Roll forming continuous fibre-reinforced thermoplastic sheets: experimental analysis

Roll forming continuous fibre-reinforced thermoplastic sheets: experimental analysis

Composites: Part A 31 (2000) 1395–1407 Roll forming continuous fibre-reinforced thermoplastic sheets: experimenta...

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Composites: Part A 31 (2000) 1395–1407

Roll forming continuous fibre-reinforced thermoplastic sheets: experimental analysis R.J. Dykes, S.J. Mander, D. Bhattacharyya* Centre for Polymer and Composites Research, Private Bag 92019, University of Auckland, Auckland, New Zealand Received 17 September 1999; accepted 10 April 2000

Abstract This paper describes the results of an experimental roll forming study performed on sheets of continuous fibre-reinforced thermoplastic (CFRT) material. The deformation length, or transitional region over which the material deforms under each roll station, is studied for various temperatures, laminate configurations, and roll settings using still photography and surface grid strain techniques. The severity of the deformation through the first forming pass has been found to be related to the fibre architecture as well as the forming temperature. The kinematic constraints imposed by the fibres are found to result in a trellising effect that is observed clearly using surface grid strain techniques. Finally, as a first step in the theoretical analysis of the process, kinematic relations are developed, which describe the rate of change of strain in a sheet through the deformation zone. 䉷 2000 Published by Elsevier Science Ltd. Keyword: Roll forming

1. Introduction Roll forming is a rapid processing operation used for transforming flat sheets of material into useful profiled sections. As shown in Fig. 1, the method employs consecutive roll stations to deform the strip progressively into some desirable shape. The continuous nature of the process coupled with its versatility and speed makes it an extremely attractive technique for producing light weight, structurally efficient components from sheet materials. In the past, the process has found favour with the metallic sheet forming industry who have utilised it to produce a variety of wide and narrow profiles. The automotive industry in particular has made extensive use of roll forming to produce body moulding and trim, door headers, window guides, bumper reinforcements, seat tracks and door impact bars. Preliminary roll forming trials on CFRTs have been reported by Cattanach and Cogswell [1] but the degree of success and the methodology employed remain unpublished. More recently Mander et al. [2] have published the results of a series of roll forming experiments conducted on Plytron 䉸 (a polypropylene/glass fibre unidirectional preimpregnated material with a nominal fibre volume fraction of 35%). Their article describes the techniques used to * Corresponding author. Tel.: ⫹64-9-3757-599 ext. 8149; fax: ⫹64-93737-479. E-mail address: [email protected] (D. Bhattacharyya). 1359-835X/00/$ - see front matter 䉷 2000 Published by Elsevier Science Ltd. PII: S1359-835 X( 00)00 076-2

increase the process reliability as well as the formed part quality. The methods used to do this involved generally controlling the temperature of the sheet in situ. The present paper focuses on the deformation length and strain characteristics of sheets formed through the initial pass. In the first part of the paper, the deformation length and its implications are discussed in the context of forming fibre-reinforced thermoplastic sheets. The next part outlines an experimental program used to study the deformation length through the initial forming pass. The forming experiments are again divided into two parts: the first utilises still photography as a means of establishing the deformation profile through the initial pass with the main emphasis on the effects of fibre architecture and roll angle. The latter part utilises a grid strain analysis (GSA) technique to investigate the in-plane deformation in the sheet. Finally, as a forerunner of analysis to come, the kinematics of the process is outlined using a large strain approach. 1.1. Deformation length Like most other sheet forming processes, roll forming can be considered as a plane stress operation in which the sheet is made to bend and stretch over some region in order to conform to the shape of the rolls. Since the transition from a flat strip to the desired shape does not occur instantaneously, the strip must deform gradually over some transitional length. This length has been coined as the deformation


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Fig. 1. A schematic illustration of the roll forming process incorporating vertical as well as horizontal rolls.

