LAMINATE STIFFNESS AND CURVATURE FOR LAMINATED CARBON FIBERCOMPOSITES, EXPERIMENTAL OBSERVATION AND MODEL VALIDATION

Theresa Vo, Russell Mailen, and David A. Jack1

Department of Mechanical Engineering, Baylor University. Waco, TX 76798

Abstract

Carbon fiber laminates are extensively used within the automotive and aerospace industry due totheir high strength to weight ratios, but their design and fabrication pose increased engineeringdifficulties over alternative manufacturing approaches. During manufacturing residual strains areintroduced due to a combination of the curing kinetics of the thermoset and the induced thermalstrains due to a coefficient of thermal expansion mismatch for the resin and carbon fiber. In thepresent paper we present results for a cross-ply (un-balanced) laminate. We use micromechanicaltheories to predict the stiffness and the coefficient of thermal expansion of an individual laminafrom the constitutive properties for the fiber and the matrix, and couple the lamina results with afinite element structural and coupled thermal-structural analysis to predict the observed stiffnessand the observed strain of a processed laminate. The finite element results are then comparedwith the measured results and we provide suggestions for improvements on the method for futuremodeling and experimental studies.

Introduction and Motivation

Carbon fiber laminates are extensively used within the automotive and aerospace industry due totheir high strength to weight ratios. The design and subsequent fabrication with these laminatedsystems pose increased engineering difficulties relative to metal forming or injection/compressionmolded products. During the manufacturing process of these composite laminates, residual strainsare introduced due to a combination of the curing kinetics of the thermoset used to bind the carbonfibers together as well as the induced thermal strains due to a mismatch of the coefficient ofthermal expansion for the resin and carbon fiber. In particular, the resin goes through a chemicalchange during the cure process and the nature of the strains will not be consistent during thechemical curing process. It is important to understand the nature of the strain build up during thisprocess (see e.g., [1–3] for several approaches to quantify the induced strain) and then managethese strains during the cure process. The motivation for this management is often attributed todimensionality arguments, and a convenient approach to mitigate the appearance of the strains isto limit the composite stack to be symmetric about the centroid (see e.g., [4]). Although limitingthe stack to be symmetric will alleviate the observed curvature of the processed composite, itwill not alleviate the formation of the internal strains and their resulting induced stresses that willvary along the thickness of the stack as demonstrated in Mailen and Jack [3]. These residualstrains influence the final shape of the laminate as well as the laminate strength, stiffness, andfailure characteristics. In this paper we couple classical micromechanical theories with a finiteelement approach to predict the observed stiffness and the induced curvature of our processedlaminates, compare our predictions with the measured results, and demonstrate the limitations ofusing classical approaches.

Stiffness PredictionsThroughout the present study, we focus our attention to laminated composites fabricated withindividual laminas (plys) of unidirectional fibers as indicated in Figure 1. It is assumed that thefibers are axisymmetric and identical in shape, as well as that the fiber and matrix are well bondedat their interface. It is assumed that the fibers are axisymmetric and identical in shape, as well asthat the fiber and matrix are well bonded at their interface [5]. These approximations have beenaccepted by researchers as valid for isotropic and transversely isotropic materials for systems

1Author to who correspondence should be addressed: david jack@baylor.edu

composed of a linear elastic fiber and matrix. As will be shown by the results for our particular resinsystem, the bulk stiffness behaves in a viscoelastic manner. The results show that our assumptionof a linear elastic matrix is not valid, and in the following work we will relax this assumption.Tucker and Liang [5] discuss several micromechanical theories for predicting the effective stiffnessof a unidirectional composite given the constitutive properties of the fibers and the matrix andthey concluded that the Lielens [6] implicit model was the most accurate micromechanical theory.Similarly, the Tandon-Weng theory [7], with the closed form of the Poisson Ratio parameter ν12suggested by Tucker and Liang [5], yielded solutions that were nearly indistinguishable from that ofLielens’ model [6]. Thus in the present study, for computational efficiency, we use the approach ofTandon and Weng to compute the three dimensional material stiffness tensor C and compliance Stensors for each lamina based on the properties of the matrix and the fiber, where the double-strikecharacter (also called a ‘bulletin-board’ character) represents a fourth-order tensor. Required forthis calculation are the Young’s moduli E and Poisson’s ratio ν for the matrix and fiber, the aspectratio of the individual fibers ar, and the volume fraction of fibers Vf . The constitutive properties ofthe matrix and fiber are used to calculate Lame’s constants, which are then used to derive the bulkstiffness of the lamina. The interested reader is encouraged to read Zhang [8] for the complete setof the Tandon and Weng equations.

