Transverse squeeze flow of concentrated aligned fibers in viscous fluids

Jour~of Non-Ncwtonian Fl.id ELSEVIER

J. Non-Newtonian Fluid Mech., 65 (1996) 47 74

Mechanics

Transverse squeeze flow of concentrated aligned fibers in viscous fluids S.F. Shuler, S.G. Advani* Center Jbr Composite Materials amt Department ~ Mechanical En,~dneering, University ~!I Delaware, Newark, DE 19717, USA Received 30 September 1995: in revised form 10 t:ebruary 1996

Abstract This paper examines the effect that aligned long fiber reinforcement has on the processing characteristics of thermoplastic composites. More specifically, the influence of fiber volume fraction on the transverse shear viscosity of unidirectional composites is explored. Through both experimental evaluation and theoretical modeling, the study fl~cuses on the flow behavior of a model material consisting of unidirectional nylon or glass tibets aligned in a clay matrix. The flow behavior of a commercially produced material (APC-2) composed of unidirectional carbon fibers aligned in a thermoplastic polyether ether ketone (PEEK) matrix is also examined. An experimental technique has been employed that characterizes both the bulk transverse shearing viscosity and the fluid mechanics of such highly filled fiber resin systems in squeeze flow. Squeeze flow experiments were performed t~r the model material containing various fiber volume fractions and, with specially designed hot platens, for the carbon fiber-PEEK composites. Flow visualization techniques have been developed to measure the velocity profile of the material during flow. A cell model is proposed to calculate the effect of fiber volume fraction on the transverse shear viscosity of aligned fiber composites and, hence, the squeeze force requirements of such materials. The cell model, which calculates individual fiber interactions based on the lubrication approach with a viscous Newtonian or Carreau fluid, demonstrates the effects of varying the fiber volume fraction, fiber diameter and the shear thinning nature of the matrix fluid on the force requirement under constant squeeze rates. Comparisons are made between the experimentally measured squeeze force and the cell model predictions. Good agreement is found at high fiber w~lume fiactions as the lubrication flow assumptions become more accurate.

Kevword,s: Fibers: Thermoplastic composites; Transverse squeeze flow: Viscous fluids

I. Motivation Application of consolidation pressure is an important part of any fiber reinforced thermoplastic composite forming process. These processes include autoclave diaphragm forming, wherein * Corresponding author. ()377-0257,'96,'$15.0() © 1996 PII S0377-02 57(96)01440-1

Elsevier Science B.V. All rights reserved

48

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47-74

the hydraulic pressure used to form the material is typically held until the part is cooled, matched metal die forming, where the solid platens provide the pressure, thermoplastic tape laying, in which the consolidation pressure is applied through a calendering follow-roller, and in filament winding, where it results from tension in the fiber tows being wrapped upon one another. In all of these processes, it is important to apply sufficient consolidation pressures that will result in void free laminates. These consolidation pressures may also lead to squeeze flow of the composite laminate. Transverse squeeze flow is the main mechanism for healing flow, permitting the local redistribution of resin and fibers and assisting in the bonding of different ply layers. However, excessive flow may induce resin or fiber migration and thus affect the mechanical properties, dimensions and integrity of the final product. Therefore, understanding the squeeze flow behavior and the actual fluid mechanics of laminate squeeze flow are important when considering the processing of sheet composites. There is an additional motivation for studying the squeeze flow of composite sheet laminates. Thermoplastic composite sheet forming involves the deformation of a laminate within and out of the plane and is characterized by the flow of resin and fibers. It is, therefore, necessary to have effective constitutive relations describing the highly anisotropic flow behavior of these laminates. Unfortunately, the shear and extensional properties of long fiber reinforced composites in their melt state are difficult to obtain using traditional rheological techniques. For these reasons, an experimental approach has been developed that utilizes squeeze flow to characterize the bulk transverse shear viscosity of these highly filled viscous resin systems and also to describe the fluid mechanics of these materials in squeezing flows.

2. Previous work with squeezing flow The healing flow or squeeze flow phase of matched platen reconsolidation may be viewed as a squeeze flow which occurs between two impermeable surfaces. The time dependent planar flow of viscous or viscoelastic fluids between two parallel surfaces due to either an applied normal force or to a constant closure rate is encountered in a testing device which has been used to study the viscosity of polymer melts and other viscous materials. These devices have been referred to as the "parallel plate plastometer" or "squeeze plate viscometer" [1,2]. The advantages of this type of test include mechanical simplicity, the very high shear rates that can be achieved, the fact that it can be used at high temperatures and the ease with which high viscosity materials can be tested [3]. Additionally, this type of flow is of interest since it is encountered within lubrication systems and the stamping of plastic sheets. Finally, it provides a technique for investigators to evaluate rheological equations under transient conditions. In the parallel plate plastometer, the test material, in the form of a cylinder, is placed between parallel circular flat plates. With the lower plate fixed, either a constant force or a constant closure rate is applied to the upper plate. If a constant force is applied, measurement of the resulting displacement for the upper plate is used to determine the rheological properties of the specimen. If a constant closure rate is applied, measurement of the resulting force on the upper plate is used. Although these experiments involve unsteady shear flows, the flow rates are usually small enough so that the analyses are based on quasi-steady state solutions. Two different sample arrangements may be used in the plastometer. In the first case, the specimen is larger

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74

49

than the plates, thus the area under compression is constant. In the second case, the plates are larger than the specimen so that the volume of the specimen under the platens is constant. Squeeze flow rheometers have been used to study both unfilled viscous and viscoelastic liquids. Experimental squeeze flow studies on polymeric liquids indicate that under slow loading rates, their behavior can be well approximated through the shear dependent viscosity [4-6]. There has been some work with fiber filled polymers as well. Barone and Caulk [7,8] investigated flow of sheet molding compounds (SMC) in compression molds. These SMC materials consisted of chopped short fibers randomly oriented within a thermosetting resin. Their experimental results showed that, during squeezing, SMC deforms in uniform extension within individual SMC layers and that at slow closing speeds slip occurs at the mold surface and between the layers of SMC. Tucker and coworkers [9-11] and Osswald and Tseng [12] have studied the flow and heat transfer as well as the fiber orientation during compression molding of SMC materials. The squeeze flow behavior of aligned fiber reinforced thermoplastic materials has been studied by Balasubramanyam et al. [13], Barnes and Cogswell [14] and Wang and Gutowski [15] who treated these composites as transversely isotropic materials. Experimental observations of aligned fiber reinforced APC-2 material by these investigators showed that the resulting squeeze flow deformation is strictly perpendicular to the fiber direction. This is due to the high ratio of extensional viscosity in the fiber direction, which is infinite in the case of continuous reinforcement, to the shear viscosity of the composite transverse to the fiber direction. Therefore, with no resin percolation out of the fiber bed, these composites have been viewed as incompressible anisotropic fluids with an effective shear viscosity transverse to the fiber direction. O'Bradaigh [16] has experimentally measured the effects of transverse squeeze flow on the cross-sectional thickness variation of diaphragm formed APC-2 components. Hull et al. [17] have examined the evolution of fiber wrinkling during these squeezing flows. The typical squeeze flow modeling assumptions made are that (l) the material is incompressible; (2) no body force acts on the material; (3) the squeeze motion is very slow; (4) there is no flow in the fiber direction; (5) the material height is smaller than its width, and (6) there is a no-slip condition on the platen surfaces. With these assumptions, Barnes and Cogswell [14] have derived a squeeze flow relation by treating the unidirectional composite laminate as having an effective Newtonian transverse shear viscosity. A comparable approach was taken by Balasubramanyam et al. [13] who also investigated the influence of slip on the platen surfaces Wang and Gutowski [15] studied the transverse flow of unidirectional laminates of APC-2 and Radel 8320 (polyarylsulfone thermoplastic with 62 vol.% of 7 /~m diameter T650-42 fibers) materials by modeling them as shear thinning power-law fluids. The deformation of a 30 ply Radel 8320 laminate was tracked by placing tracer fibers in the laminate and measuring the location of the fibers before and after the transverse spreading experiment. This technique demonstrated that, unlike random fiber reinforced SMC laminates, transverse shear takes place through the thickness of the laminate.

