Coupled compression RTM and composite layup optimization

R. Le Richea*, A. Saouabb, J. Breardb

aCNRS URA 1884/SMS, Ecole des Mines de Saint Etienne, 158 Cours Fauriel, Saint Etienne cedex 2 42023, FrancebLab. de Mecanique, Univ. du Havre, France

Received 17 December 2002

Abstract

A methodology for studying the implications of taking into account manufacturing at early design stages is presented. It isapplied to a rectangular laminate made by RTM and compression (RTCM). The plate is designed for injection time, maximummold pressure, stiffness and buckling. Semi-analytical, numerically inexpensive models of the processes and structure enable athorough investigation of the couplings between process and structure by comparing four design formulations: in decoupled pro-

blems, either the process or the structure is optimized; then, the process is optimized with targetted low or high structural perfor-mance. A globalized Nelder–Mead optimization algorithm is used. The rigour of the method and the relative simplicity of theapplication case provide a clear description of how maximum mold pressures, injection times and final structures properties are

traded-off.# 2003 Elsevier Ltd. All rights reserved.

Keywords: Optimization; C. Laminates; B. Porosity; C. Buckling; E. Resin transfer moulding

1. Introduction

After having been praised for their mechanical prop-erties, composite materials have lately received muchattention with regard to reducing their manufacturingcost. The possibility of automating composite structureproduction by processes based on resin transfer moldingor resin infusion is seen as an important factor forcontrolling costs [1].Choosing a structural design (which includes materi-

als) and a manufacturing process are coupled decisions[2]. While early composite design studies simplified theanalysis by separating structural design [3] and processtuning [4], in the last decade cost has increasingly beenconsidered at an early stage.A series of works focus on tuning a particular process,

e.g. Resin Transfer Molding (RTM), and sizing thestructure simultaneously [5,6].Other contributions estimate cost in terms of struc-

tural complexity, independent of the process. In [7,8],ease of prepreg layup is considered as one of the design

criteria for general structures along with strength. In [9],the complexity depends on the geometry of ribs andspars of a composite box.Finally, in [10] and [11,12], different manufacturing

routes are compared using cost models.In terms of optimization methods, most of the afore-

mentioned works rely on enumeration [10,5] or problemdecomposition and local searches [7,12]. Some attemptshave been made at using more robust, but numericallymore expensive, global optimization algorithms (e.g.genetic algorithms in [4,6]. Also, because many criterianeed to be traded off against one another, multi-criteriaapproaches have appeared in [12,9].This article presents a methodology to study the

effects of taking into consideration the process at earlydesign stages in a numerically well-founded fashion. Asimple, yet common, scenario is considered: a rectan-gular plate is manufactured by resin transfer moldingand compression (RTCM). Injection can be flow rate orpressure controlled and compression can follow or besimultaneous with injection, which means that there arefour processes to study. Process criteria are maximummold pressure and injection time. They affect toolingquality and time to market. They are therefore partlyrepresentative of manufacturing cost. Structural criteria

0266-3538/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0266-3538(03)00195-7

Composites Science and Technology 63 (2003) 2277–2287

www.elsevier.com/locate/compscitech

* Corresponding author. Tel.: +33-477-420074; fax: +33-477-

420249.

E-mail address: leriche@emse.fr (R. Le Riche).

are stiffness and buckling. Under these assumptions,semi-analytical models of the process and structure areavailable. These models are numerically inexpensive,which enables the use of a global optimization algo-rithm. The process parameters and the ply orientationsare the design variables. The effects of required struc-tural performance and maximum allowable mold pres-sure on optimal processes and designs are characterizedby solving coupled optimization problems.

2. Design formulation

2.1. Structural and RTCM couplings in composites

When considering the RTCM process, engineers aretypically concerned with tuning parameters such asinjection pressure/flow rate, temperature, gate/vents

positions and preform characteristics. The criteria theyobserve are the injection time, the maximum moldpressure, the temperature distribution, preform damage,void contents and degree of curing. At the same time,structural designers specify materials, geometry, fibervolumes, fiber orientations and ply thicknesses toachieve minimum mass and specified strength, stiffness,damage or other failure criteria. An overview of processand structure variables and criteria involved in compo-sites engineering is given in Fig. 1. While process andstructure engineering are traditionally separate activ-ities, the central part of the figure shows variables andcriteria that are common to both fields: preform char-acteristics, void content, polymerization and preformdamage couple manufacturing and the final structure.A method will shortly be introduced that estimates

the consequences of accounting for these couplings incomposite laminates. A channel flow model of the

Fig. 1. Process and structure couplings.