length [3] within which the strip acquires a complex threedimensional (3D) geometry. Fig. 2 illustrates the concept of the deformation length for a simple channel section formed through a single pass and then through two passes. Within this region, defects such as fibre wrinkling, edge buckling, thickening and twisting of the section can occur. The occurrence of these defects can hinder the stability of the process and lead to unacceptable product irregularities. The deformation region is therefore an important facet of the process that must be understood if the technique of roll forming is to be successfully applied to reinforced thermoplastic sheets. Not surprisingly, the analysis of the deformation length has received considerable attention in the metal forming circles. Most deformation analyses have considered two areas: modelling the deformed surface during roll forming; and modelling the strain distribution in the sheet during roll forming. By minimising the work done due to stretching and transverse bending, Bhattacharyya et al. [3] were able to derive a useful expression for the deformation length, L, of a simple channel section: s 8a3 Du …1† Lˆ 3t where a is the flange length, Du is the roll increment, and t is

the sheet thickness. Interestingly, Eq. (1) consists entirely of geometric variables even though a rigid-perfectly plastic material model was assumed in the analysis. Experimental evidence supports generally the relationship for a variety of metallic materials despite the fact that the model neglects the effects of the rolls. Since thermoplastic composites in their molten state behave more like viscous fluids containing a network of elastic fibres, there is no reason why a similar approach cannot be employed to predict the deformation in a CFRT sheet. By assuming that the sheet adopts a configuration that minimises the rate of energy being dissipated by the deforming material, a suitable solution may be achieved.

2. Experimental details An experimental study of the roll forming process was undertaken using a modified horizontal beam raft-type roll forming machine. The machine had five individual forming stations at 275 mm spacing and was powered by a 4.4 kW induction motor fitted with an infinitely variable speed controller. The gear reduction allowed linear line speeds between 0.1 and 10 m/min. The bottom roll shafts of the roll former were interconnected and driven by a sprocket and chain arrangement. An inter-locking conveyor belt drive

Fig. 2. Illustration of the deformation length for a channel section through a single forming pass and through two forming stages. Note that the forming rolls have been omitted for clarity [4].

R.J. Dykes et al. / Composites: Part A 31 (2000) 1395–1407 Table 1 Roll former specifications Type

Horizontal beam raft (lightmedium gauge)

Number of stands Shaft diameter Stand width Drive

5 40 mm 350 mm Bottom—Chain via 3 phase AC 410 V Top—conveyor belt friction drive via bottom roll Infinitely variable speed, 0.1– 10.0 m/min 275 mm Manual screw thread—75 mm 1040 Steel, Delrin娃 plastic

Speed control Roll stand pitch Vertical adjustment Roll Material

was also installed to eliminate any possibility of slip between the top and bottom rolls. A summary of the specifications for the roll forming machine are provided in Table 1. A pre-heating oven and feeding system capable of supplying long semi-continuous lengths of pre-consolidated thermoplastic composite sheet was designed and constructed specifically for this investigation. The oven, shown in Fig. 3, consisted of a series of halogen elements arranged along its length. The nine elements, capable of delivering a total of 6.4 kW, were mounted above a 115 mm wide stainless steel conveyor belt. The halogen elements were attached to the underside of a reflective stainless steel sheet, which formed the roof of the oven. The transportation of the strip through the oven and into the forming operation was facilitated by means of a conveyor belt arrangement that was connected to, and driven via the main drive motor. Teflon strips, measuring 2 mm in thickness, were fixed to the belt to prevent the conduction of heat from the strip. The roll forming experiments were conducted using Plytron sheets. The overall dimensions of the samples tested are detailed in Fig. 4. Consolidation was performed at a temperature of 200⬚C under a vacuum pressure of 6 kPa. The control axis (or 0⬚ axis) was defined in the longitudinal


