Analysis of fiber motion during wet filament winding of composite cylinders with arbitrary thickness

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ANALYSIS OF FIBER MOTION DURING WET FILAMENT WINDING OF COMPOSITE CYLINDERS WITH ARBITRARY THICKNESS A. Department

and H. TENG

AGAH-TEHRANI~

of Theoretical and Applied Mechanics. University Champaign. Urbana. IL 61801. U.S.A.

of Illinois at Urbana-

Abstract-A continuum consolidation model is proposed for macroscopic analysis of fiber motion during filament winding of thermoset composite cylinders with arbitrary thickness. The model takes account of the variation of both instantaneous stiffness and permeability of the mixture with fiber

compaction. Due to resin tillration. structure of the resulting initial-boundary value problem is similar to that of 3 moving boundary problem. Based on this analogy, a finite difference schemeis

devisedfor

the solution of the problem. For the case of winding onto a rigid mandrel, the results point to the etistence of an active and a passive zone of consolidation. The results further indicate the possibility that during hoop winding of relatively thick cylinders. the tension can be completely lost in the portion of the passive zone away from the mandrel.

I. INTRODUCTlON

The filament

winding

which

cvcntually

longitudinal

which can be used for forming composite

cover the surface of the mar&cl.

(parallel

oF the

wet

axis of the mandrel)

to the

mandrel).

winding.

angle may vary from (perpendicular

to the

resin is mixed with the fiber bundle during the winding stage, one

Alternatively.

see Shilbcy

The winding

to circumferential

The binder is ;I thcrmosrt resin which can be epoxy. polyester or vinyl if prc-impregnated

and fiber bundle is used, one has dry winding winding

stiff cylinder which is invariably

The layers may bc wrapped in adjacent bands or in repeating patterns

ester. If the thrrmosrt has

process

with diliizrcnt winding angles onto a moderately

Inycrs

circular (man&cl).

axis

is ;I manufacturing

with high strength. The csscncc of this processing method is to successively add

cylinders

(19Y2)].

B-staged form of the mixture of resin

[for an exhaustive discussion of filament

For a given composite system, the primary control variables

during winding can be identified as: the tension applied to each layer, the velocity of the machine which traverses along the axis velocitiy

of the mandrel

(Calius

stage will be the development

of the

mandrel (cross-head velocity), and the angular

and Springer,

1985). The end result of the forced winding

of a system of compressive radial and tensile circumferential

stresses within the cylinder. In the case of wet winding, one also has the consolidation

of

the fiber network. Following

the winding stage, the wound cylinder is initially

peratures so as to cause cross-linking temperature.

During

cured at elevated tem-

of the resin, and then it is cooled down to ambient

the heat build-up,

such phenomena

as micro- and macroflow

of the

binder. thermal expansion and simultaneous decrease in stiffness in the transverse direction, partial relaxation of the winding stresses, and change in the winding pressure on the mandrel due to mismatch

in thermal expansion coeflicients of the mandrel and the cylinder take place. The difrerence in the thermal expansion coefficient of a semifabricated material and the mandrel Icads to an increase in the winding pressure on the mandrel. The decrease in

stiffness and intensification second

of dissipation

phcnomcna

lead to the opposite effect with the

mechanism

usually prevailing. Upon polymerization. the increase in strength and stiffness in the transverse direction together with the chemical shrinkage take place prefcrcntially in the same direction, while slight relaxation of the winding stresses continues. The balance between tcurrently. CA 91181-OIYI.

these two competing

Lcyturer at the Department U.S.A.

mechanisms

of Mechancial

2649

leads to the winding pressure on

Engineering.

San Diego Stale University.

San Diego.

A. AGAH-TEHKANI and

2650

H. TEVG

the mandrel remaining constant during polymerization. case where curing is accomplished during winding.

Alternatively

one can envisage the

In such a case, the rate of heating the

mandrel should also be considered as one of the control variables. Upon cooling. while the processes of creep and stress shrinkage

and increase in stiffness

in the transverse

relaxation

direction

continue. thermal

take place. The

winding

pressure on the mandrel changes due to the difference in coefficients of thermal expansion between the mandrel and the cured article. This

is in contrast to that during the heat build-

up. In the present case, the nature of the competition

between the two mechanisms

is

different. The free strain caused by the change in stiffness of the cured material upon cooling under conditions contribution

of tinite stress is smaller than the thermal shrinkage strain. Therefore,

of thermal shrinkage

is dominant

the

and upon cooling the radial compressive

stresses decrease and in some cases regions of tensile radial stress develop. In order to predict the occurrence of damage in the final product which is often in the form of the fiber waviness (due to the circumferential buckling) or the transverse cracks (due to the tensile radial stress) in between the layers. there is a need for a reliable method for predicting the state of the composite cylinder at the end of each of the three stases of the manufacturing

attribute during

process: (I) winding. (3) curing and (3) final cool-down. An important

of each submodcl must bc its ability

to predict the stresses which arc genoratcd

the corresponding

stage. In the case of wet winding. one must also be able to detcrminc the radial distribution of the fiber volume fraction which arises as the result of the settling of fiber huntllcs in the liquid resin. Hy assuming

tl1;1t

succcssivc Iiiycrs

arc wrapped simultancoiisly

on the niandrcl. one

can utili02 the cnisting niodcls for stcss analysis of tape winding (Willctt Lin and Wcstmann. the fact that

and I’ocs&.

