Composites: Part A 46 (2013) 19–25

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Composites: Part A

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Three-dimensional modeling of the advanced grid stiffened structures in theco-curing process

Wen-Bin YoungDepartment of Aeronautics and Astronautics, National Cheng-Kung University, Tainan 70101, Taiwan, ROC

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 June 2012Received in revised form 7 August 2012Accepted 11 October 2012Available online 2 November 2012

Keywords:E. AutoclaveE. ConsolidationC. Computational modeling

1359-835X/$ - see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.compositesa.2012.10.013

E-mail address: youngwb@mail.ncku.edu.tw

Advanced grid stiffened (AGS) structures are made of carbon fiber reinforced polymer composites, whichconsists of a main panel with isogrid stiffeners. During the co-curing process, the autoclave pressure willcompress the surface of the AGS structure to repel excess resin and voids. Because of the thermal expan-sion of the rubber, the rubber mold can provide the necessary lateral compacting forces on the rib stiff-eners. This study constructs a three dimensional numerical model to simulate the consolidation behaviorbetween the skin prepreg and the silicone rubber under a preset temperature profile. The model couldprovide the thickness distribution and fiber volume fraction of the AGS after the co-curing process.The consolidation pressure of the silicone rubber is proportional to its CTE and the curing temperature.With the numerical model, a suitable type of silicone rubber can be selected for the tooling.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In the design of a composite panel by employing the skin andrib structures to reduce the total weight, the concept of isogridused in a metal plate was also applied. Those isogrid structuresare called advanced grid stiffened (AGS) structures and usuallymade of carbon fiber reinforced polymer composites. Because ofthe light weight and efficient of the structure, it is often used asa panel that is subjected to the loads of bending and compression.The isogrid design of the rib structures, attached on a thin skinplate, enables the AGS structure to distribute the load efficientlywithout buckling. A typical AGS structure panel is shown in Fig. 1.

The difficulty of fabricating the AGS structures is mainly on theintegrated stiffened rib structures in a co-curing process. There arethree times fiber built-ups problem at the stiffener cross-over no-dal points because the continuous fiber is used in the rib struc-tures. Special design such as offset at the nodal point [1] is oneway to avoid this problem. During the curing process, adequatecompacting forces must be applied on the fiber prepreg to elimi-nate the excess resin and air to have a void-free composite. Con-ventional vacuum bag process only provides the compactingforce on the normal direction of the tool surface. In order to consol-idate the panel with the rib stiffeners, specially designed toolincorporated with silicone rubbers is necessary to provide the lat-eral compaction force at the rib structures.

Several types of tooling designs for the compaction of the ribstructures have been employed. One type of the tools is the hybrid

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mold made of a metal mold with grooves for the rib structures [2].The metal mold defines the shape of the composite part whilesome inserted rubber material inside the grooves provides the lat-eral compaction force to the rib. The compaction force on the rib isprovided by the thermal expansion of the rubber due to the tem-perature increase during the curing process. The other type of tool-ing consists of a base tool and expansion blocks. The expansionblock is made of materials with high thermal expansion to providethe lateral compaction on the ribs. The expansion blocks are lo-cated at the surface of the base tool to form the channels for therib structures. This tooling design provides more freedom for thefabrication of AGS structures. One disadvantage is that the align-ment and assembly of the expansion blocks on the base tool needstremendous labor work.

The schematic diagram of the tooling with a metal base andexpansion blocks is shown in Fig. 2. During the curing process,the autoclave pressure will compress the surface of the AGS struc-ture to repel excess resin and voids. The rubber mold can providethe necessary lateral compacting forces on the rib structures dueto the thermal expansion of the silicone rubber. The final thicknessof the skin and rib after consolidation depends on the curing tem-perature, the autoclave compaction pressure, and the thermalexpansion of the silicone rubber. The co-curing process uses theexcess resin of the skin layer as an adhesive to the rib structures.Thus, curing and joining processes for composite structures canbe achieved simultaneously. In the co-curing process, the consoli-dation and curing of polymer composites involve chemical reac-tion, resin flow, and heat transfer at the same time. Propercontrol of the applied consolidation pressure and temperature isnecessary in order to allow the resin to flow before it gels. The resin

Fig. 1. A schematic diagram of an advanced grid stiffened panel.

20 W.-B. Young / Composites: Part A 46 (2013) 19–25

flow is the primary mechanism for removing the excess resin andvoids entrapped inside the laminate and obtaining the desired fibervolume content.