direction of the strips. Each sample was consolidated with a K-type thermocouple embedded in the centre of the laminate as shown in Fig. 4. The variables investigated in the deformation trials were the laminate architecture, forming increment (or roll angle) and forming temperature. Entry angles of 20, 30, and 40⬚ were investigated using a variety of laminates configurations with a constant flange length of 40 mm. Entry temperatures ranging from 120 to 180⬚C were investigated using [0⬚/90⬚/90⬚/0⬚]S samples and a roll angle of 30⬚. Entry speeds ranging from 5–10 m/min were also investigated. The surface strain characteristics were examined using a 5 mm square grid pattern screen printed on the upper surface of each sheet. To enable a static measurement of the deformed surface samples to be made were brought to a complete stop and quenched using chilled water dispensed from a perforated copper coil surrounding the first forming station. Each grid pattern was subsequently digitised using a 3D digitising package developed at the University of Auckland. These data were then analysed using a GSA package, the details of which have been outlined in an article by Martin et al. [4]. Two single lens reflex cameras, both fitted with fast motor drive winders, were positioned directly above (camera 1) and to the side (camera 2) of the deformation region. Fig. 5 schematically shows the positions of the cameras and the respective views from each. To assist with measuring the deformation length, a scale ruler was attached to the side of the guide tray in such a way that it was visible in photographs provided from camera 2. The cameras were activated after approximately 150 mm of sample had passed through the roll stand and 2–4 photographs were made in each view of each strip over a 5–10 s period. These photographs were then analysed and the numerical average of the deformation length was calculated.

3. Results and discussion A series of photographs, depicting the deformation

Fig. 3. A side view of the heating and transportation device used for supplying semi-continuous strips to the roll former.


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Fig. 4. Dimensions of consolidated strips used in roll forming experiments.

profiles of [0⬚/90⬚]2S laminates through the various entry rolls, were taken and a typical example is shown in Fig. 6. This image and others like it were used to measure the deformation length of various laminates through the first forming pass. In general, the results of the deformation length trials followed the same trends as those commonly observed for sheet metal. Consequently, as the roll angle was increased, the deformation length was observed to increase as shown in Fig. 7. The deformation length, as predicted for metallic sheet material of equal geometric dimensions using Eq. (1), is shown by the solid line. It is interesting to note that the theoretically derived expression involves only geometric parameters and ignores the constitutive characteristics of the material. The surprisingly good correlation between the experimental results for CFRT sheets and the theory for metallic sheet suggests that, as with metals, the deformation profile in CFRT sheets is influenced significantly by the overall kinematics of the process. By examining Fig. 7, it is difficult to discern any clear pattern pertaining to the influence of fibre architecture on the deformation length. In fact, at a roll angle of 20⬚, virtually no variation in length can be detected between the various fibre architectures examined other than for the [^15⬚]2S samples. However, if one examines the deformation profiles

Fig. 5. Schematic illustration of camera locations showing the views from camera 1 and camera 2.

from the images captured from camera 2, a number of important differences between the various laminates can be observed. Fig. 8(a)–(c) show the deformation zones of a number of different laminates as viewed from camera 2. Each photograph was taken using a constant roll angle of 40⬚. The images show that as the fibre architecture is changed from [0⬚/90⬚]2S to [^15⬚]2S to [^45⬚]2S the deformation prior to the rolls becomes more severe. In this case the severity is deemed to be directly related to the length through which transverse bending of the sheet takes place—in other words, the rate at which the width of the sheet changes, as viewed from camera 2. The most severe deformation is observed in the [^45⬚]2S laminates where it appears as if no transverse bending takes place until the sheet is in contact with the rolls. This is in contrast to the [^15⬚]2S laminate that exhibits a more gradual deformation through the transition region. The observed differences in the deformation profiles between the laminates can be reconciled by considering various aspects of the materials behaviour. The first and perhaps the most important aspect to consider is the way in which the fibres constrain the material from stretching in the reinforcing direction. Obviously, altering the fibre architecture of the sheet, for any given roll geometry, has the effect of altering the strain characteristics or forming path and hence the deformation profile. This kinematic effect gives rise to a phenomenon known as trellising, which is examined in more detail later when the surface strain characteristics of the sheet are investigated using GSA techniques. The kinematics of the process must also be considered in the context of the material’s constitutive behaviour i.e. in the simplest case, a transversely isotropic viscous fluid. As with any fluid-type problem, the kinematic quantities of most interest are the velocity gradients that control the stresses in the deforming sheet. This aspect of the process is considered in the final part of the paper that introduces the basic kinematic relations for a the steady state deformation of a continuous fibre-reinforced sheet. While the results from the still photography experiments

