NUMERICAL APPLICATIONS AND VERIFICATION OF AN INTEGRATED FLOW-STRESS MODEL IN PROCESSING OF THERMOSET COMPOSITES

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1 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Abstract Ths paper presents numercal eamples to study both aspects of flow-compacton and stress development throughout the curng process of thermoset composte materals and the effects of resn flow on the development of resdual stresses predcted by a newly developed ntegrated model. Varous numercal eamples rangng from smplfed cases represented by a sngle fnte element to undrectonal composte lamnates undergong autoclave curng process are consdered. To verfy the ntegrated model, the results obtaned from these numercal eamples are compared to those generated from establshed ndvdual models for stress development n the composte materal. The numercal eamples wll also serve to provde comparsons for predctons of stress development over the doman of the processed composte lamnate wth those obtaned by regular stress models that consder ether the ntal or fnal resn volume fractons. Introducton NUMERICAL APPLICATIONS AND VERIFICATION OF AN INTEGRATED FLOW-STRESS MODEL IN PROCESSING OF THERMOSET COMPOSITES M. Haghshenas *, R. Vazr, A. Poursartp Compostes Group, Depts. of Cvl Engneerng and Materals Engneerng, The Unversty of Brtsh Columba, Vancouver, Canada * Correspondng author (mehd@compostes.ubc.ca) Keywords: Thermoset Matr Compostes, Resn flow, Resdual stress, Process Modellng The behavour of thermoset matr compostes undergong cure may be dvded nto two dstnct stages: pre-gelaton and post-gelaton. Durng the frst stage, flow of resn through the fbre-bed plays a major role n deformaton of the part as the vscosty of the uncured resn drops sgnfcantly due to the ncrease n temperature. Durng the postgelaton phase, the gelled resn and the fbres deform as a whole, leadng to the epected sold composte materal response. In current process models, each of these two aspects s dealt wth usng separate sub-models, typcally called the flow and stress modules. The flow module s relevant to the pre-gelaton behavour of resn, whle the stress module s vald for the post-gelaton of composte materal. Typcally n process smulatons, the flow module s run from the start of the process untl resn s gelled throughout the composte lamnate. Then the geometry and volume fractons are updated and re-meshng s performed f deemed necessary. The last step nvolves runnng the stress module wth the new propertes from the begnnng to the end of the process. Therefore, only the effect of resn flow on the fnal geometry and propertes of the composte materal s consdered n the predcton of resdual stresses and the ongong nfluence of flow on the development of stresses n the composte system s overlooked. In a prevous work by the authors [1] the basc framework for the ntegraton of modelng porous flow of resn through the fbre bed wth the smulaton of stress development n the composte materal durng the processng cycle was presented n a unfed model. Such a unfed model capable of capturng both aspects of flow and stress development n an ntegrated manner enables one to seamlessly track and analyse the resn flow, deformaton, and stress development n the composte throughout the curng process. Governng Equatons The ntegrated modellng approach used n ths work s developed based on a 2-D plane stran flowcompacton FE representaton. The governng equatons for flow and deformaton of porous meda nvolve three dstnct equatons ncludng the equlbrum equaton of the flud phase, the equlbrum equaton of the two-phase system, and the mass conservaton equaton. The two-phase model s modfed so that t can capture both the stress development and resn flow. These nclude,