Figure 1: Schematic of unidirectional laminated cross-ply composite.For a unidirectional fiber distribution aligned along the x1 axis, the stiffness tensor C may be

written as the 6×6 matrix C in contracted notation as a function of the material constants obtainedfrom the selected micromechanical theory for unidirectional fibers as [4,9,10]

C−1 = S =

1E1

−ν12E1

−ν12E1

0 0 0−ν12E1

1E2

−ν23E2

0 0 0−ν12E1

−ν23E2

1E2

0 0 0

0 0 0 1G23 0 0

0 0 0 0 1G12 0

0 0 0 0 0 1G12

−1

(1)

where E11, E22, G12, G13, ν12, and ν23 are, respectively the Young’s moduli along the x1 and thex2 direction, the shear modulus of the x1 face in the x2 direction, the shear modulus of the x1 facein the x3 direction, and the Poisson’s Ratios in the x1 − x2 plane and the x2 − x3 plane. Notice inEquation (1) the stiffness tensor C is expressed as the tensor inverse of the compliance tensor S,which can be solved in either the fourth-order tensor form (see, e.g. [11]) or because of the formof the stiffness tensor it may be inverted using standard techniques in contracted 6 × 6 form asC = S−1. The resulting material compliance tensor S from the Tandon and Weng theory [7] isvalid for a unidirectional lamina oriented at zero degrees along the fiber direction. To predict thestiffness or compliance of an individual lamina in a general coordinate direction one must rotatethese tensors into the reference coordinate frame using the rotation tensor Rij for i, j ∈ {1, 2, 3}defined as

R(θ, ϕ)=

sin θ cosϕ sin θ sinϕ cos θ− sinϕ cosϕ 0− cos θ cosϕ − cos θ sinϕ sin θ

(2)2

where θ is the angle measured from the x3 axis to the unit vector along the unidirectional fiberdirection and ϕ is the projected angle onto the x1 − x2 plane measured from the x1 axis to the unitvector along the unidirectional fiber direction. For a planar laminate, the angle θ will equal π/2.The rotation tensor takes the compliance fourth-order tensor defined in the principal coordinatesystem as depicted in Equation (1) into the reference frame using the relation

Slamina = RTRTSprincipalRR (3)

where RT is the transpose of the rotation tensor. Combining Equation (3) with the assumedorthotropic form for the material stiffness as depicted in Equation (1), the effective values for thematerial coefficients E11, E22, G12, G13, ν12, and ν23 are obtained for a given reference frame.These coefficient values, expressed in the global reference frame, will be used within the finiteelement model discussed later in this text.

Coefficient of Thermal Expansion PredictionsUnfortunately the study of the coefficient of thermal expansion is not as extensive in the literatureas the body of work on stiffness predictions of a unidirectional composite. In the present studywe use the approach of Camacho et al. [12] as described in her thesis [13]. In the present studywe will assume the fibers and matrix both have isotropic coefficients of thermal expansions, andleave for the follow-up work the study of transversely isotropic fibers as is typical of carbon fibers.Schapery [14] provided the upper and lower bounds on the coefficient of thermal expansion forunidirectional composites with isotropic phases. His work was further expanded on by Halpin [15]who suggested that the mid-point of the Schapery bounds was a reasonable estimate for the axialcoefficient of thermal expansion α1 as

α1 = α+

(Kα

K− α

)1/EL − 1/E111/EL − EU

(4)

where the over-bar’s represent the volume average of a property, i.e. α = αfVf +αmVm where αfand αm are, respectively, the isotropic coefficient of thermal expansion of the fiber and the matrix,and Vf and Vm are, respectively, the volume fractions of the fiber and the matrix with Vf + Vm = 1.In Equation (4) K = KfVf +KmVm with Kf and Km being the bulk modulus of, respectively, thefiber and the matrix. The parameters EL and EU are the Schapery [14] lower and upper boundson the uniaxial stiffness modulus defined as

EL =

(VfEf

+VmEm

)−1, EU = (VfEf + VmEm) (5)

In Equation (4) the parameter E11 is the longitudional Young’s modulus obtained from Equation(1). The transverse coefficient of thermal expansion is approximated as (see e.g. [13])