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Fig. 6. A typical deformation profile of a [0⬚/90⬚]2S laminate; roll angle ˆ 40⬚.

allowed the deformation length to be measured, no information regarding the actual shape adopted by the sheets or the nature of the deformation could be obtained. For this reason, a series of samples were examined in a static condition using 3D grid strain techniques. As described earlier, it was anticipated that by rapidly stopping the process and quenching the sheet, a profile representative of the steady state configuration could be obtained. Once the parts had cooled, the forming rolls were removed and the deformation lengths were compared to those measured from images captured from camera 1. Reasonable agreement was found to exist between the deformation lengths measured in the static condition and those results presented earlier, thus providing confidence in the following GSA. Fig. 9(a)–(b) shows an example of the 3D shape adopted typically by the samples formed through the initial pass at temperatures of 160⬚C or greater. Each mesh was generated from digitised data that was processed through the GSA

program. The surfaces have been divided into three zones: region A, where the strip remains unaffected by the rolls or deformation downstream; region B, where the sheet experiences deformation but is not in contact with the forming rolls; and region C, where the sheet deforms while in contact with the rolls. Virtually all the deformations appear to take place in the transition between regions B and C, where the sheet comes in contact with the forming rolls. While region B generally encompasses a much larger portion of the sheet than region C, only a minimal amount of bending, mostly in the longitudinal direction, would appear to take place there. As illustrated by the side view in Fig. 9(a), almost all the transverse bending required to form the flange takes place as the sheet comes into contact with the rolls. This is in contrast to sheet metals that tend to bend and stretch over the entire transition region prior to contacting the rolls [3]. Zhu et al. [5] have presented experimental evidence to show that the flange

Fig. 7. Comparison of deformation lengths of various laminates (flange length of 40 mm) and the length predicted using sheet metal theory.


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Fig. 8. Deformation zones for different laminates: (a) [0⬚/90⬚]2S; (b) [^45⬚]2S; and (c) [^15⬚]2S. Roll angle ˆ 40⬚; inlet temperature ˆ 160⬚C.

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Fig. 9. A digitised surface showing the characteristic shape adopted typically through the first forming pass, ([0⬚/90⬚/90⬚/0⬚]S; forming speed ˆ 5 m/min; inlet temperature ˆ 160⬚C; flange length ˆ 40 mm. Note that only half the section is shown due to symmetry.

length remains constant and develops a definite plastic hinge through the transition region. Accordingly, the deformed flange in sheet metal can be described by some function u (x) as shown in Fig. 10. However, given the complexity of the deformation profile it is unlikely that a similar

Fig. 10. Deformed surface typically encountered in roll forming of metallic sheet.

geometric model could be used to describe the deformed surface in CFRT sheets. The characteristic shape adopted by the sheet (Fig. 9(a)– (b)) remained unaffected largely by any variation in the forming speed or flange length, although trends similar to those observed in the still photography experiments were seen. Though predictable, the forming temperature was found to have a significant effect on the overall deformation. Samples formed at temperatures of 160⬚C or greater generally adopted the shape illustrated in Fig. 9(a)–(b) while samples formed at temperatures below 140⬚C tend to deform through a more gradual profile. The difference in shape can be seen clearly in Fig. 11, which illustrates the effect of temperature on the forming process viewed from the side. As the forming temperature was lowered, the deformation length increased and the area in contact with the rolls reduced. Interestingly, the shape of the sample formed at 140⬚C (Fig. 11(b)) bears a close resemblance to that adopted by sheet metals where the flange develops over the majority of the deformation length. This result demonstrates clearly that as the forming temperature approaches


