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On deformation of woven fabric-reinforced thermoplastic composites during stamp-forming
On deformation of woven fabric-reinforced thermoplastic composites during stamp-forming Hiroshi Suemasu, Klaus Friedrich and Meng Hou The processability of woven glass fabric-reinforced composite plates by three-dimensional stamp-forming into spherical caps is theoretically considered from a geometrical point of view before and after the deformation process. For doing this, tensile properties of the material were measured under similar heat conditions as in the relevant stamp-forming process. Stretch in the fibre direction was found to be smaller than the maximum elastic extension of the glass fibres. Reduction of the angle between the crossing fibres was quite large when the satin woven fabric composite was pulled in the 45” direction. In theory, the material is assumed to attain maximum stretch along the meridian in the fibre direction on the formed dome. The curve in the flat plate before the deformation process, which will be formed into a parallel of latitude of the dome after stamp-forming, is obtained by an iteration scheme on the condition that the deformed length of the curve is equal to the circumferential length of the curve of constant latitude. A possible deformation of the spherical dome without serious wrinkling is obtained. The flexibility of the satin woven fabric composite material is found to be sufficient for the plate to be formed into a hemisphere, as long as the working condition is well controlled. A possible working condition during the stamp-forming is proposed. Keywords: contraction;
thermoplastic analysis
composites;
woven
fabrics;
stamp-forming;
INTRODUCTION Continuous fibres used in composite materials are usually thought to be inextensible. This means that continuous fibre-reinforced composites may not be ideally suited to a stamp-forming process. However, woven fabric composites are thought to allow some amount of extension owing to both the stretching of undulated fibres in the fibre direction and a fairly large amount of extension in the diagonal direction resulting from deformation of the square units of the woven fibre arrangement. Therefore, such a material can be deformed by a three-dimensional stamp-forming method’v2. However, the amount of deformation is limited by the ultimate extension of the composite. Compression stresses in the hoop direction may become significant during the process, so that wrinkling of fibres is very difficult to avoid’e3. Therefore, it seems to be very instructive, from a fundamental point of view, to estimate the forming limits of such a material from the basic material properties before the actual thermo-forming process. The ultimate extension of the material during the stamping process seems to be one of the most important factors. Robertson et ~1.~ studied the problem under an assumption of no extension in the fibre direction (net assumption). Their results show the feasibility of such a deformation process even for inextensible fabric composite materials. Wurfhorst and Horsting’ used a more generalized numerical method to obtain deformed 0956-7143/94/01
Composites Manufacturing Vol 5 No 1 1994
I0031
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@ 1994
deformation;
Poisson
shapes of the nets covering not only a spherical dome but also various complex-shaped parts. They also showed the large deformability of the net system. The fabric composite is an orthotropic material. As it is not very easy to visualize the forming mechanism, the coordinate system and curves are defined here. The x- and y-coordinates are taken in the fibre directions and the angles 6 and 4 are defined along the curves with constant longitude and latitude (meridian and parallel of latitude), respectively, as shown in Figure 1. The angle 09 refers to the position of the dome edge. Deformation of square units of the fabric composite is thought to be made up of two factors. The one is the angle reduction y between the crossing fibres and the other is stretching E of undulated fibres. Contraction in the hoop direction due to extension of the fabric in the meridian direction is one of the most important factors in the stamp-forming process. The mechanisms of this forming process and the feasibility of this manufacturing method are discussed in more detail during the course of this paper. TENSILE
EXPERIMENT
The tensile properties of a fabric composite material in the fibre directions and at 45” to the fibre directions were determined under a heating condition similar to that encountered during the stamp-forming process. The Butterworth-Heinemann
Ltd
31
fiber angle (s/Z)
- 7
t
45” direction hoop direction circumferential
meridian
J
radial direction
X
Figure
1’
Coordinate
system
of a spherical
dome
a00 160mm
+I
600
(*.*mm
Tenslle DirectIon Warp or Weft
400
-0.0 Figure Figure
2
Sketch
of the experimental
3
Relation
0.2
between
0.4
0.6
Strain
[% ]
load and expansion
0.8
in the fibre direction
apparatus 40
materials tested were 4-H and 8-H satin-woven fabric composites (glass fibres in a polyetherimide (PEI) matrix; fibre volume fraction, V, z 39%). The test apparatus is sketched in Figure 2. Specimens, tabbed at both ends, were pulled in a tensile testing machine at a speed of 5 mm min-‘, with the test portion being heated and kept at 330°C. At least two specimens were tested for each case. The experimental results were consistent. The relations between load and elongation in the fibre directions and at 45” are shown in Figures 3 and 4, respectively. For convenience, strain is defined as:
z
30
x 3
20
aI .Z E
10
Tensile Direction + 45” to Warp or Weft
8
/
0 0
,
,
,
,
,
10
20
30
40
50
60
E = (I - lo)/Zo
(1) where 1, and 1are the length before and after deformation, respectively. The inelastic stretch of the material in the fibre direction under low load was found to be very small and even less than the maximum elastic strain of the glass fibres. The contraction of plate width under initial stress was almost negligible during heating despite the fact that the plate thickness increased. In the 45” direction, the material could be stretched very much at very low stress (co.5 N mm-‘). The material became unstable and showed necking in the gauge section, see Figure 5. The necked portion grew with further stretching. However, total expansion of the whole specimen in this direction was not realized in the experiment owing to fibre pull-out
32
Strain [ % ] Figure
4
Relation
between
load and expansion
in the 45” direction
and, therefore, the rapid increase of load after stretching was no longer observed. The maximum permanent strain cp in the fibre direction was about 0.3%. The maximum angle reduction Y,,,,, measured at the necked portion of the test specimen in the 45” direction was about 55”; that is, the fibre angle changed from 90 to 35”. Owing to the finite width of the fibre bundle, the angle between the crossing fibres does not reach zero.