α2 = (1 + νf )αfVf + (1 + νm)αmVm − α1ν12 (6)

where ν12 is the Poisson’s Ratio obtained from Equation (1). The coefficient of thermal expansionsecond-order tensor for any coordinate system with the x1 axis defined by the direction along thefiber longitudinal axis, is

α =

α1 0 00 α2 00 0 α2

(7)To predict the coefficient of thermal expansion for an individual lamina in a general coordinate

direction one must rotate the stiffness tensor into the reference coordinate frame using the rotationtensor Rij from Equation (2). The rotation tensor takes the coefficient of thermal expansion tensor

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α defined in the principal coordinate system as depicted in Equation (7) into the local referenceframe using the relation

αlamina = RTαprincipalR (8)

where RT is the transpose of the rotation tensor. The result of Equation (8) is then used as inputsto the finite element model for the curvature predictions presented in the results.

Material InputsFor the T700 carbon fiber used in our studies, the manufacturer provided values of the Young’smodulus of the fiber is Ef = 33.36Msi and a fiber Poisson’s ratio of νf = 0.2. For the Pro-Set125 Resin with the Pro-Set 237 Hardener, the manufacturer stated a matrix Young’s modulus ofEm = 5.0Msi (which implicitly assumes the cured resin is linear elastic) for our selected cure cycle.The aspect ratio of the fibers ar in the present studies was somewhat arbitrarily assumed to be1,000, but this factor is nominal in the final stiffness predictions as both Agboola [16] and Zhang [8]demonstrated a negligible difference in the final stiffness predictions for fiber aspect ratios rangingfrom 100 to ∞.

The values of thermal expansion are not as well documented. Based on studies from severalsources, a typical epoxy has a coefficient of thermal expansion of approximately 50 × 10−6(1/K)and a carbon fiber was reported to have a coefficient of thermal expansion in the range of 0.5 ×10−6(1/K), a drop of two orders of magnitude relative to the epoxy system. The Poisson’s Ratio ofthe epoxy system was not provided, but based on a preliminary in-house study using a rudimentarystrain gage setup we found a Poisson’s Ratio of νm = 0.39. At the present time we have noexperimental or specific literature results for the coefficient of thermal expansion and we intend tocomplete that study prior to the upcoming work.

The volume fraction, Vf , of the laminate can be determined experimentally using the densityof the carbon fiber, ρfiber, the mass of the carbon fiber, mfiber, the mass of the completed curedlaminate, mtotal, and the density of the cured resin, ρresin. The mass of the fibers and of thecompleted composite is measured during panel construction, and the carbon fiber density mayeither be taken from the material’s technical data sheet or determined experimentally following theASTM D3800 Standard Test Method for Density of High-Modulus Fibers [17]. A nominal decreasein the mass of the resin is observed during the curing cycle (see e.g., the TGA results in [3]), butwe observed that there is a visible degree of resin shrinkage during curing. Thus the cured resindensity is not the same as the uncured resin density and must be determined experimentally. Theresin sample is weighed first in air, then submerged underwater where the density of the wateris known. These values may then be used to determine the density of a resin sample using thefollowing

ρresinρH2O

=mresin

mresin −mresin underwater(9)

where the mass of the resin, mresin, is determined from the mass of the cut carbon fiber, mfiber,and the total mass of the cured laminate plate, mtotal as mresin = mtotal − mfiber. However,this value does not take into account the voids inside the composite panel, which we have notyet characterized. We have performed optical and SEM studies in-house to quantify by visualinspection a volume fraction of voids, but based on our observations this number will be negligibledue to the small number of observed voids. The experimentally derived value for the resin densityis used in conjunction with the previous values to calculate the carbon fiber to epoxy volumefraction of the completed laminate using the following expression

Vf =Vfiber

Vcomposite=

mfiber/ρfibermfiber/ρfiber +mresin/ρresin

(10)

Using our laboratory scales on the sample we used to generate the stiffness values presented inthe results section, we measured the fiber and total laminate mass to be, respectively, mfiber =

4

22.1g and mtotal = 29.9g. The T700 specification sheet states the density of the fiber to beρfiber = 1.8g/cm3, and using our Toyo Seiki D-H100 densitometer we calculated the density of acured neat resin sample to be ρresin = 1.13g/cm3. Using Equation 10 this yields a fiber volumefraction of Vf ∼ 60%.

