3 Freedom of DesignThe sky is the limit?Limits in FORMABILITYWhich, why, where & how?
4 Composite Life lineWhat 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 qualityrecyclingjoining, welding & bondingenvironmental loadingmechanically induced stressescrack initiation & crack growthAfter life
16 Continuum Mechanics Balance Equations Conservation of mass Conservation of energyConservation of momentumMaterial ‘Laws’Constitutive equations, relating forces & fluxesFormalismScalars, vectors, tensorsDeformation theories
17 Balance EquationsConservation of mass𝜌 =−𝜌𝛁∙𝒗Conservation of momentum𝜌 𝒗 =𝛁∙𝝈+𝜌𝒃Conservation of energy (1st Law)𝜌 𝑢 =𝝈:𝑫−𝛁∙𝒒
18 Constitutive 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
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:𝑙Δ𝑙𝜖= 𝜕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 𝜕𝑥𝑑𝑥𝑑 𝑢 𝑥
30 Constitutive 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
31 Constitutive 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
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 basiswhich 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
51 Shear LockingAligned vs unaligned mesh (triangles) Force vs Displacement
52 Process Modelling INCLUDE RELEVANT DEFORMATION MECHANISMS UD laminates:Intra-ply shearInter-ply shearLaminate bending
53 Reduction of trial & error Process ModellingReduction of trial & errorProduction process simulation of wing leading edge stiffenersBenchmarkingexperiments+ analysis+ modelling
54 Recap: 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
55 Composites 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