161
NUMERICAL AND SEMI-ANALYTICAL MODELLING OF THE PROCESS INDUCED DISTORTIONS IN PULTRUSION
I. Baran*, P. Carlone**, J.H. Hattel* and G.S. Palazzo** * Technical University of Denmark, Department of Mechanical
Engineering, Denmark
**University of Salerno, Department of Industrial Engineering, Via Giovanni Paolo II, 84084, Fisciano, Italy
ABSTRACT
The geometrical changes of the processed material (process induced distortions) are a critical issue in pultrusion, since they affect the process dynamics (mainly the pull force), as well as the mechanical properties and geometrical precision of the final product. Hence, a detailed understanding of the mechanical behavior generating the distortions during the process is eventually required. In the present study, two different modelling approaches are implemented and compared to calculate the development of the distortions during the pultrusion of a graphite/epoxy composite rod. In both cases, the temperature and the degree of cure distributions are obtained from the thermo-chemical analysis, using a finite element and a finite volume approach, respectively. Process induced distortions have been computed solving a sequential stress-strain finite element model, in the former case. In the latter, the transient distortions are inferred adopting a semi-analytical procedure, i.e. post processing numerical results by means of analytical methods. The predictions of the process induced distortion development using the aforementioned methods are found to be qualitatively close to each other. Furthermore, the location of the detachment between the heating die and the part due to shrinkage is also investigated.
1. INTRODUCTION
Pultrusion is a continuous manufacturing process used to realize constant cross sectional composite profiles. In recent years, the pultrusion process experienced a remarkable growing within the composite industry, due to its cost-effectiveness, automation and quality of products. Nowadays the process is widely used to manufacture highly strengthened structures such as wind turbine blades, window profiles, door panels, and reinforcement beams for concrete. In a pultrusion process, the fibers/mats are first impregnated by the (thermoset or thermoplastic) matrix material by means of a resin bath or employing a resin injection chamber. The wet out reinforcements then enter the forming and curing die where the heat activates the exothermic Proceedings of the 34th Risø International Symposium on Materials Science:
Processing of fibre composites ± challenges for maximum materials performance Editors: B. Madsen, H. Lilholt, Y. Kusano, S. Fæster and B. Ralph
Department of Wind Energy, Risø Campus Technical University of Denmark, 2013
162 reaction p saw and depicted Process i frames a challenge distortion and Holm Gillespie (CTE) of (iii) the t the comp presented [Baran, T (2012c), Nielsen a Carlone Crouch, Arafat (1 In the pre process h thermo-c pultrusio coupling mechanic hardenin pseudo-v been use semi-ana employed material, means of formulati are expo detailed. highlight process. Th finally is in Fig. 1. Fig. 1. induced sh and fencing es in a pult ns. The tran mberg (200 e Jr (1992), f the matrix tool-part int posite thick d in the lite Tutum, and Baran, Tut and Hattel ( and Palazz and Lam ( 996)]. esent paper have been chemical as on process o of a 3D Eu cal analysis g instantan viscoelastic ed to solve b alytical proc d to derive while the p f analytical ion and the osed, while Numerical ting the mos
he cured pro cut to the Schematic v ape distorti panels due trusion proc nsient distor 01), Wisno Johnston ( x material a teraction an kness due erature, in o Hattel (201 tum, and H (2013), Car zo (2008), (2000), Tutu r, two differ proposed, c spects as w of a graphite ulerian ther s. The resin neous linear approxima both bound cedure: a c the 3D dis process ind l methods. adopted sol e in Section l outcomes st relevant c Baran, C ofile is cont desired len view of the ions may b e to their d cess is avoid rtions durin m, Gigliott 1997)]: (i) and the fibe nd (iv) the t to non-uni order to ach 12a), Baran, attel (2013a rlone, Palaz Joshi and L um, Baran rent approac compared a well as proc e/epoxy rod rmo-chemic elastic mod r elastic (CH ation of the dary value p computation stribution o duced distor The paper lution strate n 3 the im s are repor conclusions Carlone, Ha tinuously ad ngth. A sch pultrusion be importan desired high ding or red ng process a ti, Ersoy, C the mismat ers, (ii) the