I ‘)8X :

IOS9) in order to simulate the dry winding. In so doing. one is neglecting

in reality

the axial variation

the layers

arc wrappctl

scqucntially

next

to each other

rcsuiting

in

of the outer radius. The analogy can hc further complctcd by observing

that dry winding is

a

gcncrali4

plant strain problem while tape winding is

;I

plant stress

problem. Irrcspcctivcofthc

thickncssofthc

woundcylindcr,

must take account of the resin filtration

thcorotical modclling ofwet winding

through the fiber bundles which is induced by the

existing pressure gradient. Recently Calius and Springer simulating

the cntirc stages of wet winding. Their

(1990) have outlined a model for

submodcl related to the forced winding

stage, which neglects radial resistance of the fiber network, is based on considering the force equilibrium

and the conservation

of mass within

each layer. Although

this model is only

appropriate for thin cylinders where one can neglect the increase in the radial stiffness the decrease in permeability arbitrary

orientation

of the fiber bundles due to consolidation,

of the fiber bundle. Lee and Springer

submodel on a continuum

formulation

model is the assumption

u priori assumption winding filtration

by basing their stress

have extended the model by Calius and Springer

for analysing filament winding of thick cylinders. Springer

(1990)

and

it does allow for

An important

that consolidation

characteristic of the Lee-

takes place only in the last layer. This

for the tiber motion is only valid if the time scale associated with the

of each layer is exceedingly large compared to the time constant of the resin through

the fiber bundle.

models proposed by Springer

It must be mentioned that different filament winding are the only models which provide a unified

and co-workers

view of this complicated manufacturing

Within

process.

of a discretized consolidation model. one must mention the work by Bolotin L’[ (11. (I9SO) who have included the elasticity of the fiber network by introducing the term associated with the radial pliability in the equilibrium of the layers whose

the framework

thickness

is below

the critical

value at which

the radial

resistance

arises.

Any comprehcnsivc model for analysing fiber motion during wet filament winding must take account of two important aspects which arc associated with fiber compaction :

(I) increase in the radial stilTncss of the composite (Gutowski CI (I/.. I%7a.b), and (2) reduction in the pcrmcability (Gutowski 6’1al.. 1987a.b: Gcbnrt, 1990). The signifcancc of these attributes bccomcs particularly apparent when the thickness of the cylinder becomes large. The primary aim of this paper is to provide a co~~fi~z~~~~~ cotrsoiidariotr tt~d~! for analysing

the wet winding

of thermosct composite cylinders

with arbitr;try

thickness which

Fiber motion during wet filament winding

265

I

incorporates the above characteristics. Such a model is developed by utilizing the rate form of Biot’s consolidation mixture theory (Biot. 1941). In comparison with the existing discretized consolidation model purposed by Bolotin et al. (1980). the present formulation has two distinct advantages: One is that it is based on a continuous model. and the other is that it takes account of the reduction of permeability with fiber compaction. Following the description of the key features of the model, the details of the numerical calculation and interpretation of the results will be presented. The article will be concluded by a discussion on the importance of including the dependence of the radial resistance and permeability on fiber volume fraction in the analysis of wet winding. 2. DEVELOPMENT

OF

THE

MODEL

The composite cylinder is assumed to be a mixture of incompressible fluid (resin) and solid (fiber bundle). The fiber bundles are assumed to be in the hoop direction. The axial flow of the resin due to the time delay in winding along the axis of the cylinder will be neglected. i.e. it is assumed that all layers along the axis of the cylinder are laid simultaneously. Moreover. the angular variation of the outer radius due to the helical nature of the winding will also be neglected, see Fig. I. Because of these two assumptions. the only non-vanishing component of the velocity of each phase will be the radial velocity, and the problem becomes one-dimensional in space. In order to facilitate the analysis. the following non-dimensional group of parameters are introduced :

whrrc R, is the outer radius of the mandrel. r is the dimensional radial coordinate. rl, and 12~ are radial displaccmcnts of the solid and tluid phase. Cy, and Cs, arc the moduli of the fiber network which has re:lched the maximum achievable tiber volume fraction and CO, is the hoop stifTncss of an individual fiber. a’,,and d,,, are, respectively, the radial and the hoop stress components of the mixture, p is the partial pressure of the resin, I denotes time and IS“is the permeability as the fiber bundle passes through the resin bath.

Fig. I. Gcomctry

of the model.

A. AGAH-TEHKANI and H. TESG

2652

By utilizing expressions

the above non-dimensional

for the non-dimensional

relations,

one can determine the following

velocities of the fiber network and the resin denoted

respectively by ~9,and w,-t

)I’,

=

su,

I[

hi,

-~

_.

___

=

c?_vR,

,1’,

,

ct

=

c'LC,- tr CL& ~ = TV . Cl, R, i‘t

with lr = R,?/K”C~~ as the characteristic time for filtration. increasing

K” reduces the characteristic

(’-, 3)

The definition of fr indicates that

time for filtration

and hence the time for con-

solidation. The dependence of the instantaneous taneous permeability

moduli of the fiber bundle C:, and the instan-

of the fiber network on fiber volume fraction 4t are assumed to be ot

the form given below

(C,,. Cd) = F(&)(CR. G,).

cm,= c#J,c;,,.

K

=

h?G(&).

(4. 5)

The values of the moduli CF, and L’y,, and the functions F(@,) and C;(+,) must be determined from experiments

on fiber compaction (Gutowski

be noted that for simplicity

ct d., 1987b: Gebart, 1990). It should

we have assumed identical dependence of C,, and C,,,, on the

fiber volume fraction 4,. in terms of the apparent density of the phases ([J, = m,;V,.

the conservation 1976 ; Canyon, 1976)

i = s, f),

of mass for the fluid and the solid phase can bc written as (Bowcn,

with V denoting the griidicnt operator. By assuming that the ttcnsitics of both solid antI 11uid phases remain constant during deformation,

(6) and (7) can be manipulatctt

conservation

of mass in terms of the volume fractions of individual

to obtain ths following

cxprcssions

for the

phases ((b, = p,V,/rtr,.

i = t:S)

Addition

of the above two relations,

and usage of the identity 4, + 4, = I leads to the

final desired form of the conservation of mass which involves only the vclocitiy of the solid and the filtration

flux y = &(rq-

N*,) (the surface integral of q over the outer radius of the

cylinder is the amount of the fluid leaving the mixture).

