Composite Sheet Forming edited by D. Bhattacharyya 9 Elsevier Science B. V. All rights reserved.
Chapter 9
Bending of Continuous FibreReinforced Thermoplastic Sheets T.A. M A R T I N , S.J. M A N D E R , * R.J. D Y K E S and D. B H A T T A C H A R Y Y A Composites Research Group, Department of Mechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand
Contents Abstract 371 9.1. Introduction 372 9.2. Development of an idealised viscous bending model 374 9.2.1. Constraint conditions and kinematics 374 9.2.2. Stress in a constrained material 376 9.2.3. Constitutive equation 377 9.2.4. Stress equilibrium 377 9.2.5. Kinematic model for a veebend 378 9.2.6. Admissible stress fields 379 9.3. Experimental procedures 380 9.4. Results and discussion 382 9.5. Modified constant shear rate tests 392 9.5.1. Determination of transverse shear behaviour from veebending 393 9.5.2. Transverse shear viscosity tests 395 9.6. Conclusions 399 Acknowledgements 399 References 400
Abstract This chapter is primarily concerned with the rheological behaviour of continuous fibrereinforced thermoplastic ( C F R T ) materials in shear. The analysis presented in this chapter centres a r o u n d a novel piece of testing equipment which establishes veebending as a means of determining both the longitudinal and transverse shear viscosities of such materials. In the analysis presented here, no distinction is drawn between the constituents of the composite. Instead an idealised c o n t i n u u m model subject to the kinematic constraints of incompressibility and fibre inextensibility has been adopted. The first part of the chapter deals with various concepts related to the deformation and related flow mechanisms which p r e d o m i n a t e in C F R T materials. This is followed by the development of an idealised material model for an *Currently at McKinsey & Company, Sydney, NSW, Australia. 371
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incompressible viscous fluid reinforced with a single family of inextensible fibres. The analytical model leads to a straightforward interpretation of the effects of forming speed and geometry on the bending stresses expected in a real sheet during forming. An experimental programme is then outlined which details the forming rates and temperatures over which the veebending experiments are conducted. The results of these tests are discussed in two parts. Firstly, the quality of the samples is assessed with regards to fibre instability and the springback/forward phenomenon. The second part of the discussion centres around the interpretation of the material's longitudinal shear behaviour. A further modification to the bending mechanism is then introduced which allows the tests to be carried out at constant shear rates. This is then followed by the development of a method for predicting both the longitudinal and transverse shear viscosities of CFRTs. The experimentally obtained viscosity ratios from these final tests are compared to a number of alternative models which relate the longitudinal and transverse viscosities to the fibre volume fraction and the viscosity of the matrix. 9.1. Introduction
The formability of a continuous fibrereinforced thermoplastic is governed by four basis flow mechanisms: resin percolation, transverse fibre flow, interply shear and intraply shear [1]. These mechanisms play important roles in different forming operations. Resin percolation and transverse squeeze flow are normally associated with consolidation, welding and compression moulding processes, as they enable gaps to be closed or "healed," thereby ensuring a good bond between adjacent layers [2]. Whereas, shaping processes which induce single curvature and double curvature require interply and intraply shear deformations. Cogswell and Leach [3] have suggested that various flow processes are commonly present in combination according to the hierarchy shown in fig. 9.1. This chapter investigates outofplane bending of CFRT sheets. Since the fibres resist inplane deformations along their lengths [4], a singlecurvature operation necessitates interply shear. Consequently, it might be misleading to refer to this type of deformation as bending in the classical sense. The usual assumptions, that plane sections remain plane and straight lines remain straight are not valid for reinforced thermoplastics with high loadings of continuous fibres. The entire deformation is accommodated by shear. Nevertheless, it seems appropriate to use the word bending in the current context, since it correctly implies the introduction of curvature into a sheet. In the light of the mode of deformation it also seems logical to use a bend test to investigate the interply shear properties of CFRT laminates. The following text examines the subject of forming CFRT composites into veebends at elevated temperatures in order to observe the deformation mechanisms in a relatively simple singleaxis bend. The three main objectives are: (a) to observe the quality of 90 ~ Plytron veebends formed under isothermal conditions in a novel bending device, (b) to measure the experimental loads required to form the strips, and (c) to compare these results with an idealised beam bending model for an incompressible viscous fluid reinforced with a single family of inextensible fibres.
Bending of continuousfibrereinforced thermoplastic sheets DEFORMATION MODE
Consolidation
Matched Die
Ff~
373
REQUIRED FLOW MECHANISM
Resin Percolation.
plus
~///'////~
Transverse Flow
plus
Single Curvature
Interply Shear
plus Interply Rotation
o
%
[ O174174174] 174174174174174 ~ oOO174174174 ~ *,414 IOOOOO6OOOl ~!ooooooooo] IOOOOO6O'6ol lOOOOOOOOoI
t00oooooooi IOOOOOO•OOl
Double Curvature plus Intraply Shear
Fig. 9.1. Hierarchy of deformation processes.
The novel veebending test method outlined in this chapter allows the unidirectional laminates to be bent into shape without compressing them between contacting surfaces and constrains them to a known geometry. The experiments lead to a precise study of the fibre placement within the bend region, the fibre wrinkling instability, the springforward phenomenon, and the longitudinal and transverse shear viscosities as functions of the forming speed and the forming temperature. Two main methods of analysis have been used to study the interply shear properties of CFRT laminates" (a) oscillatory shear experiments [5] with a conventional rotating disc rheometer and (b) ply pullout tests [6], as a variation on the standard sliding rheometer test. Oscillatory shear experiments are useful for determining the viscoelastic properties of CFRT laminates for small amplitudes of deformation. However, the results of this linear approach are not really applicable to real forming operations with very large deformations and high strain rates. On the other hand, ply pullout tests tend to cause a gross redistribution of the fibres during testing which greatly influences the behaviour of the laminates. Even for a Newtonian fluid matrix, the experimentally determined shear viscosities obtained from this test method can be unreliable [7]. Several researchers have manufactured veebends using both cold and hot matched faced dies in order to understand the practical problems with forming such sections [810]. In Tam's paper [9], a model for bending a laminate of elastic layers separated by thin viscous fluid layers is developed; however, no attempt is made to establish an analytical load/displacement relationship based on a theoretically derived strip deflection curve. One way to simplify the analysis of CFRT
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materials is to assume an idealised behaviour by imposing suitable kinematic constraints on a continuum model. Some authors have recently published theoretical solutions for bending fibrereinforced elastic, plastic and linearly viscoelastic cantilever beams subjected to two kinematic constraints: fibre inextensibility and incompressibility [1113]. Rogers and O'Neill [14] deal specifically with a viscous fluid model using these constraints. An idealised material model for an incompressible viscous fluid reinforced with inextensible fibres is utilised in this chapter. In addition, a plane strain constraint is imposed. A purely kinematic approach is taken to generate the forming solution, which is made statically admissible, by satisfying the boundary conditions. The analytical model yields a straightforward interpretation of the effects of forming speed and geometry on the bending stresses in a real sheet during forming. It also provides an excellent basis for further work with nonlinear viscoelastic material models. 9.2. Development of an idealised viscous bending model Thermoplastic polymers are known to exhibit viscous behaviour when formed at temperatures within or above their melting ranges. The matrix material in a CFRT composite sheet may therefore be idealised as an incompressible Newtonian fluid in its molten state. In addition, the fibres can be treated as thin homogeneously distributed inextensible cords, since they severely limit the deformation along their lengths. In the current section we consider plane strain bending of an initially fiat plate with uniaxial fibre reinforcement in its plane. The fibres lie perpendicular to the bend axis when the sheet is subjected to veebending, as shown in fig. 9.2. Spencer [15] has derived many of the mathematical expressions presented in the following theory. 9.2.1. Constraint conditions and kinematics
The first two considerations regarding this idealised material concern the kinematic constraints imposed on any deformation by the assumptions of incompressibility and
!