Nomenclature

*, allow optimal, allowable valuea,b plate length and widthA1, A2, B1, B2 permeability law parameters(CL) coupled low structural performance

optimization problem(CO) coupled optimal structure

optimization problemE1,f, E2,f longitudinal and transverse fiber

Young’s moduliEm resin Young’s modulusEy transverse effective stiffnessG12,f fiber shear stiffnessGxy shear effective stiffnesshi(t) i-th ply thickness at time tH(t) laminate total thickness at time tIP % of laminate length filled before

compressionKx average longitudinal permeabilityl buckling load factor

� resin viscosity�12,f, �m fiber and resin Poisson’s ratioNx, Ny longitudinal, transverse in-plane

loadP injection pressure(PO) process only optimization problemPmax maximum mold pressureRC compression rateRF injection flow rate(SO) structure only optimization problem�, �i fiber orientation (in i-th set of plies)Tinv inversion time (RTCM-SP process

only)TP total injection timevf,i(t) i-th ply fiber volume fraction at time

tVf(t) average fiber volume fraction at

time tx generic vector of optimization

variables

2278 R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287

RTCM process [13] is used. Among all the RTCMprocesses that are described in Appendix A, four arecompared here:

RTCM-P A successive two step process, starting witha pressure imposed injection, at P constant, followedby compression at a constant rate, RC. IP% of thelaminate length is filled only by injection beforecompression starts, i.e. compression may be appliedbefore the laminate is fully infiltrated.RTCM-F Same as above, except that injection isperformed at a constant flow rate RF.RTCM-SP Pressure controlled injection and com-pression are simultaneous.RTCM-SF Flow rate controlled injection and com-pression are simultaneous.

Note that these process models assume a closed mold,i.e. no resin can be vented in the compression phase. Inthis case, when IP=100% or RC=0 m/s, the RTCMprocess degenerates into a RTM process.

2.2. Accounting for process and structure in the optimi-zation problem formulation

The process–structure couplings first take place in thephysical model of the system that is being optimized.They need to be taken into account further in the opti-mization problem formulation. This is achieved whenchoosing design variables and criteria.There are two possible formulations for a coupled

design problem. One can solve a multi-objective opti-mization problem (e.g. [12,9]),

minx2Sf1 xð Þ;and minx2Sf2 xð Þ;. . .and minx2Sfmþ1 xð Þ;

8>><>>: ð1Þ

where fi are the criteria, x is a vector of design variablesand S is the design space. Alternatively, a constrainedsingle objective problem is formulated by transformingsome criteria into inequalities, gi,

minx2Sf xð Þ;and g1 xð Þ4 0;. . .and gm xð Þ4 0:

8>><>>: ð2Þ

With respect to a criterion, a constraint requires anadditional allowable value. As an example, the criterionassociated with the maximum pressure during injection,Pmax, is

min Pmax; ð3Þ

while the constraint is

satisfy Pmax4Pallowmax : ð4Þ

The multi-objective approach, which does not need apriori allowable values, is the most general formulation.It has many—eventually, an infinite number of—solu-tions, called the Pareto set, which is all the possiblecompromises that can be struck between the criteria.However, from a numerical point of view, solving themulti-objective problem (1) is much more complex thansolving the constrained single objective problem. (2)State-of-the-art multi-objective algorithms are evolu-tionary algorithms (see [14] for a review) which have aslower convergence than other single objective algo-rithms such as the Globalized and Bounded Nelder–Mead algorithm (see Section 2.4 and [15]). Furthermore,the set of solutions need to be post-processed in order todecide which compromise should be manufactured.In the current work, a complete multi-objective opti-

mization is replaced with a series of four constrainedsingle objective problems: a process only problem (PO),a structure only problem (SO), a coupled problem withlow performing structure (CL), and a coupled problemwith optimal structure (solution of the structure onlyproblem) (CO). Because the series is composed of con-strained single objective problems, numerical resolutionis eased. Extreme regions of the Pareto set are located inthe process and structure only problems. The solutionof the structure only problem is an indication of whichallowable values can be achieved in the coupled pro-blems. From a didactic point of view, comparing thesolutions of the four problems shows how simultaneousconsideration of the process and the structure affects thefinal design and its manufacturing.