temperature iform curin hieve a bett
, Tutum, an a), Baran, T zzo and Pasq Lam (2001 and Hattel ches to the and discuss cess induce d. The first m cal model to dulus devel HILE) appr e linear vis problems. T nal fluid dy of temperatu rtion are inf
is structure egies for the mplemented
rted, compa s in Section
attel and Pal
dvanced via hematic vie domain for nt for pultru h geometric ducing the d arise due to Campbell a tch in the c chemical sh e and the d ng. Several ter understa d Hattel (20 Tutum, and quino (2006 ), Joshi, L (2013), V computatio ed. The aim ed transient model (Mod ogether wit lopment is c roach [John coelasticity The second ynamics (C ure and deg ferred post-p ed as follow e thermo-ch d procedure ared and d 5. azzo a a pulling s ew of the p the compos uded produ al precision developmen several me and Potter oefficient o hrinkage of egree of cu numerical anding of th 012b), Bara Hattel (20 6), Carlone am and W alliappan, R onal simulat m of both m t distortion del-1) is bas h a 2D qua calculated b nston (1997 . A finite e model (Mo FD) finite gree of cure processing ws: in Secti hemical bou e for distor discussed in system to th pultrusion p site rod. ucts such as n. Hence, o nt of proces echanisms [ (2006), Bo of thermal e f the matrix ure gradient l models h he pultrusio an, Tutum, a 013b), Baran and Palazz Win Tun (20 Roux, Vaug tion of the p models is t ns in a con sed on the s asi-static pla by means o 7)], which i element sch odel-2) is b volume ap e into the p numerical r ion 2 the th undary value rtion comp n Section 4 he cut-off process is s window one of the s induced [Svanberg ogetti and expansion x material, ts through have been n process and Hattel n, Tutum, zo (2007), 003), Liu, ghan, and pultrusion to predict nventional sequential ane strain f the cure is a valid heme has ased on a pproach is processing results by heoretical e problem puting are 4, briefly
163
Numerical and semi-analytical modelling of distortions in pultrusion
2. THERMO-CHEMICAL
MODEL
Even if pultrusion is conceptually a very simple process, its dynamics are affected by several important aspects, such as heat and mass transfer, resin reaction and phase changes, voids growth or dissolution, interaction between the processing material and the heating die and stress-strain development. All the aforementioned issues are strictly related to thermo-chemical phenomena, which, from a modelling point of view, can be formulated in terms of energy and species balances. In more detail, the temperature distribution into the processing material can be inferred solving the following form of the energy equations:
ߩܥቀడ்డ௧ ݑడ௫డ் యቁ ൌ ݇௫భ డమ் డ௫భమ ݇௫మ డమ் డ௫మమ ݇௫య డమ் డ௫యమ ݍ (1)
where 7 is the temperature, Wis the time, X is the pulling speed (along the x3 direction), U is the
density, &S is the specific heat and N[, N[and N[are the thermal conductivities along [, [and [
directions, respectively. Lumped material properties are used and assumed to be constant throughout the process.
The generative term T at the second member of Eq. 1 is related to the internal heat generation due to the exothermic resin reaction or conversion, i.e. the crosslinking of polymeric chains, quantified by means of the degree of cure Į. Rigorously speaking, Į is defined as the ratio between the reacted species and the total reactive species at the beginning of the process, however, for convenience, it is commonly quantified considering some reaction dependent parameter, such as the released heat. Following this assumption, the degree of cure can be written as the ratio between the amount of heat +W, evolved up to the time W, to the total heat of reaction +WU (for Į = 1). As a consequence, considering that the reinforcing fibers do not contribute to the generative term and indicating the reaction rate (first derivative of the degree of cure with respect to time) as 5U it follows:
ݍ ൌ ൫ͳ െ ܸ൯ߩܪ௧ܴ, (2)
where9I is the fiber volume fraction and UU is the resin density.