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as: F(4) = +{(ro + ?a5 ) + (To - 145 ) cos 44)
(2)
where FoOand Fb5”are the positions of A and B, obtained by a consideration of the stretch of the material in the 0” and 45” directions, respectively. In the 0” direction, no reduction of fibre crossing angle is thought to occur. Then, considering the stretch ratio sp in the fibre direction, F,,Ois given as: i’oo= MI/(1 + E&J Figure 5 The 45” specimen before (above) and after (below) tensile test. Very large expansion is realized in the 45” specimen owing to the large angle reduction between the crossing fibres. The maxima and minima are probably due to stick-slip effects within the necked region during the large deformation
ANALYTICAL
DEVELOPMENT
A fabric composite material is macroscopically orthotropic and this effect seems to be very important in the three-dimensional stamp-forming process. The square unit of the woven composite material allows large shear deformation when heated. This means that the material stretches well in the 45” direction, and this stretching is accompanied by a good amount of contraction in its normal direction. (Poisson’s ratio, v x 1 when the angle between the fibres is 90”.) This contraction tends to introduce tension in the hoop direction and plays a very important role in conducting the stamp-forming process without serious wrinkling. It is very difficult to estimate the strain distribution in the hoop direction exactly; a sophisticated numerical analysis is required. An attempt is made here, however, to develop a simple method for estimating the processability of the material from the geometrical point of view. The stress/strain relationships are thought to follow the curves shown in Figure 6. The stress values in the 0 and 45” directions are thought to be very small until the strains reach their critical values, (E&,,~ and (EJECT, respectively, where the curved fibres seem to be stretched out and the angle reduction of crossing fibres is thought to be constrained owing to the finite width of the fibre bundles. The following slope is due to elastic deformation of the fibre which, therefore, is much stiffer than in the region of permanent elongation, so that elastic deformation can be neglected during the stamp-forming process. The material is pulled into the dome during the stamp-forming process. Consider a curve ABC, which will be transferred onto a curve A’B’C’ in the dome by the deformation process, Figure 7. The curve A’B’C’ is a circle and comprises a parallel of latitude. The decrease in length of curve ABC must be realized without compressive stress so that the material be deformed without wrinkling. Reduction in the angle between the crossing fibres makes contraction of the curve possible. The feasibility of stamp-forming is studied by asking whether the line ABC can contract to A’B’C’ owing to the Poisson effect. The dome is considered first. The curve ABC must be obtained. The effect of the accuracy of the curve shape on its contraction is thought to be insignificant as long as the points A, B and C are given precisely and the curve is smooth. (The precise curve may be determined by considering various conditions, such as load history, strain distributions, etc.) Then, the line is assumed here
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(3)
where R8 is the length of the meridian from the pole of the dome to the position 8. It is not easy to estimate the value of Fd,“, because it is influenced by both the elongation in the fibre direction and the reduction in the angle of the crossing fibres. The extensional strains in the fibre directions along the meridian of 4 = 45” can be assumed to reach their maximum during the deformation process. If the material is in tension in the circumferential direction, the shear deformation of the inclined square unit is constrained and further angle reduction of the unit is limited. Thus, it is reasonable to assume that the angle reduction occurs so that the length of the deformed curve equals the length of the parallel of latitude of the dome. In fabric composites, there exists a limit in the decrease of the angle between the crossing fibres owing to the width of the fibre bundles. When it reaches the limit, the angle reduction y is assumed constant and equal to ymax. When the angle reduction y(8) is given along the meridian in the 45” direction, the relation between the lengths of the line elements after and before the deformation process can be given as: (1 + s,)dm
d&a = R de
(4)
An integration of d?450 gives the original position FaSOof the material formed into the spherical dome along the 45” direction: 145” =
6 s
R dtI
(5)
0 (1 + &,)JGG-$j Next, the possible contraction of the curve ABC in Figure 7 is estimated. If shortening of the curve length from AB to A’B’ during the deformation process is
Strain Figure 6 Anticipated stress/strain strain ap of the hot, woven fabric fibre bundles in the weft and warp
&
relation composite directions
and maximum inelastic with equal amounts of (o”, 90”)
33
Curve before Deformation
Process
Edge of Cap to be Formed
Figure 7 Transformation of a curve ABC (in the flat plate before the deformation process) into A’B’C’ (a circle of a parallel of latitude in the spherical dome) due to stamp-forming. The square unit is transformed into a parallelogram
absorbed by the Poisson effect, no compressive stress occurs in the circumferential direction and the stampforming can be done without serious wrinkling. A square element at point P before the deformation process is deformed into a parallelogram stretched by e1 and Ed in the fibre directions and deformed by y. The relation between the line elements dt and dt on the curves AB and A’B’ is given as: dt = f(4) dT f(4) = [(l + Q)’ sin2 u + (1 + &2)2cos2 c( - (1 + cl)(l + e2) sin 2a sin ~1”~
(7)
It is very difficult to estimate the strain distribution exactly along the parallel of latitude. In a fibre direction with a smaller angle to the meridian, the strain e1 may be thought to reach its limit. In the other fibre direction, the distribution of &2is obscure. It may be reasonable to assume that the strain c2 reaches its limit at 4 = 45” and is zero at 4 = 0; i.e., Ep
E2 =
EpQ(dd
0 < c$ < n/4
(10)
where min(s, t) means to take a smaller value between s and t. The distribution of y may not be very correct, but is thought to be sufficiently accurate for the first approximation. The correct distribution may be determined by consideration of the stretch along the fibre direction. The original length of the curve AB is obtained by integrating the following equation: B
n/4
d< =
i=AB=
sA J(b) = J(dr/d+)’
s0
J(4) d4
+ r2
(11) (12)
The possible length s of the curve AB after deformation is given as:
(9)
where g(4) = 2 sin’ 4. Since extension of the material in the fibre direction E, is usually small, the effect of the
34
*
(6)
The distributions of LY,Ed, &2 and y along the curve AB are the values to be determined. From Equation (2), the direction c( is given as: tan tL = F sin C$- (d?/d+) cos 4 (8) F cos 4 + (d?/d@) sin C#J
E, =
assumption of g(4) is not significant. However, the distribution of the angle reduction y on the estimation of deformation distribution is important. The fibre angle reduction is symmetric about the 45” direction and zero at C#J = 0 and ~12. When the angle reduction reaches the maximum value ymax, no further increase in the angle reduction is assumed. Then:
d4 S=
s0
f(4)J(4) W
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If the length s is equal to the length of the arc on the parallel of latitude of the dome, the material fits the dome, so that the process can be thought to be carried out without serious wrinkling. The value r,,(0) is obtained using a Newton method; that is, the calculation process is iterated at each 8 until the condition 1s- (7r/4)R sin 191c E is satisfied. In this procedure, the derivative of s is approximated as:
4.1af -J(4) dv,- s o aYo ds
d4
(14)
where 4i is 1r/4 when y. < ymaxand is the angle where ;’ reaches ymax along the parallel of latitude when The value of y. can be obtained very smoothly 70 ’ YnlSX~ after about seven to eight iterations when y. < ymax. If the solution cannot be obtained after 20 iterations when ;‘o ’ Ymw the process is determined to be impossible to carry out without compressive stress. In this case, y. is very large and the area where y = yrnaxspreads on most of the circumference. The length s almost equals the shortest possible length of the dome, which is obtained by assuming y. = yrnaxalong the edge of the dome. In the flange portion, the deformation is also predicted using a similar method, Figure 8. The original positions of the points at 4 = 0 and 45” on the curve DF which is deformed into a circle D’F’ of radius ri is calculated first:
r. = $
r,=r,+kAr.