By enforcing the condition that no

resin passes through the mandrel (y( I, y) = 0), the resultant

expression

can be intcgratcd

to obtain

where ~0,)

is the velocity of the interface between the mandrel and the composite.

assume that the mandrel undergoes a small deformation

WC

so that changes in the outer radius

of the mandrel can be neglcctcd. Conservation of mass analysis enables one to dctcrminc the evolution equation which governs the change in the fiber volume fraction. To obtain such a relation. one can simply rearrange the terms in (9) to get

Fiber motion during wet filament winding

where d/d,” represents the total derivative.

The above equation

2653

by itself does not contain

the property that the fiber volume fraction should not exceed the achievable volume fraction. In other words, if H; < 0. the above expression does not provide any upper bound on the value of 4,. However,

it is understood

4, varies with the packing geometry.

that although

the achievable fiber volume fraction

it can never exceed one. This constraint

implies that

as 4, -+ 4,, W, + 0. As will be indicated later, this constraint will be satisfied if the variation of permeability

with fiber volume fraction is taken into account.

Since during wet winding mandrel

the radial resistance of the layer just being wound on the

is very small, the resin material

can flow through

the fiber network

leading to

increases in both the fiber volume fraction and the radial stiffness of the composite. Because of the finitechanges

in the fiber volume fraction, one needs to distinguish between the initial

and the instantaneous

configuration

of the mixture.

In so doing, we can either follow the

motion of the fluid particle (resin) or the solid particle (fiber network).

Since the final radius

of the composite cylinder depends on the position of the last fiber layer which is being wound on the mandrel. the instantaneous configurations

of the mixture and the solid phase

are considered to be identical. Thus, a,,(.~, y) and B,,,,(x, y) are the stress components mixture point at “time”y

with this, we adopt an “updated-Langrangian” mixture.

formulation

to describe the motion of the

that is at each instant of time the current configuration

considered to be the initial configuration Due to the variation

of a

which is occupying the position x of the solid phase. In conjunction of the solid phase is

of the mixture.

of the outer radius with time. the constitutive

behavior of the rate form :

solid phase and Darcy’s Law are. respectively, expressed in the following

where ( ’ ) = a/Jy and p is the resin pressure in the current configuration. The presence of the term G(4,) approaches [y(y)

in (14) implies that as G(4,)

zero (fiber compaction

= 01. it immediately

comes to a halt).

+ 0. the filtration

flux

For the case of the rigid mandrel

follows that the velocity of the solid will also approach

zero.

Thus, we have shown that at least when the mandrel is rigid, considering the variation the permeability

with 41 leads to a bound on the fiber volume fraction.

on the increase in compaction

can be achieved by incorporating

stiffness with fiber volume fraction. compaction

This can be easily explained

of

Further limitation

the variation

of radial

by observing that the

will be suppressed once it has reached a level where the radial stiffness of the

fiber network is no longer negligible. Substitution direction

of (10) through

(14) into the rate equilibrium

equation

ci,,=0, -z+ dB,,

6””

(1%

.K

leads to the following

in the radial

parabolic partial differential

equation (PDE)

for w,

The value of the function F(+,) will be very small (close to zero) near the outer layer (where fiber compaction has not taken place), while in the inner layers where enough

A. AGAH-TEHRANIand H. TENG

XJSJ

compaction

has occurred it will have a finite value. Because of this, the above equation

an example of a singularly perturbed

3.BOUSDARY The formulation

of the problem

sutlicient boundary conditions

is

PDE.

CONDITIONS

will be completed

by providing

for solving (16). The first boundary

the necessary and

condition

derived will

be that associated with the rate of change of the fluid pressure at the instantaneous radius .yO. This can be accomplished by first substituting resultant equation. and enforcing the boundary condition

outer

(10) into (14). integrating the that at p(.r = x,, ,r) = 0 for all

time. F. The end result will be

Based on the above expression and (16). the material derivative of resin pressure will be

whcrc C/J:’is the hbcr volume fraction in the layer just being wound. Following Lin and Wcstmann (1989). WC Ict the amount cylinder during the winding of each layer bc c/(Vol/unit

Icngth) =

the material

added to the

Zn.r, d.r = r?,,(y) dJ,H,

(I’))

with H as the thickness of each liber network just being wound, and G,, as the speed of the last layer. In considering difTerent possibilities for the variation of GOwith “time” J’, one can recognize two special cases: one in which G”,(y) = constant, and the other in which the angular velocity is held constant. If the amount of resin filtration is small, then changes in the geometry can be neglected and (19) can be integrated to determine the outer radius as a function of time. Indeed, this is the approach adopted during the elastic or viscoelastic analysis of winding (Willett and Poesch. 1988 ; Lin and Westmann, 1989). In the current problem where resin filtration leads to changes in the position of the fiber network, instantaneous

current configuration

eqn (19) will be interpreted

of the fiber network.