Several kinetics models have been proposed and discussed in the inherent literature to describe the evolution of the cure reaction. In the present investigation the well-established n-order model has been adopted [Valliappan et al. (1996)], assuming an Arrhenius type dependence on the absolute temperature:
ܴሺߙǡ ܶሻ ൌడఈడ௧ ൌுଵ ೝ ௗுሺ௧ሻ ௗ௧ ൌ ܭ݁ݔ ቀെ ா ோ்ቁ ሺͳ െ ߙሻ . (3)
The above equations have been solved using a finite element scheme in Model-1. The evaluation of degree of cure and reaction rate has been obtained by means of iterative in-house developed routines implemented into the commercial software ABAQUS (version 6.11, 2011), until the matching of a temperature and degree of cure tolerance. In Model-2 a finite volume approach has been employed, defining the degree of cure as an additional volumetric scalar variable, whose transport equation, in the aforementioned hypothesis, can be written as:
ቀడఈడ௧ ݑడ௫డఈ
యቁ ൌ ܴ. (4)
164
Baran, Carlone, Hattel and Palazzo
3.
CALCULATION OF THE DISTORTIONS
Model-1. The evolutions of the process induced displacements in transverse directions are predicted using a 2D plane-strain model [Baran et al. (2013)]. In this model, the cross section of the part is assumed to move along the pulling direction of the process while tracking the corresponding temperature and degree of cure profiles calculated in the 3D thermo-chemical simulation. In other words, a 3D Eulerian thermo-chemical model is coupled with a 2D plane strain Lagrangian mechanical model. The corresponding transient distortions are calculated based on the temperature and the cure distributions together with the corresponding glass transition temperature (7g) of the cross section by using the quadratic plane-strain elements in
ABAQUS. The instantaneous resin elastic modulus ((U) development during the process is calculated using the CHILE approach [Johnston (1997)], as follows:
ܧ ൌ ൞ ܧ ܧ ் כି் భ ்మି்భ ܧஶ ሺܧஶെ ܧሻ݂ݎ ܶכ൏ ܶ ଵ ܶଵ ܶכ ܶ ଶ ܶכ ܶ ଶ , (5)
where (U and (Uf are the uncured and fully cured resin moduli, respectively. 7& and 7& are the critical temperatures at the onset and completion of the glass transition, respectively, 7 represents the difference between the instantaneous resin glass transition temperature (7g) and
the resin temperature, i.e. 7 = 7
g – 7 [Johnston (1997)]. The evolution of the 7 is given by:
ܶכ ൌ ܶ
െ ܶ ൌ ሺܶ ߙ்ή ߙሻ. (6)
The effective mechanical properties of the composite are calculated by using the self-consisting field micromechanics (SCFM) approach which is a well-known technique in the literature [Bogetti and Gillespie Jr (1992)]. User-subroutines in ABAQUS are used for the calculation of the transient distortions as used in [Baran et al. (2013)].
Model-2. The basic assumption of this model is that the section of the processing material varies along the pultrusion die, preserving the position of its axis of gravity (barycenter), following the approach proposed in [Joshi and Lam (2001)]. The virtual dimension (radial in this case) of L-th control volume can be computed multiplying its initial value times the correction factor:
ߜǡ ൌ ܸߜǡ ܸߜǡ, (7)
being įr,i and įf,i the variation of a unit dimension of the L-th volume entirely filled respectively
with resin or fiber materials:
ߜǡ ൌ ൫ͳ ߝሺܶ െ ܶሻ൯ ή ቀͳ െఊଵೝఈቁ ଵȀଷ
, (8)
ߜǡ ൌ ቀͳ ߝሺܶെ ܶሻቁ. (9)
In the above equation İr represents the thermal expansion coefficient and Ȗr is the percentage
volumetric shrinkage of the fully cured (Į = 1) resin. Repeating the calculation one finally has:
οǡൌ ݎ൫ߜǡെ ͳ൯. (10)
The total displacementǻU can be evaluated extending Eq. 10 to the whole radius. In particular, if the virtual section of the processing composite results greater than the internal die section, i.e.