2.3. Detailed design problems formulations

The series of four optimizations are repeated for thefour processes RTCM-P, RTCM-F, RTCM-SP andRTCM-SF. In all processes, imposed pressures, flowrates, and compression rates are time constants (butoptimization variables).Although the proposed method can be applied to any

laminate, symmetric and balanced laminates of eightlayers are considered, so there are two design variablesto describe the stacking sequence, �1 and �2. The �s arezero when the fibers are aligned with the resin flow. Idenotes the injection imposed condition, that is, either apressure P or a flow rate RF . All design variables arecontinuous.There are five design variables for processes that have

successive injection and compression,

PRF

;RC; IP; �1; �2

�� �; ð5Þ

and four design variables when injection and compres-sion are simultaneous since IP has no meaning in thiscase,

R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287 2279

x ¼PRF

;RC; �1; �2

�� �: ð6Þ

The two process design criteria are the total injectiontime, TP, and the maximum mold pressure, Pmax. Bothmeasures are representative of the manufacturing cost.TP is a small part of the total production time which isdominated by the curing time. In practice however, theinjection time is the main degree of freedom of the pro-duction time since the resin choice is limited. Pmax is anindication of the tooling price. High Pmax implies heavymetal molds while a low Pmax suggests that a compositemold (like in RTM-light, cf. [16]) may be sufficient.Note that when injection and compression are succes-sive, TP is the total time of the two phases. The modelsused to calculate TP and Pmax are presented in Appen-dix A. In the RTCM-SP process, all combinations ofpressure and compression rate are not valid. The pres-sure in the mold may become larger than the injectionpressure, which causes an inversion of the resin flow attime Tinv. This scenario is avoided by defining an inver-sion time constraint

TP4Tinv; ð7Þ

which is added to all optimization problems involvingthe RTCM-SP process. Structural design criteria con-cern the transverse effective stiffness Ey, the shear effec-tive stiffness Gxy and the buckling load factor l.Calculations of the structural criteria are based on theClassical Lamination Theory and are outlined inAppendix B. Ex, the longitudinal effective stiffness, mayalso be an important structural property. Yet, Exdoesn’t need to be included here because it is redundantwith the process criteria, TP and Pmax: Ex, like TP andPmax, drives the fiber orientation to 0�.The process only problem (PO), is

POð ÞminI;RC;IP;�1;�2TP;such that Pmax4Pallowmax ;

�ð8Þ

where Pallowmax is a user-defined: allowed maximum moldpressure.Moregenerally, allowable valuesaredenotedwiththe allow superscript. The structure only problem, (SO), is

SOð Þ

maxI;RC;IP;�1;�2;Ey;such that Gxy5Gallowxy ;and l5 1:

8<: ð9Þ

Let E�y ; G�

xy and l� denote the structural properties ata solution of the (SO) problem. A first coupled process–structure problem, where the structure has a low qual-ity, is

CLð Þ

minI;RC;IP;�1;�2TP;such that Pmax4Pallowmax ;Ey5Eallowy ;

Gxy5G allowxy andl5 1:

8>>>><>>>>:

ð10Þ

This problem has a low structural quality because theallowable values in the structural constraints are chosensuch that Eallowy < E�

y ; Gallowxy < G�xy and lallow ¼ l�:

Finally, the coupled process–structure problem with anoptimal structure, (CO), has the same expression as(CL), but the allowable values of the structural criteriacorrespond to the solution of the structure only prob-lem, i.e. Eallowy ¼ E�

y ; Gallowxy ¼ G�

xy and lallow ¼ l�:

2.4. The globalized and bounded Nelder–Mead optimiza-tion algorithm

The constrained single objective optimization pro-blems are dealt with by the Globalized and BoundedNelder–Mead algorithm ([15]), GBNM. This algorithmis based on repeated direct Nelder–Mead (sometimesalso called simplex, [17]) searches, which work on con-tinuous variables, do not need gradient informationand, individually, converge faster than probabilisticglobal search methods such as evolutionary algorithms([18]). The method is made global by restarting localsearches based on a probability density which biasesnew local searches towards unexplored regions of thedesign space S. Restarts are stopped when a user speci-fied maximum number of analyses, Nmax, has beenreached. If the maximum number of analyses is largeenough and the problem has several local optima, theGBNM algorithm typically finds many of them. Notethat degeneracy cases of the Nelder–Mead methods aredetected and induce re-initialization of the simplex.Additionally, the GBNM algorithm copes with boundson the variables by projection and general non-linearconstraints by an adaptive linear penalty scheme.