I[
Fig. 9.2. Veebending of an incompressible inextensible beam.
I
Bending of continuous fibrereinforced thermoplastic sheets
375
fibre inextensibility. In the following analysis, capital letters indicate vector quantities referred to the undeformed configuration, whereas small letters indicate vector quantities referred to the deformed state. In a Cartesian reference frame the incompressibility constraint may be expressed mathematically as Ovi Oxi
=dgiO
(9.1)
where x and v refer to the deformed coordinate vector and the velocity vector respectively, and d O9is the rate of deformation tensor defined by 1 "(OUi nt OVj~ dij  2 ~OXj OXi,]
(9.2)
In a continuum containing a family of fibres, each fibre has its own path, which may be characterised by a field of unit vectors. In the current configuration these may be represented by a unit vector field a(XR, t). The trajectories of a represent the fibres themselves and the components of a are denoted by ai. During a deformation the fibres will be convected with the continuum and the same particles will lie on a given fibre at any time. Using this definition the fibre inextensibility condition may be written as OVg __ aiajdi j a~aJ~x j
(9.3)
_ 0
The only other constraint imposed on the deformation is that of plane strain. At this point it is necessary to consider a series of nlines, which represent orthogonal trajectories to the alines. These lines are not material curves in general, because particles lying on a normal line before deformation will not necessarily lie on the same normal line after deformation. Pipkin and Rogers [16] have derived the governing equations for plane strain deformations of incompressible, inextensible materials. In the present analysis a plate of uniform thickness is considered, in which the fibres all lie in parallel surfaces in the plane of deformation. Two significant kinematic results follow from the analysis of Pipkin and Rogers: (a) If the alines are initially parallel, the nlines are initially straight and must remain straight throughout any plane strain deformation. Thus, the fibres remain in parallel surfaces. (b) The normal distance between any two adjacent fibres must be constant at all points along that pair. Therefore, the thickness of the sheet cannot change during plane strain deformation. These two requirements permit only simple shear deformations along the fibres and the amount of shear is conveniently expressed by the change in angle between two adjacent fibres, y. Consider fig. 9.3, in which an initially flat plate is deformed by shear. Rigid body rotations and translations are ignored. In the (Xl, x2) plane, the a vector represents the deformed fibres as a family of acurves and the nvector represents a family of curves normal to the acurves. These two families have the vector components a = (cos r sin r 0),
n = (  sin r cos r O)
(9.4)
T.A. Martin et al.
376
X2
a
Xl
(a)
~
xI
(b)
Fig. 9.3. (a) Undeformed element. (b) Element after deformation.
where 4~ represents the angle between the tangent fibre direction and the X1 axis. When the fibres are embedded at one end, or a line of symmetry exists along which there is no shear deformation, the shear angle, y, may be expressed simply as y = 4~
(9.5)
and the shear rate is given by ~' = q~
(9.6)
9.2.2. Stress in a constrained material The next important step in this analysis involves the introduction of a stress tensor which divides the stress into two distinct parts. The total stress in a constrained material can be thought of as the sum of a reaction stress, rij, and an extra stress, SO.. aij ~ rij % S O"  p(Sij  aiaj) + Taiaj + Sij
(9.7)
where S 0. satisfies the constraints aiajSij   0 and ninjS #. = O. In other words, S/j involves no normal stress component on surface elements normal to the adirection or the ndirection. The reaction stress does no work in a deformation and the reactions p and T arise as a result of the incompressibility and fibre inextensibility constraints respectively. T is the total tension on elements normal to the fibre direction and p represents the total pressure on elements normal to the ndirection. These scalar terms must be determined by solving the equilibrium equations. The deviatoric stress tensor, SO., needs to be specified by an appropriate constitutive relationship. If the material has reflectional symmetry in the x3 plane and the deformation is homogeneous, under plane strain conditions the only nonzero components of a/j are frO"  P(SiJ  aiaj) "q Taiaj q S(ainj + ajni) + S33kikj
(9.8)
Bending of continuous fibrereinforced thermoplastic sheets
377
where k is the vector normal to the plane of deformation. A constitutive relationship is required to define $33 and S. 9.2.3. Constitutive equation In the present analysis, the deformation of an incompressible Newtonian fluid reinforced with a single family of inextensible fibres in the (Xl, x2) plane is considered. According to Rogers [17], the constitutive relationship for a viscous fluid subjected to these two constraints is given by Sij = 2lzTd O + 2(/zL /zv) (aiakdkj + ajakdki )
(9.9)
where r/L and r/7 are the respective viscosities of the continuum along and transverse to the fibres. In a plane strain deformation, v3 = d33 = 0. Therefore, $33 = 0 and cr33 =  p . Hence, a stress must be applied normal to the plane of deformation to maintain the plane strain condition. Using eqs. (9.8) and (9.9) it may be shown that [18] S
ainjcro.  ainjSij  21zLainjd(i  l l ~ L f /  lZL(b
(9.10)
where S is the shear stress associated with simple shear along the fibre direction. For a viscous fluid model, S depends only on the shear rate. In this particular example the only material property which can be determined in the constitutive equation is the longitudinal shear viscosity,/zL. By including fibre layers in the laminate which are not aligned with the longitudinal direction, the transverse shear viscosity may also be determined as will be shown later [19]. 9.2.4. Stress equilibrium By considering the equilibrium of an elementary cross section of the deformed plate, shown in fig. 9.4, it is a simple matter to deduce that 8T*
S* = 0
(9.11)
OS* T* +   ~  roPo
(9.12)
when the only boundary traction is the pressure exerted on the inner surface. The force resultants, S* and T *, are the resultant shear and tensile forces per unit length in the k direction acting across a normal line 4~ constant. h
S*ISd ~ o
h and
T*ITd o
(9.13)
o
Using eqs. (9.10)(9.12), the shear stress and the fibre tension can be determined throughout the entire sheet, when the deformed geometry is prescribed as a continuous function of time.
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~176176176176176176176176 o~176176 Oo i
ro(~)
:
S* .~
Fig. 9.4. Quasipolar coordinate system.