(~o)~=(y~)~., , (ro-)k=Irk+R(9,sineo}(l
+EpZ-’
{rI + R(8, - sin 0,)) P
dr
JRsineo (1 + sp),/l + sin y(r)
(16)
The distribution of y(r) is also determined to fit the circumference. The length of the curve is given by Equation (13). A flow chart of the solution process is given in Figure 9. r
t
E’
0
E
F’F
r
I Edge
of the
Figure 8 Transformation due to the stamp-forming
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of curves DE and DF into D’E’ process in the flange
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and D’F’
Figure 9 Flow chart of the analysis. The upper half accounts for the analytical procedure to obtain the deformation distribution in the dome area and lower half is that for the flange portion
The present theory is a simple approximation. Only geometrical conditions about the strains in the 0” and 45” directions and the possibility of contraction of the curve to a circumference of the formed material are considered. The deformation necessary to fit curves to circumferences may not be realized due to the other constraints, such as limited elongation of the material. When the deformation for fitting the material is impossible, the process will have some difficulties. One such constraint is the length of the line element in the fibre direction. The length of the material in the fibre direction normal to the coordinate D’E’ cannot be longer than the length (1 + a,)DE (see Figure 8). A violation of this condition becomes apparent in the flange portion when the elongation of the fibre direction ap is zero. Let us consider the lines DE and DE’ in Figure 8. Movement of the material is constant along the 0“ direction. The length of the formed material D’F’ is shorter than DF. In order to satisfy the condition, the point D must be in tension in the 45” direction. The movement FF’ decreases with distance from the dome in the 45” direction and will be negative. As movement of the material is very small or negative in the 45” direction, the length D’E’ becomes longer than DE, so that the above-mentioned condition cannot be satisfied. The movement Ar,,. of the point E towards the dome must be larger than the following value
35
in order for the line length D’E’ not to exceed (1 + QDE: Ara5i > _f_ [(l - Q. + a fi - J( 1 + #r; + (1 + &Jar, - a21 (17) where ‘u’ is the movement of the plate on the line 4 = 0 in the radial direction; it amounts to (n/2 - l)R when E, = 0. Both the left- and the right-hand sides of Equation (17) are decreasing functions of fO. If the right-hand side becomes less than zero before Ar,sO becomes zero, the condition of Equation (17) will not be activated, because negative movement of the material is impossible during the stamp-forming process. When ArqgOdoes not satisfy Equation (17), the smooth deformation process to the shape obtained by the present theory becomes difficult. For example, if the position of the point of no movement in the 45” direction (Ard5” = 0) is fixed, slip between the crossing fibres becomes necessary.
30 o
0
@ Figure I1 the edges approximate
of
60” ( After
Deformation
90 o )
Comparison of the distributions ofangle reduction along the spherical dome, as obtained from the present analysis and from consideration of an ideal network4
60 1
ANALYTICAL
RESULTS
AND
DISCUSSION
First, a case of no limit in y and no expansion in the fibre directions (E, = 0) is considered. The results obtained from the present theory are compared with those obtained by Robertson et aL4, which is thought to be an exact solution for the inextensible material in the fibre direction. A curve deformed into an edge of a hemisphere is plotted in Figure 10. The result agrees well with that of Ref 4. Distributions of the angle reduction y are plotted along the edge and along the latitude in the 45” direction of the spherical dome in Figures 11 and 12, respectively. The present theory gives only a slightly larger distribution along the latitude than does that of Robertson et ~1.~. The difference can be adjusted easily by changing the distribution shape of y in Equation (10). As these results agree well with the ideal case4, the present assumption is obviously quite reasonable despite the fact
2 Present
Theory
Inextensible
5 L
Figure 10 hemisphere4
36
Net [ 4 ]
1
The shape of a flat cloth which and a fitted curve obtained by present
covers theory
exactly
a
r-
40
20
0 0
0.2
0.4
0.6
0.8
1.0
Position in Spherlcal Cap z/R Comparison of the distribution of the angle reductions Figure 12 along the latitude in the 45” direction, as obtained from the present approximate analysis and from consideration of an ideal network4
that compatibility of the length in the fibre direction normal to the coordinate is not considered. Figure 13 plots the length reduction of the curve which is formed into a circle when a hemispherical dome is produced. The contraction is a maximum at the edge of the dome and the curve length must be shortened by more than 30%. This amount can be absorbed by a Poisson effect accompanied by the stretching when the maximum angle reduction of the material is larger than 55”. The amount of length reduction decreases with distance from the dome in the flange portion. The expansion of the material in the fibre direction cp decreases the contraction only by its order. The expansion of 1% does not contribute well to the processability of the spherical dome. The contraction of the curve is not plotted for the shallower dome H/R = 0.5. Though the shape of the curve is similar to that for a hemisphere, the length reduction condition is not so tough; i.e., the length reduction is only 16% of the original length even at the edge. Elongation of the material in the fibre direction also reduces the contraction by the order of sp. The movements of the material during stamp-forming are plotted in Figures 14 and 15 for both the perfect
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H=l.O
0.8
-
0.6
’
I
0
1 Position
Figure
13
=o
&P
Contraction
I 2
3
after Deformation
of the curve
which
r/R
is formed
into
a circle
(H/R = 1)
0.8 H/R=l.O 0.6
i:::r,_ ’ Fiber Direction
0.4 45” Direction
0.2
0 1 Position Figure 14 Movement based on the condition
2
after Deformation
of the material in both the fibre and 45” directions of Equation (17) on the length of the circumferential
r/R
(H/R = 1). The lines for r > 2.35R i”, the case of the 45” direction fibre
are
direction
0.3 H / R = 0.5 0.2 Fiber Direction
45” Direction
0.1
r/R=
1.56
0 0
1
Position Figure 15 Movement of the material in both the fibre material of sp = 0 is based on the condition of Equation
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after Deformation and 45” directions (17) on the length
2 r/R
(H/R = 0.5). The line for r > 1.57R in case of the 45” direction of the circumferential
fibre
of the
direction
37
hemisphere (H/R = 1) and the shallower sphere (H/R = 0.5). Movement of the material is constant in the flange portion in the 0” direction when the material is inextensible in the fibre direction, whereas it decreases with distance from the dome in the 45” direction. Then, since the length of the line in the fibre direction must not be longer than 1 + E* times the original length (as mentioned above), movement is controlled by the condition expressed in Equation (17). Elongation of the order of 1% may be useful when a shallow dome is to be made, because the constraint of the length in the fibre direction does not become critical. If the elongation of the material is larger than 3%, the constraint is not activated even when the hemisphere is processed. In order to avoid this difficulty, the size of the material square plate should better be smaller than 2 x 2.35R;i.e., about 4.7 times the radius R for the hemisphere when the material is thought to be inextensible in the fibre direction. In the case of the dome with H/R = 0.5, although this condition may not be so strict as in the hemisphere, the size should be smaller than 2 x 1.57R; i.e., 3.14 times the radius R. The distribution of the angle reduction y along the meridian in the 45” direction is plotted against angle 0 in Figure 16 for several values of material deformability Ymaxwhen sp = 0. The value of y increases with angle 8. If the deformability ymax is less than 55”, y reaches its maximum ymax.The depth where y equals ymaxmay not be the maximum depth of the dome, because the region of Y = Ymaxmay spread in order to contract the curve. When ymax= 45”, sufficient contraction of the curve can be accomplished even for the hemispherical dome. There appears to be an area of y = ymanin the region 8 > 80”. The crosses in the figure show the limit where contraction of the material is insufficient to fit the dome even if the portion where y = ymax spreads all along the parallel of latitude. The depth may indicate a limit to the processability of the fabric composite from the geometrical point of view when the value of ymaxis not large. The effect of stretching in the fibre direction is thought to be insignificant as long as this stretching is small, as observed in the efperiment.