3. Numerical applications

A plate of length a=0.5 m and width b=0.1 m sub-ject to a bi-axial in-plane load, Nx=200 000N,Ny=20000N, is considered. Its fiber and resin proper-ties are: E1,f=80. 109 Pa, E2,f=80. 109 Pa, �12,f=0.22,G12,f=35.2 109 Pa, Em=3.45 109 Pa, �m=0.3. The initialply thickness (before compression) is hi(0)=1 mm witha fiber volume fraction �f,i(0)=0.4, i=1,8. The resinviscosity is �=0.16 Pa.s. � is assumed here to be a timeconstant, which is valid if the curing time is larger thanthe injection time. The coefficients of the permeabilityrelations (cf. Eq. (18)) are A1=1.7 10�12m2, B1=�4.12,A2=6.8 10�14 m2 and B2=�5.64.

3.1. Parametric study

Before turning to a complete optimization of the sys-tem, a parametric study is performed that gives an insightinto the design variables and criteria relationship. Tosimplify the interpretation, the set of optimization

2280 R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287

variable of Eqs. (5) or (6) is replaced with two variables,that are varied in turn: the fiber orientation � of a((

?

)2)s laminate, and the injection length beforecompression starts.For comparison purposes in the parametric study, the

imposed pressure and flow rates are chosen equivalenton average, i.e. they satisfy

RF ¼1

2�H 0ð ÞPKx; ð11Þ

where Kx is the average longitudinal permeability (cf.Appendix A) and H(0) is the initial total thickness. Theprocess where injection is pressure controlled andsimultaneous with compression is not studied herebecause under the above conditions and a compressionrate RC=1. 10�6 m/s, inversion will always occur.In Fig. 2, the variation of the design criteria as a

function of the fiber orientation � is plotted. Thefollowing observations can be done:

� The injection time (top-left plot) is independentof the fiber orientation when the flow rate isimposed, while it increases when pressure isimposed and the fibers are disoriented with respectto the injection direction (the permeabilitydecreases).

� The maximum pressure during manufacturing(top-right plot) increases when the fibers aredisoriented. The augmentation is drastic wheninjection flow rate is imposed and compression issimultaneous. The rise is much higher when flowrate is imposed than when pressure is imposed.

� The simultaneous process, with the given set-tings, yields a structure that is thicker than thatof successive processes. In other words, the finalfiber volume fraction, Vf(TP), is lower. Thisexplains why in-plane stiffnesses are lower in thesimultaneous process while the buckling loadfactor is larger. Indeed, the in-plane stiffnesses

Fig. 2. Design criteria vs. � for 3 processes, RTCM-P (+), RTCM-F (�) and RTCM-SF (o), ((��)2)s, P=3. 105 Pa, RF=5.5 10�7 m3/s, IP=80%,

RC=1.10�6 m/s.

R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287 2281

increase linearly with Vf (1/thickness) while thebuckling load increases with (thickness3). Interms of buckling, it is more efficient to sacrificefiber fraction for a better flexural strength. Notethat this observation is case dependent: for asufficiently higher compression rate or a lower IPor a lower RF, the simultaneous process yieldsthinner laminates than the successive processesdo.

� It is confirmed that the optimal fiber orientationis 90� for Ey and 45� for Gxy and buckling.

� The successive processes, with the flow rate or thepressure imposed, produce structures with thesame final thicknesses, therefore the same struc-tural criteria. Indeed, the final thickness dependsonly on the compression phase, which is the samein both cases (RC and IP are identical).

Design criteria are represented as a function of thepercentage of plate length filled by injection before

compression starts, IP, in Fig. 3 under the same materialand process conditions as in Fig. 2. The followingsimple comments can be made:

� The compression rate is slow with respect to theinjection flow rate or pressure, so the injectiontime decreases with IP. If the compression ratewere higher, opposite conclusions would bedrawn.

� For successive processes, the maximum pressuredecreases with IP as long as it occurs duringcompression. Reciprocally, when maximumpressure occurs during injection, it increases withIP. The top-right plot of Fig. 3, for RTCM-F andIP beyond 45% gives such an example.

The results of the parametric study are specific to thematerial chosen and process tuning. No monotony inthe criteria variation as a function of the variablesexists. In order to properly account for all parameter

Fig. 3. Design criteria vs. percentage of injection length before compression starts for 3 processes, RTCM-P (+), RTCM-F (�) and RTCM-SF (o),

(54�/90�)s, P=3. 105 Pa, RF=5.5 10�7 m3/s, RC=1. 10�6 m/s.