Several studies have been focused on the modeling and simula-tion on various aspects of the consolidation and cure of compos-ites, which considered the resin flow, cure kinetics, and voidgrowth. In the early stage, Loos and Springer [3] developed a modelthat assumed pressure gradients in both the vertical and horizontaldirections, and computed the flow in those two directions. Theydescribed the laminate in terms of porous media that enabledthe calculation of the vertical flow with the Darcy’s law, and as-sumed the horizontal flow as a viscous flow between two parallelplates. Dave et al. [4] also employed the Darcy’s law to model con-solidation process in both vertical and horizontal directions. Theydetermined the pressure gradients in both directions at the sametime. Gutowski [5] computed the horizontal flow based on Darcy’slaw as well and proposed a model to determine load carried by fi-ber during the consolidation of the laminate. A three dimensionalconsolidation model was used by Young [6] to include the cases

Fig. 2. A schematic diagram of a tooling assem

of angle or cross ply laminates, and the non-planar composite lam-ination. The flow and compaction was also studied by researcherssuch as Tredoux and Van der Westhuizen [7], Young [8], Costa andSousa [9], and Li et al. [10]. However, the analysis of compactionbehavior during a co-curing process with the complicated struc-tures as AGS structures has limited reports.

During the co-during process, the rib structure is compacted bythe thermal expansion force of the silicone rubber. The degree ofcompaction and the final fiber volume content depend on the ther-mal expansion of the silicone rubber and the compressibility of thefibers. In this study, a three-dimensional mathematical model isdeveloped to analyze the compaction between silicone rubberand fibers. The model will provide a means to estimate the finalrib thickness and fiber volume content after curing.

2. Mathematical model

Considering the symmetric property of the AGS structure, athree-dimensional representative unit of the AGS structure is usedto study the consolidation of the co-curing process as shown inFig. 3. The prepregs are located between the caul plate and the sil-icone rubber with Z coordinate in the thickness direction and theskin on the X–Y plane. The prepreg layup of the skin may be mul-ti-directional stacking, but the prepreg tows are assembled alongthe rib channels. In the numerical modeling, the three-dimensionalconsolidation flow is considered as the vacuum pressure is appliedalong the Z direction and the rubber expansion pressure applied onthe side wall of the rib.

A three dimensional isothermal flow model is used to model thecompacting flow in the co-curing process. It is assumed that thetemperature is uniform on the composite part in the co-curing pro-cess if the case of the small thickness is assumed. During the con-solidation of a prepreg laminated composite, pressure is applied ata specific time. The applied pressure is taken by the fiber and resinof the laminate at the same time. The pressure taken by the resinwill drive the resin to flow through the fiber reinforcements. As aresult of it, resin flows out the composite in the Z-direction tothe bleeder layers. The resin flow rate is dependent of the fiber per-meability and the resin viscosity. The permeability of the fiber

bly for the advanced grid stiffened panel.

Fig. 3. A three-dimensional representative unit cell of the AGS structure. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

W.-B. Young / Composites: Part A 46 (2013) 19–25 21

reinforcement may change during the process because of the vari-ations of the fiber content and void ratio caused by the compres-sion. As the resin flows out of the composite, the resin contentand laminate thickness decrease. At the end of the consolidation,the applied pressure is taken all by the fibrous bed, and the resinpressure drops to zero. In an autoclave curing, the applied pressureacts effectively only on the skin part while the compaction on ribstructure will rely on the thermal expansion of the silicone rubber.In practice, a small gap is left at the tip of the rib to avoid the buck-ling of the rib during the compaction process. In this way, the sil-icone rubber will expand at the Z-direction, but it does not affectthe compaction of the skin and rib.

The consolidation flow of a laminated composite is analogous tothe phenomena of the flow through a porous medium. Thus, the re-sin flow in the laminate during consolidation can be described bythe Darcy’s law. For flow in porous media containing incompress-ible fluid within the pores, the stress is given by the equilibriumequation as follows:

r ¼ pþ pf ð1Þ

where r is the applied consolidation pressure, pf is the fiber stress(i.e., the stress born by the fibrous bed of the prepreg laminate),and p is the resin pressure. If a unit cell of a laminate is considered,the total applied consolidation pressure, r, is equal to the sum ofthe fiber stress, pf, and resin pressure, p. The fiber structure of thecell can be treated as a non-linear spring, and the value of the fiberstress, pf, will be a function of the cell thickness. If a void ratio, e, isdefined as the volume of voids, which contain resins, per unit

volume of solid fibers in the laminate, the thickness can be relatedto the void ratio in the following equations:

h ¼ 1þ e1þ eo

ho ð2Þ

where ho and eo are the initial values of the thickness and void ratio.Since the thickness of the laminated fibers under compression de-pends on the applied pressure, a test can be conducted to determinethe relation between the fiber thicknesses with respect to differentcompression pressures. Thus, the void ratio can also be related tothe fiber compression pressure, pf, and it will be used in the follow-ing simulations. A expression between the fiber stress and the voidratio was derived by Young [8] from the consolidation experimentsof AS4/3501-6 prepregs conducted by Gutowski [5]. The void ratiowas expressed as a function of the fiber stress as the followingequations:

e ¼ �2:122x10�5pf þ 0:9973 for 0 6 pf 6 3587:6Pa

e ¼ �0:181log10pf þ 1:57 for 3587:6 6 pf 6 1:034� 106Pa

(

ð3Þ

At the beginning of the consolidation, pf, is equal to zero and p isequal to the applied pressure, r. As the consolidation process pro-ceeds, resin pressures decreases with respect to time because ofthe pressure gradient in the prepreg laminate. Darcy’s law can beused to determine the resin pressure of the consolidation flow.After that, the fiber stresses are calculated by using Eq. (1), and

22 W.-B. Young / Composites: Part A 46 (2013) 19–25

are used to compute the updated void ratio with Eq. (3). Change ofthe void ratio, in term, can be used to derive the new laminatethickness by Eq. (2).

For an incompressible fluid, the consolidation equation can beexpressed as [6]:

mm@p@t¼ r �

��kl� rp

!ð4Þ

where

mm ¼ �1

1þ e@e@pf

coefficient of volume compressibility ð5Þ

The permeability tensor, ��k, quantifies the flowability of the fluidin the fibrous network. The transverse permeability was given byGutowski [5] as:

k ¼r2

f

17:4

ffiffiffiffiffiffiffi0:82Vf

q� 1

� �3

0:82Vfþ 1

� � ð6Þ

where rf is the radius of the fiber filament and Vf is the fiber volumefraction. Eq. (4) is the working equation to represent the situation ofthe three dimensional consolidation flow. During the process ofconsolidation, the variations of resin pressures are calculated byusing Eq. (4). Then, the corresponding fiber stress, the void ratio,and the thickness can be determined as described earlier.

In the consolidation process, the equations of species balancecan be expressed as:

species :@a@t¼ _m ð7Þ

where a is the conversion and _m is the mass generation rate.For resin kinetics, several forms were made to describe the mass

generation rate, _m, versus the resin conversion, a, with a modifiedArrhenius type equation. In this study, the following empiricalequations derived from dynamic scanning experiments were usedto model the mass generation rate during reaction [3]:

_m ¼ ðK1 þ K2aÞð1� aÞð0:47� aÞ for a 6 0:3_m ¼ K3ð1� aÞ for a > 0:3

ð8Þ

where

Table 1The AGS structure geometry and properties of the AS4/3501-6 prepreg [3].

(a) Geometryrf = 3.8 � 10�6 mB = 5 mml = 50 mmh = 3 mms = 10 mm

(b) Parameters of resin kineticsA1 = 2.101 � 109 min�1

A2 = �2.014 � 109 min�1

A3 = 1.960 � 105 min�1

E1 = 8.07 � 104 J/molE2 = 7.78 � 104 J/molE3 = 5.66 � 104 J/mol

(c) Parameters of the viscosity modello = 7.93 � 10�14 Pa sU = 9.08 � 104 J/molK = 14.1

(d) Physical and thermal propertiesCoefficient of thermal expansion = 5810�3 l/�CInitial fiber volume fraction = 0.5Young’s modulus, Es = 5 � 106 Pa

K1 ¼ A1 expð�DE1=RTÞK2 ¼ A2 expð�DE2=RTÞK3 ¼ A3 expð�DE3=RTÞ

ð9Þ

The parameters in the above equations are listed in Table 1.The viscosity of the resin will change during consolidation due

to the change of the tool temperature. The viscosity was modeledusing the form:

l ¼ lo expURTþ Ka

� �ð10Þ

where l is the viscosity, lo is a constant, U is the activation energyfor viscous flow, and K is a constant accounting for dissipative ef-fects of the chemical reaction.

