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Journal of The Textile Institute
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A Simulation of the Draping
of Bidirectional Fabrics over
Arbitrary Surfaces
a
a
B. P. van West , R. B. Pipes & M. Keefe
a
a
Center for Composite Materials and Department
of Mechanical Engineering, University of Delaware,
Newark, DE, U.S.A.
Published online: 01 Dec 2008.
To cite this article: B. P. van West , R. B. Pipes & M. Keefe (1990) A Simulation of the
Draping of Bidirectional Fabrics over Arbitrary Surfaces, Journal of The Textile Institute,
81:4, 448-460
To link to this article: http://dx.doi.org/10.1080/00405009008658722
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A Simulation of the Draping of
Bidirectional Fabrics over Arbitrary
Surfaces
B.R Van West, R.B. Pipes, and M. Keefe
Center for Composite Materials and Department of Mechanical
Engineering, University of Delaware, Newark, DE, U.SA.
Downloaded by [Virginia Tech Libraries] at 10:07 14 July 2013
Received 12.6.1990 Accepted for publication 27.6.1990
The development of a graphical simulation to descrihe the draping of hidirectional
fabrics over arhitrary surfaces is reported. The simulation applies to any
surface, descrihed either analytically or numerically, and allows any numher
of draped configurations on a surface. A unique draped configuration results
when one warp thread and one weft thread are constrained to specific paths on
the surface. The warp and weft crossover-point locations are calculated hy
numerically solving the intersection equations ofthe surface and two spheres
that represent all possible positions of the ends of a warp and weft; segment.
Fahric-wrinkling and £abric-hridging over surface depressions are extreme
cases of deformation, which, as with thread orientation, can he controUed by
the placement of the constrained threads on the surface to be flraped. Several
surfaces are draped with the simulation, which is used as a design tool for
selecting suitahle draped configurations for specific surfaces.
1. INTRODUCTION
The draping of fabrics is of interest to the textile industry, which has long been
concerned with the conformance of fabrics to surfaces having compound curvature,
such as the human body. The draping of fabrics has recently become of interest
to the manufacturers of structural components made from composite materials,
which consist of reinforcing fibres embedded in a solid matrix. A fabric composed
of reinforcing and matrix fibres is draped over a mould surface and consolidated
under heat and pressure into a rigid component. Certain properties of the
component, such as strength, stifEhess, thermal expansion, and conductivity, are
dependent on the reinforcing-fibre paths within the component. Should the
fabric be unable to conform to an underljring surface owing to wrinkling or
bridging, the structural integrity or utility of a formed component may be
compromised. The prediction of fibre paths within a component is therefore an
advantage to composites manufacturers.
Bidirectional fabrics undergo in-plane shear deformation when forced to
conform to a surface having compound curvature. This deformation, coupled
with the kinematics of mapping paths from a flat plane onto a curved surface,
makes the prediction of thread paths on the surface non-trivial. This paper
describes a draping simulation that predicts thread paths at all points on a
surface draped with a bidirectional fabric. The simulation predicts the location
of wrinkling and bridging based on given fabric-deformation limits. The draped
surface is graphically displayed and shows thread paths and areas of wrinkling
and bridging. The simulation was used to study the draping of various surfaces
of particular interest. The ability to select the draping constraints allowed the
simulation to be \ised as a design tool with which acceptable draped configurations
for specific surfaces were selected.
J. Text. Int.. 1990. 81 No. 4 C TextiU InMtituU
A Simulation ofthe Draping of Bidirectional Fabrics over Arbitrary Surfaces
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2. BACKGROUND
Mack and Taylor^ derived differential equations for fitting fabrics to surfaces of
revolution. Tbe fitting equations were based on the shearing deformation of
fabric whereby a square cell boimded by pairs of warp and weft threads is
deformed into a rhombus having unequal angles at adjacent comers. Deformation
causes the cell to elongate along one diagonal direction while it contracts in the
perpendicular diagonal direction. The cell diagonal lines remain at right angles
for all degrees of deformation. Thus a rectangle of fabric, cut on a 45" bias to warp
and weft threads and draped onto a surface symmetrical about one axis, will have
cell diagonals that are parallel to surface longitude and latitude lines everywhere
on the surface. The total length of diagonals across the width ofthe fabric will
match the surface circumference at a latitude line. The model solves for the
intersection of trace-geodesic lines and tbe surface and is limited to analytical
surfaces of revolution draped with fabric on a 45° bias.