9.2.5. Kinematic model for a veebend
Martin et al. [20] obtained a solution for threepoint bending of an ideal viscous beam using the theory just outlined. However, the solution did not compare well with the experimentally observed behaviour of Plytron laminates. A possible explanation for this is that the elastic flexural rigidity of the fibre bundles caused them to deform by flexure, so that the deformed geometry did not match the theory. The selfweight of the samples was also ignored in the model. In the analysis presented here the deformation of the strip is constrained by using a specially designed jig, which allows the true shear response of the material to be measured. The kinematics and dynamics of this solution are detailed as follows. The following kinematic solution is proposed for the deformation of a flat plate subjected to the novel method of veebending shown in fig. 9.5. Because the bending is symmetrical about the radius bar, a geometric construction for only one half of the deformed strip is illustrated. The specimen sits on a pair of rectangular platens which are hinged at the centre of the radius bar. This mechanism causes the sheet to wrap around the radius bar and shear along its length. The platens are supported on a pair of pivot wheels and the central hinge pin is connected to a load cell. With progressive deformation, fan regions grow beneath the radius bar and at the free end, while the region between the radius bar and the free end remains straight. The halfstrip, shown in fig. 9.5, has been divided into three main sections: [ABEF] represents the fan region beneath the radius bar, [CC'D] represents the developing fan region at the free end and [BC'DE] represents the straight section between the two fan regions. Using the geometry in fig. 9.5 the angular rotation rate of the platens, 4~, can be readily related to the punch velocity, w.
Bending of continuousfibrereinforced thermoplastic sheets
L '" 0
379
~i
~X~
W h
...........
"1,t
B~ A~
RS FiE Fig. 9.5. Veebend deformation model.
cos2~
)
q~ 4v L  (Rs + b)sin 4~
(9.14)
The kinematic model leads to the result that the shear rate in sections [ABEF] and [CC'D] is zero. Consequently, these fan regions move downwards like rigid bodies and S* is zero there. In region [BC'DE] the strip remains straight and the shear strain rate, ), is equal to the rate of rotation of the strip, ~, defined in (9.14). The shear rate varies with time, when the punch speed remains constant. 9.2.6. Admissible stress fields
The foregoing kinematically admissible solution leads to an admissible stress field solution, which admits stress discontinuities across alines and nlines in order to satisfy the boundary conditions. A finite tensile force is carried by the lower surface and a finite compressive force is carried by the upper surface; however, the surface fibre layers have an infinitesimal thickness. Consequently, the upper and lower surfaces of the sheet carry infinite stresses. The tensile and compressive forces increase from zero at points D and C' in fig. 9.5 up to their maximum values at points E and B and remain constant in region [ABEF]. Such a result is characteristic for solutions involving inextensible fibres [16]. In reality, most of the stress is carried near the surface of the sheet during forming. This result is supported by Tam and Gutowski's research [9] on the forming of thermoplastic composite beams. In addition, region
380
T.A. Martin et al.
[ABEF] carries a hydrostatic pressure, which leads to transverse fibre spreading, if the sides of the sheet are not constrained during deformation. In the experimental study a radius bar collar was used to enforce the plane strain condition. Figure 9.6 shows the variation in the resultant shear force as a function of the arc length along an aline in the sheet. Clearly, the resultant shear force is discontinuous across nlines: BE and C'D. This is possible, provided point loads are applied on the surface at points B and D with force magnitudes equal to the jump in shear stress across each line of discontinuity. The shear force resultant acting on the sheet in region [BC'DE] can be obtained from eqs. (9.13) and (9.14). h
S*J" tZL i/ d o 
COS2 t~ tZ L h 'iv L  ( R ~ + b ) s i n
) 4~
(9.15)
0
The net downward load on the radius bar can be calculated by considering the equilibrium of the veebending mechanism in fig. 9.7. When the sample is bent to an angle 4~, the moment and force equilibrium equations enable the forming load per unit width, P, to be expressed in terms of the longitudinal shear viscosity of the material. p = 21ZLh fv(ls  ( R r "l h ) f l p ) c o s 4 ~b
(9.16)
(L  (Rr + h + tp + Rs)sin t~) 2
where ls is the halflength of the sheet. This concludes the development of the theoretical model.
9.3. Experimentalprocedures A series of veebend experiments were carried out with Plytron [0]8 preconsolidated laminates using the mechanism previously described. Plytron is a polypropylene/glass composite with a nominal 35% fibre volume fraction. The laminates were cut into sheets 140mm long and 40mm wide. The tests were carried out under S*
AF
BE
C'D
CD
Fig. 9.6. Resultant shear force along the length of the beam.
~:
Bending of continuousfibrereinforced thermoplastic sheets
381
P
L .....
,,
,N
I
0
Q+
RS
Q
Fig. 9.7. Force equilibrium of the veebending mechanism.
isothermal test conditions over a range of forming speeds and f o r m i n g temperatures, outlined in table 9.1. A schematic d i a g r a m of the experimental setup is shown in fig. 9.8. D u r i n g testing the forming speed was kept c o n s t a n t and the gross load was measured using a 50N load cell in an I n s t r o n 1185 testing machine. Later the tare load of the platens and the weight of the specimen were subtracted f r o m the gross load to obtain the experimental forming load. After being f o r m e d into 90 ~ bends, the
TABLE 9.1 An outline of the test parameters Crosshead speed
140~
150~
160~
170~
180~
50 mm/min 100 mm/min 200 mm/min 500 mm/min
v' r v' r
r v' r r
r r r r
r r r r
r r r v'
382
T.A. Mart& et al.
50 N Load Cell
Load Cell Amplifier ~
\
t
/z/z////////////
I
71o, Controller

' & c o n v e r s i o n A to
load D board
Fig. 9.8. Schematic representation of the experimental setup.
specimens were allowed to cool to 60~ before being removed from the test rig. At this point the final angle of each bend was measured and some samples were prepared for microscopic investigation under a scanning electron microscope. 9.4. Results and discussion
In this type of bending operation two factors are of fundamental interest: (a) the quality of the final part and (b) the shear viscosity characterising the behaviour of the material during forming. The part quality will be considered first and the effects of temperature and shear rate on the material response will be discussed later. One of the most notable features about folding unidirectional Plytron sheets into veebends is the routine shape taken up by the strips as a result of the deformation. Figure 9.9 illustrates a number of typical sections after complete forming and solidification. It is clear from these samples that the deformation has been accommodated by shear and that the strip width has remained constant. The specimens exhibit a smooth shiny surface in the bend region, where they have been compressed against the radius bar during forming. One problem which can undermine the structural integrity of CFRT components is the migration of the reinforcing fibres towards the inner radius of a bend during forming. Cogswell [1] has discussed how the mobility of the resin affects this fibre migration. A matrix with a low viscosity can lead to resinrich and resinstarved regions, as demonstrated by Martin et al. [21] when forming Plytron sheets at temperatures above 180~ whereas a matrix with a high viscosity can cause fibre wrinkling. In this study no evidence of fibre migration was observed. A micrograph of a
Bending of continuousfibrereinforced thermoplastic sheets
Fig. 9.9. Veebend sections at various forming temperatures: (i) 140~ speed = 50 mm/min.
