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Automatically generated geometric descriptions of textile and composite unit cells
Composites: Part A 34 (2003) 303–312 www.elsevier.com/locate/compositesa
Automatically generated geometric descriptions of textile and composite unit cells F. Robitaille*, A.C. Long, I.A. Jones, C.D. Rudd School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, Nottingham NG7 2RD, UK
Abstract This paper presents an algorithm that generates geometric descriptions of unit cells of textiles and composite materials. The purpose of these geometric descriptions is to act as domains for calculations preformed at the scale of the unit cell, where the heterogeneity of the material must be considered. The algorithm defines both the volumes of the tows and the empty volumes that extend between the tows within the calculation domain, for general textiles. Resulting geometric descriptions are provided as assemblies of topologically simple volumes that encompass either part of a tow or part of an empty volume. Typical applications of the geometric definitions include the calculation of local permeability values for textile preforms and the investigation of local stress distributions in textile composites. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Fabrics/textiles; C. Computational modelling; E. Resin transfer moulding (RTM); Permeability
1. Introduction Composite parts produced by resin transfer moulding (RTM) are increasingly used in high-volume applications. New textile and preform manufacturing processes make this possible through increased process automation. Existing preform-manufacturing technologies are also being developed and their possibilities enhanced. Preforms of increasing complexity are manufactured using braiding [1]. New technical textiles that are easier to drape and that can be processed in a more predictable way are regularly offered. High-volume applications require efficient software tools for process simulation and performance prediction. Such software tools are commercially available; for example, LCMflote [2] and LIMSe [3] have been used for many years to predict filling patterns in RTM whilst ABAQUSe [4] is used regularly to predict deformations under load. The difficulty lies in obtaining the required local material properties. The permeability data required by the above software is usually measured. Failure envelopes are sometimes measured but often, conservative approximations are used. Properties such as these vary over the volume of a part. The complete characterisation of a typical perform or laminate featuring different combinations of textiles as well as varying thickness and levels of in-plane shear can * Corresponding author. Fax: þ 44-115-951-3800. E-mail address: [email protected] (F. Robitaille).
represent a major task. However, in-plane shear and compaction of textiles must be accounted for as both have major effects on the local properties of the preforms and parts. The alternative is to predict local processing and performance properties using software tools. In addition to potentially large time savings, this avenue also opens the possibility of interactive simulations. For example, the local geometry of a dry preform obtained from a compaction simulation performed in a commercial finite element code such as ABAQUSe may be used to investigate the local permeability at the same position, using a CFD software package such as FLUENTe [5]. Calculation of local processing properties for textile preforms, such as the permeability, requires a geometric definition of the unit cell of the textile that includes both the tows and the empty volumes around them. These volumes are represented schematically in Fig. 1. Prediction of local performance properties and part behaviour, such as the distribution of stresses in a unit cell of composite material, also requires a geometric definition that includes the impregnated tows and the resin-rich volumes present around the tows in the unit cell [6]. The authors proposed and implemented a formalism for the geometric description of textiles, which allows the modelling of a large array of different textiles types using a set of simple criteria [7,8]. The resulting software, TexGen, provides geometric definitions for the tows only, such as those shown in
1359-835X/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-835X(03)00063-0
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Fig. 2. Typical output of TexGen: meshes defining the volume of tows and stitches for (a) a woven textile; (b) a warp-knitted textile with inserted unidirectional tows.
2. Objective Fig. 1. Schematic view of the tows and empty volumes defined around them in the unit cell. (a) Tows appear in grey; (b) empty volumes around tows appear in grey.
Fig. 2. In order to calculate local processing and performance properties an algorithm that provides geometric description of the volumes defined around the tows, within the unit cell, is needed. The descriptions must be provided in a consistent form that is suited to subsequent computation. Other authors have proposed geometric models of textiles [9 – 12] and have studied the effect of textile architecture on processing and performance properties [13 –15]. However, published geometric models and models of properties only consider one type of textiles and one property. The approach taken here consists in implementing the algorithm for a geometric model that can represent a very large array of textiles with a sufficient level of detail and precision, and for simple and universal input, and to use it to obtain any property from any calculation method, in an interactive manner. This paper introduces this algorithm. It is presented in the context of permeability prediction but the geometric descriptions provided by the algorithm can be used for the calculation of other local processing and performance properties.
Consider the case where a fluid is forced through a textile preform and one wishes to calculate the resulting flow field at the scale of the unit cell in order to assess the local permeability of that preform. Such a calculation will be done over a domain of limited extents. The domain includes constituents of textiles such as tows and threads; often their shapes and state of mutual contact are complex. The domain also includes empty volumes that extend between the tows; these volumes are usually of intricate shapes. The tows are porous, hence the fluid is likely to flow through both the tows and the empty volumes. A number of solution techniques could be envisaged. Some may involve two different equations, one being appropriate for the volumes of the tows and the other for the empty volumes. Other techniques may rely on the specification of the properties that describe the two different phases. Any solution will require some description of the textile structure as an input. The objective of the algorithm discussed in this paper consists in producing descriptions of the domains where tow volumes and empty volumes are identified in a consistent and usable manner, for the largest possible number of different textile structures. In order to the resulting geometric description to be usable, the tow volumes and empty volumes are subdivided in a systematic way that leads to the identification of a number of topologically simple basic volumes.
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Examples of basic volumes for a simple textile appear in Fig. 3. Each basic volume is defined by one upper surface, one lower surface, and a number of contiguous lateral surfaces; a basic volume encompasses either tow volume or empty volume. For the typical case of a textile that extends along a mid-plane parallel to ðx; yÞ the lateral surfaces of all basic volumes include the z axis and extend in a simply continuous way between the upper and lower surface of the basic volume. Upper and lower surfaces always correspond to either transitions between tow volume and empty volume
Fig. 3. Example of basic-E volumes defined between (a) two domain boundaries; (b) one domain boundary and one tow surface; (c) two tow surfaces.