2282 R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287

combinations, it is necessary to optimize the systemwith a global non-linear algorithm.

3.2. Optimization results

The GBNM algorithm (see paragraph Section 2.4) isapplied to 32 optimization problems. Each search isNmax=80 000 analyses long. Depending on the process,the average CPU time for one analysis ranges from 0.02to 0.2 s on a 800 MHz I686 IntelTM Pentium processor.The 32 problems correspond to the four formulations,process only (PO), structure only (SO), coupled lowstructural performance (CL) and coupled optimalstructure (CO), repeated with the four processes(RTCM-P, RTCM-F, RTCM-SP, RTCM-SF) and twomaximum pressures in the mold, Pallowmax ¼ 10: 105 and 6.105 Pa, respectively. The stacking sequence considered is(�1/�2)s. The design variables are bounded as fol-lows: 0.5 1054P46. 105 Pa, 0.5 10�74RF4100. 10�7

m3/s, 0.4IP4100.%, 0.4RC40.4 10�3 m/s,0.4�490�. Allowable criteria values are Eallowy ¼

15: 109 Pa; Gallowxy ¼ 5: 109 Pa and lallow ¼ 1: in (SO)

and (CL). By definition, mechanical allowable values in(CO) stem from the solution of the (SO) problem (seeSection 2.3). The results of the optimizations are givenin Tables 1–5.The principal significance of these results consist in

the numerical solutions to the coupled process-structureproblems, (CL) and (CO).

(CL) problem: at Pallowmax ¼ 10:105 Pa; RTCM-P isoptimal with an injection time of 326 s. Active con-straints are Pmax and Ey. RTCM-F and RTCM-SPare second best processes with a time of 390 s. Itshould be noted that RTCM-SP is a gentle processbecause it results in a lower mold pressure (6.105 Pamaximum). If Pallowmax is decreased to 6.105 Pa, a pres-sure controlled injection without compression isoptimal in 391 s. For low mold pressure, RTM is theoptimal process, and both RTCM-P and RTCM-SPcan degenerate into it.(CO) problem: RTCM-P is again the optimal processfor both pallowmax ¼ 10:105 and 6.105 Pa with injectiontimes of 785 and 912 s, respectively. All constraints

Table 1

(PO), (SO), (CL) and (CO) solutions for the RTCM-P process, Pallowmax ¼ 10:105 Pa

P (105 Pa)

Rc (10�5 m/s) Ip (%) �1 (

�)

�2 (�) Tp (s) Pmax(10

5 Pa)

Ey (GPa) l Gxy (GPa) Hf (mm)

(PO)

6. 1.702 82.88 0. 0. 233. 10. 9.55 0.87 3.71 7.178

(SO)

a a 81.75 54.46 90. [785.,1[ [P,1[ 27.64 1.3 6.89 7.125

(CL)

6. 1.147 81.31 54.71 0. 326. 10. 15. 1.31 6.89 7.103

(CO)

6. 0.461 81.75 54.46 90. 785. 10. 27.64 1.3 6.89 7.125

a All P and RC are optimal, associated process criteria are given as intervals.

Table 2

Coupled optimization with low structural performance (CL), for all processes, Pallowmax ¼ 10:105Pa

RF or P

(10�7 m3/s or 105 Pa)

RC(10�5 m/s)

IP (%)

�1 (�) �2 (

�)

TP (s) Pmax

(105 Pa)

Ey (GPa)

l Gxy (GPa) Hf (mm)

RTCM-F

6.27 1.128 81.96 58.74 0. 390. 10. 16.36 1.29 6.50 7.134

RTCM-P

6. 1.147 81.31 54.71 0. 326. 10. 15. 1.31 6.89 7.103

RTCM-SF

5.11 0. a 59.35 0. 470. 10. 15. 1.63 5.82 8.

RTCM-SP

6. 0. a 59.35 0. 391. 6. 15. 1.63 5.82 8.

a Variable not relevant to the process.

Table 3

Coupled optimization with optimum structure (CO), for all processes, Pallowmax ¼ 10:105 Pa

RF or P

(10�7 m3/s or 105 Pa)

RC(10�5 m/s)

IP (%)

�1 (�) �2 (

�)

TP (s) Pmax

(105 Pa)

Ey(GPa)

l

Gxy(GPa)

Hf

(mm)

RTCM-F

2.75 0.460 81.75 54.46 90. 905. 10. 27.64 1.30 6.89 7.125

RTCM-P

6. 0.460 81.75 54.46 90. 785. 10. 27.64 1.30 6.89 7.125

RTCM-SF

1.03 0.046 a 54.46 90. 1894. 10. 27.64 1.30 6.89 7.125

RTCM-SP

6. 0.088 a 54.46 90. 992. 6. 27.64 1.30 6.89 7.125

a Variable not relevant to the process.