Laminate FabricationIn the present study we focus on a single laminate stacking sequence to highlight the curvaturefrom an unbalanced laminate, with two identical samples generated with a stacking sequence of[0, 0, 0, 90, 90, 90]T (referred to in the present context as a cross-ply laminate). The laminates werefabricated using 9 oz. unidirectional tape cut according to the above stacking sequence so thatthe laminates would have a 3” x 6” planform area. The laminates were laid up by hand on aflat aluminum tool, and wetted by hand with Pro-Set 125 resin mixed with Pro-Set 237 hardener.Following the stacking and wetting process, the laminate was vacuum bagged with a pressuremaintained between 15 and 18 in-Hg.

The two laminates were cured according to two different cure cycles. Both laminates weresubjected to an initial room temperature cure for 15 hours. Following the room temperature cure,one laminate continued to cure at room temperature while the second laminate was subjected tothe elevated temperature post-cure recommended by the manufacturer. The post-cure cycle usedwas programmed as follows:

1. Ramp temperature to 120◦F at 0.3◦F/min

2. Isothermal hold at 120◦F for 1 hour

3. Ramp temperature to 140◦F at 0.3◦F/min

4. Isothermal hold at 140◦F for 8 hours

5. Ramp temperature to 95◦F at 0.3◦F/min

6. Isothermal hold at 95◦F for 2 hours

7. Shut off furnace and return to room temperature to end cycle

The reason for the two different cure cycles was to identify if there was any additional induced partcurvature due to the elevated cure and then the subsequent cool-down, or if the entire inductedcurvature was due to the matrix shrinkage process during curing.

Results and Discussion - Effective Stiffness

The first study investigates the effective stiffness and is used to confirm the proper implementa-tion of the finite element method with the local material stiffness results obtained from the laminamicromechanical theoretical predictions. This is performed by comparing the predicted laminatethrough thickness stiffness with experimental results obtained from our TA Instruments Q800 Dy-namic Mechanical Analyzer (DMA).

Sample Size Selection - Effective StiffnessTo perform an experiment with the DMA, the sample stiffness must fall within the acceptable rangeof the DMA’s load and displacement capabilities. In order to obtain the most accurate results,the sample should be sized such that the loading range of the DMA (0N - 18N for the TA Q800DSC) is capable of deflecting the sample to the displacement range for which the machine is mostaccurate. Based on the estimated stiffness of both the neat thermoset and the carbon-fiber epoxysystem we cut sample sizes on our low-speed Buehler saw, with dimensions of 18 mm × 11.8 mm× 1.75 mm that fell well within the acceptable range of our DMA.

5

L

δmax

Clamped Region with

Fixed Displacement

Zero Displacement

Zero Displacement

Clamped Region with

Fixed Displacement

(a) (b)

Figure 2: (a) Dual cantilever setup for experimentation along with (b) the finite element result from6 layer [0, 0, 0, 90, 90, 90] laminated composite for equivalent Young’s Modulus prediction.

Finite Element Prediction - Effective StiffnessA finite element model is created using the Solid Mechanics Module of COMSOL Multiphysicsto mimic the dual cantilever setup of the DMA depicted in Figure 2(a). The composite panel ismodeled as a stack of individual rectangular layers as depicted in Figure 2(b) with each layerrepresenting an individual lamina. A second shorter stack is attached to the right face of the finiteelement model in order to mimic the clamp conditions inside a DMA. The second half of the dualcantilever is not modeled due to symmetry and since both ends of a sample are clamped, theslope of the deflection at both ends is constrained to be zero.

A fixed displacement is applied to the surfaces of the clamping region and the left face of thelaminate stack is fixed with a zero displacement. A linear elastic material is added for each laminaorientation in the laminate stack sequence. These materials are inherently transversely isotropicwith the direction normal to the plane of symmetry being along the fiber axis. The Young’s modulus,Poisson’s ratio, and Shear modulus for each lamina orientation are calculated based on the angleof the lamina and the material properties of the fiber and the matrix as defined in Equation (1).These transversely isotropic values of the material stiffness are then entered into the finite elementsoftware.