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Fig. 11. Side profile of partially deformed sheets showing the effect of entry temperature on the forming characteristics (a) forming temperature ˆ 160⬚C, (b) forming temperature ˆ 140⬚C; [0⬚/90⬚/90⬚/0⬚]S, forming speed ˆ 5 m/min, roll angle ˆ 30⬚, flange length ˆ 40 mm.

the lower limits of the forming temperature window, the material becomes increasingly elastic and less viscous. From the results presented in the previous section, it is clear that the deformation length can be measured using two different approaches: the primary deformation length (L1), which is measured upstream from the roll centre to the initiation of transverse bending; and the secondary deformation length (L2), which encompasses L1 as well as the portion of the strip that deforms by bending in the longitudinal direction. These lengths are defined in Fig. 12 and were measured for each sample formed successfully. These deformation lengths of a number of [0⬚/90⬚/90⬚/0⬚]S samples formed over a variety of temperatures have been plotted in Fig. 13. The results show that as the temperature is increased, both the primary and secondary deformation lengths asymptote to a minimum value. As shown in the previous section, the primary deformation length for sheets formed at temperatures of 160⬚C or greater, could be predicted by considering the geometry of the roll. This is because the flange does not begin to develop until the sheet comes in contact with the forming rolls. As the temperature

is decreased, the deformation lengths increase as the elastic response of the material becomes more pronounced. The next part of the paper details the results from the large strain analysis performed on a series of samples that were fed through the first forming increment. The grid strain results presented herein are illustrated in two different ways: strain arrow diagrams, which indicate the magnitudes and directions of the principal surface strains within each deformed grid; and thickness strain plots, that provide a means of assessing the thickness variation through the sheet. Since GSA uses data from both the initial and final configurations, the results may be presented in terms of the undeformed or deformed states. While a number of authors have applied the technique to various types of thermoplastic composite systems in the past [4], none have used it to investigate the distribution of strain through a continuous process like roll forming. The novel way in which it is applied here means that a unique strain path can be determined directly from the deformed surface, which is unlike typical grid strain analyses that consider just the initial and final states.

Fig. 12. Deformation lengths, primary (L1) and secondary (L2).

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Fig. 13. Deformation lengths of [0⬚/90⬚/90⬚/0⬚]S sheets formed through 30⬚ rolls at various temperatures; forming speed ˆ 5 m/min; flange length ˆ 40 mm.

The systematic error, incurred when digitising each deformed sheet, was found to be approximately 0.01 mm. With the original grid spacing ˆ 5.0 mm, this error could have resulted in strain errors of up to 2%. A note of caution should therefore be issued regarding the areas of relatively small strains. The principal axes in such areas could be affected substantially by the slightest of errors incurred during the digitising process. Consequently, the trends exhibited in the results should be noted rather than the absolute strain magnitudes and directions in each element. Other practical difficulties were also encountered at the forming temperatures greater than 180⬚C, where the surface layer

Fig. 14. Principal strain arrow plot shown over the undeformed surface for [0⬚/90⬚/90⬚/0⬚]S sample; forming temperature ˆ 160⬚C; roll angle ˆ 30⬚; flange length ˆ 40 mm. 1max ˆ 10:9%; 1min ˆ ⫺11:1%: The undeformed surface fibres are illustrated by the solid lines, while the sub-surface layers are illustrated by a series of dotted lines.