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or to domain boundaries. In addition to these considerations a simple but important criterion applies to the upper and lower surfaces of the basic volumes: each of these surfaces is defined on the upper or the lower surface of only one tow, or on the upper or lower surface of the domain. Consider the three cases shown in Fig. 3. Case (a) features basic volume 1 which is defined between the upper and lower surface of the domain; the projection of basic volume 1 is shown in grey. This basic volume cannot extend any further inside the domain. The lateral surfaces of all basic volumes are vertical; if basic volume 1 extended any further, either its upper and/or lower surfaces would be defined partially on tows, or the basic volume would encompass both tow volume and empty volume, or some lateral surfaces would not extend in a simply continuous way between the upper and lower surfaces of basic volume 1. Case (b) shows basic volume 2 which extends between the upper surface of tow t1 and the upper surface of the domain. The lower surface of basic volume 2 corresponds to a transition between the volume of tow t1 and the empty volume above tow t1 ; the upper surface of basic volume 2 corresponds to the upper surface of the domain. Finally case (c) shows a number of basic volumes. Consider tows t1 and t3 ; the surface in plane ðx; yÞ that is common to the projections of both tows is shown in grey. Volume V is defined between tows t1 and t3 : Its lower surface corresponds to a transition between the volume of tow t1 and the empty volume above tow t1 ; and its upper surface corresponds to a transition between the volume of tow t3 and the empty volume below tow t3 : However volume V is not a basic volume as it encompasses both empty volume and volume that is defined within tow t2 : Its lateral surfaces are contiguous and include the z axis but they do not extend from its lower to its upper surface in a simply continuous manner as they intercept with the lower and upper surfaces of tow t2 : On the other hand basic volumes 3– 6 respect all criteria stated above. Basic volume 3 extends between the lower surface of tow t3 and upper surface of tow t2 ; basic volume 4 extends between the lower surface of tow t2 and the upper surface of tow t1 ; and basic volumes 5 and 6 both extend between the lower surface of tow t3 and the upper surface of tow t1 : Note that the basic volumes shown in Fig. 3 encompass empty volume only; basic volumes are also defined within tow volumes as detailed below. A simple 2D introduction to the algorithm appears in Fig. 4. In this paragraph the different sub-domains are referred to as volumes even though they appear as surfaces for this 2D domain. Fig. 4(a) shows the initial domain featuring three contacting tows. Fig. 4(b) shows vertical lines defined at the horizontally outermost points of each tow; the lines end either at transitions between tow volume and empty volume or at horizontal domain boundaries. Fig. 4(c) shows basic volumes defined over the empty volume present in the domain. At this stage all these basic volumes are identified; the purpose of the following steps consists in identifying the basic volumes that are defined over
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Fig. 5. Lateral edges, lower and upper surfaces.
Fig. 4. Simple representation of the algorithm for a 2D geometry.
the volume of the tows. Fig. 4(d) highlights zones within the previously identified basic volumes where the thickness is zero. These zero-thickness zones appear in cases of contact between two tows or between a tow and a domain boundary. Fig. 4(e) shows vertical lines defined at the horizontally outermost points of each zero-thickness zone. These lines end either at transitions between tow volume and empty volume or at horizontal domain boundaries; they extend continuously as long as some continuous tow volume is present. For example, line L generated at the right end of the zero-thickness zone defined between the upper surface of tow t2 and the lower surface of tow t3 extends upward through tow t3 to the upper surface of that tow, and downward through tow t2 to the lower surface of that tow. The lower and upper limits of line L correspond, respectively, to a transition between the volume of tow t2 and of tow t3 ; and empty volume. Fig. 4(f) shows all the basic volumes generated in this case including those defined immediately to the left and right of the lines just identified; these are the basic volumes defined over the empty volume present in the domain.
Some restrictions apply to the textiles that can be processed by the current algorithm. Consider a textile extending along a mid-plane that is parallel to ðx; yÞ: Then, consider one tow in this textile (Fig. 5). The tow features two lateral edges. The projections of these lateral edges on the ðx; yÞ plane delimit the projection of the tow. These projections of the lateral edges correspond to two of the four sides of the projected surface; the other two are the tow ends. The lateral edges can be regarded as successions of the two horizontally outermost points that are found on consecutive sections of the tow. The tow also features a lower and an upper surface; these extend between the two lateral edges and delimit the volume of the tow. Consider now two continuous relations et;1 ðx; y; zÞ and et;2 ðx; y; zÞ defining the two lateral edges of a tow t: Every combination of values ðx; yÞ defined by et;1 and et;2 can appear more than once. This means that the projection of each lateral edge is simply continuous, but one lateral edge associated to one tow can cross over itself or over the other lateral edge associated to the same tow. In Fig. 6(a), tows t1 and t3 comply with this requirement. Consider two continuous relations vt;u ðx; y; zÞ and vt;v ðx; y; zÞ defining the upper and lower surfaces of a tow t; respectively. A combination of values ðx; yÞ can appear more than once in each function; there can be more than one value of z associated with each combination. However, such
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values associated to the functions va;u and vb;v must be either higher or lower than both z values associated to the functions vb;u and vb;v : This means that the algorithm cannot successfully process textiles definitions where tow interference is present; whilst tow interference is clearly not possible in actual textiles, it may appear in softwaregenerated geometric definitions of textiles. In Fig. 6(b) the pair of tows t4 and t5 complies with this requirement but the pairs of tows t6 and t7 ; and t8 and t9 do not. The problem of tow interference is currently being addressed and a robust solution will be proposed and implemented at a later stage. One last set of restrictions applies. The boundaries of the calculation domain must form a rectangular volume with two faces parallel to the plane ðx; yÞ: The four other faces include the z axis; their projections on the ðx; yÞ plane are defined by the continuous functions d1 ðx; yÞ; d2 ðx; yÞ; d3 ðx; yÞ; and d4 ðx; yÞ: In Fig. 6(c) all tows t10 ; t11 ; t12 and t13 can be processed.
3. Algorithm The 3D algorithm consists of two major operations: the creation of the basic volumes that represent the empty volumes defined between the tows (basic empty volumes, referred to as basic-E volumes), and the creation of basic volumes that represent the tows (basic tow volumes, referred to as basic-T volumes). In each operation, 3D entities are projected on the ðx; yÞ plane to create 2D entities, these 2D entities are processed, and they are used to rebuild 3D entities in the form of basic volumes. 3.1. Creation of basic-E volumes
Fig. 6. Restrictions to current algorithm. (a) Projections of lateral tow edges cannot cross; (b) tow volumes cannot interfere; (c) lateral edges must cross boundaries twice or not at all.
