3Freedom of DesignThe sky is the limit?Limits in FORMABILITYWhich, why, where & how?
4Composite Life lineWhat is a material, what is a structure?What is a Forming Process?Micro is close to Meso is close to Macro...
5Composite life line After life residual stresses product distortions Impregnation & consolidation qualityrecyclingjoining, welding & bondingenvironmental loadingmechanically induced stressescrack initiation & crack growthAfter life
16Continuum Mechanics Balance Equations Conservation of mass Conservation of energyConservation of momentumMaterial ‘Laws’Constitutive equations, relating forces & fluxesFormalismScalars, vectors, tensorsDeformation theories
17Balance EquationsConservation of mass𝜌 =−𝜌𝛁∙𝒗Conservation of momentum𝜌 𝒗 =𝛁∙𝝈+𝜌𝒃Conservation of energy (1st Law)𝜌 𝑢 =𝝈:𝑫−𝛁∙𝒒
18Constitutive Equations Relations between Fluxes (transport of an extensive quantity)e.g. 𝑞 and 𝑣and Forces (gradient of an intensive quantity)e.g. 𝛁𝑇 and 𝑝 𝑣or, indeed, betweenstresses and strains / strain ratese.g. 𝜎=𝐸𝜖 and 𝜎=𝜂 𝛾
28Constitutive 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:𝑙Δ𝑙𝜖= 𝜕𝑢 𝜕𝑥
29Constitutive 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 𝜕𝑥𝑑𝑥𝑑 𝑢 𝑥
30Constitutive Equations definition of strain 𝑑𝑥Constitutive Equations definition of strainRigid rotation: Often non-zero axial strain Except for the “average configuration”𝑑 𝑢 𝑥𝑑 𝑢 𝑦𝜖 𝑥𝑥 = 𝜕 𝑢 𝑥 𝜕𝑋𝑑𝑋𝜖 𝑥𝑦 = 1 2 𝜕 𝑢 𝑦 𝜕𝑋
31Constitutive Equations definition of strain 𝑑𝑥Constitutive Equations definition of strainAverage configuration: But in which direction does the stress act? Should be in the Final Configuration! (considering the high anisotropy)𝜖 𝑥𝑥 = 𝜕 𝑢 𝑥 𝜕 𝑥 =0𝑑𝑋INCONSISTENCY
32Constitutive Equations definition of strain Result (tensile test simulation, E1/E2=105): Exact strain definition required
33Constitutive Equations definition of strain Large deformation theory Deformation gradient: 𝑭= 𝑑𝒙 𝑑𝑿 =𝛻𝒙 and also: 𝒂=𝑭∙ 𝒂 0
34Constitutive Equations definition of strain The usual polar decomposition:𝑭=𝑹∙𝑼=𝑽∙𝑹(R orthogonal, V & U symmetric)maintains an orthogonal basiswhich is usually wrong!
35Constitutive 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
36Continuum modelRecall incompressible isotropic viscous fluids:Now directional properties f (a,b)
37Continuum modelInextensibility:or introduceleads to
38Continuum modelIncompressibility:Combine withleads to
39Continuum modelextra stress tForm-invariance under rigid rotations: isotropic function of its argumentsAssume linearity, leads to:with
40Continuum modelFabric Reinforced Fluid (FRF) modelCan be simplified by symmetry considerations (sense of a, b, fabric symmetry)
51Shear LockingAligned vs unaligned mesh (triangles) Force vs Displacement
52Process Modelling INCLUDE RELEVANT DEFORMATION MECHANISMS UD laminates:Intra-ply shearInter-ply shearLaminate bending
53Reduction of trial & error Process ModellingReduction of trial & errorProduction process simulation of wing leading edge stiffenersBenchmarkingexperiments+ analysis+ modelling
54Recap: Formability Analysis of Composites Very high anisotropyHighly Sensitive to Fibre Directions – use exact (non linearised) strain definitionShear Locking for non-aligned meshes‘Stiff systems’ – Consistent Tangent Operators to prevent divergence
55Composites Forming Processes numerical aspects In summary:Very high anisotropyHighly Sensitive to Fibre Directions – use exact (non linearised) strain definitionShear Locking for non-aligned meshes‘Stiff systems’ – Consistent Tangent Operators to prevent divergence