The above relation brings out the second important

with respect to

In this case (19) leads to

timr scale in the problem which

is associated with the speed at which the outer layer is increased. We define this time scale to bc r, = R,/w,, with I$‘,,as the dimensional speed of the last layer. The fact that at each increment of time the evolution of the outer radius of the composite (the position of outermost fiber network) is not known u priori indicates that the wet winding problem is in some scnsc an example of a moving boundary problem, and one needs to sequentially integrate the velocity equilibrium equation (16) for each instant of time in ordtr to obtain the relevant information at the end of the winding process. TO accomplish this one needs two boundary conditions on either the velocity or the strain rate of the solid, or a mixed combination of both [recall that (I 6) was 2nd order in the radial direction]. These boundary conditions can be obtained by considering the continuity of

Fiber motion during wet filament winding

traction and velocity across the mandrel/composite cf,,(.K = 1,y) =

5 C,,

2655

interface

Rf-R;

So

R:(l-v,)+R:(i+v,)

with R, as the inner radius of the mandrel and (E,,,,v,) as the elastic constants of the mandrel and utilizing the equilibrium of the last layer keeping the hoop stress constant and equal to the applied tension k,,(x = x,,y)

=

_T,d.r, dy ’

(22)

x0

with T, as the non-dimensional applied tension. (The non-dimensionalization to C&). By utilizing (12) and (18). the boundary condition at x = x, becomes

is with respect

while the boundary condition at -1:= I becomes

-z

’ 5a< c 1’

To dx, = Ebg(y)-

F(4 ) ~-L-B

dy

4J,-wJ8w,

dw,

dt

(24)

with E:, =

R’-R,?,

Et,

C,,,R?(I-v,,)+R,?,(I+v,,)’

The fact that the variation of the outer radius with time is unknown provides an additional difficulty in wet winding analysis which is absent in the analysis of either linearelastic or viscoelastic dry winding. This difficulty is related to the fact that in wet winding satisfaction of the boundary conditions leads to an eigenvalue problem involving the outer radius. This dependence implies that, unlike the linear dry winding problems, superposition of solutions in wet winding corresponding to different outer radii leads to an error in satisfying force equilibrium.

4. NUMERICAL

FORMULATION

One major source of biased errors during integration of (16) in time is the lack of equilibrium at the end of each increment. In order to eliminate this type of error, (15) is modified in the following manner. dcf,,(.r.y + Ay) + ti,,(.~y+Ay)-ti,,&-, _--U_ v+Av) __a.r-_ _~_ .V +-

I

A*V[

LlcT,,(x.v)

d.r

+

a,,(-LI’) - ~,,o CLI’) s

1

= 0,

(25)

with the stress rates at time y+Ay being related to radial velocity at that time through the constitutive relations (12) and (13). The term inside the bracket represents the residual body force at time y+Ay due to the lack of equilibrium at time y. The above equation can also be interpreted as the force equilibrium at time yf Ay in terms of the stress rates at that time and the total stresses at the previous time.

and H. TENG

A. AGAH-TEHRANI

2656

The form of the modified

rate equilibrium

backward method (an implicit algorithm)

equation (25) readily renders itself to Euler

for integration

was then used to replace the spatial derivatives.

in time. A finite difference scheme

Since the solution of the problem requires

the ability to follow the history of winding from beginning (_r = 0) to end (_v = J.(), the size of the finite difference mesh was sequentially

increased as time increased. see Fig. 2. Due

to the change in the geometry which is induced by the resin filtration, in mesh size resulted in non-uniform (Dahlquist

and Bjorck,

the sequential increase

mesh spacing. By applying Richardson

1974) to the midpoint

extrapolation

rule, the first and second order spatial

derivatives of the interior nodes can be obtained as

a

Iz I

=

+emtL,.

A,( ),“+I +fm( )m-,

(26)

r,

a2

1I d.r? s,

=

,?L(

)m+l +h,t

L--I -(h”+h,“-,)(

Ll,

(27)

,”

where

Similar variations

cxprcssions can also bc derived that the function

F(g,)

still’. In order to rcducc the limitation stable integration, equation

for the boundary

can expericncc,

nodes. Due to the large

the governing equation

for IV, will bc

that this stitfncss imposts on the time incrcmcnt for

we proceed in the following

manner.

First, WC intcgratc the evolution

for the fiber volume fraction (I I) bctwccn the time J > y,, and )’ = 1”. The result

will bc

(29)

By utilizing the above relation, one can then determine

the following

spatial derivative of fiber volume fraction

cx2 Y,-o

Y2-Ay

yn

Yml

Fig. 2. Schematic of the finite diffcrcncc mesh.

WY

relation for the

Fiber motion during wet filament winding

The final step in obtaining

2657

the finite difference equation of the original PDE will be to

substitute (30) into (16). and then into (25). By utilizing the midpoint the resultant equation between the time _r = _r. and _v = y.+,

rule, integration

of

leads to

and the superscripts specify the time and the subscripts specify the node numbers. The boundary condition

(23) is replaced by the following

ditTerence equation

with

(7 =

211 +/I” ~~~. I’-__ II” (II” +/I,,

, I?= , ) *

The finite diffcrcnce equation condition

-“,“,“1-1.

?=/, “R

I

of the original

(32) can bc used to obtain

_ n

PDE

the following

(,Ih;h I

_ n

n

)’ I

(31). and that of the boundary

systems of equations

for the nodal

vclocitics at time .I*, +, = J”+AJ~

AW “+’ = B+R#‘+‘. where W”“‘=

[IYz...

(33)

CV,]““.

The coefficients of the tridiagonal matrix A. and the column matrices B and R are given in the Appendix. Based on the above finite difference formulation of the problem, the

A. AGAH-TEHRAW

2658

and

ti.

Ttx;

strategy for the computation is as follows :

(I)

For the first time increment. by assuming linear spatial variation of the velocity of

solid v,. the boundary conditions

(23) and (2-t) are used to determine the nodal values of

)(‘,. (2) During

the subsequent increments.

predictor-corrector (i)

(ii)

a

of:

Using (33) to obtain \V”‘I

(iii)

the nodal velocities are obtained through

method with each pass consisting

Utilizing

=

the boundary condition

Determining

’ [B + Ry” L ‘1.