cases must originate from a tow that crosses over itself. Tows such as t2 cannot fold over themselves. Another consequence is that the upper and lower surfaces of a tow cannot feature zones that are normal to the ðx; yÞ plane; hence the lateral edges are uniquely defined. The domains of points ðx; yÞ defined by functions vt;u and vt;v are identical; over a given tow section the z value associated to a point ðx; yÞ by function vt;u must be higher than the z value associated to the same ðx; yÞ point by function vt;v : Consider the upper and lower surfaces of two tows a and b identified by the continuous relations va;u ðx; y; zÞ; va;v ðx; y; zÞ; vb;u ðx; y; zÞ and vb;v ðx; y; zÞ: For all points ðx; yÞ that are common to the domain of the four relations, both z
The creation of basic-E volumes proceeds as follows. Consider a calculation domain containing nt tows labelled t1 ; t2 ; t3 ; …; tnt ; such a domain appears in Fig. 7 with nt ¼ 3: Firstly, control points are identified. Control points are ðx; yÞ points where projections of lateral two edges superimpose or where one such projection crosses the projected boundary of the domain; the four corners of the projected domain boundary are also identified as control points. Then, all projected lateral tow edges are divided into segments that extend between consecutive control points; segments are also defined between control points located on the projected domain boundary. Once all segments are identified they are characterised in the following way. Consider, for example, a segment that originates from a lateral edge of tow t1 (Fig. 8). All other tows t2 to tnt are considered in turn in order to assess if the segment from tow t1 extends inside, outside or on the boundaries of the projection of each tow t2 to tnt : Segments located on the projected domain boundary are characterised in the same way. At this stage the projected domain is fragmented into a number of surfaces identified as basic empty projected surfaces, or basic-E projected surfaces (Fig. 8).
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Fig. 7. Three tows defined in a domain, their projections, and control points.
The perimeter of a basic-E projected surface is identified by selecting a segment that is contiguous to this surface. One of the control points delimiting this segment is selected, and the other segments that are delimited by this control point are identified. Among these, the segment which is closest from the original segment in the counter-clockwise dimension forms part of the perimeter. The procedure is repeated from this newly identified segment, until it identifies the original segment. At this stage the perimeter is closed. Each basic-E projected surface defines a domain where distinct groups of tows are superimposed. Consider basic-E projected surfaces se1 and se2 in Fig. 9; the former results from the superimposition of tows t1 ; t2 and t3 whilst the latter results from the projection of tow t3 only. The tows from which a surface originates can be identified simply from the characteristics of the segments that form its perimeter.
Fig. 9. Basic-E projected surfaces se1 and se2 resulting from the superimposition of tows t1 ; t2 and t3 :
A basic-E projected surface originates from tow t1 if each segment that forms its perimeter either originates from a lateral edge or an end of tow t1 ; or if it extends inside the projection of tow t1 ; or if it extends on the boundaries of the projection of tow t1 : Surface se1 results from the superimposition of tows t1 ; t2 and t3 as all segments that form its perimeter either originate from/are defined inside all three tows. Surface se2 results from the projection of tow t3 only as some segments that form its perimeter are defined outside of tow t1 and some are defined outside of tow t2 : The information available at this stage allows the creation of the basic-E volumes. Consider basic-E projected surface se2 which is known to originate from the projection of tow t3 : The relations e3;1 and e3;2 define the lateral edges of this tow in 3D; one basic-E volume extends between the lower surface of the domain and the lower surface of tow t3 ; and one basic-E volume extends between the upper surface of tow t3 and the upper surface of the domain (Fig. 10). The segments defining the perimeter of basic-E projected surface se1 superimpose and the sequence of tows t1 ; t2 and t3 along z is trivial as t3 is located above t2 and t2 above t1 : The creation of the basic-E volumes associated to basic-E projected surface se1 is therefore straightforward. 3.2. Creation of basic tow volumes
Fig. 8. Identification of basic-E projected surfaces and characterisation of segments extending between control points.
It can be argued that in some cases the algorithm described above will create adjacent basic-E volumes of which both the lower and upper surfaces are defined on the same tow surfaces or domain boundary. Consider, for example, the basic-E projected surface se1 shown in Fig. 9;
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Fig. 10. Basic-E volumes defined below and above tow t3 :
basic-E volumes corresponding to this surface will be created between the lower surface of the domain and tow t1 ; tow t1 and tow t2 ; tow t2 and tow t3 ; and between tow t3 and the upper surface of the domain. However, a basic empty volume defined between the lower surface of the domain and tow t1 could extend over all the projected surface of tow t1 without infringing the limitations stated above. This may appear as over-fragmentation of the initial empty volume, and indeed it is if one wishes to process this volume only. On the other hand, in addition to its simplicity the algorithm has distinct advantages that appear clearly when the volume of the tows are processed. The creation of the basic-T volumes (basic tow volumes) starts by considering each individual basic-E volume (basic empty volume) in order to identify a zone where the thickness of the basic-E volume is near zero. Some basic-E volumes will feature no such zone. In the algorithm described here it is assumed that no more than one zerothickness zone can be defined in each basic-E volume and that this zone or surface is simply continuous; a single line defines its perimeter. The perimeters of the nz surfaces are defined by the relations z1 ðx; y; zÞ; z2 ðx; y; zÞ; …; znz ðx; y; zÞ: These perimeters are then projected on the ðx; yÞ plane. In practice the zero-thickness zones are identified automatically by inspection of the surfaces, using a numerical criterion. The next task performed by the algorithm consists in defining control points where the projections of the nz perimeters of the zero-thickness zones intersect; two control points appear in Fig. 11. Then, in a sequence similar to the one described above, each of the nz perimeters is divided into segments that are defined as being located either inside or outside of the other ðnz 2 1Þ projected surfaces. This makes possible the fragmentation of the nz surfaces into basic-T projected surfaces is a way similar to the one shown in Fig. 8 for basic-E projected surfaces. Fig. 11 shows five basic-T projected surfaces st1 ; st2 ; st3 ; st4 and st5 ; all of which are distinct. Surfaces st1 ; st2 and st5 result from contact between tows t1 and t2 : Surfaces st3 ; st4 and st5 result from contact between tows t2 and t3 : Surface st5 corresponds
Fig. 11. Basic-T projected surfaces st1 ; st2 ; st3 ; st4 and st5 resulting from contacts between tows t1 and t2 ; and between tows t2 and t3 :
to a section of the ðx; yÞ plane where the zone of contact between tows t1 and t2 and the zone of contact between tows t2 and t3 superimpose. The basic-T volumes are created directly from the basicT projected surfaces. Consider, for example, the basic-T projected surface st5 in Fig. 11, which identifies two superimposed zones of zero-thickness. As stated above this basic-T projected surface results from the superimposition of two zero-thickness zones, which themselves result from contacts between tows t1 and t2 ; and t2 and t3 : The volume defined above the basic-T surface st5 ; between the lower surface of tow t1 and the upper surface of tow t3 ; encloses only tows; no empty volumes are present in that space. Therefore, a basic-T volume extends between the lower surface of tow t1 and the upper surface of tow t3 ; enclosing part of tows t1 ;t2 and t3 : Basic-T projected surfaces st1 and st2 correspond to a contact between tows t1 and t2 ; therefore, a basic-T volume will extend above each of these surfaces between the lower surface of tow t1 and the upper surface of tow t2 : Basic-T projected surfaces st3 and st4 correspond to a contact between tows t2 and t3 ; therefore, a basic-T volume will extend above each of these surfaces between the lower surface of tow t2 and the upper surface of tow t3 : Finally, consider once more the basic-E surface se1 defined in Fig. 9. This surface was used in the identification of basic-E volumes defined between tows t1 ; t2 and t3 : At this stage it is used again, this time to define basic-T volumes. The above paragraph explained how basic-T volumes are created above basic-T projected surfaces st2 ; st3 and st5 : The last step consists in creating basic-T volumes over the difference between surface se1 and surfaces st2 ; st3 and st5 : These distinct volumes extend in either tow t1 ; tow t2 or tow t3 ; over zones where the tows do not contact. At the beginning of this section it was mentioned that overfragmentation of the empty domain was useful in the creation of basic-T volumes. That statement referred to
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the creation of the latter basic-T volumes, as their creation from the basic-E projected surfaces is trivial.
