Modelling generation of fibre waviness during processing

Modelling generation of fibre waviness during processing Brendon Qu and Michael Sutcliffe Department of Engineering University of Cambridge, UK ECCM-14, 7th-10th June 010, Budapest

Overview Introduction Single-fibre theory and experiments Multi-fibre experiments Multi-fibre finite element analysis Conclusions

Introduction Waviness causes reduction in compressive strength Induced during manufacturing Many causes of waviness - inherent in as-supplied material - due to draping/forming - due to fibre movement during moulding - due to compressive stresses developed during cure Larger, cheaper structures need to allow for defects 50 µm σ c = k φ Microbuckle in CFRP Laminate wrinkle RTM of CFRP Mandell et al 003

Introduction - mechanisms Focus on waviness generated during processing Experimental work identifies key factors: Coefficient of thermal expansion (CTE) mismatch between resin and fibres Cooling rate Critical part length below which no fibre waviness appears Tool plate properties (e.g. changes in CTE can reduce waviness) [Narin, Bhalerao, Chan and Wang, Kugler, Joyce and Moon, Olsen] Limited modelling work

Single fibre analysis Simple geometry Amenable to analysis Straightforward experiments Establish key factors and verify modelling approach P L P f(x) Pin-jointed with end loads P Euler beam with properties D f = EI and A dw dw Transverse force distribution f ( x) = η = 1µ dt dt where µ is the dynamic viscosity of the surrounding fluid

Beam bending equation Single fibre theory D f 4 d w d w dw + P = η 4 dx dx dt 4 d wˆ d wˆ dwˆ EI x w Dimensionless form D + = with D =, x ˆ =, w ˆ = and 4 dxˆ dxˆ dtˆ L P L d tˆ tp = ηl

Beam bending equation Single fibre theory D f 4 d w d w dw + P = η 4 dx dx dt 4 d wˆ d wˆ dwˆ EI x w Dimensionless form D + = with D =, x ˆ =, w ˆ = and 4 dxˆ dxˆ dtˆ L P L d Solve by separation of variables ( n π Dn 4 wˆ = Ane with the solution as a sum of mode n= 1 shapes n = 1 Coefficients A n depend on initial shape Exponential growth with time Exponent depends on mode number, load and beam properties Critical load for growth given by Euler buckling load π 4 )ˆ t sin ( nπ xˆ ) P cr = EI L Transverse forces do not affect whether buckling occurs, only the timescales π tˆ tp = ηl

Single fibre experiments Model experiment Fibres = fishing line, resin = honey Viscosity from dropped sphere test Fibre bending properties from beam test Dead weight loading Fibre deflection video capture

Single fibre results Transverse y (m) Initial shape Deflection (m) Theory Experiment Longitudinal x (m) measured Time (s) sin fit Transverse y (m) Deformed shape Longitudinal x (m) Initial and deformed shapes fitted by sinusoid Good agreement for modest deflections Agreement at large deflections can be found by adapting beam model Model captures essence of waviness generation

slot soy sauce Initial geometry load Multi-fibre experiments Deformed geometry fibre bundles 6 cm D array of fibres, not able to roll over each other Honey able to leak sideways Soy sauce used to visualise honey flow Similar behaviour to single-fibre results Interaction between bundles due to honey between fibres

Multi-fibre model - application to composites Darcy's law is used to relate resin flow rate q to pressure gradients in the resin k q = ρg dp dx = K dp dx k f dp = µ dx where k is the permeability or hydraulic conductivity (units m/s). Permeability is also defined as being k f. The single-fibre model can be applied to an array of fibres all moving in unison through a stationary fluid Replace the fluid viscosity term η by a micro-permeability k f where l is the gap between fibres and φ is fibre volume fraction = l k φµ f Typical composite properties gives a time of 15-30 mins to generate a waviness of 100d d = 7 µm E = 350 GPa σ = 50 MPa µ = Pa s

Multi-fibre finite element model Darcy's law used to model resin flow in Abaqus FE software Introduce permeability via 'dummy' base material [Sheu, Hector and Richmond] Base is linear elastic of negligible stiffness, permeable with void ratio = 1 Fibres modelled by linear elastic beams with properties E, I and A Each beam represents several fibres of diameter d Beams 'embedded' in base of width W = 00d and height H = 100d Explore effect of permeability and boundary conditions on deformed shape ramp displacement loading on top sides of base constrained horizontally elastic fibre bundles - all beam ends encastred bottom fixed permeable base fluid boundary conditions: - no flow - no pore pressure (fully drained)

Effect of permeability Appropriate dimensional time tˆ = L tei 4 d k ρg Reducing permeability increases resistance to flow and increases force required Stress / critical buckling stress Reducing permeability tˆ 1, 0.1, 0.01 Displacement /H (%)

Effect of flow boundary conditions No flow on top and bottom Drained on edges Drained top and bottom No flow on edges Drained on all edges Flow boundary conditions alter interactions between fibres

Effect of anisotropic permeability Permeability the same in both directions Permeability 10 along fibres Permeability 10 transverse to fibres Complex interaction between permeability and boundary conditions

Conclusions Fibre waviness gives knockdown in compressive strength Generation of waviness during processing important for large structures

Conclusions Fibre waviness gives knockdown in compressive strength Generation of waviness during processing important for large structures Buckling of single fibre depends on applied load, fibre stiffness, viscosity and time Good agreement between single-fibre experiments and model Multi-fibre experiments show important of fibre interactions Multi-fibre FE model shows importance of permeability and boundary conditions