tended to become smudged so that no meaningful data could be extracted. Fig. 14 shows the strain distribution on the undeformed surface of a [0⬚/90⬚/90⬚/0⬚]S laminate, which was formed through an initial roll angle of 30⬚ at a temperature of 160⬚C. As anticipated, the strain magnitudes are greatest at the roll end of the sheet (Region C), where the sheet adopts the shape of the forming rolls. In fact the position of greatest principal strain, denoted by 1 max in the diagram, is located on the outer portion of the flange just prior to where the sheet reaches the centre of the forming rolls. It is also interesting to note the directions of the principal strains, which are generally aligned at ^45⬚ to the fibre directions. This strain pattern is common particularly in the region C where the sheet deforms while in contact with the rolls. This result confirms the presence of inplane shearing, a trend noted in a number of forming studies to do with CFRTs [4,6]. As observed by Martin et al. [4], the fibre lengths appear to remain unchanged as the flange of the sheet develops by in-plane shearing. This mode of deformation has been described as a trellis action by Cattanach et al. [7], as the extension of a trellis in one direction is accompanied by a contraction in the other direction. The kinematics of these inextensible fibre networks, often referred to as Chebyshev nets, has been addressed by a number of authors [8,9]. Using the geometry in Fig. 15, the true principal strains, 1 1 and 1 2, can be established in terms of the extension ratios, l 1 and l 2 by:     cos u sin u 11 ˆ ln…l1 † ˆ ln 1 2 ˆ ln…l2 † ˆ ln : cos F sin F …2† When the fibre directions are orthogonal originally, i.e. F ˆ 45⬚, the principal compressive strains are always greater in magnitude than the principal tensile strains, so that an incompressible material will always thicken under an idealised trellis-type deformation. This has some reasonably important implications for roll forming where thickness variations in the sheet during the forming process may


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Fig. 15. Deformation of a trellis network.

lead to instabilities within the operation. A contour plot of the through-thickness strain variation in a [90⬚/0⬚/0⬚/90⬚]S sheet is illustrated in Fig. 16. From the plot, it is obvious that the thickness variation in the sheet is greatest at the outer edge where increases of up to 7.3% are predicted. As the roll angle was increased, the strain magnitudes in the flange increased, resulting in increase in thickness as high as 13.2% for the 40⬚ roll setting. Since the roll angle influences the magnitude of the shear deformation in the flange of the sheet, this effect obviously needs to be considered carefully in the design of the forming rolls and roll gap settings. The same concept may be investigated further by considering the deformation of a bi-directional laminate where the fibre orientations are skewed from the rolling direction. Fig. 17 shows the principal strain distribution on the undeformed surface of a [30⬚/⫺30⬚/⫺30⬚/30⬚]S laminate formed through an initial roll angle of 30⬚ at a temperature of 160⬚C. The results support the trellis model although in this case the principal strains are aligned in the longitudinal and transverse directions of the sheet. Once again, the maximum strains act on the outer edge of the flange as the sheet reaches the roll centre. While the underlying mechanism is still the same trellis-type deformation, the different lami-

nate configuration results in an entirely different strain path to that observed in the [0⬚/90⬚/90⬚/0⬚]S sample. 4. Kinematics of roll forming The first step in analysing a large deformation process like roll forming is to define kinematic relations that describe the motion or strain of the material. These are developed independent of any material specification although, as will be seen, the directional nature of the CFRTs allows certain quantities to be defined in a way that is unique to these materials. The key to analysing strain in a material undergoing large deformations is to establish two types of coordinate systems and develop a relationship between them. The first are fixed spatial coordinates (x, y, z) and the second (material coordinates) are embedded in the deforming body …n 1 ; n2 ; n3 †. The kinematics of the deforming body can then be defined by the relationship between the reference xi-coordinates and the material n M-coordinates in the deformation gradient tensor, which is written as: i ˆ FM

2xi 2nM


Fig. 16. Contour plot of through thickness strain in the deforming sheet (only half of the sheet is shown due to symmetry); laminate architecture ˆ [0⬚/90⬚/90⬚/ 0⬚]S; forming temperature ˆ 160⬚C; roll angle ˆ 30⬚; flange length ˆ 40 mm.