result is used as input. Fig. 13(g) shows the projections of the lateral tow edges on the plane x – y and finally, Fig. 13(h) shows a number of basic E-volumes created using the algorithm.
4. Examples Consider the simple first example of a unit cell where three tows are present (Fig. 12(a)); Fig. 12(b) shows the central tow which contacts both the lower tow and upper tows over surfaces which, when projected on ðx; yÞ; are circular. Fig. 12(c) shows a basic-E volume defined between the central tow and the lower surface of the domain. Fig. 12(d) shows a number of adjacent basic-E volumes defined between the upper surface of the lower tow and the lower surface of the central tow. Fig. 12(e) shows a number of adjacent basic-E volumes defined over the projection of the central tow. Finally, Fig. 12(f) shows a number of adjacent basic-T volumes; note here that axes x and y were rotated for easier visualisation. The second example features a real warp-knitted textile made from a thermoplastic monofilament. The textile appears in Fig. 13(a) and (b). The steps shown in Fig. 13(c)–(f) show the creation of the tows in the implemented software. These steps are not part of the algorithm presented here, but their
5. Implementation and applications The algorithm described in this paper produces geometric definitions of the tows and empty volumes defined between the tows for a unit cell of composite material, from an appropriate definition of the textile. The procedure is fully automated and the geometric information that it provides is predictable; a user will generally know what to expect by looking at the available textile geometry. The geometric definitions provided by the algorithm are useful for predicting processing properties and analysing performance of textile composites at the level of the unit cell. The analysis may be done using finite difference or finite element software; in this case the algorithm greatly simplifies the generation of appropriate meshes by providing a set of adjacent and topologically simple basic volumes. The analysis may also be done by analytical methods and in that case the algorithm precisely located
Fig. 12. (a) Unit cell featuring three tows in contact with themselves and domain boundaries; (b) central tows with contact surface; (c) basic-E volume between lower boundary and central tow; (d) basic-E volumes between lower and central tows; (e) basic-E volumes defined over the projection of the central tow; (f) basic-T volumes (rotated axes).
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Fig. 13. (a,b) Warp-knitted monofilament; (c–f) steps in the definition of geometry for tows; (g) projected lateral edges; (h) some basic-E volumes.
the different phases present in the unit cell. The way in which the information is used beyond that stage depends on the nature of the analytical method. The algorithm has the potential to offer major productivity improvements for such work, regardless of the actual analysis method used. The work described in this paper is defined within a general framework where different textiles and textile combinations can be considered. The textile modeller proposed by the authors [7,8] was developed with that philosophy in mind, and the current algorithm is designed in order to process the largest possible array of textiles. In time it is hoped that an increasing number of software tools will be developed and integrated with the existing ones; commercial software is also used whenever it is most appropriate. The algorithm presented here constitutes an important step toward the integration of simulations done at the scale of the unit cell in order to predict local properties, and those
done at the scale of the part in order to model the process and the performance of the final item. The tools developed will be increasingly used to investigate the effect of the textile structure and performing process on processing and performance properties such as the permeability, local stiffness and rupture envelopes.
6. Conclusion The algorithm presented in this paper provides geometric definitions of the tows and empty volumes defined in a repeating unit cell. This is achieved by identifying the lower and upper surfaces and the lateral edges of individual tows from an initial definition of the textile. The lateral edges are projected on a plane, their intersections are defined, and the individual surfaces that they encompass are identified. These surfaces are used in the creation of basic empty
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volumes, which together represent all the volume defined between the tows. Then, zones of zero thickness present within the basic empty volumes are identified and the perimeters of these zones are projected on a plane. The intersections of the projected perimeters are defined, and the individual surfaces that they encompass are identified. These surfaces are used in the creation of basic tow volumes, which together represent all the volume corresponding to the tows.
Acknowledgements The authors gratefully acknowledge the support of the EPSRC and their industrial partners: Dowty Propellers, ESI Group, Ford Motor Company and Formax UK.