60 &P =o
i
-
+i Y F-
Ymax
40 -_-_-_
20
> 55” 450
----_
40”
___.____
35”
-
300
I i
:
,
I
0
0 30
60 0
(dw~e
90 1
Figure 16 Effect of deformability yrnnx on processability of the material. x shows the point where the possible contraction along the longitude becomes insufficient to fit the longitude
38
K .
h
X/R Figure 17 A sketch of the deformed shape of the plate in the flange portion when a hemisphere is made (Ed = 0, ymax > 55”). The curve AP is formed into the edge of the dome. Outside the line DS, the deformation process is dominated by the condition of Equation (17), and deformation without compressive stress in the hoop direction is difficult
Whenymaxis larger than 55” and the deformation is well controlled in the meridian direction in the dome and also in the radial direction in the flange, the fabric composite can be deformed into a hemisphere. When the maximum angle reduction is less than the critical value of 55”, some difficulties appear. However, expansion of the region of the maximum angle reduction ymaXmay make the process possible. When ymaxis less than 45”, it becomes impossible to deform the material into a hemisphere without circumferential wrinkling. The maximum possible depth of the dome becomes smaller with a decrease in deformability of the material, in particular a decrease of ymax. In Figure 17, the deformed shape of the square element in the flange portion is sketched for the material with E, = 0 and ymax> 55”. The shape is obtained only from the condition s = 4r,/rr. In the 0” and 90” directions, the material moves by 0.57 times the radius of the dome (PP’, QQ’ = 0.57R).On the contrary, the material movement becomes smaller with the distance from the dome in the 45” direction owing to strain (AA’ > BB’ > CC’). The strong deformation of the material in the 45” direction is well understood from this figure. The curves plotted with broken lines (r > 2.2R)can be realized only when the material is pulled outwards. This, of course, is impossible to be executed in the stamp-forming process. Restriction of the deformation D’S’ < DS, for example (i.e., that movement of the material in the 45” direction cannot be determined only from the condition to fit the circumferential length when the flexibility of the material in the fibre direction is poor), can be understood well from this figure. The condition begins to cause a problem at point D’ (ra5” - 2.33R)or D @a50- 2.45R),where the distance between the points D’ and S’ is equal to the length DS. The values of positions, however, are not very accurate. They can be given more precisely when the distribution of the shear deformation along the hoop direction is determined accurately by considering the Composites
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relation between the lengths in the fibre directions before and after the deformation process. The position C’ is determined so as to fit the circumference from the bottom of the dome up to the point C’ in the flange. The length C’Q’ can no longer be equal to CQ (C’Q’ > CQ). In the real stamp-forming process, stretching in the 45” direction will be constrained; that is, sufficient contraction in the hoop direction cannot be realized owing to tension in the direction C’Q’. This problem can be relaxed by the existence of elongation in the fibre direction.
form should be cut so that the length from the pole to the pre-material edge in the fibre direction is smaller than the distance from the pole to the point. If the circle passing through the area is complete, compression arises the circumferential direction and some other technical assistance should be introduced to eliminate this effect. 4) A high pressure should be applied at the final stage to smooth out the small wrinkling and thickness fluctuations. ACKNOWLEDGEMENTS
CONCLUSION The present work gives some ideas about processability and the problems encountered during stamp-forming of fabric composite materials. Fabric composites allow very small expansions in the fibre directions and very large stretching capability in the +45” directions, due to the angle reduction between crossing fibres under a heated condition. The large deformability of continuous fibre-reinforced fabric composites (angle reduction between the crossing fibres) indicates the great potential of the stamp-forming process for these materials. A simple method is proposed to estimate the deformation in both dome and flange portions. Despite its rough approximation, the shortening of the curve being formed into the hemisphere edge agrees well. with the ideal case4. The Poisson’s effect of the material is found to play a very important role in the stamp-forming process, where the material must be in tension in both the meridian and the parallel of latitude directions in the dome and in the radial and hoop directions in the flange during the process. The present theory can be applied to the forming of more complex shapes. From these results, several comments on the pressure required for fixing the material in the flange portion during stamp-forming can be deduced. 1) As the material stretches very little in the fibre direction, it must be allowed to slide towards the dome during stamp-forming without strong resistance in the 0” direction. Large friction is not necessary in the fibre direction. 2) In the 45” direction, however, the material must be stretched and thus slide into the dome. Large friction is not necessary in these directions, either. Nevertheless, in order to cause sufficient stretching, the end portions of the material in these directions should be tightly fixed. The optimal positions of fixation are thought to be located near the points of no movement in the present analysis. 3) Movement of the material in the 45” direction is governed by the length in the fibre directions normal to the axis outside this point, so that stretching in the 45” direction is restricted there and sufficient contraction of the length in the hoop direction is impossible to attain. Wrinkling is thought to take place during the process. Therefore, the pre-material