The consolidation at the rib structure does not rely on the auto-clave pressure. Instead, the thermal expansion of the silicone rub-ber mold provides the lateral compaction force. The thermal stressof the silicone rubber can be expressed as:

rs ¼ EsðasDT � esÞ ð11Þ

where rs is the stress applied normal to the rib surface, Es is theYoung’s modulus, as is the coefficient of thermal expansion (CTE),DT is the temperature variation, and es is the strain in the directionnormal to the rib surface. Since the deformation of the prepreg atthe rib will be the same as that of the silicone rubber, the strainof the silicone rubber can be correlated to the void ratio of the pre-preg laminate.

es ¼Bl

eo � eeo þ 1

ð12Þ

where B is the thickness of the rib, and l is the length from the cen-ter of the silicone rubber to the edge as shown in Fig. 3.

3. Numerical formulations

If X represents the entire domain of the laminate with Cb andCt being the boundaries on the bleeder and tool surfaces respec-tively, initial and boundary conditions are mathematically ex-pressed on the following set of equations:initial conditions:

pðz;0Þ ¼ r for z 2 X

pf ðz;0Þ ¼ 0 for z 2 Xð13Þ

boundary conditions:

pðz; tÞ ¼ pbc for z 2 Cb

@pðz; tÞ@n

¼ 0 for z 2 Ct

ð14Þ

Integration of Eq. (4) with respect to a control volume results in:

mm

ZV

@p@t¼Z

Sn*�

��kl� rp

!dS ð15Þ

In the numerical formulation, the cavity is discretized into afamily of six-node wedge-type elements shown in Fig. 4 with lin-ear shape functions. The centroid of each element is connected tothe center-points of its surfaces to form a control volume. The con-nection envelopes control volumes around the grid points at whichseveral elements meet. Thus, the control volumes are chosen bydividing each element into six equal sub-elements, and then con-necting all the sub-elements that have the common node. Thepressure gradient can thus be expressed as:

rp ¼

@p@x@p@y

@p@z

2664

3775 ¼ ½J�1�½N�½P� ð16Þ

Fig. 4. A typical wedge type element and the corresponding part of a controlvolume.

(a)

(b)

Fig. 5. (a) The temperature profile set for the autoclave and (b) the conversion ofthe resin in prepreg during the consolidation. (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of thisarticle.)

W.-B. Young / Composites: Part A 46 (2013) 19–25 23

where

½J� : Jacobian matrix½N� : matrix of derivatives of the shape functions

and

½N� ¼f� 1 1� f 0 �f f 0f� 1 0 1� f �f 0 f

nþ g� 1 �n �g 1� n� g n g

264

375 ð17Þ

where n, g, and f are the local coordinates defined in an element asshown in Fig. 4. The matrix [N] is the derivatives of the linear shapefunctions of the wedge type element. Substituting Eq. (14) into Eq.(13) and discretizing the time derivative result in:

Xm

i¼1

mmVi

3pnþ1 � pn

Dt

� �¼Z

S

1l½n*�½k�½J�1�½N�½Pnþ1�dS ð18Þ

where Dt is the time step, the superscript of the resin pressure is todesignate the time step n, Vi is the volume of the element, and m isthe number of elements in a control volume. The Eq. (18) can be ap-plied to the prepreg laminate, which yields a set of linear algebraicequations. Together with the appropriate boundary conditions, thetransient resin pressures in the laminate during consolidation canbe solved by employing a simple iteration method.

The conversion over each time step increment during the simu-lation is based on the conversion and polymerization rate at theprevious time step. Thus, the discrete equation of the conversionis written as:

anþ1 ¼ an þ _mnDt ð19Þ

where _mn is calculated based on conversion at the previous timestep.

The compaction pressure on the skin is the applied autoclavepressure, but it depends on the rubber expansion at the rib part.In determining the compaction pressure by the silicone rubberand lateral fiber pressure, the following steps are conducted.

1. With the known temperature difference and resin pressure, cal-culate the strain of the silicone rubber, es, using Eq. (11).

2. If es is less than or equal to zero, the rib is not under any com-paction force. The fiber pressure equals to zero and the voidratio is presumed the initial value.

3. If es is greater than zero, calculate the strain of the silicone usingEq. (12) and the void ratio. With the es, one calculates the com-paction pressure using Eq. (11) and corresponding fiber pres-sure by Eq. (1).

4. Determine the new void ratio with the calculated fiber pressurewith Eq. (3). Check for convergence of the fiber pressure and goback to Step 3 if the convergence is not achieved.

4. Results and discussions

The AGS structure is made of AS4/3501-6 carbon fiber rein-forced epoxy prepregs with the compressibility shown in Eq. (3)and the permeability by Eq. (6). The corresponding geometry andproperties used in this study are listed in Table 1. The autoclavepressure is set to 1.379 MPa (200 psi) and the temperature profileshown in Fig. 5 is set to rise with two intermediate holding stagesfrom the room temperature to 178 �C. A one sixth finite elementmodel of the AGS structure unit cell is shown in Fig. 6. Symmetryboundary conditions will be applied on the symmetric surfaceson the model during the analysis. The corresponding conversionof the resin in the prepreg is also shown in Fig. 5.