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[5] Fluent Inc., Fluent 6.0 user’s manual; 2001. [6] Crookston JJ, Souter BJ, Long AC, Jones IA. The application of a general fibre architecture model for composite mechanical property prediction. Proceedings of ICCM-13 Conference, Beijing; June 2001. [7] Robitaille F, Clayton BR, Long AC, Souter BJ, Rudd CD. Geometric modelling of industrial preforms: woven and braided textiles. Proc Inst Mech Engrs, Part L 1999;213:69– 83. [8] Robitaille F, Clayton BR, Long AC, Souter BJ, Rudd CD. Geometric modelling of industrial preforms: warp-knitted textiles. Proc Inst Mech Engrs, Part L 2000;214:71–90. [9] Kuhn JL, Charalambides PG. Modelling of plain weave fabric composite geometry. J Compos Mater 1999;33(3):188–220. [10] Chen X, Potiyaraj P. CAD/CAM for complex woven fabrics. Part II. Multi-layer fabrics. J Text Inst 1999;90(1):73–90. [11] Lin HY, Newton A. Computer representation of woven fabric by using b-splines. J Text Inst 1999;90(1):59– 72. [12] Vandeurzen P, Ivens J, Verpoest I. A three-dimensional micromechanical analysis of woven-fabric composites. Part I. Geometric analysis. Compos Sci Technol 1996;56:1303 –15. [13] Lundstrom TS, Gebart BR. Effect of perturbation of fibre architecture on permeability inside fibre tows. J Compos Mater 1995;29(4): 424 –43. [14] Sozer EM, Chen B, Graham PJ, Bickerton S, Chou TW, Advani SG. Characterisation and prediction of compaction force and preform permeability of woven fabrics during the resin transfer moulding process. Proceedings of FPCM-5 Conference, Plymouth; July 1999. [15] Dunkers JP, Phelan FR, Zimba CG, Flynn KM, Peterson RC, Li X, Fujimoto JG, Parnas RS. Flow prediction in real structures using optical coherence tomography and lattice Boltzmann mathematics. Proceedings of FPCM-5 Conference, Plymouth; July 1999.
Automatically generated geometric descriptions of textile and composite unit cells F. Robitaille*, A.C. Long, I.A. Jones, C.D. Rudd School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, Nottingham NG7 2RD, UK
Abstract This paper presents an algorithm that generates geometric descriptions of unit cells of textiles and composite materials. The purpose of these geometric descriptions is to act as domains for calculations preformed at the scale of the unit cell, where the heterogeneity of the material must be considered. The algorithm defines both the volumes of the tows and the empty volumes that extend between the tows within the calculation domain, for general textiles. Resulting geometric descriptions are provided as assemblies of topologically simple volumes that encompass either part of a tow or part of an empty volume. Typical applications of the geometric definitions include the calculation of local permeability values for textile preforms and the investigation of local stress distributions in textile composites. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Fabrics/textiles; C. Computational modelling; E. Resin transfer moulding (RTM); Permeability
1. Introduction Composite parts produced by resin transfer moulding (RTM) are increasingly used in high-volume applications. New textile and preform manufacturing processes make this possible through increased process automation. Existing preform-manufacturing technologies are also being developed and their possibilities enhanced. Preforms of increasing complexity are manufactured using braiding [1]. New technical textiles that are easier to drape and that can be processed in a more predictable way are regularly offered. High-volume applications require efficient software tools for process simulation and performance prediction. Such software tools are commercially available; for example, LCMflote [2] and LIMSe [3] have been used for many years to predict filling patterns in RTM whilst ABAQUSe [4] is used regularly to predict deformations under load. The difficulty lies in obtaining the required local material properties. The permeability data required by the above software is usually measured. Failure envelopes are sometimes measured but often, conservative approximations are used. Properties such as these vary over the volume of a part. The complete characterisation of a typical perform or laminate featuring different combinations of textiles as well as varying thickness and levels of in-plane shear can * Corresponding author. Fax: þ 44-115-951-3800. E-mail address: [email protected] (F. Robitaille).
represent a major task. However, in-plane shear and compaction of textiles must be accounted for as both have major effects on the local properties of the preforms and parts. The alternative is to predict local processing and performance properties using software tools. In addition to potentially large time savings, this avenue also opens the possibility of interactive simulations. For example, the local geometry of a dry preform obtained from a compaction simulation performed in a commercial finite element code such as ABAQUSe may be used to investigate the local permeability at the same position, using a CFD software package such as FLUENTe [5]. Calculation of local processing properties for textile preforms, such as the permeability, requires a geometric definition of the unit cell of the textile that includes both the tows and the empty volumes around them. These volumes are represented schematically in Fig. 1. Prediction of local performance properties and part behaviour, such as the distribution of stresses in a unit cell of composite material, also requires a geometric definition that includes the impregnated tows and the resin-rich volumes present around the tows in the unit cell [6]. The authors proposed and implemented a formalism for the geometric description of textiles, which allows the modelling of a large array of different textiles types using a set of simple criteria [7,8]. The resulting software, TexGen, provides geometric definitions for the tows only, such as those shown in
1359-835X/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-835X(03)00063-0
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Fig. 2. Typical output of TexGen: meshes defining the volume of tows and stitches for (a) a woven textile; (b) a warp-knitted textile with inserted unidirectional tows.
2. Objective Fig. 1. Schematic view of the tows and empty volumes defined around them in the unit cell. (a) Tows appear in grey; (b) empty volumes around tows appear in grey.
Fig. 2. In order to calculate local processing and performance properties an algorithm that provides geometric description of the volumes defined around the tows, within the unit cell, is needed. The descriptions must be provided in a consistent form that is suited to subsequent computation. Other authors have proposed geometric models of textiles [9 – 12] and have studied the effect of textile architecture on processing and performance properties [13 –15]. However, published geometric models and models of properties only consider one type of textiles and one property. The approach taken here consists in implementing the algorithm for a geometric model that can represent a very large array of textiles with a sufficient level of detail and precision, and for simple and universal input, and to use it to obtain any property from any calculation method, in an interactive manner. This paper introduces this algorithm. It is presented in the context of permeability prediction but the geometric descriptions provided by the algorithm can be used for the calculation of other local processing and performance properties.
Consider the case where a fluid is forced through a textile preform and one wishes to calculate the resulting flow field at the scale of the unit cell in order to assess the local permeability of that preform. Such a calculation will be done over a domain of limited extents. The domain includes constituents of textiles such as tows and threads; often their shapes and state of mutual contact are complex. The domain also includes empty volumes that extend between the tows; these volumes are usually of intricate shapes. The tows are porous, hence the fluid is likely to flow through both the tows and the empty volumes. A number of solution techniques could be envisaged. Some may involve two different equations, one being appropriate for the volumes of the tows and the other for the empty volumes. Other techniques may rely on the specification of the properties that describe the two different phases. Any solution will require some description of the textile structure as an input. The objective of the algorithm discussed in this paper consists in producing descriptions of the domains where tow volumes and empty volumes are identified in a consistent and usable manner, for the largest possible number of different textile structures. In order to the resulting geometric description to be usable, the tow volumes and empty volumes are subdivided in a systematic way that leads to the identification of a number of topologically simple basic volumes.
