Composites Forming Analysis Remko Akkerman www.utwente.nl/ctw/pt r.akkerman@utwente.nl 26th September 2013

Introduction Scope Mechanisms Constitutive Models Implementation

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

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...

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

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

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

Forming Mechanisms

Forming Mechanisms

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

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

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

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

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

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

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

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

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

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

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

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)

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.

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

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

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

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

Fibre Directions deformation a' b' a b

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

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

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

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 1 2 =0 𝑑𝑋 INCONSISTENCY

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

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

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!

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

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

Continuum model Inextensibility: or introduce leads to

Continuum model Incompressibility: Combine with leads to

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

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

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

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

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

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

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

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

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

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!

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

Shear Locking Aligned vs unaligned mesh (quads)

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

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

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

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

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