3. Squeeze flow modeling The fiber structure within the composite materials strongly affects their flow response. Consider the oriented fiber assembly lamina of Fig. 1. The individual fibers make the material

50

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74

inhomogeneous and a continuum model for this material is only appropriate on a scale much larger than this inhomogeneity. Therefore, the first question to be answered is if, considering a length scale on the order of an entire laminate, the fiber reinforced composite in squeezing flow can be viewed as a continuum with an effective bulk transverse shearing viscosity, r/23. With this assumption, it is possible to view the material as transversely isotropic and to formulate the governing squeeze flow equations for an incompressible anisotropic viscous fluid subjected to slow squeeze rates and creeping flow. Applying the appropriate boundary conditions, the solution to these governing equations can be found for shear thinning fluids. The force calculated from these models can then be compared to the experimentally obtained force/material deformation response of actual composite laminate test samples. In developing the following models, there are several assumptions made concerning the nature of the composite materials. First, the composites are assumed to be unidirectionally reinforced with aligned fibers (Fig. 1) and are assumed to behave macroscopically as incompressible anisotropic viscous fluids in which there is no flow parallel to the fiber direction (this assumption is verified with experimental evidence). The platens are additionally assumed to be initially fully filled with material and any material that flows outside the platens is not considered (our experiments are consistent with this scenario). This results in a condition where there is always a constant material area under the platens. Finally, no-slip boundary conditions are assumed on the upper and lower platen surfaces and the squeeze flow rates are assumed slow enough to neglect inertia effects and assume "quasi-steady state" at each time step. The criterion used to assess creeping flow is often the dimensionless Reynolds number, Re, defined as Re = p [dh (t)/dt]h (t)

,

(1)

where p is the fluid density, dh(t)/dt is the platen closure rate, r/is the fluid viscosity and h(t) is the sample thickness or distance between the platens at the instant of time. A Reynolds number smaller than unity indicates creeping flow conditions. The creeping flow assumption will be shown to be valid based on Reynolds number in the subsequent squeeze flow experiments discussed.

X3

X1 Fig. 1. Oriented fiber assembly (60% fibers) of long aligned fibers (LID ~ 104).

S.F. Shuler, S.G. Advani / J . Non-Newtonian Fluid Mech. 65 (1996) 47 74

51

Applied Force "F" or Applied Closure Speed "dh(t)/dt"

,5 i Y

ht

~. . _. ; . . , . ~.1 ..*.II..~*a.%

( ) ~, =;,,,,;,,,..,%,;

I--

2,

o,=,~,-

-

x

--llw

Fig. 2. Unidirectional laminate within parallel square squeeze platens.

3. I. Governing equations Consider the unidirectional laminate depicted in Fig. 2 between two square parallel platens. Since flow only occurs perpendicular to the fibers, transverse squeezing flow of these laminate types can be modeled similarly to two-dimensional hydrodynamic lubrication systems. The pertinent equation of motion is in the x-direction: +

+

+

when q is the viscosity of the fluid. An order of magnitude analysis shows that the terms on the ]eft side of Eq. (2) are negligible for creeping flow (where Reynolds number is much less than unity) as the viscous forces dominate. Also, the body force term on the right is insignificant and is neglected. Since the flow is assumed two-dimensional with no flow in the fiber direction, ?2 V,/Oz2 is eliminated. Finally, the sample's length is much greater than its height. ]'his length scale difference means that the flow will primarily be in the x-direction so that V,. << V, from continuity and also ©K,./Sx <<8Vx/Sy. This makes it possible to ignore 8zv~/'Sx 2. Consequently, the x-direction equation of motion reduces to

(~A--:-= n \ ay- / "

(3)

Furthermore, recognizing the symmetry of the physical situation about the x = 0 and y = 0 planes, only one quarter of the sample geometry needs to be considered. The boundary conditions of flow symmetry about the x = 0 and y = 0 planes and no-slip at the upper and lower surfaces m a y be expressed as follows: Symmetry: -~-I/, = 0 at y = 0, ~v

No-slip: V,.=0aty=

h(t) 2

(4)

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996)47-74

52

From the incrompressibility assumption the expression relating the flow rate at any cross-section, Q(t), to the platen closure speed, dh(t)/dt, can be derived from mass conservation which results in dh(t)

Q(t) = - x d---~

(5)

The negative sign is present since a negative closure rate yields a positive flow rate. The final flow field and pressure distributions across the platen depend on the rheological characterization used for the fluid. The results for Newtonian and Carreau fluid types are presented in the following sections.

3.2. Newtonian fluid The simplest fluid description is the one parameter, or constant viscosity, Newtonian fluid, where the viscosity is independent of the shear rate of the fluid: r/=/t.

(6)

Inserting (6) into Eq. (3), integrating twice and applying the no-slip and symmetry boundary conditions of (4) yields the following fluid velocity distribution as a function of position and time 1 ~P~ 2 Vx(x'y't)=f~-~x~ y _ (~_))2} .

(7)

The volumetric flow rate at any particular x-position may then be obtained by integrating this velocity distribution from y = 0 to y---h(t)/2. In order to determine the relation between the force on the platen, F(t), and the platen closure rate, dh(t)/dt, one first finds the pressure distribution by equating material continuity with the integral of Eq. (3) and then integrates the pressure distribution on the total platen surface [6]. The result is L

F(t)=2W f edx=

. , dh(t)F L 3 j] ,

(8)

o

where W is the width of the platens and L is one half of their length. It is instructive to note that the force is a linear function of viscosity and closure rate, it is an even stronger (inversely cubic) function of sample height.