2 but are not lmted to, modfcatons to the mass conservaton equaton to produce bulk elastc propertes consstent wth the stress models for cured compostes; ntroducton of a modfed concept of effectve stress; and modfcatons to the numercal soluton technque. Incorporatng the above modfcatons n the governng equatons of general two-phase meda leads to [1] u, v, vs P, fd σs p, j bp, j (1) as the set of governng equatons for the ntegrated model. The three equatons n (1) are the mass conservaton equaton, the equlbrum equaton of the resn matr, and the total equlbrum equaton, respectvely. In dervng (1), the effect of body forces s neglected. u, v, and P are the man varables of the above dfferental equatons. u s the dsplacement of the sold structure whch s representatve of the dsplacement feld of the system, v s the relatve velocty feld of resn, whle P represents the resn pressure. s the rate of volumetrc stran of the two-phase system, and s defned by v s 1 K c tr σ t s v s (2) so that the bulk behavour of the two-phase system s consstent wth the mcromechancs formulaton of choce for the modul of the cured composte. σ s s the total stress of the two-phase system, and K c s the bulk modulus of the system obtaned from the relevant mcromechancs formulaton. denotes the volume fracton of resn, and b s the Bot coeffcent [2] defned by b 1 K K fb f (3) where K fb s the bulk modulus of the fbre-bed (sold skeleton n general) and K f s the bulk modulus of ndvdual fbres (sold grans n general). f d are the components of the drag force between the sold and flud phases that, f defned as follows f 1 d S j v j (4) lead to the Darcy s law [3]. μ s the vscosty of the flud phase, and S s the permeablty matr of of the porous fbre-bed. σ are the components of s p j the modfed fbre-bed stress tensor. For sotropc materals, the elastc modul that relate the modfed fbre-bed stress tensor to the stran feld are defned by G K s p s p K fb, G fb G c () where G fb s the shear modulus of fbre-bed, and G c s the shear modulus of the composte materal obtaned from the selected mcromechancs scheme. For brevty, the formulaton of the ntegrated approach for transversely sotropc materals s not dscussed n ths paper. More theoretcal detals of the ntegrated approach may be found n Haghshenas et al. [1] and Haghshenas [4]. The ntegrated formulaton s ntroduced nto a Q 1 P blnear soparametrc element wth 4 nodes for the system dsplacement and relatve velocty of the flud phase, and only one central node assgned to pressure of the flud phase. Ths element, whch s depcted schematcally n Fg. 1, wll also be referred to as the 4-1 element. The stress development model used n ths work s based on a pseudo-vscoelastc model presented by Zobery et al. []. Numercal Applcatons The ntegrated model was mplemented n a MATLAB code developed n-house. The stress model based on the pseudo-vscoelastc formulaton was also separately mplemented n MATLAB to compare ts results wth those reported by the ntegrated approach. The fnte element chosen for the stress model s a 4-noded blnear soparametrc element n 2D plane stran. Clearly n ths case the

3 NUMERICAL APPLICATIONS AND VERIFICATION OF AN INTEGRATED FLOW-STRESS MODEL IN PROCESSING OF THERMOSET COMPOSITES DOFs are only of the dsplacement type. In terms of dscretzng the dsplacement feld ths element s essentally dentcal to the 4-1 element. The composte materal n all of the eamples consdered n ths work s assumed to be AS4/31-6. The materal propertes pertanng to the flowcompacton behavour of AS4/31-6 are adapted from Hubert et al. [6]. The thermo-mechancal propertes of the fbres and resn are presented n Table 1. We assume that both the shear and bulk modul, G, and K, of the resn evolve wth changes n temperature and degree of cure throughout the processng. CTE g and CTE r are resn s glassy and rubbery coeffcents of thermal epanson, respectvely. CSC s the volumetrc cure shrnkage coeffcent of resn. The relaaton tmes and ther assocated weght factors for 31-6 resn are obtaned from Km and Whte [7]. Subscrpts u and r denote the un-relaed and relaed values of the modul respectvely. Eample 1. Fully constraned undrectonal [ ] lamnate In ths eample, the stress development n a fully constraned composte materal undergong cure s studed. The sample s assumed to be a undrectonal [ ] lamnate wth a length of 1 mm and a heght of 4 mm (Fg. 2). The problem s modeled by a sngle 4-1 element and the tme-step sze chosen for the numercal soluton s 3 seconds. All boundares are assumed to be mpermeable, and snce all the knematc DOFs are located on the boundary of the sngle element, resn flow s completely