A

(34)

(74) to obtain the interface velocity ,q”+,.

the nodal velocities from (34). and hence the increments of stress.

pressure and tiber volume fraction. We used the total equilibrium

equation in the radial direction to obtain the following

estimate of the error associated with the numerical calculation

5. DISCUSSION

AND

RESULTS

In this section by assuming specific forms for the functions

F(&)

and C(+,),

we

will

nnalysc the response of the proposed model. These forms arc given as follows

F(cP,) = F,,(~p,-~p:I)~+E”,

with 9: as the: initial mandrel,

E”C:,

constant r’ maximum

G(g,)

= G”

(J )? (P.,

--I

.

(P,

(36, 37)

fiber volume fraction of the layer which is being wound onto the stifl‘ncss of that layer. and G”K” as its permeability. The

as the initial

will be the radial stiffness

is such that F($,)CE

of the fiber network with the

possible volume fraction.

The form for G(c),), see Fig. 3, is chosen based on the expression (1990) who showed that the variation

0.5

0.6

of the transverse permeability

0.8

0.7

dcrivcd by Gebart of the mixture

0.9

$5

Fig. 3. Variation

of the functions G(q5.J and F(rb.1 wvlth fiber volume fraction

with

Fiber motion dunng wet filament wmding Table

I. The constants used in the numerical

Velocity of deposition H‘,, Thickness of the last layer H Applied tension Outside radius of the mandrel R, Hoop stiffness of the fiber C:;, c’:,. C” &.”

0.256 m se’ 0.25 mm IO MPu 0.5 m 222 GPa 0.204 ‘x,O-I’m:pd-Is-’

Initial fiber volume fraction I$:’

6.5

fiber volume fraction is fundamentally The form of the function F(+,)

26.59

solution

different from the variation of the axial permeability.

is chosen so that the radial resistance will initially

remain

almost constant. see Fig. 3. and eventually equals CZr. The above forms have the interesting feature that the drop in the permeability

is faster than the rise in the radial stiffness. This

implies that based on the assumed forms for the functions F(&,) compaction

can be significantly

and G(4,)

even though

reduced due to the decrease in the permeability.

stiffness can still be small. The material constants and the value of the parameters

the radial

associated with the winding

process are given in Table I. In order to rcciuce the amount of computation inrolred, we hare

only considered winding onto a rigid mcmdrel. Initially. see Figs 4-8,

the variation of the radial stiffness with fiber volume fraction was neglected, however the pcrmcability

of the mixture was allowed

to vary with 4,. This

was done so as to investigate the influence of the radial stiffness on fiber compaction. time incrcmcnt for each calculation to signilicant error accumulation 2 prcscnt the maximum

The

was Ar = 0.2 s. Time incrcmcnts greater than this lcad

and eventual instabilitiy

errors in equilibrium

of the result. The results in Table

which happen to occur in the outer mode.

The results in Fig. 4 show the variation

of the libcr volume fraction

through

the

thickness. As expected, this figure indicates that increasing the radial stitTness of the fiber network supprcsscs fiber compaction.

The interesting fcaturc of this figure is that for cases

where the initial resistance of the fiber network is sutlicicntly small so as to allow significant compaction,

there are two boundary-layers

: one close to the mandrel-composite

interface

and the other close to the outer layer. In between these two layers fiber volume fraction is nearly constantt.

Figure 5 presents the details inside the boundary layer close to .Y = I. As

the results indicate,

the boundary

increases. In interpreting

layer becomes more diffused as the radial

this figure, it is important

stiffness

to note that the length of the boundary

layer is of the order of the thickness of an individual composite layer. Thus keeping in mind that fiber diameter is of the order of IO pm, adopting the continuum

approach in the manner

presented in this study should be viewed with caution. Apart from the inner boundary layer which is dependent on the stiffness of the mandrel,

the results presented in Figs 4 and 5

imply that the profile of the fiber volume fraction can be separated into a passive and an active zone of consolidation.

Inside the passive zone the compaction

due to the winding

Table 2. The maximum error for the cases whcrc the radial stillilcss was considered constant Radial stiiTncssC;, 147.5 Pa I.475 kPa 14.75 kPa I.475 MPa 0. I475 GPa

Error IO.56 % 1.78 % 0.217 % 5.02 x IO-’ -4.52x lo-’

t Numerical results indicate a slight linear variation of rhe fiber volume fraction with radial position. The slope of this inner layer increases as the radial resistance of Ihe fiber increases. LIl 29-21-E

A. ACAH-TEHRAZI

2660

and

H. TENG

0.7

2

I

.I ..,.

0.65

/ ____________-_-_____._~~ ** _______-___________.________________ ‘:; 0.6 9s

OS5 4

\

‘---_.-.-_-.--.--

0.5

0

3

.__.__._.___

0.2

0.4

- ._.___

0.6

0.8

I

6 = ( R - R,M R,- R,) Fig. 4. Radial distribution of fibcr volume fraction for the following CXXS: (1) CL’, = 147.5 Pa. (2) C:, = I.475 kPa. (3) C” = 14.75 kPa. (4) C;, = I.475 MPa.

0.X5 0.R

0.65 0.6

:. . L.‘..’ “‘...‘ 1

I

0

0.002

O.MM

O.OC4

5 = ( R - R,M

Ro-

0.008

R,)

Fig. 5. Profile of liber volume frxtion in the boundary layer at the rigid interbce belween the mandrel and the composite for the following cases: (I) C;, = 147.5 Pa. (2) C;, = 1.475 kPa. (3) C;, = IA.75 kPtI. C,, = constant.

I

“‘........,._

‘.

,.,,

H4

.,...

“..O

,,,.,......~

“.‘.’

/

--._

0.11

0.6 w

T. 0.4

\ =._-. 1 -.. \ *--__ \ \ \ \

i

0.2

0

a.24

i

0

“y,

\

_..*

5

Ii

.’f i

\

\\

.-__

0.2

0.4

0.6

0.8

5=(~-q)/tR,-R>

Fig. 6. Radial variation of the non-dimensional hoop stress for the followmy cases: (I) CC = i-l.75 kPa and n = 100 layers. (2) C:, = I.475 MPa and n = 50 Inycrs. (3) Cz, = I.475 MPa and n = IO0 layers. (4) C:. = 0. I475 GPJ and n = 50 layers. (5) Cr. = 0. I475 GPa and n = IO0 layers.