3.3. Carreau fluid model Newtonian fluids have a simple molecular structure wherein interactions between molecules occur only over short distances. For most thermoplastic polymers, however, entanglement of the polymer macromolecules produces a shear rate dependent non-Newtonian shear thinning viscosity [18]. At higher platen closure rates, this shear thinning nature of the matrix resin may become apparent. The three-parameter Carreau fluid model [19,20] describes the viscosity as

S.F. Shuler, S.G. Advani /J. Non-Newtonian Fluid Mech. 65 (1996) 47-74

53

Newtonian behavior at low shear rates followed by shear thinning power-law behavior at higher shear rates and is commonly expressed as q = qo[l + (2~,)2](" iv2,

(9)

where qo is the "zero shear rate" viscosity, i.e. the viscosity the fluid would have when deformed infinitely slowly, n is the shear thinning exponent that adjusts the slope of the shear thinning region and 2 is a time constant that adjusts the position of the "knee" in the q.,-;~ curve where there is an onset of shear thinning behavior. The Newtonian model is recovered for n = 1 and the power-law expression is approached for large values of ,~. The Carreau model consititutive relation is r,.,, = ~/o[1 + (2~")2]{"- 1}/2aVx

(10)

Substituting this constitutive relation into the x-direction equation of motion, the following expression is obtained:

Due to the non-linearity of this expression, explicit solutions for the velocity and force are not possible. However, the velocity distribution and platen forces may be found through an interative numerical solution technique. The first step in the numerical solution procedure is to divide the region contained within 0 < x < L and 0 <_y <<_h(t)/2 into an appropriate number of discrete sections. Then, for a given Y/o,2, n and x-position, a guess is made as to the correct pressure gradient at the particular x-position. Eq. (11) may then be solved for the velocity gradient, ?,V~/~y, for every y-position

0 < y <_h(t)/2. The total flow rate is found by integrating by parts over the velocity profiles in both regions: h(t)/2

h(t)/2

Vx(y, t)dy= 2[y Vx(t)]l~;-~(')/2- 2

O(t)-- 2 0

y--~y dy.

(12)

0

Due to the no-slip boundary condition at y = h(t)/2, the first term is zero and this integration is simplified to h(t)/2

y-~-vdY.

Q(t)= -2

(13)

0

Therefore, once the velocity gradients are obtained, the flow rate at the x-position is obtained by numerically integrating this equation over 0 < y <_h(t)/2. This computed flow rate is then compared with the actual flow rate of Eq. (5) determined from continuity. The initial guess for the pressure gradient is repeatedly increased until the flow rates in Eqs. (5) and (13) agree within 0.5%. This procedure is repeated for each x-position. Once the pressure gradients are calculated, the force on the platens can be numerically integrated. This integration is simplified by integrating by parts and neglecting the atmospheric pressure at x = L as follows:

S.F. Shuler, S.G. Advani /J. Non-Newtonian Fluid Mech. 65 (1996) 47-74

54 L

F(t)

=

2W

L

" - -- 2W P(x,t) dx = 2W[xP(x, t )][~-~ 0

L

X-~xdX=2W 0

x

--~

dx.

(14)

0

Repeating this procedure, the force on the platens is found at desired time steps. A flow chart summary of this solution process is shown in Fig. 3. In all cases, Simpson's formula is used for the numerical integration and the procedure was verified by comparing it with the limiting cases of the analytic solutions for Newtonian and power-law fluids.

I

I

NP :

qo,~.,n, ho, L,W,a~att [ ] Discretize h(t)/2 and L by dx end dy ]

+

IIndax on the time step ( [index on the X-position J

-- [Index on the Y-position 1

SOLVE:

no ] + ;~

"3-U=~y

for, bVx/~, at a given bP/~, x, y.

..~Forsolved ~Vx/Oy,numerically integrate for flow rate "Q" at X-position by indexing on Y-position: Q =-2.0x -~.-~xx y xdy Compare flow rate, Q, with ~ continuity flow rate: Q =_ x × ~llf

J

they agree, within some error, E, proceed. If they do ] not, return to index on ~P/&.

+ I

Calculate force on area ~.

I

'JF(t)=-2.0xWX~xXXX~I l Index on time step [ Fig. 3. Flow chart of the solution process for the squeeze flow experimental apparatus.

S.F. Shuler, S.G. Advani / J. Non-Newtonian [:hdd Mech. 65 (1996) 47 - 74

55

eaters (4x) pie (4x) mple Video :amera

Fig. 4. Schematic of the squeeze flow experimental apparatus.

4. Experiments

4. I. Squeeze flow apparatus The experimental apparatus consists of a servohydraulic test frame coupled with a personal computer used for data collection. Specially designed hot platens fit within the test frame. The testing arrangement, as depicted in Fig. 4, consists of two square steel platens fastened to two specially constructed round steel fixtures that attach to the test frame's hydraulic rams. Four cartridge heaters, mounted within these fixtures, provide the platen heat and are controlled through feed back controllers that receive temperature information from four thermocouple probes mounted within the platens. Each probe is positioned as close as possible to the platen surfaces in order to attain accurate surface temperature readings. A video camera is mounted perpendicular to the sample which records the material flow during the experiments. Although it is not a traditional rheometric device, an advantage of using the servohydraulic system is that it allows for either controlled load or controlled displacement experiments. For steady squeeze flow experiments, either a constant force may be applied and the change in height measured over time, or a constant closure rate may be applied with the change in Ibrce measured over time through appropriate load cells. If desired, special oscillatory loading, dynamic testing or exponential loading rates may also be applied through the servohydraulic system. Fig. 5 shows the detailed description of the squeeze platens and material samples. The platens are 1.25 cm high and are 5 cm square. The fixtures and platens have been carefully machined to be aligned with the center and to be both parallel and square when fitted in the test frame.

cm 1¢

I-

5 cm

Fig. 5. Squeeze platens and material geometry.

56

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996)47-74

Fig. 6. Photograph of test platens with unidirectional APC-2 laminate ready for testing.

The photograph in Fig. 6 shows the square hot platens fastened to the hydraulic rams along with a thermoplastic material sample ready for testing. The thermocouple probes can also be seen at the right of Fig. 6. The anisotropic nature of aligned fiber filled material systems constrains the material to two-dimensional flow perpendicular to the fiber direction. This is not the case, however, for unfilled or non-directionally reinforced viscous materials. In order to use the same two-dimensional squeeze flow analysis models for these systems, the flow must be physically constrained to two-dimensional flow. To account for this situation, a channel flow platen was developed to be used with the hot rams described above. Filled or unfilled material samples may be placed in the 2.5 cm wide, 7.8 cm long, 1 cm high channel section detailed in Fig. 7. After the carefully machined squeeze ram is placed on top of the sample, the squeeze flow experiment may be carried out. The analysis for this channel flow is identical to that detailed previously for the square platen/unidirectionally reinforced sample squeeze flow situation.

~ Cartridge Heaters

~__~,~)Q

(4X)

pFor

.oj

Sample~

Thefmocouples

(4x)

f

~

2L=7.8c ~ ~

SqueezeRam

Fig. 7. Schematic of experimental setup for channel flow.