R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287 2283

are active (structural constraints are active byconstruction of the (CO) problem).

Comparing (CL) with (CO) results shows that highstructural performance more than doubles the injectiontime. In other words, when aiming at low structuralperformance, an uncoupled design procedure where,first, the layup is optimized and, second, the process istuned, doubles the injection time with respect to a cou-pled procedure (CL).Solutions of the process only and structure only pro-

blems are familiar to many experts. Yet, in order to becomprehensive, they are now commented. Secondarily,these classical solutions validate the numerical optimizersimplemented.

Process only problem: fibers are oriented along theresin flow, �1=�2=0. Compression is useful to suc-cessive processes, except RTCM-P at low Pallowmax .Compression never speeds up simultaneous processesbecause of the associated loss of permeability.Structure only: loss of solution uniqueness for all pro-cesses, the structure only problems (SO) have aninfinite number of solutions, corresponding to allinjection and compression settings that produce thesame final thickness. These solutions are not equiva-lent in terms of injection time or mold pressure, butthese criteria are not considered in (SO). It can beseen for the RTCM-P process in Table 1 on the (SO)line. For RTCM-P and a fixed IP, any combination ofP and RC results in the same final thickness, which is aresin conservation property. Indeed, IP sets the amountof resin injected in the mold which subsequently

determines how much compression is needed to fillthe plate.

Finally, when interpreting the optimization results interms of process features, the following well-knowntraits underlie the solutions.

Compression: compression improves in-plane struc-tural stiffnesses and reduces injection times at theexpense of larger mold pressures. Therefore, whenhigh structural performance is not required and moldpressure should remain low ((CL) problem,Pallowmax ¼ 6:105 Pa), all optimal processes but RTCM-F have no compression. In the simultaneous pro-cesses, since compression induces fast mold pressuregrowth (it reduces permeability), it is absent in allsimultaneous processes for low structural quality. Inthe (CO) formulations, compression is necessary toachieve high structural performance.Flow control: flow controlled injection is not optimalin our numerical applications because, for anequivalent injection time, it yields higher mold pres-sure than pressure controlled injection. RTCM-Foptimal settings all use compression in a way suchthat Pallowmax is reached during both injection and com-pression steps. Nevertheless, flow controlled processesregain competitiveness for shorter plates.Simultaneous processes:RTCM-SP and RTCM-SF arenot optimal for theproblems consideredhere.However,for low mold pressures and high structural properties,RTCM-SP has a strong potential. When Pallowmax dropsfrom10.105 Pa to 6.105 Pa, its time excesswith respect tothe optimal process shrinks from to 20 to 9%.

Table 4

Coupled optimization with low structural performance, (CL), for all processes, Pallowmax ¼ 6:105 Pa

RF or P

(10�7 m3/s or 105 Pa)

RC(10�5 m/s)

IP (%)

�1 (�) �2 (

�)

TP (s) Pmax

(105 Pa)

Ey (GPa)

l Gxy(GPa)

Hf (mm)

RTCM-F

3.80 0.643 80.65 59.35 0. 654. 6. 16.71 1.25 6.49 7.071

RTCM-P

6. 0.a 100.00 59.35 0. 391. 6. 15. 1.63 5.82 8.

RTCM-SF

3.06 0. b 59.35 0. 783. 6. 15. 1.63 5.82 8.

RTCM-SP

6. 0. b 59.35 0. 391. 6. 15. 1.63 5.82 8.

a RC arbitrarily set to 0 because there is no compression (Ip=100).b Variable not relevant to the process.

Table 5

Coupled optimization with optimum structure (CO), for all processes, Pallowmax ¼ 6:105 Pa

RF or P

(10�7 m3/s or 105 Pa)

RC(10�5 m/s)

IP (%)

�1 (�) �2(

�)

TP (s) Pmax

(105 Pa)

Ey (GPa)

l Gxy(GPa)

Hf (mm)

RTCM-F

1.65 0.276 81.76 54.46 90. 1508. 6. 27.64 1.30 6.89 7.125

RTCM-P

6. 0.276 81.76 54.46 90. 912. 6. 27.64 1.30 6.89 7.125

RTCM-SF

0.621 0.028 a 54.46 90. 3156. 6. 27.64 1.30 6.89 7.125

RTCM-SP

6. 0.088 a 54.46 90. 992. 6. 27.64 1.30 6.89 7.125

a Variable not relevant to the process.