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Examples of basic volumes for a simple textile appear in Fig. 3. Each basic volume is defined by one upper surface, one lower surface, and a number of contiguous lateral surfaces; a basic volume encompasses either tow volume or empty volume. For the typical case of a textile that extends along a mid-plane parallel to ðx; yÞ the lateral surfaces of all basic volumes include the z axis and extend in a simply continuous way between the upper and lower surface of the basic volume. Upper and lower surfaces always correspond to either transitions between tow volume and empty volume
Fig. 3. Example of basic-E volumes defined between (a) two domain boundaries; (b) one domain boundary and one tow surface; (c) two tow surfaces.
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or to domain boundaries. In addition to these considerations a simple but important criterion applies to the upper and lower surfaces of the basic volumes: each of these surfaces is defined on the upper or the lower surface of only one tow, or on the upper or lower surface of the domain. Consider the three cases shown in Fig. 3. Case (a) features basic volume 1 which is defined between the upper and lower surface of the domain; the projection of basic volume 1 is shown in grey. This basic volume cannot extend any further inside the domain. The lateral surfaces of all basic volumes are vertical; if basic volume 1 extended any further, either its upper and/or lower surfaces would be defined partially on tows, or the basic volume would encompass both tow volume and empty volume, or some lateral surfaces would not extend in a simply continuous way between the upper and lower surfaces of basic volume 1. Case (b) shows basic volume 2 which extends between the upper surface of tow t1 and the upper surface of the domain. The lower surface of basic volume 2 corresponds to a transition between the volume of tow t1 and the empty volume above tow t1 ; the upper surface of basic volume 2 corresponds to the upper surface of the domain. Finally case (c) shows a number of basic volumes. Consider tows t1 and t3 ; the surface in plane ðx; yÞ that is common to the projections of both tows is shown in grey. Volume V is defined between tows t1 and t3 : Its lower surface corresponds to a transition between the volume of tow t1 and the empty volume above tow t1 ; and its upper surface corresponds to a transition between the volume of tow t3 and the empty volume below tow t3 : However volume V is not a basic volume as it encompasses both empty volume and volume that is defined within tow t2 : Its lateral surfaces are contiguous and include the z axis but they do not extend from its lower to its upper surface in a simply continuous manner as they intercept with the lower and upper surfaces of tow t2 : On the other hand basic volumes 3– 6 respect all criteria stated above. Basic volume 3 extends between the lower surface of tow t3 and upper surface of tow t2 ; basic volume 4 extends between the lower surface of tow t2 and the upper surface of tow t1 ; and basic volumes 5 and 6 both extend between the lower surface of tow t3 and the upper surface of tow t1 : Note that the basic volumes shown in Fig. 3 encompass empty volume only; basic volumes are also defined within tow volumes as detailed below. A simple 2D introduction to the algorithm appears in Fig. 4. In this paragraph the different sub-domains are referred to as volumes even though they appear as surfaces for this 2D domain. Fig. 4(a) shows the initial domain featuring three contacting tows. Fig. 4(b) shows vertical lines defined at the horizontally outermost points of each tow; the lines end either at transitions between tow volume and empty volume or at horizontal domain boundaries. Fig. 4(c) shows basic volumes defined over the empty volume present in the domain. At this stage all these basic volumes are identified; the purpose of the following steps consists in identifying the basic volumes that are defined over
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Fig. 5. Lateral edges, lower and upper surfaces.
Fig. 4. Simple representation of the algorithm for a 2D geometry.
the volume of the tows. Fig. 4(d) highlights zones within the previously identified basic volumes where the thickness is zero. These zero-thickness zones appear in cases of contact between two tows or between a tow and a domain boundary. Fig. 4(e) shows vertical lines defined at the horizontally outermost points of each zero-thickness zone. These lines end either at transitions between tow volume and empty volume or at horizontal domain boundaries; they extend continuously as long as some continuous tow volume is present. For example, line L generated at the right end of the zero-thickness zone defined between the upper surface of tow t2 and the lower surface of tow t3 extends upward through tow t3 to the upper surface of that tow, and downward through tow t2 to the lower surface of that tow. The lower and upper limits of line L correspond, respectively, to a transition between the volume of tow t2 and of tow t3 ; and empty volume. Fig. 4(f) shows all the basic volumes generated in this case including those defined immediately to the left and right of the lines just identified; these are the basic volumes defined over the empty volume present in the domain.
Some restrictions apply to the textiles that can be processed by the current algorithm. Consider a textile extending along a mid-plane that is parallel to ðx; yÞ: Then, consider one tow in this textile (Fig. 5). The tow features two lateral edges. The projections of these lateral edges on the ðx; yÞ plane delimit the projection of the tow. These projections of the lateral edges correspond to two of the four sides of the projected surface; the other two are the tow ends. The lateral edges can be regarded as successions of the two horizontally outermost points that are found on consecutive sections of the tow. The tow also features a lower and an upper surface; these extend between the two lateral edges and delimit the volume of the tow. Consider now two continuous relations et;1 ðx; y; zÞ and et;2 ðx; y; zÞ defining the two lateral edges of a tow t: Every combination of values ðx; yÞ defined by et;1 and et;2 can appear more than once. This means that the projection of each lateral edge is simply continuous, but one lateral edge associated to one tow can cross over itself or over the other lateral edge associated to the same tow. In Fig. 6(a), tows t1 and t3 comply with this requirement. Consider two continuous relations vt;u ðx; y; zÞ and vt;v ðx; y; zÞ defining the upper and lower surfaces of a tow t; respectively. A combination of values ðx; yÞ can appear more than once in each function; there can be more than one value of z associated with each combination. However, such
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values associated to the functions va;u and vb;v must be either higher or lower than both z values associated to the functions vb;u and vb;v : This means that the algorithm cannot successfully process textiles definitions where tow interference is present; whilst tow interference is clearly not possible in actual textiles, it may appear in softwaregenerated geometric definitions of textiles. In Fig. 6(b) the pair of tows t4 and t5 complies with this requirement but the pairs of tows t6 and t7 ; and t8 and t9 do not. The problem of tow interference is currently being addressed and a robust solution will be proposed and implemented at a later stage. One last set of restrictions applies. The boundaries of the calculation domain must form a rectangular volume with two faces parallel to the plane ðx; yÞ: The four other faces include the z axis; their projections on the ðx; yÞ plane are defined by the continuous functions d1 ðx; yÞ; d2 ðx; yÞ; d3 ðx; yÞ; and d4 ðx; yÞ: In Fig. 6(c) all tows t10 ; t11 ; t12 and t13 can be processed.