S.F. Shuler, S.G. Advani / J . Non-Newtonian Fluid Mech. 65 (1996) 47-74

57

4.2. Materials 4.2. I. Model composite materials Since similar thermoplastic matrix composites containing aligned fibers are not readily available commercially in a suitable range of fiber volume fractions, a model fiber reinforced composite material system was fabricated to investigate the influence of fiber volume fraction on the transverse viscosity and squeeze flow behavior of unidirectional laminates. A matrix material of modeling clay was filled with various volume percentages of either nylon or glass fibers and tested under constant closure rate squeeze flow conditions. Modeling clay was chosen for the matrix material since it can be easily shaped and tested at room temperature. In order to construct composite materials that were similar in structure to commercial material systems but with lower fiber volume fractions, 1 m m diameter nylon bristles were cut into 5 cm lengths and placed in the clay matrix. A transversely isotropic structure was achieved by carefully rolling the clay into thin 5 cm square sheets and aligning the fibers parallel to one another in between the rolled sheets. The nylon fibers were stiff enough so that when the sheets were stacked and consolidated, the fibers became homogeneously distributed throughout the sample. The clay nylon fiber test samples were 5 cm square and approximately 1 cm tall and were constructed with 20, 40, 50 and 60 vol.% fiber contents. Clay matrix-glass composite samples were also constructed by adding 60 vol.% of 10 /~m diameter glass fibers. These glass fiber reinforced samples were constructed to the same 5 cm x 5 cm x ~ 1 cm dimensions as the nylon fiber composite samples and were assembled using the same method described above, first cutting the glass fibers into 5 cm lengths and then laying them between thin rolled 5 cm square clay sheets. A 68 wt.% fiber content was needed to achieve a 60 vol.% fiber composite. After carefully weighing out the appropriate fiber content, the fibers were laid in between the clay sheets. However, because the glass fibers were much less stiff than the thick nylon fibers they did not disperse as well into the clay. This resulted in a layered effect with several fiber-rich and clay-rich areas. 4.2.2. Unidirectional A P C - 2 laminates In order to also study a mercantile composite material system, unidirectional thermoplastic carbon fiber reinforced laminates were constructed from sheet ply APC-2 material which is manufactured by ICI Ltd. and consists of PEEK thermoplastic polymer reinforced with a reported 61 vol.% of continuous, aligned, 7 /~m diameter AS4 carbon fibers [21]. These test samples were constructed by cutting and stacking the individual plies, sealing them in Kapton film~ and then consolidating them at approximately 1 MPa (150 psi) and 380°C, in a picture frame mold/hot press arrangement. The fiber filled test samples are 5 cm square preconsolidated, unidirectional and multi-angle laminates approximately 1 cm high. The APC-2 samples were preconsolidated prior to the squeeze flow experiments to ensure that consolidation effects were not observed during the experiments [14]. The unidirectional (0 °) laminates were constructed by laying up and consolidating 70 plies of APC-2 sheet. The laminates were originally constructed as 30.5 cm x 10.2 c m x 1.0 cm plates that were then slot cut with a diamond blade into the appropriate size test samples. Since similar test laminates were cut from the same panels, they had experienced the same prior processing

58

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996)47-74

history. Micrographs of the postconsolidated samples were used to estimate the final fiber content of the samples at 64 vol.% . This can be attributed either to resin percolation out of the fiber bed caused by the combined effects of the consolidation pressure and the picture frame mold which confined the material from any transverse flow or to possibly inaccurate reporting from the material supplier. Since the same platen configuration of Fig. 5 was used to test these materials, the unidirectionally reinforced APC-2 test laminates were cut 5 cm square and approximately 1 cm high. In order to follow the material flow during the experiments, the sides of the samples were silk screened with high temperature resistant paint to impart a 2 mm-square grid suitable for video imaging. A typical unidirectional test sample is shown marked with the 2 mm paint grid in Fig. 6. The video camera mounted perpendicular to the face of the sample records the material flow during the experiments. The captured video tape image is later imported into a personal computer using image analysis software and the material flow followed during the squeeze experiment. The flow visualization technique involves tracking the material grid on the face of the test samples, allowing measurement of the developing velocity profiles and material deformation during the experiment. These may then be compared to predicted velocity fields and deformations determined for a fluid with certain Carreau fluid parameters r/0, 2, and n. Therefore, by recording the time dependent material deformation the rheological parameters can be independently determined from either the force/displacement measurements or the material grid deformation analysis. This flow visualization technique can also verify certain assumptions made in the modeling effort. One question typically raised is whether or not squeeze flow is actually shear flow or if it mainly consists of extensional stretching. Another issue is whether or not the no-slip boundary condition on the platen surfaces is valid. Examination of the grid deformation provides new information on these important aspects of squeezing flow.

4.3. Experimental technique The squeeze flow experiments discussed here have all been conducted under the condition of constant area of material under the platens. Therefore, the material samples were constructed so that they would initially fill the entire platen area. Additionally, all the experiments discussed in this chapter were performed by imposing a constant closure rate and measuring the change in force over time. This test mode was chosen since it is much easier to impart an instantaneous constant closure rate than to produce an instantaneous constant force on the platens. The experiments were stopped when the samples reached one half of their original starting height. The fiber reinforced thermoplastic samples were all tested at their melt flow processing temperature of 370°C. These preconsolidated samples were placed in the room temperature platens under a small pre-load. The heaters were then turned on and the processing temperature was reached within 20 min. Due to the thermal expansion of the rams and platens, the ram heights were continually adjusted during the platen heating to maintain the constant compressive pre-load on the sample. To insure thermal equilibrium, the samples were then kept at the test temperature for an additional 10 min before the constant rate squeeze flow was started. At the start of the experiment, an indicating marker was removed from the front of the material

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74

59

Table 1 Carreau viscosity parameters determined from experimental data for unfilled and fiber filled clay Fiber vol. percentage

Platen closure rate (cm min I)

~1~ (Pa. s)

2

tl

0 20% Nylon fibers 40'!/, Nylon fibers 5~"/, tl o Nylon fibers 60'~/,, Nylon fibers 60% Glass fibers

0.0254-0.762 0.254 0.254 0.254 0.254 0.254

220 000 I ll)0 000 1 700 000 4 400 000 12 000 000 21 000 000

10.1 1N 20 _~ ~" 24 20

0.1 0.1 0.1 0.1 0.1 0.1

sample so as to provide a starting reference for the videotape flow visualization. Platen force and displacement data were recorded through a c o m p u t e r interface with the test frame at a rate of l0 points s ~.

4.4. ln[tuenee of.fiber volume fraction 4.4.1. Unfilled clay Since the unfilled clay is h o m o g e n e o u s with no preferred flow directions, the channel flow platen apparatus depicted in Fig. 7 was used to characterize its r o o m temperature viscosity. Several platen closure rates ranging from 0.0254 to 0.762 cm m i n I were used. The modeling clay was found to be adequately modeled as a shear thinning Carreau fluid within this range of squeeze flow rates. Using the analysis of Section 3.3 the Carreau model viscosity parameters listed in Table 1 were found to describe the viscosity of the unfilled clay. Fig. 8 shows a typical experimental loading curve for the unfilled clay compared to the predicted load curve.

300

~ . .~ ~ ~ ~ ~._ ._-_~ _. _.. , . . . , . . .

250

..

2oo

Data]

220,000 Pa-s

Z 0~a) 150 u.

Experimental C a ~

o

~ : 10 sec. ~~ _ -h l ~

~

I

I

/

I

1

,~

I

.

.

100 5O 0

.J~*~-,

0

,

I

20

,

,

,

I

40

,

,

,

I

H

i

I

60 80 T i m e (sec.)

,

~

I

100

,

,

,

20

Fig. 8. Experimental loading curve for unfilled clay compared to predicted loading curve using a CaJreau fluid.

60

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47-74 5000

....

, ....

, ....

, ....

Experimental 4000

z

o

Carreau

.... f=0.6 de

, ....

i

Data

Model

Characterlzatlo_n /

aooo

G

2000 U.

.;7 1000

_,¢~

f=O.5

-." 0.2-

_J°f=O.4

0 0

10

20

30

Percent

40

50

60

Compression

Fig. 9. Force vs. percent compressionfor clay-nylon fiber compositeswith various fiber volume percentages.