(ii) 160~
383
(iii) 180~
forming
typical specimen in fig. 9.10 shows how the fibres are evenly distributed through the sheet thickness. This lack of fibre movement can be attributed to the novel bending mechanism used to fold the specimen. Another material defect which can cause concern in structural applications is the presence of fibre buckles. This phenomenon occurs in regions where the fibres are subjected to compressive loading causing instability. In the detailed theoretical analysis presented by Martin et al. [19], the model predicts a maximum compressive stress at the inside of the bend radius where the sample meets the radius bar. This stress stays at its maximum value in the fan region [ABEF], which is where fibre wrinkling is most likely to occur. On all samples formed at 150~ and above there was no evidence of fibre wrinkling. However, the samples formed at 140~ did exhibit fibre wrinkling beneath the radius bar which continued 25 mm into the straight section of the sample (see fig. 9.11). Furthermore, the wrinkle amplitudes were greater in the samples formed at 50 mm/min as opposed to 500 mm/min. This may be explained by considering the postbuckling behaviour of the fibres. Once the fibres reached their critical buckling load they wrinkled. The slower crosshead speeds allowed more time for these defects to grow as the sheet continued to deform. The preceding results indicate that fibre wrinkling is more likely to occur in a matrix resin with a higher viscosity. This finding is in agreement with Cogswell's comments, but contradicts research by Martin et al. [20] on deep drawing hemispherical shells. Martin's results demonstrated that at forming temperatures greater than 180~ the fibres in Plytron sheets wrinkled in the plane of the sheet during deformation, whereas below 180~ the sheets buckled outofplane. The key difference in this
384
T.A. M a r t i n et al.
Fig. 9.10. Micrograph of the bend radius region in a typical veebend. Temperature = 180~ speed = 50 mm/min.
forming
Fig. 9.11. Veebend section showing fibre wrinkling.
study is that the sheet itself was not subjected to an inplane compressive load, so gross buckling did not occur. Consider the micrograph shown in fig. 9.12. At forming temperatures greater than or equal to 150~ the deformed samples exhibit interply slip. The degree of shear deformation is indicated by the angle of slip between the
Bending of continuous fibrereinforced thermoplastic sheets
385
Fig. 9.12. Micrograph of free end showing interply and intraply deformation. Temperature = 150~ forming speed  500 mm/min.
fibre layers at the free end. This behaviour is consistent with the continuum theory presented earlier in this chapter. In contrast fig. 9.13 shows a sample formed at 140~ which exhibits predominant slip in the resinrich layers between the lamina, characterised by distinct steps at the free end. Deformation of this type leads to severe fibre wrinkling and is probably associated with the lack of shear within each prepreg layer. The low forming temperature and forming speed allowed the partial recrystallisation of the material before it was completely moulded; thereby inhibiting a uniform shear deformation across the thickness of the sheet. The fibre wrinkling occurs in the surface layer underneath the radius bar, as predicted by the theory. These findings indicate that fibre wrinkling occurs when the shear deformation is restricted in the laminate. Another interesting feature of the micrographs in fig. 9.12 and fig. 9.13 is the lack of curvature at the free end of the sheet. There is no fan region there as predicted by the theoretical model. This is understandable, since the elastic flexural stiffness of the fibre bundles is several orders of magnitude greater than the shear viscosity of the matrix. In reality the flexural rigidity of the fibres counters the formation of a fan region at the free end, so that the fibres completely restraighten during the relaxation phase of the forming process. This type of response could be accounted for by including an elastic bending stiffness term in the constitutive equation, but it would greatly complicate the kinematic solution of the bending problem, while hardly affecting the outcome. It is presumed that the violation of the theoretical model in the small region near the end of the sheet does not represent a gross error.
386
T.A. M a r t & et al.
Fig. 9.13. Micrographof a free end showingmostlyinterply shear. Temperature = 140~ forming speed  500 mm/min. A final point to consider regarding the quality of the formed parts is the included angle of the bends after solidification. A welldocumented phenomenon occurring during the processing of CFRT sheets is the springforward effect. Upon cooling from its melt temperature a moulded thermoplastic composite laminate decreases its included angle in the final part. This effect is primarily due to the anisotropic thermal contraction of the material. Zahlan and O'Neill [22] have investigated this phenomenon and derived a simple theoretical expression for the change in angle of an Lsection with differential inplane to thickness contraction. A0  (c~R  or0)0
(9.17)
where an and c~0 are the respective radial and circumferential coefficients of thermal expansion for a linear thermoelastic material. 0 is the mould angle and A0 is the difference between the forming temperature and the ambient temperature. Their experimental results agree favourably with eq. (9.17). Hou et al. [10] have demonstrated the effect of the forming speed on the magnitude of the springforward in a series of matched die stamp forming tests. Their results indicate a marked increase in the degree of springforward as the forming rate is increased. However, this anomaly has been attributed to fibre migration and squeeze flow as a result of contact between the dies. Table 9.2 shows the effect of the forming speed and the forming temperature on the degree of springforward in Plytron veebends in the current study. There appears to be little correlation between the forming parameters and the springforward. Considering the temperature range over which the samples were deformed (140~ T~< 180~ and the linear relationship between A0 and AT in (9.17), these
Bending of continuous fibrereinforced thermoplastic sheets TABLE
387
9.2
Experimental springforward for 90 ~ Plytron veebends Crosshead speed
140~
50 m m / m i n
7~
5.5 ~
6.5 ~
6.5 ~
200 mm/min
7.5 ~
6.5 ~
500 m m / m i n
6.5 ~
Average
6.8 ~
100
mm/min
150~
160~
170~
180~
5.5 ~
5~
5.5 ~
5.5 ~
5.5 ~
5.5 ~
5~
5~
5.5 ~
6~
5.5 ~
5.5 ~
5.5 ~
6.1 ~
5.4 ~
5.3 ~
5.5 ~
results appear to contradict Zahlan's theory, but it should be remembered that PP does not behave like a linear thermoelastic material when cooling from its melt state. Polypropylene remains molten until cooling to 125~ when rapid recrystallisation occurs. At this point the polymer dramatically decreases its volume causing the springforward effect. Such a result would be expected for any crystalline or semicrystalline thermoplastic composite and is supported by the observations made while the experimental samples were being cooled to room temperature. There was no observable springforward effect until the sample temperature reached 120130~ whereupon the bends rapidly changed their shape by a few degrees. Samples which were cooled rapidly from their forming temperature also developed a considerable amount of curvature across the width of the strip, due to thermal gradients through the sheet thickness. The experiments suggest that moderate controlled cooling leads to the best quality parts. Figure 9.14 shows the first set of experimental results for the loads required to form Plytron sheets at various temperatures and a constant crosshead speed. In this 
~
140~
0
150~
5 
0 160~
4
  O   180~
w
Z .~
,,,
170~
O
~3
1
0...
I
I
., I
t
0
5
10
15
20
......
I
I
25
30
Time (sec) F i g . 9.14.