165 the pultr shrinkage thermal c procedur used, unt The 3D t carried o symmetry wall tem length be are defin exterior temperatu (Lpost-die i Material respectiv pulling sp resin bath is assum epoxy res Fig. 2. S mm. Materi Resin Fiber Nu ruded is co e effect pre contact resi re connectin til a tempera thermo-chem out. Only a y. Instead mperature [V etween 0-91 ned in which surface of ure depend in Fig. 2) properties vely. The pa peed is set h temperatu med to be un sin is assum Schematic v ial ȡ [kg n 126 r 179 umerical an ompressed evails, indu istance (TC ng the therm ature criteri
4.
mical simul a quarter o of using th Valliappan e 15 mm (i.e h no heat fl the pultrud dent convec is determin and the r arameters us to 5 mm/s. ure of 38 oC ncured (D = med to be 4% view of the Table 1. M m-3] [J k 60 1 90 Table 2 . [s-1] 19.14 E+ d semi-anal into the d cing the de CR) is induc mochemical on is satisfiRESULT
lation of the of the pultr he die and t et al. (1996 . inside the low is allow ded rod is e tive heat tr ned to be a resin kineti sed in the C The inlet te C [Valliappa = 0). In the % [Joshi and pultrusion Material ph cp kg-1 K-1] [ 1255 712 2. Resin kin ] ( [J m +4 60.5 E lytical mode die, perfect etachment o ced betwee l model wit ied.TS AND D
e pultrusion rusion dom the heaters 6)] is appli e die). At th wed across exposed to ransfer coef approximat ic paramet CHILE appro emperature an et al. (19 e present st d Lam (200 domain for hysical prop kx3 [W m-1 K-1] 0.2 66 netics and rh mol-1] E+3 1 elling of dis thermal c of the mate en the die a th the dimeDISCUSS
n of a UD g main, seen in the num ed to the o he symmetr the bounda ambient te fficient. Th tely 1370 m ters are lisoach for Mo of the com 96)] and the tudy, the to 01)]. r the compo perties and c Kx1,x2 [W m-1 K 0.2 11.6 heological p Q + 1.69 3 stortions in contact is a rial from th and the com nsional cha
IONS
graphite/epo in Fig. 2, merical mod outer surfac y surfaces aries. At the emperature e length of mm [Valliap ted in Tab odel-1 are g mposite part e matrix ma tal volumet osite rod. A concentratio K-1] İr [°C 4.5E 7.2E parameters +WU [J kg-1] 23.7 E+3 pultrusion assumed. W he die, an a mposite. An ange model oxy compos is modelle del, the mea ce of the ro adiabatic b e post die re (27 oC) as f the post d appan et al. ble 1 and given in Tab is assumed aterial at the tric shrinka All dimensio on C-1] fraV E-5 0 E-6 0 When the additional n iterative has been site rod is ed due to asured die od for the oundaries egion, the ssuming a die region . (1996)]. Table 2, ble 3. The d to be the e die inlet age of the ons are in Vol. action 0.378 0.622166 ( The calcu that the p al. (1996 and conv the die w generatio The max region, th after the of cure i correspon F The tran distributi The resu composit analytica developm calculate However post die Model-1 the mech the fiber controlle end of th which ag carbon/ep the transv Table 3. P (U[MPa] 3.447 ulated centr predicted re 6)]. This sho verged to a r wall tempera on of the ep ximum com he degree o die exit. Th is increased nds to a per Fig. 3. The nsient distor ions in Mod ults are dep
te rod, i.e. th al model (M ments are su ed by the f r, there is a region in M which calc hanical resp s dominate ed by the ma he process a grees well poxy with a verse direct Parameters (Uf[MPa 3.447E+3 reline tempe esults match
ows that the reliable solu ature after a poxy resin. A mposite temp
of cure is in his fact was d from 0.83 rcentage inc centerline t rtions are c del-1 (2D p picted in Fi he top poin Model-2) w ufficient to finite eleme significant Model-1 an ulates the e ponse of the the longitu atrix materi are found to with typic a fiber volu tion in Mod Baran, C used in the a] 7&[oC 3 165 erature and h quite well e present nu ution. The t approximate At that poin perature is ncreased sli also observ 38 (at the d crease of app temperature calculated plane strain g. 4 for the t in Fig. 4. I with access o obtain a t ent method deviation b nd Model-2. ffective com e composite udinal prop ial. For insta
be approxim cal values ume fraction del-2, homo Carlone, Ha CHILE mo C] 7& 215 d degree of c with the av umerical sc temperature ely 380 mm nt the peak calculated ightly which ved in [Vall die exit) to proximately e (OHIW) and d using the o FE model) e displacem It is seen th to the ca trend for th d (FEM) (M between the . As aforem mposite mec quite well perties, on t ance, the lo mately 130 given in n of 60%. F ogenization