3. Algorithm The 3D algorithm consists of two major operations: the creation of the basic volumes that represent the empty volumes defined between the tows (basic empty volumes, referred to as basic-E volumes), and the creation of basic volumes that represent the tows (basic tow volumes, referred to as basic-T volumes). In each operation, 3D entities are projected on the ðx; yÞ plane to create 2D entities, these 2D entities are processed, and they are used to rebuild 3D entities in the form of basic volumes. 3.1. Creation of basic-E volumes
Fig. 6. Restrictions to current algorithm. (a) Projections of lateral tow edges cannot cross; (b) tow volumes cannot interfere; (c) lateral edges must cross boundaries twice or not at all.
cases must originate from a tow that crosses over itself. Tows such as t2 cannot fold over themselves. Another consequence is that the upper and lower surfaces of a tow cannot feature zones that are normal to the ðx; yÞ plane; hence the lateral edges are uniquely defined. The domains of points ðx; yÞ defined by functions vt;u and vt;v are identical; over a given tow section the z value associated to a point ðx; yÞ by function vt;u must be higher than the z value associated to the same ðx; yÞ point by function vt;v : Consider the upper and lower surfaces of two tows a and b identified by the continuous relations va;u ðx; y; zÞ; va;v ðx; y; zÞ; vb;u ðx; y; zÞ and vb;v ðx; y; zÞ: For all points ðx; yÞ that are common to the domain of the four relations, both z
The creation of basic-E volumes proceeds as follows. Consider a calculation domain containing nt tows labelled t1 ; t2 ; t3 ; …; tnt ; such a domain appears in Fig. 7 with nt ¼ 3: Firstly, control points are identified. Control points are ðx; yÞ points where projections of lateral two edges superimpose or where one such projection crosses the projected boundary of the domain; the four corners of the projected domain boundary are also identified as control points. Then, all projected lateral tow edges are divided into segments that extend between consecutive control points; segments are also defined between control points located on the projected domain boundary. Once all segments are identified they are characterised in the following way. Consider, for example, a segment that originates from a lateral edge of tow t1 (Fig. 8). All other tows t2 to tnt are considered in turn in order to assess if the segment from tow t1 extends inside, outside or on the boundaries of the projection of each tow t2 to tnt : Segments located on the projected domain boundary are characterised in the same way. At this stage the projected domain is fragmented into a number of surfaces identified as basic empty projected surfaces, or basic-E projected surfaces (Fig. 8).
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Fig. 7. Three tows defined in a domain, their projections, and control points.
The perimeter of a basic-E projected surface is identified by selecting a segment that is contiguous to this surface. One of the control points delimiting this segment is selected, and the other segments that are delimited by this control point are identified. Among these, the segment which is closest from the original segment in the counter-clockwise dimension forms part of the perimeter. The procedure is repeated from this newly identified segment, until it identifies the original segment. At this stage the perimeter is closed. Each basic-E projected surface defines a domain where distinct groups of tows are superimposed. Consider basic-E projected surfaces se1 and se2 in Fig. 9; the former results from the superimposition of tows t1 ; t2 and t3 whilst the latter results from the projection of tow t3 only. The tows from which a surface originates can be identified simply from the characteristics of the segments that form its perimeter.
Fig. 9. Basic-E projected surfaces se1 and se2 resulting from the superimposition of tows t1 ; t2 and t3 :
A basic-E projected surface originates from tow t1 if each segment that forms its perimeter either originates from a lateral edge or an end of tow t1 ; or if it extends inside the projection of tow t1 ; or if it extends on the boundaries of the projection of tow t1 : Surface se1 results from the superimposition of tows t1 ; t2 and t3 as all segments that form its perimeter either originate from/are defined inside all three tows. Surface se2 results from the projection of tow t3 only as some segments that form its perimeter are defined outside of tow t1 and some are defined outside of tow t2 : The information available at this stage allows the creation of the basic-E volumes. Consider basic-E projected surface se2 which is known to originate from the projection of tow t3 : The relations e3;1 and e3;2 define the lateral edges of this tow in 3D; one basic-E volume extends between the lower surface of the domain and the lower surface of tow t3 ; and one basic-E volume extends between the upper surface of tow t3 and the upper surface of the domain (Fig. 10). The segments defining the perimeter of basic-E projected surface se1 superimpose and the sequence of tows t1 ; t2 and t3 along z is trivial as t3 is located above t2 and t2 above t1 : The creation of the basic-E volumes associated to basic-E projected surface se1 is therefore straightforward. 3.2. Creation of basic tow volumes
Fig. 8. Identification of basic-E projected surfaces and characterisation of segments extending between control points.