Graph of forming load versus time for Plytron veebends. Forming speed
=
500
mm/min.
388
T.A. M a r t i n et al.
case the load is plotted against time so that the relaxation effects in the material may be seen. It is not surprising that the highest peak load of ~6 N occurs at the lowest forming temperature, while the lowest peak load of ~ I N occurs at the highest forming temperature. In all cases the forming loads are small. This demonstrates the ease with which thermoplastic composites may be moulded when molten. The load profile can be described by three distinct stages: (b) a rapid rise in the forming load as the sheet starts to deform, (c) a quasisteadystate forming load which decreases as forming proceeds, (d) a rapid decay in the forming load at the cessation of forming. The first stage corresponds to the transient load response at the initiation of flow. This lasts for a very short time before the load reaches its maximum value and thereafter slowly declines. The gradual decrease in load after reaching a peak value corresponds to a decrease in the shear rate with the increasing punch depth. A rapid drop in the load follows at the completion of forming, as would be expected for a viscous fluid. This is followed by a gradual relaxation in the load which demonstrates some viscoelasticity in the material. The load tends towards zero at a very slow rate. A note of caution is given here regarding the interpretation of the results obtained from this type of test. Given the viscoelastic nature of the response, it is unwise to draw conclusions regarding the elongational behaviour from the shear behaviour. Lodge [23] has demonstrated how the mode of deformation substantially affects the theoretical viscosity for the simplest rubberlike liquid constitutive equation. In steady shear flow the rubberlike liquid exhibits a viscosity which is independent of the shear rate. In steady elongational flow it has an indefinitely increasing viscosity. Therefore, shear type experiments are not sufficient on their own to universally validate a constitutive model. The results make it tempting to apply linear timedependent material models to characterise the observed viscoelastic behaviour, but the large strains and strain rates encountered during forming make this a pointless curvefitting exercise. Instead, a generalised nonlinear viscoelastic model is needed to account for the finite strains and high strain rates, if mathematical sense is to be made of the rheological behaviour exhibited here. The instantaneous drop in the load at the cessation of forming shown in fig. 9.14 is in contrast to the results obtained from a series of threepoint bend tests performed by Martin et al. [19]. Under similar test conditions, the Plytron sheets showed a much smoother load decay at the completion of forming, due to the elastic recovery of the curved fibres between the load supports. It is interesting to note the effect of forming speed and temperature on the load profile in fig. 9.15. When the platen angle reaches 45 ~ the forming stops and thereafter the load curves are plotted against time. At a forming temperature of 140~ there is a marked change in the shape of the load curve for the two different forming speeds. This effect is not evident in the specimens formed at 180~ This result clearly shows the effect of recrystallisation on the material behaviour as the temperature is lowered. At 140~ and 50 mm/min as the material recrystallises, the matrix behaves more like a viscoelastic solid than a viscoelastic liquid. Under these conditions the
Bending of continuous fibrereinforced thermoplastic sheets
389
9  0   ! 80~ 50 ham/min 0 180"(2 500 mm/min 140~ 50 mm/min X
140~ 500 mm/min
~4
.~3
o~ 0o
I
[
15~
30~
[t,.o~
"'
t, = 4 0 sl
45 ~
Fig. 9.15. Graph of load versus platen angle for Plytron veebends.
constitutive response of the material becomes more dependent on the shear magnitude than the shear rate. The relaxation response is also altered by the change. Using the aforementioned experimental load profiles and eq. (9.16), the apparent longitudinal shear viscosity, /zL, was calculated as a function of temperature and forming speed. During forming the sheets did not experience a steady shear flow, since the shear rate was not constant; therefore, the apparent shear viscosity is illustrated with the shear rate in fig. 9.16 as a function of the platen angle. The actual variation in strain rate over the duration of the forming period is quite 9000
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20
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I
35
40
Platen Angle ~ (degrees) []
50mm/min
•
100mm/min
0
200mm/min
O
500mm/min
Strain r a t e
Fig. 9.16. Longitudinal shear viscosity and shear rate versus platen angle,f.
390
T . A . M a r t i n et al.
small at around 0.02 rad/s. The profile of/ZL is seen to rise and fall in a similar manner to the shear rate. It is also evident from this figure that the samples exhibit a shear thinning phenomenon: a decrease in the apparent viscosity with increasing shear rate. Clearly the material response is nonNewtonian. In the particular case of simple shear flow of an incompressible viscous fluid, the only nonzero strain rate invariant can be expressed as a function of the shear rate [24]. Given the nature of the results, the shear stress may be expressed as a function of the shear rate using the "powerlaw" relationship described by r = m~ n
(9.18)
where r is the shear stress and m is the consistency index. When n < 1 the material is said to be shear thinning and when n > 1 the material is said to be shear thickening. A Newtonian response is obtained by setting n  1, so that r is a linear function of the shear rate. The apparent (Newtonian) viscosity of the material,/ZL is defined by "~ I~ L   
= my
(9.19)
n1
This model has been commonly applied to molten polymer solutions. Using the forming load curves in the range 15 ~ < 4~ < 30 ~ an average /ZL was calculated for each average shear rate. The averaged data is shown on a loglog plot in fig. 9.17 for sheets formed at 180~ Figure 9.17 represents a fairly good fit between the powerlaw model and the experimental data. A linear regression analysis was performed to determine the m and n coefficients for the Plytron samples at various test temperatures. The numerical results from the linear regression are given in table 9.3 along with the residual squared error of the best fit. The n values increase with increasing temperature in accordance with the general trend for polymer melts. A typical n value for PP in the temperature range 200230~ is 0.30.4. This compares favourably with the result for the sample formed at 180~ On the other hand, the consistency index shows a wide variation with increasing temperature. This finding
~ 10000
~o o,, t~ O O
\ N~
xx,
>
<
1000 0.01
0.1
Log Shear Strain Rate dr
1
(tad/s)
Fig. 9.17. Loglog plot of/z L versus shear rate for Plytron samples.
391
Bending of continuous fibrereinforced thermoplastic sheets
TABLE 9.3 Powerlaw parameters for the longitudinal shear viscosity of Plytron laminates Temperature (~
m (Pa sn)
n (dimensionless)
R2
140 150 160 170 180
165 510 395 285 445
0.335 0.172 0.198 0.144 0.353
0.9993 0.9697 0.995 0.9897 0.9904
is contrary to the rapid decrease which would be expected for a polymer. Further experimental work is needed to verify this result. A further comparison is made between the powerlaw model and the experimental results in fig. 9.18. By determining the instantaneous apparent shear viscosity at each platen angle, based on the current shear rate and the powerlaw coefficients from table 9.3, the theoretical load was calculated using eq. (9.16). The theoretical load curve compares well with the measured load profile. This result indicates that a powerlaw relationship can be used with confidence to predict the shear stresses in C F R T sheets at various strain rates. Using table 9.3 and eq. (9.16), the apparent longitudinal shear viscosity of the unidirectional Plytron sheets was calculated in the temperature range 140~ T ~< 180~ The results showed a variation in /ZL from 5,00055,000 P a s depending on the forming rate. The shear viscosity increased with decreasing forming temperature, as expected. At 140~ and 50 mm/min the shear viscosity reached its highest value, which was far greater than most of the calculated values at higher forming temperatures and faster forming speeds. The increased viscosity associated with
Theoretical load Experimental
z
~ 2 t~ et0
O
z
Ol 0
i ................ 5
I 10
I 15
i 20
Time (sec) Fig. 9.18. Theoretical and experimental forming loads versus time. Temperature = 170~ forming speed = 200 mm/min.