Robertson et al^ developed a computational procedure for draping a spherical
surface that involved finding the intersection points of three spheres. One
sphere represents the surface being draped, and the other two spheres represent
all possible positions ofthe ends of a warp segment and a weft segment whose
other ends already lie on the surface at known points that coincide with the
centres ofthe two spheres. The intersection equations are solved analytically,
and one ofthe two sphere-intersection points is chosen as a warp/weft crossover
point. In a second paper, Robertson et al. ^ extended their results to the treatment
of a conical surface with a spherical apex. The cone and sphere are treated
separately, and special conditions are required where yam segments cross the
boundary between cone and sphere. Although the formulation was specifically
restricted to cones and spheres, the method may be applied to an arbitrary
surface provided that at every point on the surface an analytical description of
the surface can be derived.
Smiley and Pipes* applied Robertson's sphere-intersection method of a draping
simulation to numerically defined surfaces of revolution. A cubic curve is
interpolated through specified points and a surface generated by sweeping the
curve around a central axis. The intersection equations are solved numerically.
A simulation was developed by Bergsma and Huisman^ for draping fabrics
over numerically generated surfaces approximated by fiat triangular facets. The
authors assumed pivoted joints and trellis type deformation in the fabric.
Draping starts at tbe point of greatest elevation on the surface, where four fabric
nodal points are specified that determine the orientation ofthe fabric. Subsequent
points are found by iteratively seeking the intersection of warp and weft
segments and the surface triangles. A feature ofthe simulation is that fabric-cell
deformation angles are continuously summed and minimized by iteration to
minimize total fabric deformation. This minimization of deformation at each step
ofthe draping process is the constraint that allows a unique draping configuration
to be achieved. Experiments were performed by physically draping symmetrical
surfaces, and good agreement with simulations was foimd. The use of triangular
facets in generating a curved surface simphfies the intersection equations but
introduces an error in proportion to the size of the facets. The deformationminimization feature permits only one draped configuration to be determined for
each geometry and results in only symmetrical draping of symmetrical surfaces.
Minimizing the total deformation at each step of the draping process does not
necessarily preclude the formation of local wrinkling. (It is possible that, by
inducing a high degree of deformation, such as stretching at some step in the
draping process, the formation of a wrinkle at a later step can be avoid(xi*.)
Experiments were performed only on symmetrical surfaces draped in symmetrical
configurations.
J. nxt Irut.. 1990. at No. 40 TextiU InstituU
449
Van West, Pipes, and Keefe
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Heisey and Haller^ developed a computational method forfittingwoven fabric
to non-analytical surfaces by using numerical-analysis techniques. An advantage
of their method is that the thread segments follow the contour ofthe surface and
are not straight between thread-crossover points as assumed by other methods.
Equations are derived for two lines on the surface inscribed about two known
thread-crossover points. The lines are loci of points equidistant over the surface
from the known crossover points. One ofthe two intersection points ofthe lines
IS a new crossover point. Tofindthe equations ofthe two locus lines, the equation
for the surface must be known. The authors give an example of a surface in a
cylmdrical co-ordinate system where four non-linear equations are derived by
integration and solved by iteration. The surface illustrating the method was
selected so that the results could be compared with those of Mack and Taylor^.
The derivation of intersection equations for more complicated surfaces could
prove difficult. Although it is stated that the method applies to non-analytical
surfaces, no examples with such surfaces were given.
3. SIMULATION
3.1 Assumptions
The following assumptions of Mack and Taylor^ conceming the kinematics of
draping are adopted in the draping simulation.
(a) The threads are inextensible.
(6) Crossover points of warp and weft threads act as pivoting joints.
(c) No slippage of warp and weft threads relative to one another occurs at the
pivotal points.
(d) The length of a thread segment between adjacent warp and weft threads
IS much less than the radius of curvature ofthe surface.
(e) Thread segments are straight between pivotal points.
(f) The fabric is everywhere in contact with the surface.