attel and Pal
odel (Eq. 5) [oC] 7J 0 cure profile vailable exp chemes in M of the com m from the d increase rat approximat h indicates iappan et al o 0.867 (at y 3.3%. degree of cu obtained te and Mode ment evolut hat the mech alculated tem he transient Model-1) fo e calculated mentioned, chanical pro [Bogetti an the other ha ongitudinal a GPa and 1 [Zenkert an For the virt
of the fiber azzo [Baran et a J[oC] es are shown perimental d Model-1 and mposite part
die inlet due te of degree tely as 205 that the cu l. (1996)]. T the end of ure (ULJKW) d emperature l-2 (1D sem ion of the hanical assu mperature distortions or the circu d displaceme the SCFM operties. Th nd Gillespie and the tran and the tran 0 GPa, resp nd Battley tual displac s and the ep al. (2013)]. D7J 380 n in Fig. 3. data in [Vall d Model-2 becomes hi ue to the inte e of cure is oC. At the uring still ta The centerli f the proces distributions and degree mi analytica outer surfa umptions in and degree s similar to ular compo ents especia approach i his approach e Jr (1992)] nsverse pro nsverse mod pectively in (2009)] f cement calcu poxy resin v It is seen liappan et are stable igher than ernal heat obtained. e post die akes place ne degree ss) which s. e of cure al model). ace of the the semi-e of cursemi-e the ones osite part. ally at the is used in h captures such that operties is duli at the Model-1, for a UD ulation in variations
167 is consid are calcu resin is c cause a l inside of predict th die-part i from Fig Model-2 top point Model-2, In the pr (Model-1 distributi models. evolution using the expansio to the on between distortion Model-2. This wor (EC) und Nu dered (Eq. 7 ulated relativ changed afte larger devia f the heating he transient interface, w g. 4. The de which is a t in Fig. 4 i , respective Fig. 4. resent work 1) and a 1D ions are firs Using thes n of the com e two mode on and chem nes obtained the two re n behavior . rk is a part der FP7 prog umerical an 7) in terms o vely smalle er vitrificati ation in the g die. Insid t distortions which can b tachment is good agree is calculate ly. The proces k, the transie D semi-analy st calculated se temperat mposite surf els since the mical shrink d in Model esults has in the trans of DeepWin gram platfo d semi-anal of fiber volu er in Model-ion accordin displaceme de the heatin s with reaso e defined a s found to b ement betw d approxim s induced tr
5.
C
ent distortio ytical mode d at steady ture and d face was pr e process in kage. The di l-1 inside th become lar sverse direc ACKNO nd project w orm Future E lytical mode ume fractio -2 if compa ng to the CH ent evolution ng die, the onable accu as the “zero be approxim ween the two mately as -0 ransient dispCONCLU
ons were pr el (Model-2 state and go degree of c redicted. A nduced var isplacement he heating rger since ction is mor OWLEDGE which has b Emerging T elling of dis on. Hence, th ared to Mod HILE appro n at the pos simplified uracy. The o” displacem mately 540 m o models. T .025 mm an placementsUSIONS
redicted usi 2). The temp ood agreem cure profile similar tren iations wer t levels in M die. For thethe effect re pronounc EMENTS been grante Technology. stortions in he transient del-1. The st oach (see Eq st-die region 1D model virtual deta ment during mm in Mod The residual nd -0.017 m in the [-di ing a 2D pl perature an ment was fou
s, the tran nd of develo e defined b Model-2 we e post die r of the mat ced in Mod d by the Eu pultrusion t displacem stiffness of t q. 5), hence n as compa (Model-2) achment po g the proces del-1 and 58 l displacem mm for Mo irection. lane strain F nd the degre und betwee nsverse disp opment was based on th ere found to region, the trix materia del-1 as com uropean Co ment levels the epoxy e this may red to the is able to oint at the ss, is seen 80 mm in ent of the del-1 and FE model ee of cure en the two placement s obtained he thermal o be close deviation al on the mpared to mmission
168
Baran, Carlone, Hattel and Palazzo
REFERENCES
Baran, I., Tutum, C.C., and Hattel, J.H. (2012a). Optimization of the thermosetting pultrusion process by using hybrid and mixed integer genetic algorithms. App. Compos. Mat. DOI: 10.1007/s10443-012-9278-3.