4.4.2. Nylon fiber filled clay All the nylon fiber filled composites were tested at a room temperature of approximately 22°C using a platen closure rate of 0.254 cm rain 1. Similarly to the unfilled clay, the filled samples were found to behave as shear thinning Carreau fluids. Again, using the analysis of Section 3.3 the Carreau viscosity parameters listed in Table 1 were found to best describe the transverse shear viscosity of the nylon fiber filled clay. Fig. 9 compares the experimental load vs. percentage of sample compression curves for the filled samples to the predicted curves using the Carreau parameters of Table 1. Fig. 10 plots the viscosity of the unfilled and nylon fiber filled clay up to the maximum shear rate of approximately 1.0 s - ~ experienced within the composite samples during the tests. Although the unfilled clay will flow in three dimensions if placed under the platen arrangement, an interesting experimental observation was that the addition of as little as 20 vol.% of

10s

'

'

'

•

' | " 1

,

,

'

'

' ' " 1

~

'

'

"

"

' ' '

f=0.6

?IQ a.

107 1 06

>,

f = O . 2 ' ~ ' f ~

4"*

o

0.5

1 0s

Unfil/l~led

._~ 1 04 10:

1 04

.

.

.

.

.

.

.

.

I

10a

Shear

,

Rate

,

,

,

, , . . I

1 0"1 (sec. "1)

0o

Fig. lO. Carreau model transverse shear viscositiesof clay-nylon fiber composites.

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74 10000

'

•

'

'

I

'

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'

'

I

'

"

'

"

I

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o

Experimental Data Cerreau Model I

Carresu Parameters:I ~" Z,

/

~o: 21,ooo,ooo p H i

6000

~,:

/

I

20 sac.

.o o u.

4000

2000

0

..o,---'~T 0

,

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~

,

,

I

. . . .

100

i 150

. . . . 200

Time (sec,) Fig. l 1. Carreau fluid modeling for the transverse shear viscosity of glass fiber filled clay. aligned fibers caused the flow of the composite to be entirely two-dimensional transverse to the fiber direction. Table 1 reveals the decidedly non-linear relationship between the increase in the composite's transverse shear viscosity and the increasing percentages of fibers present. At low shear rates, the transverse shear viscosity of the clay-60% nylon fiber composite is approximately 55 times greater than the viscosity of the unfilled clay. With the bulk transverse shear viscosity of the nylon fiber filled clay quantified, the question as to the correctness of the creeping flow assumption that was made in interpreting the experimental results can now be addressed. Considering the most extreme test conditions of the (lowest viscosity) 20 vol.% fiber sample tested at a squeeze rate of 0.254 cm m i n - 1 (4.23 × 10 5 m s-1), at its shortest sample height (greatest shear rate) of 0.005 m, and a material density estimated at 1620 kg m 3 (PNylon~'~ 1.14 g cm-3, PClay~ 1.74 g cm-3), the Reynolds number for flow can be estimated as Re = 1.25 x 10 --9 which certainly meets the criteria for creeping flow.

4.4.3. Glass fiber filled clay The previous sections's work on nylon fiber reinforced clay gives insight into how the fiber volume fraction can affect the transverse shear viscosity of aligned unidirectional composite material systems. However, a question remains as to how the diameter of the reinforcing fibers may affect the squeeze flow behavior. The 1 mm diameter nylon fibers are very large as compared to the typical fiber size in commercial composite material systems (7/tm carbon fibers are present in APC-2). Composite samples were, therefore, constructed using the same clay matrix but adding 60 vol.% of 5 cm long, 10 ~ m diameter glass fibers. The glass fiber-clay composites were tested under a constant platen closure rate of 0.254 cm m i n - 1. Using the analysis of Section 3.3 and, again, comparing the experimental loading curves to the predicted curves, the Carreau fluid viscosity parameters listed in Table 1 were found to best describe the transverse shear viscosity of the glass fiber filled clay. Fig. 11 compares the experimental load for a glass fiber filled clay sample to the predicted curves using the Carreau fluid viscosity parameters in Table 1.

62

S.F. Shuler, S.G. Advani / J . Non-Newtonian Fluid Mech. 65 (1996) 47-74 3000

I

I

I

I

P

I

I

0.5 cm/mln, o Carreau ModelData I 0.2 cm/mln. Characterization ~ ~ Carreau Parameters:l 1] o = 2.5e6 Pa-s I = 50.0 sec. I n = 0.65 ]

2500 ,-. 2000 Zv w 1500 o 1000

~

Experimental

c ~ 0 " 0 5

cm/mi~

500

0~ 0

100

200

300

400

500

600

700

T i m e (sec.) Fig. 12. Experimental force data for unidirectional APC-2 laminates compared to the Carreau model characterization.

Adding 60 vol.% of the larger diameter nylon fibers resulted in a transverse shear viscosity 55 times greater than the viscosity of the unfilled clay while 60 vol.% of the glass fibers resulted in a transverse shear viscosity 95 times greater. From this information, it seems that adding the same volume percentage of smaller diameter fibers has the effect of increasing the transverse shear viscosity even more than the increase found using larger diameter fibers. It should be noted that the squeeze flow deformation of the glass fiber composites was also entirely two-dimensional transverse to the fiber direction. 4.5. Unidirectional A P C - 2 laminates The unidirectional APC-2 laminates were tested at the four different closure rates of 0.05, 0.1, 0.2, and 0.5 cm m i n - i, covering a full order of magnitude range of closure rates. Three samples were tested at each closure rate and little variation was found between the platen loading response of the samples. A representative platen load vs. time response of the unidirectional laminates for each platen closure rate is shown in Fig. 12. The loading response is shown up to the point where the samples reached one half of their original height.

Table 2 Carreau viscosity parameters for PEEK at 370°C Parameter

Value

~/o 2 n

687 Pa. s 0.0932 s 0.787

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74

63

Table 3 Carreau model parameters for the transverse shear viscosity o f APC-2 material at 370°C Parameter

Value

Jh, )~ n

2.5 × l0 ~' Pa. s 50 s 0.65

4.5.1. Rheological parameters The reinforcing PEEK matrix polymer in the APC-2 laminates is known to behave as a shear thinning Carreau type fluid with the viscous parameters listed in Table 2 [22]. Viewing the transverse shear flow of the fiber filled fluids as viscous flow with a lubricating fluid between the fibers, the Carreau model was investigated as perhaps an appropriate model to describe also the shear thinning viscous behavior of the APC-2 laminates. Using the numerical procedure described in Section 3.3, the platen force data for the four closure rates was least-squares fit using the Carreau model for the bulk transverse shear viscosity of the laminates and the viscous Carreau parameters of Table 3 were found to produce loading curves that closely match the experimental platen force data collected. The loading curves based on this particular Carreau model characterization for the transverse shear viscosity are compared to the experimental loading curves in Fig. 12. The Carreau model curve for the transverse shear viscosity of the APC-2 material at 370°C based on the parameters listed in Table 3 is plotted in Fig. 13 up to the maximum shear rate experienced withing the samples. The Carreau model viscosity of the neat PEEK matrix polymer at 370°C is shown in Fig. 13 as well. It is noteworthy that the transverse shear viscosity of the composite is approximately 3600 times greater than the viscosity of the PEEK matrix resin. Also, the onset of shear thinning behavior occurs at a lower shear rate than for the neat resin. Because consolidation flow or squeeze flow encompasses such a broad range of shear rates I

I

T

1 06

? m ,--

10 s

P e r a 11o= 2.5e6 Pa-s ~ "

APC-2 Cerreau

=

370°C

~

~. = 50.0 sec. n = 0.65

o 104 >

PEEK Carreau Parameters:

= 687 Pa-s ~. : 0 . 0 9 3 s e c . ~ n 0.787

1"I°

1 03 1 03

I

I

1 0 .2

1 0 "1

Shear

rate

0°

(sec. 1)

Fig. 13. Carreau model transverse shear viscosity for APC-2 c o m p a r e d to neat P E E K at . 70 C.