392
T.A. M a r t & et al.
recrystallisation in crystalline and semicrystalline CFRTs should be considered in general sheet forming processes. 9.5. Modified constant shear rate tests
This part of the chapter introduces a slight modification to the veebending mechanism detailed earlier as well as establishing the method as a means of determining the transverse shear behaviour of CFRTs. The veebending mechanism described earlier demonstrated itself to be very useful in terms of constraining the deformation of the strip and allowing the longitudinal shear response of the material to be isolated and studied. Unfortunately, one of the drawbacks of the setup was that the rate of angular rotation of the platens and consequently the shear rate of the sample, y, were not constant. While the actual variation in strain rate in the sample was very small compared with the strain magnitude, it would seem preferable to test at constant rates of shear. An effective way of doing this is to replace the circular pivot wheels, upon which the platens sit, with a support profile which manipulates the rate of angular rotation of the platens to be constant and proportional to the speed of the punch. This modification, illustrated in fig. 9.19, has no effect other than X2
L
,, 0
.
.
.
.
.
X~
j
~1
i ]
W
p B N
AN
Fig. 9.19. Kinematic model of the modified veebending mechanism.
Bending of continuous fibrereinforced thermoplastic sheets
393
to alter the relationship between the speed of the punch and the rate of rotation of the platens which after some manipulation reduces simply to ~i,_ s  L 4' cos,/,
(9.20)
where ep is the orthogonal distance between the centre of the radius bar and the point of contact between the platen and the support, ep and cos 4~vary with time but the ratio of the two remains constant and equal to L for all 4~.
9.5.1. Determination of transverse shear behaviour from veebending The transverse shear viscosity of CFRTs is often very difficult to determine using methods such as oscillatory shear and plypullout tests. An alternative technique for gaining some insight into the transverse shear response of such materials is to subject a laminated strip, which possesses both longitudinal and transverse layers, to the same type of singleaxis bending operation as described earlier. The introduction of transverse layers, in which the fibres are aligned with the x3 axis, poses somewhat of a dilemma as they are not subject to the same kinematic restrictions which govern the plane strain deformation of the longitudinal layers. In fact, the only kinematic effect these layers have on the strip is to enforce the plane strain condition already assumed. It therefore becomes necessary to make a further assumption regarding the deformation of the transverse layers so that useful solutions may be obtained. It is assumed that the thickness of the transverse layers in a laminated strip remains constant during any plane strain deformation. In other words, they exhibit the same kinematic behaviour as the longitudinal layers. By way of an example, consider the shear deformation of an initially flat laminated plate consisting of three layers, as shown in fig. 9.20. The outer two (longitudinal) layers possess fibres which lie in the plane of deformation while the reinforcement in the central (transverse) layer is aligned in the direction of the x3 axis. In this example, the unit vector b represents the shearing direction such that ab = 1 in the longitudinal
x~
X2
b
iiiiiliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii] iiiiii!!i!!ii!ii v
Xl
(a) Fig. 9.20. (a) Undeformed plate. (b) Deformed plate.
v
(b)
Xl
394
T.A. Mart& et al.
layers, and ab = 0 in the transverse layer. Again a represents the fibre direction in the material and the unit vector n is defined to be orthogonal to both the shearing direction and the x3 axis. The shear strain and strain rate of the entire plate can then be specified as before using eqs. (9.5) and (9.6). By once again applying the same constitutive relationship (eq. (9.10)) as that adopted earlier leads to the result: S L  binjo'ij  binjSij = 21ZLbinjd(i  t X L Y   tXL~
(9.21)
S T  binjo~i = binjSij = 21ZTbinjdij   ~ T Y   ~ T ~
(9.22)
where S L and S T a r e the shear stresses associated with the shear deformation in the longitudinal and transverse layers respectively. According to eqs. (9.21) and (9.22) the shear stress through a laminated strip, possessing both longitudinal and transverse layers, varies discontinuously at the boundary surfaces and interfacial surfaces between the differently orientated layers. Initially it would appear as though equilibrium between such layers is therefore unachievable. However, there exists the possibility that a sheet of fibres can support a finite force and hence an infinite stress across the layers. In this analysis it is necessary to accommodate discontinuities in S at the outer surfaces of the strip and also at the interfacial surfaces. We may achieve this by using step and delta functions that allow T to take infinite values in the fibres in the material adjacent to the boundary surfaces, and in the fibres adjacent to, and on either side of, interfacial surfaces. The effect of this is to introduce simple shear stress jump discontinuities. For further clarification of this stress solution the readers are referred to Rogers and Pipkin [25] where the discontinuous stress condition is used to satisfy the shear traction boundary conditions in plane strain bending problems. Spencer [26] has also shown how the same property can be used to admit shear stress discontinuities for the more general case in which the individual layers of an elastic laminated beam can assume any orientation oblique to the plane of deformation. By considering the equilibrium of the entire modified bending mechanism shown in fig. 9.21, it becomes a straightforward task to determine the net downward bending load on the punch. When the sample is bent to an angle 4~, the moment and force equilibrium equations enable the forming load per unit width, P, to be expressed in terms of both the longitudinal shear and transverse shear viscosities of the material. p = 2r
 qb(Rr + h))(tZLhL + tZThT)
L2
(9.23)
where h L and hr are the combined thicknesses of the longitudinal and transverse layers respectively. For the case in which the strip possesses no transverse layers (i.e. hr = 0), it becomes a simple matter of rearranging eq. (9.23) to yield an expression for the longitudinal viscosity. Once/xL has been established for a particular forming condition, the subsequent introduction of transverse layers into the laminated beam may then be used to determine/Zr.
Bending of continuous fibrereinforced thermoplastic sheets
395
s*, . . . .
Q+S*
Q
Fig. 9.21. Equilibrium of the modified veebending mechanism. Note the introduction of transverse layers into the laminate.
9.5.2. Transverse shear viscosity tests
A series of experiments were performed using the modified veebending mechanism in an attempt to establish its usefulness as a means of determining both the longitudinal and transverse shear viscosities of CFRTs. These tests were performed using exactly the same procedures as those outlined for the earlier case. However for these tests, two different preconsolidated laminates were constructed from Plytron; [0~ and [0~176176176 s. The temperature and shear rate dependency observed for the longitudinal viscosity in the earlier experiments were also noticeable in the results of the tests performed on the modified apparatus. One apparent discrepancy though, that would seem to be of concern were the magnitudes of the viscosity results which are shown in fig. 9.22. The longitudinal viscosity results obtained from the modified testing jig were found to be significantly greater than those observed earlier on. This discrepancy can, however, be somewhat accounted for by considering the effect of the modifications on the shear rate of the strip for a given forming speed. It is to be noted that, although these set of tests were performed at identical punch speeds to the previous experiments, the actual forming rate, under the modified conditions, was slightly higher than the average rate encountered using the earlier setup. In other words, for the same punch speed, the entire forming operation was completed in a shorter period of time. Bearing in mind the shear thinning behaviour of the material, and the slightly
396 b ,,
T.A. M a r t & et al.