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Fig. 17. Principal strain arrow plot shown over the undeformed surface for [30⬚/⫺30⬚/⫺30⬚/30⬚]S sample; forming temperature ˆ 160⬚C; roll angle ˆ 30⬚; flange length ˆ 40 mm. (1max ˆ 12:3%; 1min ˆ ⫺7:4%†:

As shown in Fig. 18, we choose the n M-coordinate directions to coincide with certain features of the undeformed fibre-reinforced sheet: n 1 is aligned with the fibre direction; n 2 is aligned in the plane of the sheet; and n 3 is normal to the sheet surface. In the undeformed sheet the n M-coordinates form an orthogonal coordinate system. From the deformation gradient, the following Lagrangian quantities can be

…n † CMN ˆ

2xi 2xi ˆ FT F 2nM 2nN

…n † ˆ EMN

1 2

…n † …CMN ⫺ dMN †

…4† …5†

where terms are summed over repeated indices. CMN is termed the Green deformation tensor or right CauchyGreen tensor. C possesses material coordinate (upper case) indices indicating that this tensor is only dependent on material coordinates and is therefore independent of any rigid body motions. Here a bracketed superscript is used to indicate the particular set of axes that the quantity is referred to. EMN is the Lagrangian strain tensor, the components of which are zero when the material is undeformed. For small strain, E becomes the strain tensor of classical linearised elasticity theory. Similar Eulerian strain and deformation tensors can just as easily be developed in terms of the spatial xi-coordinates using the inverse of the deformation gradient tensor. For viscous fluids or rate dependent materials, it is important to consider the time rate of change of deformation. Typically, these quantities are developed in terms of velocity gradients referred to the current configuration. However the Lagrangian equivalent can be evaluated as the material, or total time derivative of E (in a spatial sense). Thus, …n † …n† …n † dEMN 2EMN 2EMN …n † ˆ ⫹ ˆ ni E_ MN dt 2t 2xi


where n i is the contravariant component of a velocity vector for a material particle at a fixed point in space on the deformed body. The first term on the right hand side of

Fig. 18. The undeformed and deformed sheet with spatial (x1, x2, x3) and material (n 1, n 2, n 3) coordinate systems. Note that the shaded region represents the cross-section of the circular female forming roll.


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Eq. (6) gives the time rate of change of EMN at a fixed position in space. This is termed as the local rate of change EMN and for a steady state deformation this term is equal to zero. The second term on the right hand side of Eq. (6) results from the particles changing position in space and is referred to as the collective rate of change of EMN. As indicated by the index, the components of velocity n i at a point on the deformed sheet are referred to as spatial coordinates but can be related back to the undeformed configuration via the deformation gradient in the same way that dx ˆ F d X: It is emphasised that this transformation is valid only under steady state conditions. Expressed in component form, this transformation may be written as

ni ˆ

2xi M V 2nM


where V M is the component of velocity for a point on the undeformed body referred to the fibre coordinates. Substituting Eq. (6) into Eq. (5) allows the Lagrangian strain rate tensor to be written as 1 n† V n† ˆ C_ …MN ˆ 0 E_ …MN 2 2

…n † …n † 2CMN 2CMN cos f ⫹ sin f 2n1 2n2

! …8†

where V0 is the velocity of the sheet and f is the angle between the fibres and the rolling direction. Note that Eq. (7) assumes that the undeformed sheet has no velocity in the x3 direction. Nefussi and Gilormini [10] used a similar kinematic model for predicting the deformed shape and deformation length in a metallic cold roll forming operation. Their choice of shape functions however, placed theoretical restrictions on the predicted surface. With the addition of certain kinematic constraints, the tensor may be utilised within the framework of the ideal fibre-reinforced material model. A suitable finite element scheme could then be employed to solve for the deformed configuration subject to the appropriate boundary conditions. The most significant advantage of this approach, is that the problem can be formulated using the same methodologies used for finite elasticity problems. The entire theory pertaining to the computational analysis of the deformation length has been derived and will be presented with numerical examples in a separate paper by the authors [11]. 5. Conclusions The experimental analysis of the deformation length has been performed by utilising still photography and surface grid strain techniques. The characteristic shape adopted by the sheets through the first pass has been divided into three regions: region A, where the strip remains unaffected by the rolls or deformation downstream; region B, where the sheet experiences deformation but is not in contact with the forming rolls; and region C, where the sheet deforms while in contact with the rolls. The deformation profile has been