nhbted. The temperature hstory and the resultng change n the degree of cure and vscosty of resn are presented n Fg. 3 and Fg. 4, respectvely. Fg. depcts the tme-hstory of σ usng the ntegrated model and compares the results wth those obtaned from the stress model. Fg. 6 presents the comparson of the through-thckness or transverse stress, σ z wth the results obtaned from the stress model. It can be seen that predctons from both models are vrtually dentcal. Eample 2. Un-aally constraned [ ] lamnate under pressure loadng appled on the permeable boundary In ths eample, the stress development response of a un-aally constraned composte materal undergong cure s studed. The geometry and B.C. of the problem are presented n Fg. 7 where the appled pressure on the top surface s f =4 kpa. The sample s a undrectonal [ ] lamnate wth an assumed wdth of 1 mm and a heght of 4 mm. The top surface of the sample s assumed to be permeable. The temperature hstory and the resultng degree of cure and vscosty of resn are the same as the prevous eamples n Fg. 3 and Fg. 4. The problem s modeled by a sngle 4-1 element. Fg. 8 presents the tme-hstory of the developed resn pressure n the system. Fg. 9 depcts the changes n the vertcal velocty of resn at the top surface of the sample durng the process. The evoluton of the resn volume fracton, φ, obtaned by the ntegrated model s presented n Fg. 1. The predctons of the stress model wth ntal and fnal volume fractons are supermposed for comparson. Note that the fnal volume fractons are obtaned from the ntegrated model. The changes n the transverse stran of the system are presented n Fg. 11 and compared wth the results of the stress model assumng both ntal and fnal volume fractons. At the early stages of the process, the resn undergoes epanson due to temperature rse and, n the presence of a permeable boundary, resn flows out of the system to releve some of the developed pressure. In the frst few mnutes however, an nsgnfcant amount of resn flows out of the system as at that stage not enough pressure has bult up n the resn. Ths s evdent n the ntal agreement between the predctons of transverse stran n the sample obtaned by the ntegrated model and the stress model. Fg. 12 depcts the tme-hstory of the stress along the -as, and compares the results wth those obtaned by the stress model assumng both ntal and fnal volume fractons. The appled pressure provdes a sgnfcant motvaton for the resn flow and leads to a mamum resn velocty that s one order of magntude larger than that n the prevous eample. The large amount of resn flow, leads to an apprecable dfference between the ntal and fnal values of resn volume fracton. Ths causes the trend of the post-gelaton behavour of the system obtaned by the ntegrated model to be dfferent from the one predcted by the stress model based on the ntal volume fractons. Fg. 12 clearly shows the dfference between the stress predctons of the 3

4 ntegrated model and the results obtaned from the stress models based on the ntal and fnal volume fractons. The flow of resn s responsble for the nsgnfcant amount of longtudnal stress predcted by the ntegrated approach whle the stress models predct sgnfcant stresses n the -drecton. After the gelaton of resn, the stress tme-hstory predcted by the ntegrated model essentally becomes an offset of the predctons of the stress model wth fnal volume fracton. Eample 3. Flat undrectonal lamnate undergong cure Here, we consder a flat undrectonal lamnate fully bonded to a rgd tool. All materal propertes reman the same as n the prevous two eamples. The geometry and BC of the problem are depcted n Fg. 13. Due to symmetry, only half of the length of the lamnate s analysed. Fg. 14 shows the autoclave temperature cycle. It also presents the assumed thermal hstory of the part and the resultng degree of cure of resn throughout the processng. The predctons obtaned usng three approaches wll be analysed and compared here: () the ntegrated model, () the stress model usng ntal dstrbuton of resn volume fracton, and () the stress model usng the fnal dstrbuton of resn volume fracton. We nvestgate three dfferent drectons for the placement of the fbres: [ ] and [9 ]. The