The assumption of inextensibility ofthe threads is justified by the high elastic
modulus of most fibres, resulting in insignificant axial strain under the loads
experienced dunng draping. It was found previousl/^ that warp and weft
thread-crossover points may be treated as pivoting joints except at fabric-shear
deformations near the locking limit. The thread-segment length is arbitrarily
selected m the simulation and does not necessarily match the spacing between
threads, which m most fabrics is very small. The thread-segment length must be
selected so that it is much smaller than the surface local radius of curvature
«r f,-^^°^^^® **^ draping may lead to wrinkling or bridging in the fabric
WrmkUng is defined in this context as an excess of fabric over that needed to
cover the surface, resulting in the excess fabric's not being in contact with the
surface. Bndging occurs when the fabric cannot deform sufficiently to drape into
a concavity in the surface, which results in the fabric's covering the concavity
without contacting the surface.
3.2 Uniqueness Constraints
A surface may be draped in an infinite number of configurations, depending on
where the drapmg process begins, how the fabric is initially oriented relative to
450
J. Text. iTut., 1990, 81 No. 4 €> TextiU tnstitutt
A Simulation ofthe Draping of Bidirectional Fabrics over Arbitrary Surfaces
the surface, and how it is manipulated to conform to the surface. To achieve a
unique draped configuration, constraints must be imposed tbat uniquely determine
all thread paths. If one warp and one weft thread are selected as in Fig. 1 to follow
specific paths on the surface, the remainder ofthe thread paths on the surface
are uniquely determined given the assumptions listed previously. An intersecting
warp/weft pair divide the fabric into quadrants, which drape independently of
one another. By specifying the paths ofthe constrained threads, the severity of
deformation and the existence and location of wrinkling and bridging can be
controlled.
_ constrainec
warp yarn
draping _
start point
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\
s /
\
A
B
y^^
s
C
constrained
weft yarn
D
Fig. 1
Fabric with constrained yamB and quadrants A, B, C, and D
Fig. 2
A parametric sui-face patch with tangent vectors'"
3.3 Surface Generation
The draping simulation is performed by generating a surface, computing fabricnodal-point locations, and displaying the draped surface. The surface is generated
by using commonly applied geometric modelling techniques, such as those
described by Mortenson"* and by Rogers and Adams'^ The surface is defined by
a set of co-ordinate points, which lie in the surface, and the tangent and twist
vectors at those points. The surface is divided into patches, each patch containing
four comer points, as shown in Fig. 2. Locations on the patch surface are
described by the parameters u and w, in a co-ordinate system lying in the surface,
which range between 0 and 1, with the 0 and 1 values representing the pateh
boundaries. The tangent and twist vectors ofthe surface boundary points iu*e
known. (A tangent vector expresses the slope of a line in terms of magnitude and
direction. A twist vector is the cross derivative of a slope, also expressed as a
magnitude and direction.) For simplicity, the vector magnitudes are normalized.
Text. IrM.. 1990. 81 No. 4 C TextiU InatituU
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Van West, Pipes, and Keefe
Tangent and twist vectors at points between the boundaries are calculated by
cubic-spline interpolation, the co-ordinates and vectors of a patch are used to
develop parametric bicubic polynomial expressions locating any point on a
patch. (A bicubic polynomial is the simplest mathematical expression that can
describe changes in slope, inflexion points, and twist in a surface, since the
existence of a second derivative is assured.) A discontinuity in slope must not
occur witliin a particular patch owing to the inability of a bicubic polynomial to
describe it but may occur at a patch boimdary if the surface and draping models
are developed to accommodate it.
The geometrical form of a bicubic-surface patch equation is:
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(1)
where U = [u^ u^ul]andVf
= [w^ w^ w 1],
u and w are parameters of a co-ordinate system that lies on the surface
patch,
M is a transformation matrix that depends on the surface model and
continuity constraint,
B is the boundary-condition matrix for a patch, and
p is a position vector with components x, y, and z.
Expanded, the bicubic-surface equation for the x co-ordinate is:
2-2
-3
1
r
3-2-1
x(,u,w) =
0 0
1 0
•^00 "^Ol
00
"10
U
* 0 0 ^ O l -^OO ^ O l
0 0 0 0
2-3
0 1
-2 3
0 0
1-2
1 0
.1-1
0 0
(2)
w
1
where x^^ is a patch-corner co-ordinate, with u and w equal to 0 or 1,
X" is the tangent vector, or slope, 6x/6u, in the u direction,
x^ is the tangent vector, or slope, 6x/6w, in the w direction,
X""' is the twist vector, or change in slope, 5^x/5u6w.