2284 R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287

4. Concluding remarks

The effects of accounting, at the first design stages, forthe couplings between process and structure have beeninvestigated in the case of an injected and compressedlaminated plate.The study is based on four design formulations, two

where the couplings are ignored and either the structureor the process are optimized, and two coupled formula-tions aiming at obtaining low and high structural per-formances, respectively. Four processes, which havepressure or flow controlled injection and simultaneousor successive injection and compression phases arecompared. Solutions analyses is based on semi-analy-tical process and structure models, whose numericalefficiency authorizes a global optimization by theGBNM algorithm.The optimizations exhibit the fundamental trade-offs

that exist between injection time, maximum mold pres-sure, structural in-plane and exural stiffnesses. Forexample, the manufacturing of a low performancelaminate by the optimal RTCM-P process needs anextra minute when the maximum pressure decreasesfrom 10 to 6.105 Pa. If a high structural performance istargetted, the injection time increases by 2 min.In terms of perspectives, this work lays the founda-

tions for an approach to select, based on a final struc-ture, a resin molding techniques from a wider panel ofprocesses (Advanced RTM, Liquid Resin Infusion,Resin Film Infusion, . . .).

Appendix A. RTCM process analysis

A.1 A family of RTCM processes

The generic resin transfer and compression moldingscenario considered in this study is the case of a rectan-gular plate injected on one side and uniformly com-pressed (cf. Fig. 4). If one further assumes a channel-likeflow, the analysis is one-dimensional.Many combinations of the RTCM process settings

are possible. They make up a family of processes, ofwhich four are optimized in the body of this article: theinjection is either pressure or flow rate controlled atconstant P or RF respectively. Compression eventuallyoccurs at a constant rate, RC. Compression and injec-

tion may be simultaneous (the RTCM-S. . .processes) orsuccessive. During pure compression, a vent may beopened at x=0, allowing resin to flow out of the mold,or not. Finally, constant or time varying viscosities �can be accounted for. All these RTCM processes areanalyzed in [13], resulting in 16 models to calculateinjection time and maximum mold pressure.

A.2 Calculation of injection time and mold pressure

The one-dimensional flow of Fig. 4 can be analyzedusing Darcy’s law and a continuity equation whichtogether yield ([19])

r: �Kx�

rp

� �¼ �

RCH

; ð12Þ

where Kx is the macroscopic longitudinal permeability(more details are given in paragraph A.3), � is the resinviscosity, p the resin pressure, and H the total platethickness. If H, �, Kx and Vf depend only on time t, Eq.(12) can be integrated for p(x,t). The pressure field isquadratic if compression is applied, linear otherwise.The differential equation describing the resin frontposition also stems from Darcy’s law and is

dxFdt

¼ �1

H 1� Vf� RCxF �

RFb

� �; ð13Þ

where Vf is the average fiber volume and b is the platewidth. A total thickness increment ~H is shared amongthe N plies by looking at the stacking sequence assprings of stiffness E2,i in series. If ~H is applied to thelaminate, the i-th ply thickness, hi, changes by

Dhi ¼hiDH

E2;i

XNj¼1

hj=E2;j

; ð14Þ

where E2,i is the i-th ply transverse Young’s modulus.Injection and compression times are calculated by inte-grating Eq. (13). For example, if the injection is pressurecontrolled at P, the front position follows,

H 1� Vf�

xF ¼ 2

ðtti

HHxP

�ds

þ 1� Vf 0ð Þ�

H 0ð ÞxF tið Þ2: ð15Þ

In the case of the pressure controlled RTM with simul-taneous compression (RTCM-SP), the process time TPis solution of the non-linear equation ([13])

a2 ¼2P

RCTP þ 1� Vf 0ð Þ�

H 0ð Þ� � H 0ð ÞI1 TPð Þ þ RCI2 TPð Þð Þ; ð16Þ

Fig. 4. RTCM scenario considered.

R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287 2285

where a is the plate length and

I1 tð Þ ¼Ð t0

Kx�ds; I2 tð Þ ¼

Ð t0sKx�ds: ð17Þ

Eq. (16) is solved by numerical dichotomy.