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Professor H. Suemasu gratefully acknowledges the support of the Alexander von Humboldt Foundation, Bonn, Germany, for his stay at the Institute for Composite Materials, Kaiserslautern, Germany. Professor K. Friedrich expresses his thanks to the Deutsche Forschungsgemeinschaft (DFG FR675-7-2) for support of his project on thermoforming of thermoplastic-matrix composites. Further thanks are due to the Fonds der Chemischen Industrie, Frankfurt, for support of his personal research activities in 1994. The authors also acknowledge Professor Debes Bhattacharyya (Auckland, New Zealand) for his very helpful comments on the present paper. REFERENCES Hou, M. ‘Zum Thermoformen und WiderstandsschweiBen von Hochleistungsverbundwerkstoffen mit thermoplastischer Matrix’ IVW Bericht No 93-2 (University of Kaiserslautern, Germany, 1993) Okine, R.K. ‘Analysis of forming parts from advanced thermoplastic sheet materials’ SAMPE J 25 No 3 (1989) pp 9-19 Martin, T.A., Bhnttacharyya, D. and Pipes, R.B. ‘Deformation characteristics and formabilitv of fibre-reinforced thermoplastic sheets’ Composites Manufartuthg 3 No 3 (1992) pp 165-172 Robertson. R.E.. Hsiu. , ES.. Sickafus. E.N. and Yeh. G.S.Y. ‘Fiber rearrangements during the molding of continuous fiber composites. I: Flat cloth to a hemisphere’ Polym Composites 2 No 3 (1981) pp 1266131 Wurfltorst, B. and H&sting, K. ‘Rechnergestiitzte Simulation der Drapierbarkeit von Geweben aus HL-Fasern fir Faserverbundwerkstoffe’ Industrie-Testilien Chemifasern/Textilindustrie 40192 (1990) pp 118-123
AUTHORS H. Suemasu, to whom correspondence should be addressed, is with the Department of Mechanical Engineering, Faculty of Science and Technology, Sophia University, 7-l Kioicho, Chiyodaku, Tokyo 102, Japan. K. Friedrich is with the Institute for Composite Materials, University of Kaiserslautern, Erwin Schroedinger Street 58,675O Kaiserslautern, Germany. M. Hou is with the Department of Mechanical and Mechatronic Engineering, University of Sydney, New South Wales 2006, Australia. (Received 1 July 1994; revised 19 November 1993)
thermoplastic analysis
composites;
woven
fabrics;
stamp-forming;
INTRODUCTION Continuous fibres used in composite materials are usually thought to be inextensible. This means that continuous fibre-reinforced composites may not be ideally suited to a stamp-forming process. However, woven fabric composites are thought to allow some amount of extension owing to both the stretching of undulated fibres in the fibre direction and a fairly large amount of extension in the diagonal direction resulting from deformation of the square units of the woven fibre arrangement. Therefore, such a material can be deformed by a three-dimensional stamp-forming method’v2. However, the amount of deformation is limited by the ultimate extension of the composite. Compression stresses in the hoop direction may become significant during the process, so that wrinkling of fibres is very difficult to avoid’e3. Therefore, it seems to be very instructive, from a fundamental point of view, to estimate the forming limits of such a material from the basic material properties before the actual thermo-forming process. The ultimate extension of the material during the stamping process seems to be one of the most important factors. Robertson et ~1.~ studied the problem under an assumption of no extension in the fibre direction (net assumption). Their results show the feasibility of such a deformation process even for inextensible fabric composite materials. Wurfhorst and Horsting’ used a more generalized numerical method to obtain deformed 0956-7143/94/01
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I0031
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@ 1994
deformation;
Poisson
shapes of the nets covering not only a spherical dome but also various complex-shaped parts. They also showed the large deformability of the net system. The fabric composite is an orthotropic material. As it is not very easy to visualize the forming mechanism, the coordinate system and curves are defined here. The x- and y-coordinates are taken in the fibre directions and the angles 6 and 4 are defined along the curves with constant longitude and latitude (meridian and parallel of latitude), respectively, as shown in Figure 1. The angle 09 refers to the position of the dome edge. Deformation of square units of the fabric composite is thought to be made up of two factors. The one is the angle reduction y between the crossing fibres and the other is stretching E of undulated fibres. Contraction in the hoop direction due to extension of the fabric in the meridian direction is one of the most important factors in the stamp-forming process. The mechanisms of this forming process and the feasibility of this manufacturing method are discussed in more detail during the course of this paper. TENSILE
EXPERIMENT
The tensile properties of a fabric composite material in the fibre directions and at 45” to the fibre directions were determined under a heating condition similar to that encountered during the stamp-forming process. The Butterworth-Heinemann
Ltd
31
fiber angle (s/Z)
- 7
t
45” direction hoop direction circumferential
meridian
J
radial direction
X
Figure
1’
Coordinate
system
of a spherical
dome
a00 160mm
+I
600
(*.*mm
Tenslle DirectIon Warp or Weft
400
-0.0 Figure Figure
2
Sketch
of the experimental
3
Relation
0.2
between
0.4
0.6
Strain
[% ]
load and expansion
0.8
in the fibre direction
apparatus 40
materials tested were 4-H and 8-H satin-woven fabric composites (glass fibres in a polyetherimide (PEI) matrix; fibre volume fraction, V, z 39%). The test apparatus is sketched in Figure 2. Specimens, tabbed at both ends, were pulled in a tensile testing machine at a speed of 5 mm min-‘, with the test portion being heated and kept at 330°C. At least two specimens were tested for each case. The experimental results were consistent. The relations between load and elongation in the fibre directions and at 45” are shown in Figures 3 and 4, respectively. For convenience, strain is defined as:
z
30
x 3
20
aI .Z E
10
Tensile Direction + 45” to Warp or Weft
8
/
0 0
,
,
,
,
,
10
20
30
40
50
60
E = (I - lo)/Zo
(1) where 1, and 1are the length before and after deformation, respectively. The inelastic stretch of the material in the fibre direction under low load was found to be very small and even less than the maximum elastic strain of the glass fibres. The contraction of plate width under initial stress was almost negligible during heating despite the fact that the plate thickness increased. In the 45” direction, the material could be stretched very much at very low stress (co.5 N mm-‘). The material became unstable and showed necking in the gauge section, see Figure 5. The necked portion grew with further stretching. However, total expansion of the whole specimen in this direction was not realized in the experiment owing to fibre pull-out
32
Strain [ % ] Figure
4
Relation
between
load and expansion
in the 45” direction
and, therefore, the rapid increase of load after stretching was no longer observed. The maximum permanent strain cp in the fibre direction was about 0.3%. The maximum angle reduction Y,,,,, measured at the necked portion of the test specimen in the 45” direction was about 55”; that is, the fibre angle changed from 90 to 35”. Owing to the finite width of the fibre bundle, the angle between the crossing fibres does not reach zero.
