastic low modulus, O(1 MPa)
Fibre response: Elastic high modulus, O(100 GPa)
Prepreg/laminate response: Elastic/high modulus - in fibre direction Visco-elastic/low modulus - transverse dir.

23. Composites Forming Processes constitutive equations
Concluding:
Very high anisotropy
Large rotations & deformations possible except in the fibre direction
woven fabric
ud ply

24. Reinforcement structures … some terminology
Unidirectional
Biaxial (weft & warp)
Triaxial ….

25. Textiles: Woven Fabrics
warp
fill 1 2
plain
3x1 twill
2x2 twill
5H satin

26. deformation gradient F rate of deformation D
Fibre Directions
unit vectors a, b
deformation gradient F
rate of deformation D a b

27. Fibre Directions
deformation a' b' a b

28. Constitutive Equations definition of strain
Strain definition: strain= length increase length 𝜖= Δ𝑙 𝑙 Frame of reference: Which “l” ? Total Lagrange or Updated Lagrange? Differential calculus: 𝑙 Δ𝑙
𝜖= 𝜕𝑢 𝜕𝑥

29. Constitutive Equations definition of strain
3D Strain definition: 𝜖 𝑖𝑗 = 1 2 𝜕 𝑢 𝑖 𝜕 𝑥 𝑗 + 𝜕 𝑢 𝑗 𝜕 𝑥 𝑖 Good for linear elasticity But does it work for Composites Forming?
𝜖 𝑥𝑥 = 𝜕 𝑢 𝑥 𝜕𝑥 𝑑𝑥
𝑑 𝑢 𝑥

30. Constitutive Equations definition of strain𝑑𝑥
Constitutive Equations definition of strain
Rigid rotation:  Often non-zero axial strain Except for the “average configuration”
𝑑 𝑢 𝑥
𝑑 𝑢 𝑦
𝜖 𝑥𝑥 = 𝜕 𝑢 𝑥 𝜕𝑋 𝑑𝑋
𝜖 𝑥𝑦 = 1 2 𝜕 𝑢 𝑦 𝜕𝑋

31. Constitutive Equations definition of strain𝑑𝑥
Constitutive Equations definition of strain
Average configuration: But in which direction does the stress act?  Should be in the Final Configuration! (considering the high anisotropy)
𝜖 𝑥𝑥 = 𝜕 𝑢 𝑥 𝜕 𝑥 =0 𝑑𝑋
INCONSISTENCY

32. Constitutive Equations definition of strain
Result (tensile test simulation, E1/E2=105):  Exact strain definition required

33. Constitutive Equations definition of strain
Large deformation theory Deformation gradient: 𝑭= 𝑑𝒙 𝑑𝑿 =𝛻𝒙 and also: 𝒂=𝑭∙ 𝒂 0

34. Constitutive Equations definition of strain
The usual polar decomposition:
𝑭=𝑹∙𝑼=𝑽∙𝑹
(R orthogonal, V & U symmetric)
maintains an orthogonal basis
which is usually wrong!

35. Constitutive Equations definition of strain
Solution: multiplicative split 𝑭=𝑹∙𝑮 (R orthogonal, G non-symmetric), knowing 𝒂=𝑭∙ 𝒂 0 such that 𝒂= 𝑙 𝑙 0 𝑹∙ 𝒂 0 and hence 𝑮∙ 𝒂 0 = 𝑙 𝑙 0 𝒂 0 leading to 𝜖= 1 2 𝑙 𝟐 − 𝑙 0 𝟐 𝑙 0 𝟐 = 1 2 𝑙 0 𝟐 𝒂 0 𝒂 0 : 𝑪−𝟏 as the scalar fibre strain ϵ in direction a with 𝑪= 𝑭 𝑇 ∙𝑭= 𝑮 𝑇 ∙𝑮

36. Continuum model
Recall incompressible isotropic viscous fluids:
Now directional properties f (a,b)

37. Continuum model
Inextensibility:
or introduce
leads to

38. Continuum model
Incompressibility:
Combine with
leads to

39. Continuum model
extra stress t
Form-invariance under rigid rotations: isotropic function of its arguments
Assume linearity, leads to:
with

40. Continuum model
Fabric Reinforced Fluid (FRF) model
Can be simplified by symmetry considerations (sense of a, b, fabric symmetry)

41. Constitutive Modelling
Continuum mechanics
Alternative: Discrete approach (resin + fibre + structure)
for instance using mesoscopic modelling

42. Shear response from FE model
Mesoscopic modelling
Composite property prediction from mesostructure
Shear response from FE model

43. Composite property prediction from mesostructure
Mesoscopic modelling
Composite property prediction from mesostructure 3D
Biaxial 2D
Triaxial 2D
Multiaxial 2D (NCF)
Knit
TexGen, WiseTex, etc

44. Implementation issues
Accuracy especially concerning fibre directions
Consistent tangent (as above)
Shear locking (due to large stiffness differences)

45. Shear Locking
Linear triangle (N1, N2, N3)
Linear strains & rotations

46. Shear Locking
Fibres in x and y direction (inextensibility)
Eliminate rigid body displacements

47. Shear Locking x y N1 N3 N2
N1 in the origin (0,0)
Remaining d.o.f.s

48. Shear Locking
Suppress a single node Ni (i=2,3)
 Shear locking !
Unless: xi=0 or yi=0 (i=2,3)
 Edge coincides with fibre direction!

49. Example: bias extension
Shear Locking
Result of locking:
Far too high stiffness
Spurious wrinkles
Incorrect deformations
Example: bias extension

50. Shear Locking
Aligned vs unaligned mesh (quads)

51. Shear Locking
Aligned vs unaligned mesh (triangles) Force vs Displacement

52. Process Modelling INCLUDE RELEVANT DEFORMATION MECHANISMS
UD laminates:
Intra-ply shear
Inter-ply shear
Laminate bending

53. Reduction of trial & error
Process Modelling
Reduction of trial & error
Production process simulation of wing leading edge stiffeners
Benchmarking
experiments
+ analysis
+ modelling

54. 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

55. Composites Forming Processes numerical 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