For a dual cantilever beam, the relationship between the total deflection and the applied forcecan be readily obtained using standard beam theories through the fourth-order differential equation(see e.g., [18])

d2

dx2

(EI

d2v

dx2

)= w (x) ∼=

ab

26(11)

where EI is the equivalent stiffness, E is the equivalent transverse Young’s modulus (in the case ofan isotropic and homogeneous material this is the actual Young’s modulus), I is the cross-sectionalmoment, v is the beam deflection, and w (x) is the distributed load. For the dual cantilever beam,the boundary conditions can be expressed simply as v(0) = dvdx

∣∣x=0

= 0, v(L) = δmax, dvdx∣∣x=L

= 0

and d3v

dx3

∣∣∣x=L

= P where P is the applied load and δmax is the deflection of the fixed displacementclamp of the cantilever beam. Applying the boundary conditions, the equivalent Young’s Moduluscan be expressed in terms of the maximum deflection and applied load as

Eequiv = −PL3

12δmaxI(12)

Using Equation (12) we can predict the effective Young’s modulus for the cross-ply laminate thatwould be returned by the DMA which will measure the applied load and the measured displace-ment to return a value for the Young’s modulus. The finite element model shown in Figure 2

6

µµ(a) (b)

Figure 3: Effective stiffness results, (a) DMA results for neat cured epoxy at different loading rates,fixed loading rate, and (b) DMA and FEA results for composite system, fixed strain rate.

using the material properties for the fiber-matrix system presented in the Material Inputs sectionwas given a fixed deflection and the resulting total reaction force to impose this deflection wasobtained.

DMA and FEA Results - Effective StiffnessFlex bars were made of the neat cured epoxy system to obtain the actual stiffness of the curedthermoset with the published value and the results are presented in Figure 3(a). The first thing tonote is that the stiffness modulus, which is typically called the Young’s modulus, is not a constant,either with loading rate or with displacement. This is not surprising as we do not have a linearelastic system, but a viscoelastic polymer. It is not surprising that the measured Young’s modulus(if it were defined as the instantaneous slope of the stress-strain curve) would decrease withincreasing displacement as this is typical of viscoelastic materials. With polymers, the publishedYoung’s modulus value is in reality the slope of the stress strain curve at a zero load for a givenstrain rate. As such, we will have to assume that the Young’s modulus for the thermoset at a givendisplacement and loading condition for the neat system and the carbon fiber system is identical.Thus we input the observed instantaneous Young’s modulus into the lamina stiffness predictionsand subsequently the lamina stiffness into the FEA model for each lamina. This approach is aquasi-static assumption where the real system will require the local strain rates to properly predictthe effective response. The results from the predicted system with the quasi-static assumptionand the measured DMA system are given in Figure 3(b). The carbon fiber system was run undera fixed strain rate whereas the neat thermoset measurements were at a fixed loading rate, andwe will fix this discrepancy in the forthcoming work. For the carbon fiber system, the effectiveloading rate was approximately 1.5-2.5 N/min, and based on the observation that it appears theviscoelastic response of the neat thermoset is nearly constant from the 0.5 N/min to 1 N/min testswe will assume that these results can reasonably be used for the inputs of the FEA model at afixed strain rate. The important observation from the figures is that the FEA results over predictthe stiffness by approximately 10%. We believe this is quite reasonable in the present study as wehave only recently created the ability to fabricate samples in-house. We are currently developingan in-house infusion process with a low viscosity resin which will lead to more consistent partswith fewer defects and we anticipate the stiffness of the fabricated parts to increase. In the currentmodel we do not account for void content within the matrix, variations in the volume fraction offibers due manufacturing quality control issues, nor do we account for variations in ply orientationsdue to manufacturing variabilities. We have not quantified the variability within a single laminatefor the hand-made wet lay-up parts used in the present study, but this is known to exist based onvisual observations of the fabricated parts and we are moving to an infusion process.

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Results - Curvature of Unbalanced LaminateFollowing the complete cure cycle, deformation was visible in both cross-ply laminates as can beobserved in Figure 4(a) of a fabricated laminate. To quantify the out-of-plane deformation in thispreliminary study, a grid of measurement points was laid out on the cross-ply laminates in 1 inchsquares as depicted in Figure 4(c). The grid-points were offset from the edges of the laminateto avoid measurements in regions of ply drop-offs and manufacturing nuances. Measurement ofthe laminate deformation was accomplished at each grid-point using dial calipers mounted in atabletop vice as shown in Figure 4(b). We are currently in the process of developing an automatedprocess using linear transducers mounted on an x-y translation table, and we will include theseresults in the future work.

(a)

(b) (c)

Figure 4: (a) Fabricated cross-ply laminate indicating curved nature from the unsymmetrical stack-ing sequence. (b) Gage used to measure the curvature of the laminated ply stacks, currentlyplaced at the 9th grid point. (c) Grid pattern for curvature measurements.