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as: F(4) = +{(ro + ?a5 ) + (To - 145 ) cos 44)
(2)
where FoOand Fb5”are the positions of A and B, obtained by a consideration of the stretch of the material in the 0” and 45” directions, respectively. In the 0” direction, no reduction of fibre crossing angle is thought to occur. Then, considering the stretch ratio sp in the fibre direction, F,,Ois given as: i’oo= MI/(1 + E&J Figure 5 The 45” specimen before (above) and after (below) tensile test. Very large expansion is realized in the 45” specimen owing to the large angle reduction between the crossing fibres. The maxima and minima are probably due to stick-slip effects within the necked region during the large deformation
ANALYTICAL
DEVELOPMENT
A fabric composite material is macroscopically orthotropic and this effect seems to be very important in the three-dimensional stamp-forming process. The square unit of the woven composite material allows large shear deformation when heated. This means that the material stretches well in the 45” direction, and this stretching is accompanied by a good amount of contraction in its normal direction. (Poisson’s ratio, v x 1 when the angle between the fibres is 90”.) This contraction tends to introduce tension in the hoop direction and plays a very important role in conducting the stamp-forming process without serious wrinkling. It is very difficult to estimate the strain distribution in the hoop direction exactly; a sophisticated numerical analysis is required. An attempt is made here, however, to develop a simple method for estimating the processability of the material from the geometrical point of view. The stress/strain relationships are thought to follow the curves shown in Figure 6. The stress values in the 0 and 45” directions are thought to be very small until the strains reach their critical values, (E&,,~ and (EJECT, respectively, where the curved fibres seem to be stretched out and the angle reduction of crossing fibres is thought to be constrained owing to the finite width of the fibre bundles. The following slope is due to elastic deformation of the fibre which, therefore, is much stiffer than in the region of permanent elongation, so that elastic deformation can be neglected during the stamp-forming process. The material is pulled into the dome during the stamp-forming process. Consider a curve ABC, which will be transferred onto a curve A’B’C’ in the dome by the deformation process, Figure 7. The curve A’B’C’ is a circle and comprises a parallel of latitude. The decrease in length of curve ABC must be realized without compressive stress so that the material be deformed without wrinkling. Reduction in the angle between the crossing fibres makes contraction of the curve possible. The feasibility of stamp-forming is studied by asking whether the line ABC can contract to A’B’C’ owing to the Poisson effect. The dome is considered first. The curve ABC must be obtained. The effect of the accuracy of the curve shape on its contraction is thought to be insignificant as long as the points A, B and C are given precisely and the curve is smooth. (The precise curve may be determined by considering various conditions, such as load history, strain distributions, etc.) Then, the line is assumed here
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(3)
where R8 is the length of the meridian from the pole of the dome to the position 8. It is not easy to estimate the value of Fd,“, because it is influenced by both the elongation in the fibre direction and the reduction in the angle of the crossing fibres. The extensional strains in the fibre directions along the meridian of 4 = 45” can be assumed to reach their maximum during the deformation process. If the material is in tension in the circumferential direction, the shear deformation of the inclined square unit is constrained and further angle reduction of the unit is limited. Thus, it is reasonable to assume that the angle reduction occurs so that the length of the deformed curve equals the length of the parallel of latitude of the dome. In fabric composites, there exists a limit in the decrease of the angle between the crossing fibres owing to the width of the fibre bundles. When it reaches the limit, the angle reduction y is assumed constant and equal to ymax. When the angle reduction y(8) is given along the meridian in the 45” direction, the relation between the lengths of the line elements after and before the deformation process can be given as: (1 + s,)dm
d&a = R de
(4)
An integration of d?450 gives the original position FaSOof the material formed into the spherical dome along the 45” direction: 145” =
6 s
R dtI
(5)
0 (1 + &,)JGG-$j Next, the possible contraction of the curve ABC in Figure 7 is estimated. If shortening of the curve length from AB to A’B’ during the deformation process is
Strain Figure 6 Anticipated stress/strain strain ap of the hot, woven fabric fibre bundles in the weft and warp
&
relation composite directions
and maximum inelastic with equal amounts of (o”, 90”)
33
Curve before Deformation
Process
Edge of Cap to be Formed
Figure 7 Transformation of a curve ABC (in the flat plate before the deformation process) into A’B’C’ (a circle of a parallel of latitude in the spherical dome) due to stamp-forming. The square unit is transformed into a parallelogram
absorbed by the Poisson effect, no compressive stress occurs in the circumferential direction and the stampforming can be done without serious wrinkling. A square element at point P before the deformation process is deformed into a parallelogram stretched by e1 and Ed in the fibre directions and deformed by y. The relation between the line elements dt and dt on the curves AB and A’B’ is given as: dt = f(4) dT f(4) = [(l + Q)’ sin2 u + (1 + &2)2cos2 c( - (1 + cl)(l + e2) sin 2a sin ~1”~
(7)
It is very difficult to estimate the strain distribution exactly along the parallel of latitude. In a fibre direction with a smaller angle to the meridian, the strain e1 may be thought to reach its limit. In the other fibre direction, the distribution of &2is obscure. It may be reasonable to assume that the strain c2 reaches its limit at 4 = 45” and is zero at 4 = 0; i.e., Ep
E2 =
EpQ(dd
0 < c$ < n/4
(10)
where min(s, t) means to take a smaller value between s and t. The distribution of y may not be very correct, but is thought to be sufficiently accurate for the first approximation. The correct distribution may be determined by consideration of the stretch along the fibre direction. The original length of the curve AB is obtained by integrating the following equation: B
n/4
d< =
i=AB=
sA J(b) = J(dr/d+)’
s0
J(4) d4
+ r2
(11) (12)
The possible length s of the curve AB after deformation is given as:
(9)
where g(4) = 2 sin’ 4. Since extension of the material in the fibre direction E, is usually small, the effect of the
34
*
(6)
The distributions of LY,Ed, &2 and y along the curve AB are the values to be determined. From Equation (2), the direction c( is given as: tan tL = F sin C$- (d?/d+) cos 4 (8) F cos 4 + (d?/d@) sin C#J
E, =
assumption of g(4) is not significant. However, the distribution of the angle reduction y on the estimation of deformation distribution is important. The fibre angle reduction is symmetric about the 45” direction and zero at C#J = 0 and ~12. When the angle reduction reaches the maximum value ymax, no further increase in the angle reduction is assumed. Then:
d4 S=
s0
f(4)J(4) W
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If the length s is equal to the length of the arc on the parallel of latitude of the dome, the material fits the dome, so that the process can be thought to be carried out without serious wrinkling. The value r,,(0) is obtained using a Newton method; that is, the calculation process is iterated at each 8 until the condition 1s- (7r/4)R sin 191c E is satisfied. In this procedure, the derivative of s is approximated as:
4.1af -J(4) dv,- s o aYo ds
d4
(14)
where 4i is 1r/4 when y. < ymaxand is the angle where ;’ reaches ymax along the parallel of latitude when The value of y. can be obtained very smoothly 70 ’ YnlSX~ after about seven to eight iterations when y. < ymax. If the solution cannot be obtained after 20 iterations when ;‘o ’ Ymw the process is determined to be impossible to carry out without compressive stress. In this case, y. is very large and the area where y = yrnaxspreads on most of the circumference. The length s almost equals the shortest possible length of the dome, which is obtained by assuming y. = yrnaxalong the edge of the dome. In the flange portion, the deformation is also predicted using a similar method, Figure 8. The original positions of the points at 4 = 0 and 45” on the curve DF which is deformed into a circle D’F’ of radius ri is calculated first:
r. = $
r,=r,+kAr.