40 35
,.d
30 0~
25
+
50mm/min
~
;~
lOOmm/min
20
200mm/min "~
15
0
~
~
10
~
5
I
.
0
f
t
I
I
0.2
0.4
0.6
0.8
500mm/min
Platen Angle, ~ (rads) Fig. 9.22. Apparent longitudinal viscosity of Plytron versus platen angle. Forming temperature = 180~
different shear rates in this set of experiments, it would be inappropriate to make direct comparisons between these results and those obtained earlier. Using the method outlined in the previous section, a further series of tests were performed under identical forming conditions on bidirectional laminates which possessed both longitudinal and transverse layers. The results of these tests were used in conjunction with the longitudinal viscosity results, shown in fig. 9.22, to determine the transverse shear response of the material. The transverse shear viscosity results are shown in fig. 9.23. It is interesting to note that the same shear thinning effect observed for the longitudinal response is also apparent in the 3O t~
25
::3.
.~ 20
lOOmm/min /l 200mm/min
0
0
500mm/min
~> 10
<
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Platen Angle, ~ (fads) Fig. 9.23. Apparent transverse viscosity of Plytron versus platen angle. Forming temperature = 180~
Bending of continuousfibrereinforced thermoplastic sheets
397
transverse viscosity. Another interesting feature of the curves presented in fig. 9.23 is how flat each of the viscosity curves remain throughout the deformation. The results obtained from these experiments allow a direct comparison to be made between the transverse and longitudinal shear viscosities. This provides an important opportunity to verify a number of theoretical models which have been proposed in an attempt to relate #T and IzL to the fibre volume fraction f and the matrix viscosity #M. A summary of the models that have been proposed by Pipes [26], Christensen [27], and Binding [28] is given in table 9.4. These models are based around geometric arguments and assume somewhat of an idealised behaviour. The models of Pipes and Christensen predict that IZT > IZL for all fibre volume fractions, while the model of Binding predicts only IzL as a function o f f and #M. Furthermore, the longitudinal viscosity, as predicted by Binding, is larger than either of the transverse viscosity terms predicted by Pipes and Christensen. It should be noted that the viscosity results obtained for Plytron indicate that tXT < /XL for all the temperatures and forming speeds investigated, which is clearly not in agreement with the models of Pipes and Christensen. However, it is interesting to note that, by combining the model of Binding with those of Pipes or Christensen to eliminate/XM, expressions for the viscosity ratio #T/IZL may be readily obtained which are in line with the experimentally observed results: Pipes/Binding /s /zL
1 V~ 1 f
Christensen / B in din g (1  0.193f)3(1 _ f ) 2 /XL
(1  0.5952f)3/2(1 f)3/2(1  f )
Assuming that Plytron has a hexagonally packed arrangement of fibres and a nominal fibre volume fraction of 35 %, the above expressions would suggest that the ratio TABLE 9.4 Theoretical models relating ]s and ].LL to the fibre volume fractionf and ]s l.t T / l~tM
Pipes [27]
I~tL / l~tM
1
2(1  v/f) Christensen [28] Binding [29]
(1  0.193f) 3 (1  0.5952f) 3/2 (1 _f) 3/2
1 + 0.873f (1  0.8815f) 1/2 (1 f),/2 1f (1 V~)2 
T.A. Martin et al.
398
lZT/i~L is approximately 0.58 using the combined Pipes/Binding theory, and 0.54 using the alternative Christensen/Binding theory. These values seem to slightly underestimate the experimentally obtained results shown in fig. 9.24. The experimental results would tend to indicate that at 180~ the ratio of the two viscosity terms remains within a narrow band for the various deformation rates studied in this investigation. This would appear to be in agreement with the idealised theory presented above, in which no ratedependent terms arise. The results would also tend to suggest that the shear thinning, or ratedependent, behaviour observed in both the longitudinal and transverse directions, is largely attributable to the nonlinear behaviour of the molten matrix material. It should also be noted that the idealised models presented in table 9.4 fail to adequately take account of the resinrich layers that form between the individual plies. It is the authors' belief that the presence of these thin layers, coupled with a slight amount of fibre misalignment, accounts for the discrepancy between the theoretical and actual results. Earlier on in the chapter, it was shown how a powerlaw expression could be used to give a reasonably good description of the shear thinning behaviour observed for the longitudinal shear viscosity. Given a similar type of trend for the transverse shear response, it would not seem unreasonable to assume the same sort of general relationship for the transverse shear viscosity. We therefore assume that both the longitudinal and transverse shear viscosities can be approximated by powerlaw expressions which take the same form as eq. (9.19). Using this these types of relationships enables the ratio of the two to be written as [A T
m T f/n T
= ~
#L
(9.24)
m L ~ 'nc
where the constants m and n are the same as those defined earlier. If eq. (9.24) is set to a constant, as the results shown in fig. 9.24 would suggest, then it can be readily 2
A
1.8
A
0 50mm/min
7
/t 200mm/min
1.6
~
1.4
lOOmm/min
1.2
0.8 0.6
 ....
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.\
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ol 0
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ding
o.2
. 0.1
.
.
. 0.2
.
.
.
. 0.3
.
.
. 0.4
.
.
. 0.5
,
,
,
0.6
0.7
0.8
Platen a n g l e , ~ (rads) Fig. 9.24. T r a n s v e r s e to l o n g i t u d i n a l viscosity ratios for v a r i o u s f o r m i n g rates. F o r m i n g t e m p e r a t u r e = 180~
Bending of continuous fibrereinforced thermoplastic sheets
399
shown that n L = n T = n. From this result it becomes obvious that the shear thinning, or ratedependency, of both viscosity terms is almost certainly attributable to the nonlinear matrix material behaviour. Upon reflection, this might seem unsurprising as the shear flow of such materials is largely dominated by the thin resinrich layers which form between the individual plies.