Fiber motion dunng wet filament wmding

Fig. 7. Radial variation of the non-dimensional o,, for the following cases: (I) C;, = 14.75 kPa and n = 100 layers. (1) C” = I.475 MPa and n = SO layers. (3) Cy, = 1.475 MPa and n = 100 layers. (4) CY, = 0. I475 GPa and n = 50 layers. (5) C:: = 0.1475 GPa and n = IOU layers.

tension has reached its saturation

limit. while inside the active zone the compaction is still

taking place. In cases (I) and (2) where the radial resistance of the fiber network assumed to be small. the length of the active consolidating

was

region came out to be seven to

eight times the thickness of each ply. Figure 6 shows the variation of the hoop stress in the composite cylinder. The results indicate that the drop in the hoop stress (loss in tension) incrcascs by increasing the thickness of the cylinder. The loss in tension is more dramatic when the radial stiffness of the fiber is small. The drop in hoop stress can bc explained by observing that each time a layer is added to the cylinder thcrc will bc ;I compressive hoop strain increment and a comprcssivc radial strain

increment. These compressive strain

incrcmcnts

result in comprcssivc hoop and

ritdial stress increments. The litrger the magnitude of itnisotropy compressive the hoop stress increment. Thus.

(the larger A.). the more

ils the degree of anisotropy increases, the loss

of the initial winding tension increases. The results in Fig. 6 also indicate that in the case where the radial stillness

is very small (case I). the hoop stress will bc almost zero in the

interior of the cylinder. Figures 7 and 8, respectively, present the variation of the radial stress and the pressure through

the thickness of the cylinder. The results in general indicate that increasing the

Fig. 8. Radial variation of the resin prcssurc for the following casts: (I) C:, = 14.75 kPa and n = 100 layers. (2) CL = I.475 MPa and n = 50 layers. (3) Cy, = I.475 MPa and n = 100 layers. (4) CZ = O.l47S GPa and n = SO layers. (5) Cy, = 0.1475 GPa and n = 100 layers.

A.

‘66’

AGAH-TEHKANI

and

H. TE,YG

Table 3 lmt~al radial stiffness I.175 kPa I.475 kPu I 475 kPa 1171; kPa

Final

CY, (6:‘)

radial stffncss CF, (4.) I.475 \fPa 11.75 41Pu 147.5 \I Pa I.475 GP2

Jr 5 0.2 0.7 0.2 0.1

0.95

0.6

OS5

-_. __._. _ _._..___.._. _.._.._..- -._ ._.--..-.._

1

0

O.lXX!

.-.- -..-

-..- ..-..-. -..-

---.

-..-..-t..

i

O.CW O.lXX 5 = ( R - R,M R,- R,)

OSBU

0.01

Fig. 9. Prolilc of’ lihcr volume wlthln the inner boundar) Lcycr for the following cats : (I) C’;, = 1.47X kP:t = conamt. (2) 1.47X kP;t < C’,, Q 1.476 MPJ. (3) 1.47X kPa < C,, $ 14.76 MPa. (4) 1.47X kP;i G C’,, G 147.6 MPa. (5) 1.47X kPa < C,, $ 1.476 Cif’;~. The numhcr of Iaycrs n is held consl;Int a1 IOU.

thickness

of the composite Icads to incrcascs in the level of the compressive radial stress

and the resin pressure. Although

the pressure distribution

only shows the existence of the

outer boundary layer, the actual numerical results do indicate both boundary layers. The results also indicate that for a given number of layers, increasing the radial stiffness

leads

to a drop in the resin pressure and an increase in the level of the compressive radial stress. When the radial resistance is sufficiently small that significant fiber compaction can take place, increasing the thickness of the cylinder leads to the expansion of the passive zone of consolidation.

This

can bc seen from Fig. 8 where increasing the number of layers leads to

the enlargement of the region in which pressure is constantt. Figures 9-14 present the results of the numerical simulation W;IS

in which the radial stitrness

allowed to vary with the fiber volume fraction. The details pertaining to the limits

the radial stiffness,

on

the time increment and the maximum error of each calculation is shown

in Table 3. Figure 9 indicates that inclusion

in the analysis of the variation of the radial stiffness

with the fiber volume fraction leads to an increase in the length of the inner boundary layer. Furthermore. as Fig. I4 indicates, considering such an etl’ect causes a non-monotonic dependenceof the resin pressure on the attainable radial stitfnrss. This is such that increasing the maximum radial stiffness from 1.476 MPa to 147.6 MPa will actually increase the pressure inside the passive zone of the consolidation. Aside from these differences, inclusion of the variation of the instantaneous

elasticity

does not alter any of the other characteristics For example, similar

of the fiber network with fiber compaction of the solution

with constant radial stiffness.

to the case of constant radial resistance, the thickness of the active

region of consolidation was found to be seven to eight times the thickness of each fiber bundle. Another important common characteristic between the two classes of solutions is t Note that based on Darcy‘s hence the consolidation stops.

law (14) when the pressure is constant.

the filtration

flux drops to zero and

2663

Fiber motion during wet filament winding

0s 0.x 0.5 - I

7---------“-\ \\,

I

0

0.2

I

0.4

0.6

5 = ( R - RJ(

I

0.8

I

R.- R,)

Fig. IO. Radial distribution of fiber volume fraction for the following cases: (I) C”, = 1.478 kPa = constant. (2) I.478 kPa < C,, < 1.476 MPd. (3) 1.478 kPa 6 C,. Q 14.76 MPd. (4) 1.478 kPu $ C,, < 147.6 Mk’d. (5) 1.478 kPa $ C_ < I.476 GPa. The number of layers is held constant at 100.

