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**Composites Forming Analysis**

Remko Akkerman 26th September 2013

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Introduction Scope Mechanisms Constitutive Models Implementation

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

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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...

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**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

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**Interrelations: Processing, Properties & Performance**

process settings product properties fibre orientation fibre/matrix properties composite geometry hypothese, experiment, conclusie

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

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Forming Mechanisms

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Forming Mechanisms

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

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

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**Material Characterisation**

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

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**Material Characterisation**

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

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**Material Characterisation**

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

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

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**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

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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

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**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

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

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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

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**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)

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**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.

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**Composites Forming Processes constitutive equations**

Concluding: Very high anisotropy Large rotations & deformations possible except in the fibre direction woven fabric ud ply

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**Reinforcement structures … some terminology**

Unidirectional Biaxial (weft & warp) Triaxial ….

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**Textiles: Woven Fabrics**

warp fill 1 2 plain 3x1 twill 2x2 twill 5H satin

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**deformation gradient F rate of deformation D**

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

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Fibre Directions deformation a' b' a b

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**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

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**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

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**Constitutive Equations definition of strain**

𝑑𝑥 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

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**Constitutive Equations definition of strain**

𝑑𝑥 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

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**Constitutive Equations definition of strain**

Result (tensile test simulation, E1/E2=105): Exact strain definition required

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**Constitutive Equations definition of strain**

Large deformation theory Deformation gradient: 𝑭= 𝑑𝒙 𝑑𝑿 =𝛻𝒙 and also: 𝒂=𝑭∙ 𝒂 0

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**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!

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**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

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

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Continuum model Inextensibility: or introduce leads to

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Continuum model Incompressibility: Combine with leads to

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

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

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**Constitutive Modelling**

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

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**Shear response from FE model**

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

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**Composite property prediction from mesostructure**

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

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**Implementation issues**

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

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

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

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

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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!

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**Example: bias extension**

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

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Shear Locking Aligned vs unaligned mesh (quads)

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

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**Process Modelling INCLUDE RELEVANT DEFORMATION MECHANISMS**

UD laminates: Intra-ply shear Inter-ply shear Laminate bending

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**Reduction of trial & error**

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

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**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

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**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

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