64

S.F Shuler, S.G. Advani/J. Non-NewtonianFluidMech. 65(199~ 4 7 - ~ 0 sec.

40 sec.

80 sec.

Fig. 14. Flow visualization of velocity distribution during squeeze flow of an APC-2 laminate (0.2 cm min ' closure rate). throughout the volume of the material, a simple Newtonian or power-law fluid model can not adequately describe the squeeze flow behavior of a fluid that, depending on the shear rate, follows both Newtonian and shear thinning behavior. Since the Carreau model captures both the Newtonian behavior at low shear rates and the shear thinning behavior at higher shear rates, this three-parameter model is able to describe the viscous behavior of the unidirectional laminates for all the closure rates. It is instructive again to note that during squeeze flow the material experiences a range of shear rates that varies both over time and throughout the volume of material. Because consolidation flow or squeeze flow encompasses such a broad range of shear rates throughout the volume of the material, a simple Newtonian or power-law fluid model can not adequately describe the squeeze flow behavior of a fluid that, depending on the shear rate, follows both Newtonian and shear thinning behavior. 4.5.2. Flow visualization

As described previously in Section 4.1, all the unidirectional APC-2 samples were videotaped during the squeeze flow experiments. This documents the material flow by recording the deformation of the 2 m m grid on the surface of the sample perpendicular to the fiber direction. An example of the velocity distribution in a unidirectional APC-2 laminate during a 0.2 cm min 1 squeeze flow experiment is shown in Fig. 14. In this figure, all the reinforcing fibers are oriented into the page. There are several observations that can be made from this video picture sequence. First, the material grid stays visible throughout the experiment and can be easily tracked. Second, the developing parabolic velocity and parabolic material deformation profiles can be clearly observed as the bottom platen moves upward. These parabolic profiles verify that the material deformation mainly consists of shear flow, not extensional flow as has been suggested. Third, the symmetry of the material deformation about both the x = 0 and y = 0 planes is evident. Finally, it can be noted that the material grid points near the platen surfaces remain nearly stationary. Post-experiment examination of the samples also revealed very little movement of the composite

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74

65

material that was in contact with the steel platen surfaces. These two observations serve to validate the assumption of the no-slip boundary condition at the platen surfaces. An important material flow observation not evident in the pictures of Fig. 14 was that there was no material flow parallel to the fiber direction. This two-dimensional flow phenomenon is due to the presence of the fiber reinforcement which induces a material viscosity in the fiber direction that is several orders of magnitude greater than the shear viscosity transverse to the fibers. Also, the post-experiment examinations showed no significant resin percolation out of the sample fiber beds. In the case of a Carreau type fluid, the developing velocity profiles are a function of the time constant, 2, and the shear thinning exponent, n. Changing either of these parameters will affect the shape of the velocity profiles. Therefore, by recording and tracking the developing velocity profiles withing the material during squeeze flow, information is gained as to the nature of the viscous parameters ), and n. In this way, the velocity profiles can provide an independent check on the ), and n values determined from the platen force/material deformation data. As seen in Fig. 14, the positions of the material grid points serve to define the material deformation profiles during the squeeze flow. These data points were collected by using image analysis software on the videotape image. Absolute positions of the grid points were established by measuring from a known marked position on the upper platen which remained stationary throughout the entire experiment. Since the viscous parameters 2 and n had been previously determined from the platen force/material deformation data, it is possible to predict the material deformation as a function of time. A comparison can then be made between the predicted material deformation and the measured material grid deformation. The computational approach for determining the x- and y-material locations over time utilizes a Eulerian time scheme with small time steps. The first step in the computational approach is to discretize both the x- and y-directions in the sample flow field from 0 <_y < hi2 and 0 _< x < L. Next, a starting x-location which corresponds to one of the vertical grid lines on the sample is selected. A small time step is then chosen and the velocity gradient, 8V~/83,, at the incremented time is found for each discretized x- and ),-position using the computational technique detailed in Section 3.3 and Fig. 3. The velocity gradients are then used to find the complete velocity profile and the Eulerian relations are used to compute the new x- and y-positions. The time steps chosen depend on the platen closure rate being studied. However, to maintain accuracy with the Eulerian stepping procedure, the time steps are limited so that less than 5% of the total material deformation occurs within a single step. The values in Table 3 for the viscous Carreau parameters 2 and n were used in the flow simulation to obtain the predicted material displacement results in Figs. 15(a)-15(d) and 16. Figs. 15(a)-15(d) compare the predicted and measured material displacements at selected starting x-positions for samples squeezed at 0.05, 0.1, 0.2, and 0.5 cm min ~, respectively. Fig. 16 summarizes the displacement results at three different starting x-positions distributed across a sample that was squeezed at 0.2 cm min i. Good general agreement is seen at every closure speed and across the length of the samples. This similarity between the measured material deformations and the predicted deformations gives additional credibility to the concept of viewing the composite in transverse shear flow as a viscous fluid with a bulk viscosity that can be appropriately described using the Carreau fluid viscosity model. The agreement also independently verifies the accuracy of the Carreau parameters of Table 3 in describing the bulk transverse shear viscosity of the composite.

66

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47-74

5, S q u e e z e

flow cell model

The experimental results of Section 4 indicate that a strong non-linear relationship exists between the transverse shear viscosity of aligned fiber composites and their fiber volume fraction. Since the transverse shear viscosity will affect the consolidation and processing characteristics of these types of composite materials, understanding the nature of this relationship is vital and a workable model that include fiber volume fraction dependence is desirable. However, continuum models, such as were developed in Section 3, are no longer applicable when seeking to describe a fiber volume fraction dependence. Therefore, a unit cell model consisting of a single fiber surrounded by an incompressible matrix fluid has been developed based on a lubrication approach. Previously, Lindt [23] created a consolidation model for unidirectional composite materials containing a square packing arrangement of cylindrical fibers in a Newtonian fluid by develop....

1 L"'~

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, '"

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....

; ....

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~"

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|| I

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(cm)

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(cm)

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

- -

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.

,,

0 se

30 n c . sac

i

t~o sac.

1.3

X-Position

..

X-PosiUon (c)

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1.5

Predicted Diaplacemente i : --O - - Measured Displacements I

o

'

I

(cm)

,:.,., I

~

~

o ....

Z

I '' '' Displacements

I ....

Predicted

0.8

1.1

1

,

f~l~ ~

\

e

~

~: ! O 20 sac. ,, o l S aec, 6 sac. 1o sac. . '... I. L. i... '... Io.. 1.04

1.12

1.2

1.28

X-Position

1.36

o..o

I ... 1.44

I, 1.52

, 1.6

(ca)

(d)

Fig. 15. Predicted vs. experimentallymeasured APC-2 material displacements: (a) 0.05~ (b) 0.1; (c) 0.2, (d) 0.5 cm rain- ~.