The equations for the y and z co-ordinates are similar:
462
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A Simulation ofthe Draping of Bidirectional Fabrics over Arbitrary Surfaces
2 -2
-3
1
1
y
^01
3 -2 -1
V
y..
V **
VI Ul
w
y{u,w) = (u^ u^ u 1)
0
0
1
0
0
0
0
0
\i
"
V
"
^01 •' 0 0 -y 01
I
UW
UUPa
yio >!,"•'10 -y 11
2
-3
0
1
-2
3
0
0
1
-2
1
0
1
-1
0
0
J
(3)
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X
'2 - 2 1
z iu,w) =
-3
r
0
0
0
0
1~
^00 ^0.
1
Z^i
'o"
Z ^ ir
Ui
( -2 -1
^10 ^11
1
0
^oo" z "
0
0
z " z " *10
2
-3
0
1
-2
3
0
0
1
-2
1
0
w
1
-1
0
0
1
1)
01
*
UtC
w'~
(4)
X
A limitation of a bicubic poljTiomial expression for a surface patch is that it does
not exactly describe all surfaces. However, to compensate, the number of coordinate points describing the surface can be increased, with a subsequent
reduction in patch size, until the desired accuracy is attained.
3.4 Constrained-thread Paths
Constrained-thread paths are defined by the co-ordinates of points on the surface
through which the threads pass. The distance between adjacent co-ordinate
points is held constant and thereby determines the thread-segment length and
fabric-mesh cell size. The point where the two constrained threads intersect
becomes the starting point ofthe draping process. Two intersecting constrained
threads must be selected, and these divide the fabric into a maximum of four
quadrants, as shown in Fig. 1. Fewer quadrants may result. For instance, if the
constrained threads are located at two edges ofthe fabric, a single quadrilateral
results.
The choice of con strained-thread paths is arbitrary £md is a means of
controlling fibre orientation and fabric-shear deformation. Once the paths are
chosen, the co-ordinates ofthe points defining the paths must be calculated. This
requires an equation for the surface where each point is located. If the surface
itself is defined by point co-ordinates, equations for each of the surface patches
must be generated before constrained-thread-path co-ordinates can be calculated.
J. Text. Inat., 1990.81 No. 4 0 TextiU ln*tituU
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Van West, Pipes, and Keefe
The technique is similar to that used to generate interpolated co-ordinate points
as described in the previous section, requiring nujnerical-solution methods.
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3.5 Draping
The mathematical modelling of the draping process consists in calculating the
co-ordinates of all warp- and weft-crossover points on the surface. The process
is graphically depicted in Fig, 3. The constrained threads are shown as heavier
lines along which all crossover, or nodal, points are known. The first points from
the starting point on the constrained-thread paths, shown as p(i, j - 1) and
p(i - 1, j), are the centres of spheres having radii equal to a thread-segment
length. The spheres and the surface intersect at two points, one of which is the
starting point and the other the desired nodal point, shown as p(i, j). The new
nodal point is then coupled with another known point, either on the constrainedthread path or previously calculated, tofindthe next nodal point. The procedure
is repeated until all points are found in all quadrants.
constrained yarns
P(U-V
V
sphere center
- drape Starting point
sphere center
warp-weft
node
—-— yarn segment
length, r
P(U)
sphere-surface
intersection
Fig. 3
Determination of sphere/surface interBectdon
To find the new point co-ordinates, the intersection equations of the two
spheres and the surface are solved. The equation for a sphere centred at point
p(i-lj)is:
j-^i~Ajr = r'
(6)
and that for the sphere centred at point p(ij - 1 ) is:
wherex,y, and 2 are the co-ordinates of points on the spheres, and r is the threadsegment length.
Equations (2), (3), and (4) for the surface point {x.^., y.., z..) are substituted into
Equations (5) and (6), the result being two inter'seciion equations with two
unknowns, u and w, the surface-patch length parameters. The two equations are
solved numerically for u and w by using a non-linear solution technique
'*^'*
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A Simulation ofthe Draping of Bidirectional Fabrics over Arbitrary Surfaces
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producing two u and w root pairs, one for each intersection point. The root pairs
are back-substituted into Equations (2), (3), and (4) to result in two (x, y, z) coordinate sets for point p(i, j). The co-ordinate set locating point p(i, j) furthest
from the starting point of draping is selected.