(~o)~=(y~)~., , (ro-)k=Irk+R(9,sineo}(l
+EpZ-’
{rI + R(8, - sin 0,)) P
dr
JRsineo (1 + sp),/l + sin y(r)
(16)
The distribution of y(r) is also determined to fit the circumference. The length of the curve is given by Equation (13). A flow chart of the solution process is given in Figure 9. r
t
E’
0
E
F’F
r
I Edge
of the
Figure 8 Transformation due to the stamp-forming
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of curves DE and DF into D’E’ process in the flange
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and D’F’
Figure 9 Flow chart of the analysis. The upper half accounts for the analytical procedure to obtain the deformation distribution in the dome area and lower half is that for the flange portion
The present theory is a simple approximation. Only geometrical conditions about the strains in the 0” and 45” directions and the possibility of contraction of the curve to a circumference of the formed material are considered. The deformation necessary to fit curves to circumferences may not be realized due to the other constraints, such as limited elongation of the material. When the deformation for fitting the material is impossible, the process will have some difficulties. One such constraint is the length of the line element in the fibre direction. The length of the material in the fibre direction normal to the coordinate D’E’ cannot be longer than the length (1 + a,)DE (see Figure 8). A violation of this condition becomes apparent in the flange portion when the elongation of the fibre direction ap is zero. Let us consider the lines DE and DE’ in Figure 8. Movement of the material is constant along the 0“ direction. The length of the formed material D’F’ is shorter than DF. In order to satisfy the condition, the point D must be in tension in the 45” direction. The movement FF’ decreases with distance from the dome in the 45” direction and will be negative. As movement of the material is very small or negative in the 45” direction, the length D’E’ becomes longer than DE, so that the above-mentioned condition cannot be satisfied. The movement Ar,,. of the point E towards the dome must be larger than the following value
35
in order for the line length D’E’ not to exceed (1 + QDE: Ara5i > _f_ [(l - Q. + a fi - J( 1 + #r; + (1 + &Jar, - a21 (17) where ‘u’ is the movement of the plate on the line 4 = 0 in the radial direction; it amounts to (n/2 - l)R when E, = 0. Both the left- and the right-hand sides of Equation (17) are decreasing functions of fO. If the right-hand side becomes less than zero before Ar,sO becomes zero, the condition of Equation (17) will not be activated, because negative movement of the material is impossible during the stamp-forming process. When ArqgOdoes not satisfy Equation (17), the smooth deformation process to the shape obtained by the present theory becomes difficult. For example, if the position of the point of no movement in the 45” direction (Ard5” = 0) is fixed, slip between the crossing fibres becomes necessary.
30 o
0
@ Figure I1 the edges approximate
of
60” ( After
Deformation
90 o )
Comparison of the distributions ofangle reduction along the spherical dome, as obtained from the present analysis and from consideration of an ideal network4
60 1
ANALYTICAL
RESULTS
AND
DISCUSSION
First, a case of no limit in y and no expansion in the fibre directions (E, = 0) is considered. The results obtained from the present theory are compared with those obtained by Robertson et aL4, which is thought to be an exact solution for the inextensible material in the fibre direction. A curve deformed into an edge of a hemisphere is plotted in Figure 10. The result agrees well with that of Ref 4. Distributions of the angle reduction y are plotted along the edge and along the latitude in the 45” direction of the spherical dome in Figures 11 and 12, respectively. The present theory gives only a slightly larger distribution along the latitude than does that of Robertson et ~1.~. The difference can be adjusted easily by changing the distribution shape of y in Equation (10). As these results agree well with the ideal case4, the present assumption is obviously quite reasonable despite the fact
2 Present
Theory
Inextensible
5 L
Figure 10 hemisphere4
36
Net [ 4 ]
1
The shape of a flat cloth which and a fitted curve obtained by present
covers theory
exactly
a
r-
40
20
0 0
0.2
0.4
0.6
0.8
1.0
Position in Spherlcal Cap z/R Comparison of the distribution of the angle reductions Figure 12 along the latitude in the 45” direction, as obtained from the present approximate analysis and from consideration of an ideal network4
that compatibility of the length in the fibre direction normal to the coordinate is not considered. Figure 13 plots the length reduction of the curve which is formed into a circle when a hemispherical dome is produced. The contraction is a maximum at the edge of the dome and the curve length must be shortened by more than 30%. This amount can be absorbed by a Poisson effect accompanied by the stretching when the maximum angle reduction of the material is larger than 55”. The amount of length reduction decreases with distance from the dome in the flange portion. The expansion of the material in the fibre direction cp decreases the contraction only by its order. The expansion of 1% does not contribute well to the processability of the spherical dome. The contraction of the curve is not plotted for the shallower dome H/R = 0.5. Though the shape of the curve is similar to that for a hemisphere, the length reduction condition is not so tough; i.e., the length reduction is only 16% of the original length even at the edge. Elongation of the material in the fibre direction also reduces the contraction by the order of sp. The movements of the material during stamp-forming are plotted in Figures 14 and 15 for both the perfect
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H=l.O
0.8
-
0.6
’
I
0
1 Position
Figure
13
=o
&P
Contraction
I 2
3
after Deformation
of the curve
which
r/R
is formed
into
a circle
(H/R = 1)
0.8 H/R=l.O 0.6
i:::r,_ ’ Fiber Direction
0.4 45” Direction
0.2
0 1 Position Figure 14 Movement based on the condition
2
after Deformation
of the material in both the fibre and 45” directions of Equation (17) on the length of the circumferential
r/R
(H/R = 1). The lines for r > 2.35R i”, the case of the 45” direction fibre
are
direction
0.3 H / R = 0.5 0.2 Fiber Direction
45” Direction
0.1
r/R=
1.56
0 0
1
Position Figure 15 Movement of the material in both the fibre material of sp = 0 is based on the condition of Equation
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after Deformation and 45” directions (17) on the length
2 r/R
(H/R = 0.5). The line for r > 1.57R in case of the 45” direction of the circumferential
fibre
of the
direction
37
hemisphere (H/R = 1) and the shallower sphere (H/R = 0.5). Movement of the material is constant in the flange portion in the 0” direction when the material is inextensible in the fibre direction, whereas it decreases with distance from the dome in the 45” direction. Then, since the length of the line in the fibre direction must not be longer than 1 + E* times the original length (as mentioned above), movement is controlled by the condition expressed in Equation (17). Elongation of the order of 1% may be useful when a shallow dome is to be made, because the constraint of the length in the fibre direction does not become critical. If the elongation of the material is larger than 3%, the constraint is not activated even when the hemisphere is processed. In order to avoid this difficulty, the size of the material square plate should better be smaller than 2 x 2.35R;i.e., about 4.7 times the radius R for the hemisphere when the material is thought to be inextensible in the fibre direction. In the case of the dome with H/R = 0.5, although this condition may not be so strict as in the hemisphere, the size should be smaller than 2 x 1.57R; i.e., 3.14 times the radius R. The distribution of the angle reduction y along the meridian in the 45” direction is plotted against angle 0 in Figure 16 for several values of material deformability Ymaxwhen sp = 0. The value of y increases with angle 8. If the deformability ymax is less than 55”, y reaches its maximum ymax.The depth where y equals ymaxmay not be the maximum depth of the dome, because the region of Y = Ymaxmay spread in order to contract the curve. When ymax= 45”, sufficient contraction of the curve can be accomplished even for the hemispherical dome. There appears to be an area of y = ymanin the region 8 > 80”. The crosses in the figure show the limit where contraction of the material is insufficient to fit the dome even if the portion where y = ymax spreads all along the parallel of latitude. The depth may indicate a limit to the processability of the fabric composite from the geometrical point of view when the value of ymaxis not large. The effect of stretching in the fibre direction is thought to be insignificant as long as this stretching is small, as observed in the efperiment.