0.62

0.6 - ------------

O.SR -

0.56 'p, 0.54 -

1.2.3

0.52 -

0.5

1

0.9

1

0.92

I

0.94 0.96 5 = ( R - R,)/( R,,- R,)

0.98

I

Fig. I I. Profile of fiber volume fraction within the outer boundary layer for the following cases : (I) Cy, = 1.478 kPa = constant. (2) 1.478 kP4 Q C,, 6 I.476 MPa. (3) 1.478 kPa < C,. < 14.76 MPa. (4) 1.478 kPa < C,, < 147.6 MPa. (5) 1.478 kPa < C,, < 1.476 GPa. The number of layers II is held constant at 100.

0.6

$$I i

'Be/?3 0.4

J

4

1

-0.21 0

1 0.2

I 0.4

I 0.6

I 0.8

I

5=('-Ri)/(RdRi) Fig. I2 Radial variation of the hoop stress for the following cases: (I) C”, = I.478 kPa = constant. (2) 1.478 kPa Q C,. Q I.476 MPa. (3) I.478 kPa < C,. < 14.76 MPa. (4) 1.478 kPa < C,. < 147.6 MPa. (5) I.478 kPa ,< C,. ,< 1.476 GPa. The number of layers is held constant at 100.

364

A. AUH-TEHWANI

and

I 0.2

-3s 1 0

H. TENC

I 0.4

I 0.6

5 = (R - R,)/(&,-

1 0.8

I

R,)

Fig. I?. Radial variation of 0,. for the following caes: (I) C;, = 1.~78kpa = c”nstilnt. (2) 1.~78 kPa < c,, < 1.476 i%lPa. (3) 1.474 kPa < C,, < 14.76 MPa. (4) 1.474 kPa < C,, < 147.6 MPa. (5) 1478 kPa g C,, < I.476 GPa. The numhcr of layers is held constant ;II I(H)

6

5

Fq)

x104, 3

0

Fig.

I-I

Radi;ll v;lri:llion

I

0

I

0.2

of the resin pwssurz

I

I

I

0.4 0.6 6- (R - R,)/(%-RJ

1

08

for the following

casts:

(I)

C;, = I.478

kP;I =

conslrrnl. (2) I.478 kl’lr < C,, $ I.476 MPa. (3) I.178 kPa C C,, $ 11.76 MPa. (3) 1.478 kPa $ C,, < 147.6 MPa. (5) I.478 kPa Q C,, < 1.376 GPa. The number of layers is held consun 31 I(H).

the existence stage. This been wound The

of ;I radial

implies onto

in Figs

15-18.

time to the winding observing

does

mandrel composite

within

is assumed interface.

I5 shows

time increases,

fiber

of I~ to I, imply

not have enough

fractions

following process

will

of fl-/fW on the various

Figure

that large ratios

the resin

gradient

the completion continue

time

the cylinder.

to be rigid.

some

One consequence

compaction

winding

the last layer

radius

the fiber

has

consolidation

takes

is the decrease in the tension

interface

there is some tension

process

(consolidation)

can be explained

of the cylinder network

leading

by

increases. to small

due to the fact the

place close

of the delay in consolidation

bear in mind

one must

This

the case under consideration,

speed or decrease in pcrmcability 16. Of course

of the winding

of the filtration

decreases.

that as the outer

to pass through For

characteristics

that as the ratio

see Fig.

mandrel-composite

of the forced

even after

the cylinder.

effect of the variation

is shown

volume

pressure

that the consolidation

to the mandrel--

with increase in winding

drop

in the radial

that due to the consolidation drop in that region.

Another

direction.

close

to the

consequence

Fiber motion during wet fikment winding 0.7

”

0.2

0.4 0.6 5 = ( R - R,X(R,- R)

0.a

Fig. IS. Radiaf vwiation of the fiber volume fraction for diflixent ratios of f,/l,. C,, < 14.76 MPJ and n = 100 layers.

I I.478 kti

d

0.6

0.4

0.2

0 0.2 1 0

I 0.2

I 0.4

, 0.6

, O.ii

I

C=(R-R,)/(R,-R~) Fig. 16. Radial vtlriation of the hoop stress for dilferent ratios of f,/~,. 1.478 kPa C C’,, < 14.76 MPd and n = 100 layers.

(a,/TJ

x IO’

f=(R-R@%_Ri) Fig.

17.Radialwialion ofa,, for different ratios of q/t,. I.478 kPa Q C,, < 14.76 Mpd and n = 1~0 layers.

2666

A. AGAH-TEHRASI

of the delay in consolidation

and

H. TENG

is the increase in the resin pressure and hence the radial stress

as seen in Figs I7 and 18. Figure IY indicates that for small ratios of I,- to I, the active zone ofconsolidation is rcstrictcd to ;I small region close to the outer layer. As this ratio increases the Icngth of the nctivc zone incrcascs and correspondingly dccrcascs. This consolidation

is such that for substantially

the extent of the passive zone

large values of t,/t*

the passive zone o[

altogcthcr disappears. which is a highly undcsirablc fcaturc since

the time

by

the winding stage of the m~~nul’~tcturing process is finished, one hopes that ;I major portion ofconsolitlatiorl

is complctcd. Altcrnativcly

the results prescntcd in f:ig. IX indicate that in

order to incrcasc the length of the passive zone of consolidation,

one can oithcr dccrcasc

the speed of winding or incrcasc tho pcrmcability. The dcpcndcncc of the Icngth of active /.onc of consolidation

on the ratio of r,/fW

indicates that in principle any forced winding stage belongs to one of the following categories. First:

Slow winding, where the consolidation

[Lee and Springer

model;

Lee and Springer

there are two circumstances permeability

(1990)).

where either the permeability

For a given choice of fiber system,

under which one can have this type of winding,

is very large or the speed of winding

three

takes place only in the last layer

is very small. Second:

is very small (dry winding)

either the

Fast winding,

or the speed of winding is very

fast such that resin does not have enough time to filter through the fiber network.