Each surface patch has a unique bicubic description owing to its unique
location on the surface and boundary-condition matrix. The content of the
sphere/surface-intersection equations is therefore dependent on the patch ia
which the nodal point is located. A containment algorithm is used to determine
the identity ofthe patch in which the nodal point is expected to lie, and, after
calculating the point's co-ordinates, the patch in which the point lies is identified,
and the co-ordinates are recalculated if necessary.
3.6 Examples
Several surfaces were draped both to demonstrate the functioning of the
simulation and to provide guidance on the control of fabric deformation. The
surface were draped in more than one configuration to study the eflfect on fabric
deformation.
The hemisphere is a common surface adopted by many of the previous
researchers'•^••'•^'"^'' as a means of comparing theoretical and experimental
results. Part of a simulation was superimposed on a photograph ofthe physiciil
model of a draped hemisphere, and the result is shown in Fig. 4. The comparison
shows good correlation of thread orientation. The spacings between threads of
the physical model and those of the simulation do not match, so only thread
orientations and not locations are comparable.
The draping of a comer has been simulated to demonstrate control of fabric
deformation by the choice of constrained-thread paths. If the constrained
threads are positioned parallel to the edges, then, as can be seen in Fig. 5, the
underlying surface for the comer quadrant D is much smaller than areas covered
by the other fabric quadrants A, B, and C. One would expect severe fabric
deformation to occur in the comer quadrant D, with a possible wrinkle, and this
is illustrated in Fig. 5. To exercise control over fabric deformation, the constrained
Fig. 4
Superimposed view of physical model and Bimulation of draped hemisphere
J. Ttxt. liut.. 1990, 81 No. 4 © T^tik Institute
455
-n nrli Bi^f
ii •
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Van West, Pipes, and Keefe
Fig. 5
Simulation of draped comer with wrinkle
i^^^Wi
Fig. 6
456
Simulation of draped comer with evenly spaced constrained yams
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A Simulation ofthe Draping of Bidirectional Fabrics over Arbitrary Surfaces
threads were positioned on the surface in such a way that the surface area
between them was approximately evenly distributed. The intention was to
minimize the maximum shear deformation occurring in the fabric. Fig. 6 shows
that the maximum deformation in all quadrants is approximately equal and
wrinkling has been eliminated. The simulation results were confirmed with
physical models'^.
The draped surface simulated in Fig. 7 is of interest to the manufacturers of
composite-material structures and represents the end of a stiffening bead in a
plate. This example demonstrates the use of the simulation to develop an
acceptable draped configuration, which, for the purpose ofthe demonstration, is
defined as having no wrinkles. The surface can be draped without wrinkling if
one ofthe constrained threads Ues on the crest ofthe bead and if the fabric can
accommodate the induced deformations. If the surface is draped in such a way
that one constrained thread lies entirely on the plane to one side ofthe bead, a
wrinkle forms on the other side ofthe bead, as shown in Fig. 8. In many cases,
there are several stiffening beads in a plate, which the manufacturer may wish
to drape sequenticdly. Fig. 9 illustrates the result of draping across two beads.
Draping is terminated at the second bead when the fabric is stretched to the point
at whicb fabric-bridging begins to occur. Along the line between a and b in Fig.
10, an enlarged view ofthe bridging area, the fabric has been stretched to its
deformable limit so that the sides of a rhombic cell have become colUnear. To the
right of the line ab, the numerical rootsolver is unable to find roots for the
intersection equations, since the spheres representing segment-end locii no
longer intersect. The conclusion is that a multi-beaded surface may be draped
without wrinkling if each bead is draped with a constrained thread along its
crease £ind with the required amount of fabric allotted to drape the surface
between beads.
surface profiles
si
pi 1
constrained yarns
1 ft
f f 1 I 1^
I I I I I
f L-- f
-
Fig. 7
Simulation of draped staffening bead with constrained yam on creet
J. TkxL Inat.. 1990. 81 No. 4 O TtxtiU InttttuU
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Van West, Pipes, and Keefe
constrained yarns
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_,=- wrinkle
Fig. 8
Simulation of draped stiffening bead with constrained yam on plane
constrained yarns
bridging
Fig. 9
Simulation of two stiffening beads draped sequentially
Other examples of surfaces draped by means ofthe simulation may be found
in earlier work of one ofthe present authors'^.