A.3 Permeability estimation

An average plate longitudinal permeability Kx is esti-mated from the plies permeability tensors. Firstly, per-meabilities in the principal direction of the i-th ply arecalculated from the fiber volume fraction using empiri-cal laws

k1;i ¼ A1�B1f;i ; k2;i ¼ A2�

B2

f;i : ð18Þ

Next, the ply longitudinal permeability of the i-th ply isobtained by rotating the permeability tensor in theprincipal direction along the x-axis,

kx;i ¼k1;i þ k2;i

2þk1;i � k2;i

2cos 2�ið Þ; ð19Þ

where �i is the i-th ply orientation. Finally, ply long-itudinal permeabilities are averaged based on the plythicknesses,

Kx ¼1

H

XNi¼1

hikx;i: ð20Þ

Appendix B. Structure analysis

B.1 Micromechanical analysis

Including micromechanical relations in our analysisallows us to study the effect of fiber and matrix proper-ties changes, first on the ply properties, and next onlaminate properties through the Classical LaminationTheory outlined in Appendix B. Ply mechanical prop-erties are calculated from micromechanical dataaccording to the relations given in [20], where the fibersare assumed to be transversly isotropic. Denoting by fand m fiber and matrix properties, respectively, by EYoung’s moduli, by G shear stiffness, by � Poisson’sratio and by vf the ply fiber volume (a simplification ofthe �f,i notation of the text for the i-th ply), the micro-meso relationships are,

E1 ¼ �fE1;f þ 1� �f�

Em; ð21Þ

E2 ¼Em

1�ffiffiffiffiffi�f

p1� Em=E2;f

� ; ð22Þ

G12 ¼Gm

1�ffiffiffiffiffi�f

p1� Gm=G12;f

� ; ð23Þ

�12 ¼ vf�12;f þ 1� vf�

�m: ð24Þ

B.2 Classical Lamination Theory

Classical Lamination Theory ([21]) is used to calculatelaminate properties, i.e. we apply Kirchhoff plate the-ory, and neglect through-the-thickness stresses andstrains. Under those conditions, the forces N andmoments M per unit length of the cross section actingon a laminate (cf. Fig. 5) are linked to the strains e andcurvatures k through,

NM

� �¼

NxNyNxy

� TMxMyMxy

� T( )

¼A BB D

� �"

� �

¼A BB D

� �"x"y"xy� Txyxy� T

( ): ð25Þ

The coefficients of the 3�3 extensional, coupling andexural stiffness matrices, A, B, D, respectively, arefunctions of the material properties, the fiber orienta-tions in each ply, and the position of ply interfaces. Thecoefficients of the A, B and D matrices can be written aslinear combinations of material invariants Uis andlamination parameters Vis. As the names suggest, thematerial invariants depend only on the unidirectionalmaterial properties E1, E2, G12 and �12 of Eqs. (21–24),while the lamination parameters are only a function ofthe ply orientations and thicknesses. Detailed expres-sions of A, B and D using lamination parameters can befound in [22].The stiffness matrix of Eq. (25) can be inverted, which

yields the laminate compliance matrix,

A BB D

� ��1

¼a bbT d

� �: ð26Þ

The effective Young’s moduli of the laminate in thelongitudinal, and transverse direction, Ex, Ey, and shearmodulus Gxy respectively, are expressed as,

Ex ¼a11H

; Ey ¼a22H

; Gxy ¼a66H

; ð27Þ

Fig. 5. Laminate conventions.

2286 R. Le Riche et al. / Composites Science and Technology 63 (2003) 2277–2287

:

where the index 6 points to the third row/column of therelevant matrix.

B.3 Buckling analysis

A closed-form solution exists to predict the onset ofbuckling under the following conditions: the plate is at,symmetric (B=0), quasi-orthotropic (D16�D26�0),simply-supported and subject to bi-axial in-plane loads(Nx and Ny only). The plate buckles with m and n half-waves in the x and y directions respectively, when theload reaches (lmnNx,lmnNy), with

lmn ¼ ��2 D11 m=að Þ

4þ2 D12 þ 2D66ð Þ m=að Þ

2 n=bð Þ2þD22 n=bð Þ

4� �

m=að Þ2Nx þ n=bð Þ

2Ny

ð28Þ

The critical load factor, l, corresponds to the combi-nation of m and n that minimizes lmn. If 0<l<1, theplate fails under the (Nx,Ny) loading.

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