appled load on the top and sde surfaces of the sample s 4 kpa. The problem s modeled by 12 6, 16 8, and 2 16 meshes of 4-1 elements to ensure the convergence of results. Tme-step szes of 6, 12, and 6 seconds are used and the convergence of results wth refnng the tme-steps was confrmed. The 2 16 mesh and tme-step sze of 6 seconds are chosen to report the results and comparsons. To verfy the stresses predcted by the stress model, the beta verson of the commercally avalable COMPRO CCA n ABAQUS was used [8] ncorporatng the varable tme pseudo-vscoelastc (PVE) formulaton of Zobery et al. [3]. A 3D equvalent of our 2D plane stran was modeled usng a mesh of 2 16 elements for [ ] and [9 ] lay-ups. Fg. 1 shows the comparson of profles of σ at secton AB for [ ] and [9 ] lay-ups. It can be seen that the stress results of the MATLAB plane stran and ABAQUS 3D models compare very well. Fg. 16 shows the through-thckness profle of σ at secton AB of the [ ] lamnate obtaned by the three approaches. It s observed that the two stress model approaches lead to a rather large change n the amount of developed stress n the drecton n the neghbourhood of the constraned boundary AE. These sudden changes are accompaned by some oscllatons that de down qute rapdly as we move away from the constraned boundary. The sudden development of stress n the vcnty of the boundary occurs because the constrant on the bottom surface effectvely constrans the fbres located at the bottom of the lamnate from any movement n the drecton. Ths combned wth the thermal epanson of resn pror to gelaton leads to sgnfcant development of stress n that regon. The response obtaned from the ntegrated model (sold lne) s dfferent n that ths boundary layer effect s not observed. The predctons of the stress model wth the fnal volume fractons agrees very well wth the one obtaned from the ntegrated model as we move away from the constraned boundary. Ths shows that the dfference between the two results (whch s located n the vcnty of the constraned bottom surface) s n fact due to stresses that are predcted by the stress model to develop before the nomnal gelaton of resn whle n the ntegrated model no such stresses develop snce the ecess amount of resn can flow out of the system as t undergoes thermal epanson. In the case of [9 ] layup, as t was observed earler n Fg. 1, there are no sgns of sudden changes n σ predcted by stress models snce the fbre drecton s transverse to the -as. Fg. 17 and Fg. 18 present the fnal dstrbuton of aal forces and bendng moments, respectvely, along the length of the lamnate. As a general rule, the results predcted by the stress model wth fnal volume fractons are closer to those predcted by the ntegrated model than the regular stress model approach based on ntal volume fracton. Ths s especally more notceable n the aal force dagrams. Ths dfference could be attrbuted to the flow of resn through and out of the system n the ntegrated approach. Note that there s a sudden change observed n the dstrbuton of aal forces and moments n a regon close to the sde EF of the lamnate. These anomales, whch are clearly vsble n Fg. 18, are the artfacts of the nherent sngularty

5 Temperature ( C) Degree of cure NUMERICAL APPLICATIONS AND VERIFICATION OF AN INTEGRATED FLOW-STRESS MODEL IN PROCESSING OF THERMOSET COMPOSITES at pont E where the assumed fully constraned BC at the bottom meets the free BC on the sde EF. Concluson Varous numercal eamples rangng from smplfed 1D cases represented by a sngle fnte element to a more comple flat composte lamnate undergong cure were presented and analyzed usng the ntegrated flow-stress model. The results obtaned for these numercal eamples were used to verfy the predctons of the ntegrated model as compared to establshed ndvdual models for stress development n the composte materal. The numercal eamples also served to provde comparsons of stress development n the processed composte lamnate wth those obtaned by regular stress models consderng both ntal and fnal resn volume fractons. z Fg. 1. Schematc representaton of Q 1 P (4-1) lnear soparametrc element z u / v nodes Pressure node Acknowledgements The authors gratefully acknowledge fundng from the