3.7 Liiiiitations
The simulation has limitations, which may introduce inaccuracies and restrict
the application, for which the user must compensate.
(o) The bicubic-polynomial-surface expressions do not accurately describe
all surfaces, such as spheres. The inaccuracy may be reduced by increasing
the number of surface-definition points, which has the effect of reducing
the arc length. For arc angles ofless than 45", the error 6R IR is less than
5 X 1(H, where R is the spherical radius^**.
458
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A Simulation ofthe Draping of Bidirectional Fabrics over Arbitrary Surfaces
Fig. 10 Enlarged view of thread-bridging area
(6) The surface to be generated has restrictions on the locations of slope
discontinuities. A bicubic equation cannot describe a surface patch
containing a slope discontinuity. However, a surface can be divided at a
slope discontinuity into adjoining patches.
(c) The assumption that no slippage occurs at warp- and weft-crossover
points is not realistic at extreme fabric deformations. Near the locking
limit, slippage occurs at the intersections, which changes the spacing
between adjacent threads. A slippage parameter, such as that proposed
earlier^, could be included in the draping simulation, increasing the
accuracy of the simiilation when greater-than-moderate fabric-shear
deformations occur.
(d) The assumption that the local surface radius of curvature is much larger
than a thread-segment length is necessary because the thread segments
are assumed straight between nodal points and do not follow the surface
arc. Thus, surfaces with a small local radius of curvature cannot be
draped accurately in such areas, unless the thread-spacing is reduced.
(e) An additional factor affecting the accuracy ofthe simulation is fabricmesh size. The work of Smiley and Pipes* includes a study of the
convergence of deformation angles and data point co-ordinates relative
to their convergent values as a function of the distance between nodal
points. It was found that the angle between intersecting yams at the
border of a fabric quadrant and the equator of a hemisphere is withia
0.1% of the convergent value for a nodal-point spacing of 17% of the
hemispherical radius. The study also revealed that a nodal point on a
hemisphere, halfway between the pole and equator and halfway betweea
quadrant boimdaries, is within 0.1% ofthe convergent location when the
nodal-point spacing is within 10% ofthe hemisphere radius. This spacing
occurs when there are about sixteen nodal points between the hemisphere a
pole and its equator.
J. "Dnct Int.. 1990. 81 No. 40 TextiU IruUlutt
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Van West, Pipes, and Keefe
4. CONCLUSIONS
A mathematical model and graphical simulation ofthe draping of bidirectional
fabrics over arbitrary surfaces has been developed. The model calculates the
locations on the draped surface of warp and weft crossover points from which
fibre orientations and fabric-shear deformation can be determined. By specifying
the limiting angles of shear deformation in a fabric, the presence and location of
fabric-wrinkling and fabric-bridging can be predicted. The simulation models an
infinite number of draped configurations, dependent on the selected constrainedthread paths.
Defining the constrained-thread paths results in a unique draped configuration
and allows control of fibre paths and the existence and location of fabricwrinkHng. The quadrants created by defining constrained-thread paths drape
independently of one another.
The simulation has advantages over previously published simulations.
(a) The surface is defined as a series of patches, each with a unique bicubicpolynomial description, providing greater accuracy in surface definition
than a surface defined by triangular facets.
(6) An infinite number of draped configurations may be generated.
(c) The simulation applies to arbitrary surfaces that do not necessarily have
known analj^ical expressions. The surface is defined numerically by the
co-ordinates of points lying on the surface.
The simulation may be used as a design tool allowing draped configurations
to be selected that best meet the designer's needs in terms of thread paths, fabric
deformation, and, in the case of structural components, material and mechanical
properties. A useful development of the simulation would be to begin with a
draped configuration, sufficiently described to meet the designer's needs and to
ensure mathematical uniqueness, and to perform tbe simulation in reverse,
determining the constrained-thread paths necessary to drape as required.
ACKNOWLEDGEMENT
The authors would like to thank BASF Structural Materials Inc., of Charlotte,
N.C., U.S.A., for sponsoring the work reported in this paper.
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