Baran, I., Tutum, C.C., and Hattel, J.H. (2012b). Reliability estimation of the pultrusion process using the first-order reliability method (FORM). App. Compos. Mat. DOI: 10.1007/s10443-012-9293-4.
Baran, I., Tutum, C.C., and Hattel, J.H. (2012c). Probabilistic thermo-chemical analysis of a pultruded composite rod. In: Proceedings of the 15th European Conference on Composite Materials (Venice, Italy, 24-28 June 2012).
Baran, I., Tutum, C.C., and Hattel, J.H. (2013a). The internal stress evaluation of the pultruded blades for a Darrieus wind turbine. Key Eng. Mat. 554-557, 2127-2137.
Baran, I., Tutum, C.C., and Hattel, J.H. (2013b). The effect of thermal contact resistance on the thermosetting pultrusion process. Compos. Part B: Eng. 45, 995-1000.
Baran, I., Tutum, C.C., Nielsen, M.W., and Hattel, J.H. (2013). Process induced residual stresses and distortions in pultrusion. Compos. Part B: Eng. 51, 148-161.
Bogetti, T. A., and Gillespie Jr, J. W. (1992). Process-induced stress and deformation in thick-section thermoset composite laminates. J. Compos. Mater. 26 (5), 626–660.
Carlone, P., Palazzo, G.S., and Pasquino, R. (2006). Pultrusion manufacturing process development by computational modelling and methods. Math. Comput. Model. 44, 701-709. Carlone, P., and Palazzo, G.S. (2007). Pultrusion manufacturing process development: Cure
optimization by hybrid computational methods. Comput. Math. Appl. 53, 1464–1471.
Carlone, P., and Palazzo, G.S. (2008). Viscous pull force evaluation in the pultrusion process by a finite element thermo-chemical rheological model. Int. J. Mater. Form. 1, 831–834.
Johnston, A. (1997). An Integrated Model of the Development of Process-Induced Deformation in Autoclave Processing of Composites Structures. (Ph.D. thesis, The University of British Columbia, Vancouver).
Joshi, S.C., Lam, Y.C., and Win Tun, U. (2003). Improved cure optimization in pultrusion with pre-heating and die-cooler temperature. Compos. Part A: Appl. S. 34, 1151-1159.
Joshi, S.C., and Lam, Y.C. (2001). Three-dimensional finite-element/nodal-control-volume simulation of the pultrusion process with temperature-dependent material properties including resin shrinkage. Compos. Sci. Technol. 61, 1539–1547.
Liu, X.L., Crouch, I.G., and Lam, Y.C. (2000). Simulation of heat transfer and cure in pultrusion with a general-purpose finite element package. Compos. Sci. Technol. 60, 857-864.
Svanberg, J.M., and Holmberg, J.A. (2001). An experimental investigation on mechanisms for manufacturing induced shape distortions in homogeneous and balanced laminates. Compos. Part A: Appl. S. 32, 827–838.
Tutum, C.C., Baran, I., and Hattel, J.H. (2013). Utilizing multiple objectives for the optimization of the pultrusion process. Key Eng. Mat. 554-557, 2165-2174.
Valliappan, M., Roux, J.A., Vaughan, J.G., and Arafat, E.S. (1996). Die and postdie temperature and cure in Graphite/Epoxy composites. Compos. Part B: Eng. 27, 1-9.
Wisnom, M.R., Gigliotti, M., Ersoy, N., Campbell, M., and Potter, K.D. (2006). Mechanisms generating residual stresses and distortion during manufacture of polymer–matrix composite structures. Compos. Part A: Appl. S. 37, 522-529.
Zenkert, D., and Battley, M. (2009). Laminate and sandwich structures: Foundations of fibre composites (Polyteknisk Forlag, Denmark, 2nd Edition).