Intermediate

winding,

where

the length

of the active

zone

of consolidation

Third : is several

thicknesses of each fiber bundle wide.

6. CONCLUSION

By utilizing

the rate form of the continuum

winding of circumferentially

reinforced composite

mulation contains two important

characteristics

mixture

theory, fiber motion during wet

cylinders

is studied.

of this manufacturing

The proposed forprocess, namely the

variation of instantaneous permeability and elasticity of the mixture with fiber compaction. In contrast to the existing models, the present formulation does not make any u priori assumption concerning the extent of the consolidating region. However, the model in its present form lacks the generality for analysing fiber motion and associated change in stress components in filament winding processes where the angle of winding changes. For the case of winding onto a rigid mandrel, the numerical results, which are obtained based on a finite difference approximation, indicate that when variation of the permeability with the libcr volume fraction is taken into account there will be two distinct zones of consolidation. One in which consolidation has ceased to take place and the other inside which the fiber bundles are continuing to compact under the influence of the applied tension. Numerical results further indicate that the existence of these two regions does not depend on whether or not the increase in the radial stiffness of fiber network is taken into account.

Fiber motion during wet filament winding

2667

The results further indicate that for realistic choice of the control parameters the length of the outer boundary layer within which there is active consolidation is larger than the thickness of each fiber bundle. Based on the present results, an undesirable feature of this manufacturing process is the loss of tension in the passive zone of consolidation away from the mandrel. However, the results indicate that this drop in tension (hoop stress) decreases as the radial stiffness increases. Thus, if one is able to increase the radial stiffness through the frontal curing of the layer close to the mandrel, one will be able to reduce the drop in tension. The results for the cases when the radial stiffness was allowed to increase from a relatively small value to a large value show that one can achieve substantial resin pressure by the end of the winding process. This is a desirable outcome since it prevents void formation during the curing process. Further improvements on the present analysis can be achieved by developing more effective algorithms for taking into account singularity of the velocity equilibrium equation. This enables one to analyse the winding problem for arbitrary evolution of the elasticity of the network. ArX-no~k~,r!c/cm~nr-The authors are indebted to Professors C. L. Tucker. S. White and Dennis Parsons for valuable discussions and suggestions. The authors would also like to acknowledge helpful suggestions made by the two reviewers The work was sponsored by the National Center for Composite Materials Research at University of Illinois at.Urh;ln;l-Ch;lmpaign through the ONR prime contract NOOOl4-86-K-0799.

REFERENCES Biot. M. A. (1941). Gcncral theory ol’ three dirncnsional consolidation. 1. Appl. Pltvs. 12. IS5--165. Bolotin, V. V.. Vorontsov. A. N. and Murztkhanov. R. Kh. (IYHO). Analysis oi the technological stresses in wound components matlc out ofcompositcs during the whole duration of the fabrication process. M&/r. Polinr. 3. 361 36X. Bowcn. R. M. (lY76). Theory of rnixturcs. In C’rjnrinuurn f’lt~.vics /‘trr/ 111 (Edited by Ccm;d Cringcn). pp. I .l?7. Acalcnuc Press, New York. c’alius. E. I’. ;d Springer. G. S. (IYXS). Modcling the lilamcnt winding process. PrucccJ;nq.~ uf rhc Fi/rh Inrcrncrrbtntrl (;w/i*rcrtcr on C’wtrpo.uirr Mu/rr,criub (Edited by W. C. I larrigan. Jr.. J. Strife and A. K. Dhingra. The Metallurgical Society). pp. 1071 -IOXX. Caliur. E. P. and Springer. G. S. (IYYO). Modeling of filament-wound thin cylinders. fnr. J. SolirLr Srrucrttrrs 26. 271-2’37. Duhlquist, G. and Bjorck. A. (1974). Nunlericul Mrrhouk Prentice-Hall. Englewood Cliffs, NJ. Gebart. 8. R. (1990). Permeability of unidirectional reinforcement for RTM. Swedish Institute for Composites, Report No. 90-006. Gutowski. T. G.. Cai. 2.. Bauer, S., Boucher. D.. Kingery. J. and Winman, S. (1987~). Consolidation experiments for laminate composites. J. Camp. Muter. 21.650-669. Gutowski. T. G., Morigaki. T. and Cai, Z. (1987b). The consolidation of laminate composites. /. Camp. Mufer. 21. 173-188. Kenyon, D. E. (1976). Thermostatics of solid-fluid mixtures. Arch. Rur. Mech. And. 62. 117-129. Lee, S. Y. and Springer, G. S. (1990). Filament winding cylinders : 1. Process model. J. Contp. Murer. 24, 12701298.. Lin. J. Y. and Westmann. R. A. (1989). Viscoelastic winding mechanics. J. Appt. Mech. 56, 821427. Shilbey. A. M. (1982). Filament winding. In Hundbook of Cornpurim (Edited by G. Lubin). pp. 449478. Nostrand-Rheinhold, New York. Willett. M. S. and Poach. W. L. (1988). Dctcrmining the stress distribution in wound reels of magnetic tape using a nonlinear finite-diliercnce approach. J. Apppl.Mech. 55, 365-371.

APPENDIX The elcmcnts of the tridiagonnl

matrix A, and the column vectors B and R are given as follows:

2668

A. AGAH-TEHRW

and

H. TESG

where

The above expresstons are valid for all the nodal pos~tlons except the boundary nodes where: one hds to take account of the boundary conditions. Utilizing (32). the equivalent expressions for the nodes m = :! and m = n are

(A?)