The raw results for the curvature measurements for the cross-ply laminates are provided inTable 1 and is plotted in Figure 5. It is anticipated that there may be some curvature from anunbalanced laminate as in the cross-ply stack that may be attributed to the reduction in volume ofthe epoxy during the curing process. This reduction will cause a homogeneous local shrinkage ofthe thermoset that will induce on both the top 90 degree stacks and the bottom 0 degree stacks astrain casing the laminate to curve. This is highlighted in Figure 5(a) where the deflection of theunloaded laminate is presented for the room temperature cure. The continuous surface is obtainedby a 2D interpolation of the data points provided in Table 1 and the deflection is normalized by thelargest measured deflection. Notice in Figure 5(a) that the largest deflections (indicated by the redcolor) occur near the center of the laminate with the deflection dropping off in all directions, i.e.,a local maximum. Conversely, the normalized deflection of the laminate made with an elevatedtemperature cure is plotted in Figure 5(b) and it does not have the maximum deflection occurringin the center of the laminate but instead this occurs on both of edges along the centerline in the y-direction. In other words, the room temperature cure exhibits concavity down whereas the elevatedcured sample exhibits a saddle point. This implies that the two samples are in a significantlydifferent state of internal stress. This also implies that the curvature is not purely due to thermaleffects but is a combination of thermal and chemical processes inducing volumetric changes.

It is clear from Figure 5 that there are at least two competing factors to contribute to the defor-mation of the laminated composite. In the finite element study depicted in Figure 6 we investigate

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Location # 1 2 3 4 5 6Deflection (in), Room Cure 0.08 0.08 0.08 0.08 0.08 0.06

Deflection (in), Elevated Cure 0.1 0.11 0.13 0.12 0.1 0.07Location # 7 8 9 10 11 12

Deflection (in), Room Cure 0.08 0.09 0.09 0.09 0.08 0.05Deflection (in), Elevated Cure 0.1 0.12 0.13 0.12 0.1 0.06

Location # 13 14 15 16 17 18Deflection (in), Room Cure 0.08 0.09 0.09 0.09 0.08 0.05

Deflection (in), Elevated Cure 0.11 0.13 0.14 0.13 0.11 0.06

Table 1: Laminate deflections of processed cross-ply laminate for elevated temperature cure.

length (in)

wid

th (

in)

Normalized Displacement

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

1

1.5

No Disp.

Max Disp.

length (in)

wid

th (

in)

Normalized Displacement

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

1

1.5

No Disp.

Max Disp.

(a)

(b)

Figure 5: Normalized laminate deflection of the unloaded processed composite, (a) room temper-ature cure cycle, (b) elevated temperature cure cycle.

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Figure 6: Laminate deflection prediction from micromechanical inputs to the FEA model, elevatedcure sample.

only the strains due to thermal expansion/contraction effects. For the finite element study, a sampleis assumed to have a zero stress state at the elevated cure temperature of 140 degrees Fahrenheitand it is uniformly cooled to room temperature. The thermal coefficients of expansion are derivedusing Equation (8) for each lamina and are provided as inputs to the FEA model. The entire do-main is free and allowed to flex unhindered. To prevent a free-spinning of the model, several nodesare fixed in select dimensions. As can be observed in Figure 6 the deformed shape is quite similarto that shown in Figure 5(b). Unfortunately it is not possible to directly compare numerical resultsas the deflection data at each point needs to be offset to the un-deformed state. We are currentlyin the process of performing this and will include it in the future work.

Conclusion

In the present work we use micromechanical theories to predict the stiffness and the coefficient ofthermal expansion of an individual lamina from the constitutive properties for the fiber and the ma-trix, and couple the lamina results with a finite element structural and thermal-structural analysis topredict the observed stiffness and the observed strain of a processed laminate. The finite elementresults are then compared to the measured results of the stiffness and the predicted results arein reasonable agreement demonstrating the usefulness of the approach. Future work will needto incorporate the viscoelastic effects as the present model must assume a quasi-static Young’smodulus based on DMA results. Qualitatively, the deformed laminate predicted due to thermalcooling from the elevated cure temperature is in agreement with experimental observations, butwe are limited in both the true deformation of the cured laminate as well as not having a properunderstanding of the curvature induced by the room temperature cured laminate. It is also unclearif this curvature is linearly added to the thermal cooling curvature or if there will be relaxation ofthe initial curing induced strains during the elevated curing cycle.

Acknowledgments

The authors gratefully acknowledge financial support from L-3 Communications.

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