9.6. Conclusions The novel veebending mechanism discussed in this chapter allows the longitudinal and the transverse shear viscosity of CFRT sheets to be isolated and measured. The simple shear deformations between layers of fibres in CFRT laminates are characterised particularly well by the bending model. By reducing the forming temperature from 180~ while remaining above the recrystallisation temperature, the degree of elasticity in the Plytron laminates is increased. However, this leads to additional fibre loads and increases the potential for fibre wrinkling. Transverse fibre spreading can be avoided, when bending unidirectional laminates along an axis normal to the plane of the fibres, by providing side constraints. The loads needed to shape Plytron laminates are small and these are of minor importance when designing tools for manufacturing CFRT products. A model for predicting the behaviour of an idealised viscous beam has been developed, which provides an analytical expression for the forming load as a function of the forming speed, the die geometry and the sheet thickness. These factors all affect the stresses in CFRT materials as they are deformed. The model also establishes a useful basis for further theoretical work on kinematically constrained models, which take account of the highly nonlinear viscoelastic nature of the matrix during forming. Molten unidirectional Plytron laminates exhibit a viscoelastic liquid behaviour when they are deformed. Future theoretical models should reflect this behaviour. The theoretical model provides a means to determine the apparent longitudinal shear and the transverse shear viscosity of the material as a function of the shear rate and the forming temperature. The apparent shear viscosity increases with decreasing temperature and rises dramatically when recrystallisation of the matrix commences. As the physical structure of the polymer changes, its rheological behaviour changes from that of a viscoelastic liquid to a viscoelastic solid. The rate dependence, or shear thinning behaviour, of Plytron laminates can be suitably modelled by a powerlaw relationship. Finally, it would appear that the shear thinning effect observed for both the longitudinal and transverse shear viscosities depends solely on the rheological behaviour of the matrix, as the ratio l z T / l Z L remains constant.
Acknowledgements The authors wish to thank the Foundation for Research, Science and Technology (New Zealand) for providing the funds to carry out this research. They are also thankful to Borealis (Norway) and MitsuiToatsu (Japan) for their support.
400
T.A. M a r t & et al.
References [1] Cogswell, F.N., "The Processing Science of Thermoplastic Structural Composites", Int. Polymer Processing, 4, pp. 157165, 1987. [2] Wang, E.L., Gutowski, T.G., "Laps and Gaps in Thermoplastic Composites Processing", Composites Manufacturing, 2, pp. 6978, 1991. [3] Cogswell, F.N., Leach, D.C., "Processing Science of Continuous Fibre Reinforced Thermoplastic Composites", SAMPE Journal, May, pp. 1114, 1988. [4] Martin, T.A., Bhattacharyya, D., Pipes, R.B. "Deformation Characteristics and Formability of FibreReinforced Thermoplastic Sheets", Composites Manufacturing, 3/3, pp. 165172, 1992. [5] Groves, D.J., Bellamy, A.M., Stocks, D.M., "Anisotropic Rheology of Continuous Fibre Thermoplastic Composites", Composites, No. 2, pp. 7580, 1992. [6] Scherer, R., Friedrich, K., "Inter and IntraPly Slip Flow Processess during Thermoforming of CF/ PPLaminates", Composites Manufacturing, 2/2, pp. 9296, 1991. [7] Goshawk, J.A., Jones, R.S., "Structure Reorganization during the Rheological Characterization of Continuous FibreReinforced Composites in Plane Shear", Composites, 27A, pp. 279286, 1996. [8] Soil, W., Gutowski, T.G., "Forming Thermoplastic Composite Parts", SAMPE Journal, 24/3, pp. 1519, May 1988. [9] Tam, A.S., Gutowski, T.G., "PlySlip during the Forming of Thermoplastic Composite Parts", Journal of Composite Materials, 23, pp. 587605, June 1989. [10] Hou, M., Friedrich, K., "Stamp Forming of Continuous Carbon Fibre/Polypropylene Composites", Composites Manufacturing, 2/1, pp. 39, 1991. [11] Rogers, T.G., Bradford, I.D.R., England, A.H., "Finite Plane Deformations of Anisotropic ElasticPlastic Plates and Shells", Journal of Mech. Phys. Solids, 40/7, pp. 15951606, 1992. [12] Bradford, I.D.R., England, A.H., Rogers, T.G., "Finite Deformations of a FibreReinforced Cantilever: PointForce Solutions", Acta Mechanica, 91, pp. 7795, 1992. [13] Evans, J.T., "A Simple Continuum Model of Creep in a Fibre Composite Beam", Journal of Applied Mechanics, Trans. ASME, 60, pp. 190195, 1993. [14] Rogers, T.G., O'Neill, J.M., "Theoretical Analysis of Forming Flows of FibreReinforced Composites", Composites Manufacturing, 2, 3/4, pp. 153160, 1991. [15] Spencer, A.J.M., "Deformations of FibreReinforced Materials", Oxford University Press, London, 1972. [16] Pipkin, A.C., Rogers, T.G., "Plane Deformations of Incompressible Fibre Reinforced Materials", Journal of Applied Mechanics, Transactions of the ASME, Sept., pp. 634640, 1971. [17] Rogers, T.G., "Rheological Characterization of Anisotropic Materials", Composites, 20/1, pp. 2127, 1989. [18] Dykes, R.J., Martin, T.A., Bhattacharyya, D., "Determination of Longitudinal and Transverse Shear Behaviour of Continuous FibreReinforced Composites from VeeBending", Proc. 4th International Conference on Flow Processes in Composite Materials, The University of Wales, Aberystwyth, Wales, 1996. [19] Martin, T.A., Bhattacharyya, D., Collins, I.F., "Bending of FibreReinforced Thermoplastic Sheets", Composites Manufacturing, 6, pp. 177187, 1995. [20] Martin, T.A., "Forming Fibre Reinforced Thermoplastic Composite Sheets", Ph.D. Thesis, Dept. Mech. Engineering, The University of Auckland, New Zealand, 184 pp., July 1993. [21] Martin, T.A., Bhattacharyya, D., Pipes, R.B. "Computeraided Grid Strain Analysis in Fibrereinforced Thermoplastic Sheet Forming", In: Computer Aided Design in Composite Material Technology III, ed. S.G. Advani et al., pp. 143163, 1992. [22] Zahlan, N., O'Neill, J.M. "Design and Fabrication of Composite Components; the Spring Forward Phenomenon", Composites, 20/1, pp. 7781, 1989. [23] Lodge, A.S., "Elastic Liquids", Academic Press, New York, pp. 101122, 1964. [24] Bird, R.B., Armstrong, R.C., Hassager, O., "Dynamics of Polymeric Liquids", Vol. 1: "Fluid Mechanics", John Wiley and Sons, New York, 1987.
Bending of continuousfibrereinforced thermoplastic sheets
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[25] Rogers, T.G., Pipkin, A.C., "Small Deflections of FibreReinforced Beams or Slabs", Journal of Applied Mechanics, Trans. ASME, pp. 10471048 Dec., 1971. [26] Spencer, A.J.M., "Plane Strain Bending of Laminated FibreReinforced Plates", Quarterly Journal of Mechanics and Applied Mathematics, 25, Part 3, pp. 387400, 1972. [27] Pipes, R.B., "Anisotropic Viscosities of an Orientated Fibre Composite with PowerLaw Matrix", Journal of Composite Materials, 26, pp. 15361552, 1992. [28] Christensen, R.M., "Effective Viscous Flow Properties for Fibre Suspensions under Concentrated Conditions", Journal of Rheology, 37, pp. 103121, 1993. [29] Binding, D.M., "Capillary and Contraction Flow of Long (Glass) Fibre Filled Polypropylene", Composites Manufacturing, 2, pp. 243252, 1991.