S.F. Stluler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74

I

'

'

I

. . . .

67

l

0.9

"~O

0.8

0.7

0.6 Z

0.5

. . . . . . . .

0.5

. . . .

'

1 1.5 X - P o s i t i o n (cm)

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2

'

'

Fig. 16. Predicted vs. experimentally measured APC-2 material displacements at several locations across the sample width (0.2 cm min i).

ing an analytical solution based on the lubrication approximation for the consolidation ttow of circular cylinders. Hjellming and Walker [24] later rederived Lindt's solution using an asymptotic analysis for the squeeze flow generated by the fiber motions and considered the case where a consolidation force is applied only to the top row of fibers and the case where each fiber experiences a uniform consolidation force, as would be the case in filament winding where the consolidation force is derived from tension in the fiber tow. Both consolidation processes were simulated by examining the situation where rigid fibers move vertically through a Newtonian matrix fluid. In these models, the fibers are constrained from any lateral movement and remain aligned in columns while the matrix fluid is squeezed out from between the moving fibers and is allowed to pass around the fibers out of the fiber bed. The squeeze force developed between each approaching fiber pair and the shear force generated by the fluid as it flows up and past each fiber are then determined. In all cases, the consolidation force is applied and the resulting fiber motions are calculated. The present work builds upon this concept of considering the squeeze flow interactions between individual fibers. With this idea, a unit cell model is developed which describes the transverse shear flow and consolidation flow behavior of aligned fiber composites. Although the present model uses the same basic squeeze force relation between the individual fibers, the squeeze flow idea is extended by not permitting fluid flow out of the fiber bed and, instead, allowing the fibers to translate laterally along their axis direction as observed in the experiments. This is consistent with the assumption that the composite materials behave as incompressible fluids with a constant fiber volume fraction. The original work is also expanded through the incorporation of a Carreau fluid model to describe the matrix flow behavior.

68

S.F. Shuler, S.G. Advani / J . Non-Newtonian Fluid Mech. 65 (1996) 47-74

5.1. Newton&n fluid The present unit cell squeeze flow model considers the flow depicted in Fig. 17. At the start of the squeeze flow, the material volume is idealized as consisting of a square array of unit cells which contain a fiber of diameter, D, and are an equal spacing, S, apart. For a given fiber diameter, the fiber spacing is adjusted to achieve the desired composite fiber volume fraction. Starting with a similar square packing array, the approach of both Lindt and Hjellming and Walker was to apply the consolidation force and, from that, determine the resulting vertical motion of each fiber. In the current model, however, the individual x- and y-direction velocities of the fibers are imposed and the resulting y-direction force component is resolved. The horizontal, x-direction, velocity components are assumed from the overall flow behavior of the material volume. The vertical, y-direction, velocity components must match the end boundary conditions of symmetry (zero-flow) and imposed platen velocity and are presupposed to be linearly distributed. Therefore, each fiber in the array has both an x- and y-velocity component that results in the motion represented in the bottom of Fig. 17. With the velocity components of each fiber determined, the motion of each fiber relative to another can be calculated and the squeeze force between individual fibers found.

Unit Cell

(~.

VX(i'I)

" -

2H

-

-

If

i)Vx(I+I)[ v VY(i+l)

Fig. 17. Cell model fiber motion.

4"~S )

I

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47 74

69

Fig. 18. Cell model fiber interactions.

The squeeze flow between two fibers is again viewed as a lubrication type flow. Therefore, by neglecting fluid inertia, assuming two-dimensional fluid flow with no flow in the fiber direction, and taking the fiber diameter to be much greater than the separation distance, 2H(t), the x-direction equation of motion of Eq. (2) reduces in the same manner described in Section 3.1. In addition, the cylindrical fibers are assumed rigid and move in pure translation, the flow is slow and dominated by viscous forces, and a no-slip boundary is assumed at the fluid-fiber interface. The idea of the cell model approach is that the individual squeeze forces generated by the fibers sum to the total force experienced by the platens. The x-direction fiber motion results in interactions with the surrounding fibers as represented in Fig. 18. The y-direction force component of these interactions is found by resolving the relative closure velocity between the fibers and then determining the y-direction component of the squeezing force generated. Only interactions with nearest-neighbor fibers within a distance of two fiber diameters are considered. Due to hydrodynamic screening effects, any influence from fibers outside this distance will be negligible. Additionally, any fibers that flow out from between the platens during the simulation are neglected. Once the y-direction force components have been resolved for each fiber pair interaction, they are summed to determine the total force on the squeeze platens The squeeze force due to two rigid cylindrical fibers of length, W, suspended in a Newtonian fluid of viscosity, /L, separated by a distance of 2H and approaching at the rate, 2dH/dt, is as follows: Fsqueeze

=

- -

3nit W (dHi/dt) 2x/-~ (H]R)3/2

(I 5)

5.2. Carreau fluid The unfilled clay and PEEK matrix materials have been shown to be well modeled as shear thinning Carreau fluids. Therefore, the cell model was updated to incorporate a Carreau model description for the lubricating fluids. The solution technique for a Carreau model matrix fluid within the cell follows the same basic iterative procedure described in Section 3.3 and outlined in Fig. 3. The difference in the solution procedures involves accounting for the curved upper and lower no-slip boundaries. These boundaries require discretization of the region from 0 < y
70

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47- 74

The effect of incorporating the Carreau model for the matrix viscosity is demonstrated in Fig. 19 for the case of two fibers approaching one another. The force generated by the approaching fiber pair is similar for both the Newtonian and Carreau fluids at the start of the squeeze motion. The significant difference between the Newtonian and Carreau matrix fluids only appears as the fibers become close. This variance arises since, as the fibers approach one another, the shear stress in the fluid is increased and the shear thinning aspects of the Carreau fluid become evident.

5.3. Comparison to experiments 5.3.1. Model composite with varying fiber volume fraction The experimental squeeze flow data of Section 4.4 on nylon fiber filled modeling clay demonstrated the interdependence between the composite's fiber volume fraction and its transverse shear viscosity. This relationship can now be investigated in terms of the proposed cell model. Using the Carreau viscosity parameters in Table 1 for the unfilled clay as the lubricating fluid, the generated squeeze force predicted from the cell model is compared to the measured force in Fig. 20. Better agreement is seen at the higher volume fiber fractions. This may be a result of the cell model's assumptions about lubrication flow and fiber interactions being more accurate when the fibers are more densely packed. The cell model prediction curves of Fig. 20 for the model materials were subsequently curve fit using the Carreau fluid model. The resulting Carreau parameters are listed in Table 4. The discrepancy between the zero shear viscosities and time constants determined from the experimental data and the cell model arise from the differences between the measured squeeze forces and the forces predicted by the cell model. 5.3.2. APC-2 laminates The cell model prediction of the squeezing force at various closing speeds is compared to the experimental APC-2 data in Fig. 21. Since PEEK serves as the matrix polymer in these 0.005.

~

~

,

,

,.--. 0,004 t __

I

/

o.o. I-/

0

I'

10

Newlonlan Fluid:l ,,-,

20

I-'---*1 '

30 40 Tlme (sec.)