found to differ from that encountered with sheet metals although similar effects have been observed for the deformation length when the roll angle was increased. Both the fibre architecture and the forming temperature have been found to affect the severity of the deformation through the first pass. At forming temperatures greater than 160⬚C, virtually no transverse bending takes place until the sheet comes into contact with the rolls. From then on the sheet conforms to the geometry of the forming rolls by deforming in a trellis-like action. As the temperature is reduced, the transverse bending required to form the flange takes place through a greater length as the materials response becomes more like an elastic solid. Increasing the roll angle has been found to have several important consequences. As the roll angle is increased from 20 to 30 to 40⬚, both the primary and secondary deformation lengths are found to increase. More importantly, as the roll angle is increased, the strain magnitudes in the flange increase resulting in the predicted thickness increases of up to 13.2% for the 40⬚ roll setting. Each roll increment obviously would need to be considered when designing roll gap settings. The kinematics of the process has been considered also in the context of a finite strain analysis. An expression for the strain rate in a steady state deformation process has been developed, which is independent of time. This approach leads to a convenient expression that can be utilised with the framework of an idealised fibre-reinforced material model. Acknowledgements The authors gratefully acknowledge the support of the New Zealand Foundation for Research, Science and Technology. We also acknowledge the free supply of material from Mitsui-Toatsu (Japan). References [1] Cattanach JB, Cogswell FN. Processing with aromatic polymer composites. In: Pritchard G, editor. Developments in Reinforced Plastics, vol. 5. Amsterdam: Elsevier, 1986. p. 1–38. [2] Mander SJ, Panton SM, Dykes RJ, Bhattacharyya D. Roll forming of sheet materials. In: Bhattacharyya D, editor. Composite sheet forming, Composite materials series, vol. 11. Amsterdam: Elsevier, 1997. p. 473–516 (chap. 12). [3] Bhattacharyya D, Smith PD, Yee CH, Collins IF. The prediction of deformation length in cold roll forming. J. Mech. Work. Technol. 1984;9:181–91. [4] Martin TM, Christie GR, Bhattacharyya D. Grid strain analysis. In: Bhattacharyya D, editor. Composite sheet forming, Composite materials series, vol. 11. Amsterdam: Elsevier, 1997. p. 217–46 (chap. 11). [5] Zhu SD, Panton SM, Duncan JL. The effects of geometric variables in roll forming channel section. Proc Inst Mech Engng, Part B: J Engng Manufact 1996;210:127–34. [6] McGuinness GB, O’Bradaigh CM. Characterisation of the processing behaviour of unidirectional continuous fibre reinforced thermoplastic

R.J. Dykes et al. / Composites: Part A 31 (2000) 1395–1407 sheets, Presented at the 4th International Conference on Flow Processes in Composite Materials (FPCM96), University of Wales, Aberystwyth, 1996. [7] Cattanach JB, Cuff GFN, Cogswell. The processing of thermoplastics containing high loadings of land and continuous reinforcing fibres. J Polym Engng 1986;6:345–61. [8] Rivlin RS. Networks of inextensible coords, Non-linear problems of engineering. New York: Academic Press, 1964. p. 51–64.


[9] Green AE, Adkins JE. Large elastic deformations and non-linear continuum mechanics. Oxford: Clarendon Press, 1960. [10] Nefussi G, Gilormini P. A simplified method for the simulation of cold roll forming. Int J Mech Sci 1993;35:867–74. [11] Bhattacharyya D, Dykes RJ, Hunter PJ. Numerical and experimental analyses of roll forming fibre reinforced thermoplastic sheets, to be presented at the Second Australasian Conference on Composite Materials, Seoul, August, 2000.