It can be argued that in some cases the algorithm described above will create adjacent basic-E volumes of which both the lower and upper surfaces are defined on the same tow surfaces or domain boundary. Consider, for example, the basic-E projected surface se1 shown in Fig. 9;
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Fig. 10. Basic-E volumes defined below and above tow t3 :
basic-E volumes corresponding to this surface will be created between the lower surface of the domain and tow t1 ; tow t1 and tow t2 ; tow t2 and tow t3 ; and between tow t3 and the upper surface of the domain. However, a basic empty volume defined between the lower surface of the domain and tow t1 could extend over all the projected surface of tow t1 without infringing the limitations stated above. This may appear as over-fragmentation of the initial empty volume, and indeed it is if one wishes to process this volume only. On the other hand, in addition to its simplicity the algorithm has distinct advantages that appear clearly when the volume of the tows are processed. The creation of the basic-T volumes (basic tow volumes) starts by considering each individual basic-E volume (basic empty volume) in order to identify a zone where the thickness of the basic-E volume is near zero. Some basic-E volumes will feature no such zone. In the algorithm described here it is assumed that no more than one zerothickness zone can be defined in each basic-E volume and that this zone or surface is simply continuous; a single line defines its perimeter. The perimeters of the nz surfaces are defined by the relations z1 ðx; y; zÞ; z2 ðx; y; zÞ; …; znz ðx; y; zÞ: These perimeters are then projected on the ðx; yÞ plane. In practice the zero-thickness zones are identified automatically by inspection of the surfaces, using a numerical criterion. The next task performed by the algorithm consists in defining control points where the projections of the nz perimeters of the zero-thickness zones intersect; two control points appear in Fig. 11. Then, in a sequence similar to the one described above, each of the nz perimeters is divided into segments that are defined as being located either inside or outside of the other ðnz 2 1Þ projected surfaces. This makes possible the fragmentation of the nz surfaces into basic-T projected surfaces is a way similar to the one shown in Fig. 8 for basic-E projected surfaces. Fig. 11 shows five basic-T projected surfaces st1 ; st2 ; st3 ; st4 and st5 ; all of which are distinct. Surfaces st1 ; st2 and st5 result from contact between tows t1 and t2 : Surfaces st3 ; st4 and st5 result from contact between tows t2 and t3 : Surface st5 corresponds
Fig. 11. Basic-T projected surfaces st1 ; st2 ; st3 ; st4 and st5 resulting from contacts between tows t1 and t2 ; and between tows t2 and t3 :
to a section of the ðx; yÞ plane where the zone of contact between tows t1 and t2 and the zone of contact between tows t2 and t3 superimpose. The basic-T volumes are created directly from the basicT projected surfaces. Consider, for example, the basic-T projected surface st5 in Fig. 11, which identifies two superimposed zones of zero-thickness. As stated above this basic-T projected surface results from the superimposition of two zero-thickness zones, which themselves result from contacts between tows t1 and t2 ; and t2 and t3 : The volume defined above the basic-T surface st5 ; between the lower surface of tow t1 and the upper surface of tow t3 ; encloses only tows; no empty volumes are present in that space. Therefore, a basic-T volume extends between the lower surface of tow t1 and the upper surface of tow t3 ; enclosing part of tows t1 ;t2 and t3 : Basic-T projected surfaces st1 and st2 correspond to a contact between tows t1 and t2 ; therefore, a basic-T volume will extend above each of these surfaces between the lower surface of tow t1 and the upper surface of tow t2 : Basic-T projected surfaces st3 and st4 correspond to a contact between tows t2 and t3 ; therefore, a basic-T volume will extend above each of these surfaces between the lower surface of tow t2 and the upper surface of tow t3 : Finally, consider once more the basic-E surface se1 defined in Fig. 9. This surface was used in the identification of basic-E volumes defined between tows t1 ; t2 and t3 : At this stage it is used again, this time to define basic-T volumes. The above paragraph explained how basic-T volumes are created above basic-T projected surfaces st2 ; st3 and st5 : The last step consists in creating basic-T volumes over the difference between surface se1 and surfaces st2 ; st3 and st5 : These distinct volumes extend in either tow t1 ; tow t2 or tow t3 ; over zones where the tows do not contact. At the beginning of this section it was mentioned that overfragmentation of the empty domain was useful in the creation of basic-T volumes. That statement referred to
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the creation of the latter basic-T volumes, as their creation from the basic-E projected surfaces is trivial.
result is used as input. Fig. 13(g) shows the projections of the lateral tow edges on the plane x – y and finally, Fig. 13(h) shows a number of basic E-volumes created using the algorithm.
4. Examples Consider the simple first example of a unit cell where three tows are present (Fig. 12(a)); Fig. 12(b) shows the central tow which contacts both the lower tow and upper tows over surfaces which, when projected on ðx; yÞ; are circular. Fig. 12(c) shows a basic-E volume defined between the central tow and the lower surface of the domain. Fig. 12(d) shows a number of adjacent basic-E volumes defined between the upper surface of the lower tow and the lower surface of the central tow. Fig. 12(e) shows a number of adjacent basic-E volumes defined over the projection of the central tow. Finally, Fig. 12(f) shows a number of adjacent basic-T volumes; note here that axes x and y were rotated for easier visualisation. The second example features a real warp-knitted textile made from a thermoplastic monofilament. The textile appears in Fig. 13(a) and (b). The steps shown in Fig. 13(c)–(f) show the creation of the tows in the implemented software. These steps are not part of the algorithm presented here, but their
5. Implementation and applications The algorithm described in this paper produces geometric definitions of the tows and empty volumes defined between the tows for a unit cell of composite material, from an appropriate definition of the textile. The procedure is fully automated and the geometric information that it provides is predictable; a user will generally know what to expect by looking at the available textile geometry. The geometric definitions provided by the algorithm are useful for predicting processing properties and analysing performance of textile composites at the level of the unit cell. The analysis may be done using finite difference or finite element software; in this case the algorithm greatly simplifies the generation of appropriate meshes by providing a set of adjacent and topologically simple basic volumes. The analysis may also be done by analytical methods and in that case the algorithm precisely located
Fig. 12. (a) Unit cell featuring three tows in contact with themselves and domain boundaries; (b) central tows with contact surface; (c) basic-E volume between lower boundary and central tow; (d) basic-E volumes between lower and central tows; (e) basic-E volumes defined over the projection of the central tow; (f) basic-T volumes (rotated axes).
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Fig. 13. (a,b) Warp-knitted monofilament; (c–f) steps in the definition of geometry for tows; (g) projected lateral edges; (h) some basic-E volumes.
the different phases present in the unit cell. The way in which the information is used beyond that stage depends on the nature of the analytical method. The algorithm has the potential to offer major productivity improvements for such work, regardless of the actual analysis method used. The work described in this paper is defined within a general framework where different textiles and textile combinations can be considered. The textile modeller proposed by the authors [7,8] was developed with that philosophy in mind, and the current algorithm is designed in order to process the largest possible array of textiles. In time it is hoped that an increasing number of software tools will be developed and integrated with the existing ones; commercial software is also used whenever it is most appropriate. The algorithm presented here constitutes an important step toward the integration of simulations done at the scale of the unit cell in order to predict local properties, and those
done at the scale of the part in order to model the process and the performance of the final item. The tools developed will be increasingly used to investigate the effect of the textile structure and performing process on processing and performance properties such as the permeability, local stiffness and rupture envelopes.
6. Conclusion The algorithm presented in this paper provides geometric definitions of the tows and empty volumes defined in a repeating unit cell. This is achieved by identifying the lower and upper surfaces and the lateral edges of individual tows from an initial definition of the textile. The lateral edges are projected on a plane, their intersections are defined, and the individual surfaces that they encompass are identified. These surfaces are used in the creation of basic empty
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volumes, which together represent all the volume defined between the tows. Then, zones of zero thickness present within the basic empty volumes are identified and the perimeters of these zones are projected on a plane. The intersections of the projected perimeters are defined, and the individual surfaces that they encompass are identified. These surfaces are used in the creation of basic tow volumes, which together represent all the volume corresponding to the tows.
Acknowledgements The authors gratefully acknowledge the support of the EPSRC and their industrial partners: Dowty Propellers, ESI Group, Ford Motor Company and Formax UK.
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