50

l /

60

Fig. 19. Comparison between Newtonian and Carreau model matrix fluids in cell model.

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996) 47- 74

,i~,~~

6000

....

I ....

I ....

5000

I Experimental Data I I ---~----Cell Modal Prediction I

"-'Z 4000 m P

o u.

71

z ' ' '/' f=o.

3000 2000 ~ "

.

~

f=0.4

1000 0 0

50

100 T i m e (sec,)

150

200

Fig. 20. Comparison between experimentally measured platen forces for the model composite and the cell model predictions using a Carreau matrix fluid.

composites, the lubricating matrix fluid is taken as a Carreau fluid with the PEEK viscosity parameters at 370°C listed in Table 2. The cell model simulation was performed assuming a fiber diameter of 7 / t m and a volume fraction of 64%. Reasonable agreement is found between the cell model prediction and the experimental loading curves. The discrepancies may be due to the idealizations made in formulating the cell model. These simplifications include the idealized square cell packing arrangement with equal fiber spacing, the idealized round and rigid fiber structure, the assumed flow paths of the fibers and neglect of any fiber bunching or entanglements that are likely to occur within such a dense packing of pliant fibers. The cell model prediction curves for the unidirectional APe-2 laminates were also curve fit using the Carreau fluid model and the resulting rheological parameters are listed in Table 4.

6. Discussion and conclusions

The main focus of this paper concerns the effects of aligned fiber reinforcement on the processing characteristics of concentrated suspensions. This was motivated by the forming of high volume fraction aligned fiber composites. The study of the squeeze flow behavior has Table 4 Carreau viscosity parameters determined from cell model predictions for fiber filled clay and A P e - 2 laminates Material Clay with Clay with Clay with Clay with APe-2

20% 40% 50% 60%

nylon nylon nylon nylon

fibers fibers fibers fibers

qo (Pa. s)

).

n

400 3 000 6 000 12 000 3 000

1 5 10 20 20

O. 1 O. I O. 1 O. 1 0.8

000 000 000 000 000

72

S.F. Shuler, S.G. Advani / J. Non-Newtonian Fluid Mech. 65 (1996)47-74 3000

I~

'~ r0.5 clm/min.

2s°°l"/I

//l

I

I

/ ~I / ~1 2 0 0 0 I" 11

g

'

10.2 cm/min, f , ~ ! ]'

/i

I

'

' ' Experimental

•

-

' Data

Cell Model Prediction I Matrix Fluid 11o = 687 Pa-s I Parameters'. ~ : 0.093 s.c I

I

1500t[! ]/ J/'

cmlmin.

.05 cm/min.

u. 1 0 0 0 ~ 7 5OO

Ooc 0

0

100

200

300

400

500

600

700

Time (sec.)

Fig. 21. Experimentalsqueezeforce for unidirectionalAPC-2 laminatescompared to cell model predictionsusing a Carreau matrix fluid. provided insight into both the actual material deformation that occurs during the critical laminate consolidation stage of processing and the nature of the transverse shear viscosity of these composite forms. These are important issues since they impact the ability to shape the laminates effectively. Awareness of the material deformation characteristics while subjected to a consolidation pressure will assist in determining the final dimensions and mechanical properties of processed laminates while better understanding of the transverse shear viscosity will assist in predicting the laminate response during actual forming operations. Construction of an experimental hot platen squeeze flow apparatus has allowed the transverse shear viscosity to be characterized for both a model composite material system composed of clay filled with either nylon or glass fibers and a commercially available carbon fiber reinforced thermoplastic laminate. The flow visualization technique developed for this system has assisted in the viscosity detemination and has provided qualitative information about the macroscopic flow. The influence of fiber volume fraciton on the transverse shear viscosity of unidirectional composites was experimentally investigated by constructing model composites out of clay and nylon fibers and testing them at room temperature in the squeeze platen apparatus. Both the unfilled clay and the fiber filled composites were found to be adequately described as shear thinning Carreau fluids and experiments with the 20, 40, 50, and 60 vol.% fiber composites revealed a non-linear increase in transverse shear viscosity as the fiber content was increased. Filling the clay with 60 vol.% of smaller diameter glass fibers doubled the transverse shear viscosity over that of the 60% nylon fiber filled samples. The difference in squeeze force between the larger and smaller diameter fiber samples may be attributed to the fact that the smaller diameter fibers have a larger ratio of surface area to volume. In terms of the cell model view, this allows a greater squeeze force to be generated for the same volume percentage of smaller

S.F. Shuler, S.G. Advani / J . Non-Newtonian Fluid Mech. 65 (1996) 47-74

73

diameter fibers. The glass fiber reinforced clay did not, however, exhibit the same magnitude of viscosity enhancement seen in the APC-2 samples even though the fiber diameters were of the same order of magnitude. This may be ascribed to the method in which the glass fiber-clay composites were assembled and to the stratified resin-rich and fiber-rich areas that existed. If all the fibers were not sufficiently surrounded and separated by the clay, some of the generated squeeze flow force will be lost. Similar to their PEEK matrix fluid, the transverse shear viscosity of the APC-2 laminates was found to be adequately described by the Carreau viscous fluid model. Although the shear thinning exponents of the matrix fluid and composite were similar, the zero shear rate viscosity and time constant of the composite were both found to be approximately three orders of magnitude greater than those of the matrix polymer at the same temperature. The time constant and shear thinning exponent of the composite were independently confirmed from the flow visualization analysis. The flow visualization also verified that the material deformation mainly consists of shear flow, not extensional flow, and that a no-slip boundary condition was appropriate in the modeling analysis. Additional observation of the laminate deformation revealed the flow to be entirely transverse to the fiber direction. This is due to the high viscous anisotropy resulting from the presence of the fibers. The influence of fiber volume fraction on the transverse shear viscosity of aligned fiber composites was also examined through a proposed cell model based on the individual fiber interactions within the matrix fluid. The effects of varying the fiber volume fraction and of including a non-Newtonian viscosity model for the matrix fluid have been demonstrated. Reasonable agreement was found when comparing the platen loads predicted by the cell model to the clay nylon fiber experimental results with better correlation as the fiber volume fraction increased. Reasonable agreement was also found over a broad range of squeeze rates when comparing the platen loads predicted by the cell model to the APC-2 experimental results. The greatest deviations occurred towards the end of the squeeze flow experiments. The disagreement at this point can be attributed to the cell model assumptions that the rigid fibers remain aligned and equally spaced. As greater pressures build up under the platen, the fibers may tend to become displaced from their original parallel alignments and spacings. This distortion would result in a decreased platen force as compared to the idealized fiber approach mechanics assumed in the cell model. This squeeze flow study has been successful in its main goals of developing a simple and versatile method of determining the transverse shear viscosity of long aligned fiber reinforced composites, the influence of fiber content and diameter on the transvers shear viscosity, and developing a workable cell model with which to investigate the influence of fiber content and matrix fluid viscosity on the transverse shear viscosity of these laminates. The modeling and experimental studies revealed here indicate that, although greater mechanical properties can be gained by increasing the fiber volume percentage of the composite, by doing so the processing and forming ease of the laminate may be compromised. References [l] G. Dienes and H. Klemm, Theory and application of the parallel plate plastorneter, J. Appl, Phys., 17 (1946) 459 471.

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