The resin pressures at the end of consolidation are shown inFig. 7. During the compaction, the resin in the skin layer is drivenout of the prepreg to the bleeders. Finally, the compacting pressureis supported by the fiber entirely and the resin pressure reduces tozero. Therefore, the skin layer reaches the maximum compaction ofapplying pressure. On the other hand, the compaction on the rib

Fig. 6. A one sixth finite element model of the composite unit cell. (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this article.)

Fig. 7. Distribution of the resin pressure at the end of consolidation. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

24 W.-B. Young / Composites: Part A 46 (2013) 19–25

structure is not uniform. Since the fiber has a low value of perme-ability along the transverse direction, the flow along the rib to thebleeder layer is quite small due to the long flow distance comparedto the skin layer. Thus, resin pressure gradient is still preserved

Fig. 8. Distribution of the fiber volume content at the end of consolidation. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

until the resin gelation. High resin pressure can be observed atthe tip of the rib structure.

The resulting fiber volume content of the AGS structure afterconsolidation is shown in Fig. 8. Uniform fiber volume content isobserved at the skin layer, whereas the fiber volume content isquite different along the rib structure. At the junction area of theskin and rib, the fiber volume content has the highest value. Dueto the thermal expansion of the silicone rubber, the compactingforce is the same along the rib structure. However, with the exis-tence of the resin pressure, the force applied on the fiber is lessat the tip of the rib, resulting in a low value of fiber volume contentat this place. The corresponding deformation of the AGS structureafter consolidation is shown in Fig. 9. Since the difference of the fi-ber volume content along the rib is not large, the variation of thethickness at this region is not obvious in the figure. At the currentcase, a quite uniform thickness is obtained for the AGS structurecomposite.

Besides the effect of the tool temperature, it is known that thecompaction of the rib structure depends on the silicone rubber,and skin layer depends on the applied compacting pressure. Thecoefficient of thermal expansion (CTE) for the silicone rubber listedin Table 1 is 5 � 10�3 l/�C. In order to study the effect of the siliconerubber, a lower value of CTE, 1.8 � 10�3 l/�C, is used for the analy-sis. The resulting final shape of the AGS structure from simulation

Fig. 9. Deformation of the AGS structure after consolidation showing the angle andY–Z plane views (CTE = 5 � 10�3 l/�C). (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Deformation of the AGS structure after consolidation showing the angleand Y–Z plane views (CTE = 1.8 � 10�3 l/�C). (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Distribution of the fiber volume content at the end of consolidation(CTE = 1.8 � 10�3 l/�C). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

W.-B. Young / Composites: Part A 46 (2013) 19–25 25

is shown in Fig. 10. It is noticed that the thickness is not uniformalong the rib and that at the tip of the rib structure is larger thanother places. The compacting force provided by the silicone rubberis smaller than that in the previous case due to the reduced CTE. Atthe tip of the rib, a higher resin pressure is retained in the process,

which causes a less force acting the fibers, leading to a less com-paction and larger thickness. The corresponding fiber volume con-tent after consolidation is shown in Fig. 11. The fiber volumecontent is low at the tip of the rib structure because of the lesscompaction. A larger difference of fiber volume content is observedalong the rib structure that reveals the problem of insufficient con-solidation in this case due to the usage of improper silicone rubber(CTE is too small).

5. Conclusions

For the co-curing process of an AGS structures, the part thick-ness and fiber volume fraction strongly depends on the autoclavepressure and selection of silicone rubber. The autoclave pressureaffects the thickness and the fiber volume fraction of the skin panelwhile those of the skin part are controlled by the thermal expan-sion of the silicone rubber. This study constructs a three dimen-sional numerical model to simulate the consolidation behaviorbetween the skin prepreg and the silicone rubber under a presettemperature profile. The model could provide the distribution ofthe thickness and fiber volume fraction in the AGS structure afterthe co-curing process. Based on the distribution of the fiber volumecontent, the mechanical properties of the AGS structure as modu-lus and strength can be estimated. Non-uniform thickness is pre-dicted along the rib structure if the consolidation pressure of thesilicone rubber is not large enough to compact the fibers. The con-solidation pressure of the silicone rubber is proportional to its CTEand the curing temperature. By applying the numerical model, asuitable type of silicone rubber can be selected for the tooling.

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