Natural Scences and Engneerng Research Councl of Canada (NSERC). We would also lke to thank Dr. A. Arafath of Convergent Manufacturng Technologes for hs help wth the COMPRO CCA smulatons. h L Fg. 2. Schematc representaton of a fully constraned [ ] composte sample (Eample 1) Table 1. Mechancal propertes of AS4/31-6 Property AS4 fbre Property 31-6 resn.2 K u (GPa) K r (GPa) E 1 (GPa) 27 G u (GPa) 1.18 E 2 (GPa) 2.7 G r (kpa) 1 G 12 (GPa) 27.6 CTE g (/ C) Temperature Degree of cure.4.2 CTE 1 (/ C) CTE r (/ C) CTE 2 (/ C) CSC (vol.c) Fg. 3. Tme-hstory of the appled temperature and the resultng degree of cure n Eamples 1 and 2

6 σ z (MPa) Resn velocty (mm/s) σ (MPa) P (MPa) Temperature ( C) Vscosty (Pa.s) 2 1E+3 z 1 1E+2 f 1 1E+1 Temperature Resn vscosty 1E+ h 1E Fg. 4. Tme-hstory of the appled temperature and the resultng resn vscosty n Eamples 1 and Fg. 7. Schematc representaton of a un-aally constraned [ ] composte lamnate, wth permeable BC and pressure loadng on the top surface (Eample 2).6 L Stress model Fg.. Tme-hstory of the development of longtudnal stress n Eample 1 predcted by the ntegrated and stress models. The predctons are essentally dentcal Fg. 8. Tme-hstory of the development of resn pressure n Eample Stress model 1.4E-3 1.2E-3 1.E Fg. 6. Tme-hstory of the development of transverse stress n Eample 1 predcted by the ntegrated and stress models. The predctons are essentally dentcal. 8.E-4 6.E-4 4.E-4 2.E-4.E Fg. 9. Tme-hstory of the changes n resn velocty n the vertcal drecton at the top surface of Eample 2

7 σ (MPa) z (mm) Temperature ( C) Degree of cure φ NUMERICAL APPLICATIONS AND VERIFICATION OF AN INTEGRATED FLOW-STRESS MODEL IN PROCESSING OF THERMOSET COMPOSITES Intal value Fnal value Fg. 1. Tme-hstory of the changes n resn volume fracton n Eample 2 z Fg. 13. Geometry and BC of the flat composte lamnate undergong cure (Eample 3) 2 B A Permeable BC 63 mm D C F E f 4.96 mm ε z Stress model - ntal φ Stress model - fnal φ Fg. 11. Tme-hstory of the transverse stran n Eample 2 4 Autoclave Temp. Part Temp. Degree of cure Fg. 14. Tme-hstory of the autoclave and part temperatures and the resultng degree of cure n Eample Stress model - ntal φ Stress model - fnal φ [ ] Stress model [ ] ABAQUS [9 ] Stress model [9 ] ABAQUS Fg. 12. Tme-hstory of the development of n Eample σ (MPa) Fg. 1. Fnal profles of σ predcted by the stress model at secton AB of the [ ] and [9 ] flat lamnates, and ther comparson wth the correspondng predctons usng 3D elements n ABAQUS 7

8 Moment (N.mm/mm) Aal force (N/mm) z (mm) Stress model - ntal φ Stress model - fnal φ σ (MPa) Fg. 16. Fnal profle of σ at secton AB of the [ ] flat lamnate Stress model - ntal φ Stress model - fnal φ (mm) Fg. 17. Fnal dstrbuton of aal force along the length of the [ ] flat lamnate References [1] M. Haghshenas, R. Vazr and A. Poursartp Integratng the smulaton of flow and stress development durng processng of thermoset matr compostes. Proceedngs of ICCM-17, Ednburgh, Scotland, 29. [2] K. Terzagh, Theoretcal Sol Mechancs. Wley, New York, [3] C.L. Tucker and R.B. Dessenberger, "Governng equatons for flow and heat transfer n statonary fber beds," n Flow and Rheology n Polymer Compostes Manufacturng, S. G. Advan, Ed., Elsever, Amsterdam, pp , [4] M. Haghshenas, Integratng resn flow and stress development n process modelng of thermoset compostes, PhD thess, UBC, Vancouver, 212. [] N. Zobery, R. Vazr and A. Poursartp Computatonally effcent pseudo-vscoelastc models for evaluaton of resdual stresses n thermoset polymer compostes durng cure. Compostes Part A: Appled Scence and Manufacturng, Vol. 41, No. 2, pp , 21. [6] P. Hubert, R. Vazr and A. Poursartp A Two Dmensonal Percolatng Flow Model for the Process Smulaton of Comple Shape Composte Lamnates. Internatonal Journal of Numercal Methods n Engneerng, Vol. 44, No. 1, pp 1-26, [7] Y.K. Km and S.R. Whte, "Stress relaaton behavor of 31-6 epoy resn durng cure," Polymer Engneerng & Scence, Vol. 36, No. 23, pp , [8] A.R.A. Arafath, Convergent Manufacturng Technologes Inc., Prvate communcatons, Stress model - ntal φ Stress model - fnal φ (mm) Fg. 18. Fnal dstrbuton of bendng moment along the length of the [ ] flat lamnate