60 &P =o
i
-
+i Y F-
Ymax
40 -_-_-_
20
> 55” 450
----_
40”
___.____
35”
-
300
I i
:
,
I
0
0 30
60 0
(dw~e
90 1
Figure 16 Effect of deformability yrnnx on processability of the material. x shows the point where the possible contraction along the longitude becomes insufficient to fit the longitude
38
K .
h
X/R Figure 17 A sketch of the deformed shape of the plate in the flange portion when a hemisphere is made (Ed = 0, ymax > 55”). The curve AP is formed into the edge of the dome. Outside the line DS, the deformation process is dominated by the condition of Equation (17), and deformation without compressive stress in the hoop direction is difficult
Whenymaxis larger than 55” and the deformation is well controlled in the meridian direction in the dome and also in the radial direction in the flange, the fabric composite can be deformed into a hemisphere. When the maximum angle reduction is less than the critical value of 55”, some difficulties appear. However, expansion of the region of the maximum angle reduction ymaXmay make the process possible. When ymaxis less than 45”, it becomes impossible to deform the material into a hemisphere without circumferential wrinkling. The maximum possible depth of the dome becomes smaller with a decrease in deformability of the material, in particular a decrease of ymax. In Figure 17, the deformed shape of the square element in the flange portion is sketched for the material with E, = 0 and ymax> 55”. The shape is obtained only from the condition s = 4r,/rr. In the 0” and 90” directions, the material moves by 0.57 times the radius of the dome (PP’, QQ’ = 0.57R).On the contrary, the material movement becomes smaller with the distance from the dome in the 45” direction owing to strain (AA’ > BB’ > CC’). The strong deformation of the material in the 45” direction is well understood from this figure. The curves plotted with broken lines (r > 2.2R)can be realized only when the material is pulled outwards. This, of course, is impossible to be executed in the stamp-forming process. Restriction of the deformation D’S’ < DS, for example (i.e., that movement of the material in the 45” direction cannot be determined only from the condition to fit the circumferential length when the flexibility of the material in the fibre direction is poor), can be understood well from this figure. The condition begins to cause a problem at point D’ (ra5” - 2.33R)or D @a50- 2.45R),where the distance between the points D’ and S’ is equal to the length DS. The values of positions, however, are not very accurate. They can be given more precisely when the distribution of the shear deformation along the hoop direction is determined accurately by considering the Composites
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relation between the lengths in the fibre directions before and after the deformation process. The position C’ is determined so as to fit the circumference from the bottom of the dome up to the point C’ in the flange. The length C’Q’ can no longer be equal to CQ (C’Q’ > CQ). In the real stamp-forming process, stretching in the 45” direction will be constrained; that is, sufficient contraction in the hoop direction cannot be realized owing to tension in the direction C’Q’. This problem can be relaxed by the existence of elongation in the fibre direction.
form should be cut so that the length from the pole to the pre-material edge in the fibre direction is smaller than the distance from the pole to the point. If the circle passing through the area is complete, compression arises the circumferential direction and some other technical assistance should be introduced to eliminate this effect. 4) A high pressure should be applied at the final stage to smooth out the small wrinkling and thickness fluctuations. ACKNOWLEDGEMENTS
CONCLUSION The present work gives some ideas about processability and the problems encountered during stamp-forming of fabric composite materials. Fabric composites allow very small expansions in the fibre directions and very large stretching capability in the +45” directions, due to the angle reduction between crossing fibres under a heated condition. The large deformability of continuous fibre-reinforced fabric composites (angle reduction between the crossing fibres) indicates the great potential of the stamp-forming process for these materials. A simple method is proposed to estimate the deformation in both dome and flange portions. Despite its rough approximation, the shortening of the curve being formed into the hemisphere edge agrees well. with the ideal case4. The Poisson’s effect of the material is found to play a very important role in the stamp-forming process, where the material must be in tension in both the meridian and the parallel of latitude directions in the dome and in the radial and hoop directions in the flange during the process. The present theory can be applied to the forming of more complex shapes. From these results, several comments on the pressure required for fixing the material in the flange portion during stamp-forming can be deduced. 1) As the material stretches very little in the fibre direction, it must be allowed to slide towards the dome during stamp-forming without strong resistance in the 0” direction. Large friction is not necessary in the fibre direction. 2) In the 45” direction, however, the material must be stretched and thus slide into the dome. Large friction is not necessary in these directions, either. Nevertheless, in order to cause sufficient stretching, the end portions of the material in these directions should be tightly fixed. The optimal positions of fixation are thought to be located near the points of no movement in the present analysis. 3) Movement of the material in the 45” direction is governed by the length in the fibre directions normal to the axis outside this point, so that stretching in the 45” direction is restricted there and sufficient contraction of the length in the hoop direction is impossible to attain. Wrinkling is thought to take place during the process. Therefore, the pre-material
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Professor H. Suemasu gratefully acknowledges the support of the Alexander von Humboldt Foundation, Bonn, Germany, for his stay at the Institute for Composite Materials, Kaiserslautern, Germany. Professor K. Friedrich expresses his thanks to the Deutsche Forschungsgemeinschaft (DFG FR675-7-2) for support of his project on thermoforming of thermoplastic-matrix composites. Further thanks are due to the Fonds der Chemischen Industrie, Frankfurt, for support of his personal research activities in 1994. The authors also acknowledge Professor Debes Bhattacharyya (Auckland, New Zealand) for his very helpful comments on the present paper. REFERENCES Hou, M. ‘Zum Thermoformen und WiderstandsschweiBen von Hochleistungsverbundwerkstoffen mit thermoplastischer Matrix’ IVW Bericht No 93-2 (University of Kaiserslautern, Germany, 1993) Okine, R.K. ‘Analysis of forming parts from advanced thermoplastic sheet materials’ SAMPE J 25 No 3 (1989) pp 9-19 Martin, T.A., Bhnttacharyya, D. and Pipes, R.B. ‘Deformation characteristics and formabilitv of fibre-reinforced thermoplastic sheets’ Composites Manufartuthg 3 No 3 (1992) pp 165-172 Robertson. R.E.. Hsiu. , ES.. Sickafus. E.N. and Yeh. G.S.Y. ‘Fiber rearrangements during the molding of continuous fiber composites. I: Flat cloth to a hemisphere’ Polym Composites 2 No 3 (1981) pp 1266131 Wurfltorst, B. and H&sting, K. ‘Rechnergestiitzte Simulation der Drapierbarkeit von Geweben aus HL-Fasern fir Faserverbundwerkstoffe’ Industrie-Testilien Chemifasern/Textilindustrie 40192 (1990) pp 118-123
AUTHORS H. Suemasu, to whom correspondence should be addressed, is with the Department of Mechanical Engineering, Faculty of Science and Technology, Sophia University, 7-l Kioicho, Chiyodaku, Tokyo 102, Japan. K. Friedrich is with the Institute for Composite Materials, University of Kaiserslautern, Erwin Schroedinger Street 58,675O Kaiserslautern, Germany. M. Hou is with the Department of Mechanical and Mechatronic Engineering, University of Sydney, New South Wales 2006, Australia. (Received 1 July 1994; revised 19 November 1993)